In this book, Paul Franceschi provides us with an introduction to analytic philosophy. In a concrete way, he chooses to describe forty paradoxes, arguments or philosophical issues that represent so many challenges for contemporary philosophy and human intelligence, for some paradoxes of millennial origin—such as the Liar or the sorites paradox—are still unresolved in the present day. Some other philosophical puzzles, however—such as the Doomsday argument—appeared only recently in the literature. The author strives to introduce us clearly to each of these problems as well as to major attempts that have been formulated to solve them.
“I’m really impressed by this very neat and stimulating book. I highly recommend it both to students for pedagogy and general culture (prisoner’s dilemma, twin-earth, etc.), and to professionals as well for the reference tool and even more generally to those who like to think.”
Julien Dutant, Philotropes, Philosophical blog
The Kindle version is also available.
An Introduction to Analytic Philosophy
Copyright (c) Paul Franceschi
All rights reserved
From P. to T.
Introduction to the English edition
This book aims to provide an introduction to analytic philosophy. It is primarily intended for readers who want to be initiated into this philosophical style. The approach that I have chosen to offer the reader for that introduction is the same as that by which I myself was introduced to analytic philosophy: the presentation of the most famous problems and paradoxes. An advantage of this approach is that there is no need for any prerequisites. This book aims thus at presenting a significant number of contemporary issues in analytic philosophy. It consists then of illustrating how the approach that is pursued involves the accurate description of problems, clearly identified, and whose presentation does not suffer from ambiguity. The approach adopted throughout this book will therefore consist of the description of a large number of contemporary philosophical problems, thus illustrating the methodology used in analytic philosophy, which consists in describing with precision—often step by step—a number of problems for which there exists, at present, no consensual solution. It may be useful for this purpose to classify contemporary philosophical problems into three distinct categories: paradoxes, arguments and problems per se. Each of these three types of problem is exposed, in what follows, and usually accompanied by one or more solutions that have been proposed in the contemporary literature.
I will endeavor first to describe a number of paradoxes. The most famous of them are rooted in antiquity and remain still unresolved: the Liar, the sorites paradox, etc. Paradoxes are arguments based on premises and reasoning that seem well-founded, but whose conclusion leads to a contradiction. An excellent definition is provided by Mark Sainsbury, in his book Paradoxes, published in 1995: “paradoxes are unacceptable conclusions drawn from seemingly true premises and correct reasoning”.
I shall also present a number of arguments that are widely discussed within contemporary philosophical literature. Such arguments are often made up of reasoning whose premises and the deductions that accompany them seem quite acceptable, but whose conclusion proves counter-intuitive. Problems of this type distinguish themselves from the paradoxes in that they do not truly lead to a contradiction. Unlike the paradoxes, we do not observe in this type of argument a contradiction per se, but only a conclusion that proves contrary to common sense. Arguments whose conclusion appears counter-intuitive are close to the paradoxes in the sense that it is very likely that the underlying reasoning is misleading. However, they differ from the paradoxes in the sense that one cannot rule out at the start the possibility that it is our intuition that is at fault. In this case, the solution to the problem posed by this type of argument has to explain why one’s conclusion appears at first sight counter-intuitive.
Lastly, I shall describe a number of problems per se that have led to recent discussions in analytic philosophy. Among these problems based on reasoning, some have a very ancient origin, while others have only recently been described.
Analytic philosophy is essentially characterized by a requirement for clarity in the exposition of ideas and a marked concern for rigor at the stage of the argument. Clarity of ideas is intended to avoid ambiguity and difficulties in the interpretation of texts. It also allows a better critical evaluation of the ideas. This necessity for rigor may sometimes require the use of a mathematical formalism, which should not, however, go so far as to require advanced knowledge in mathematics. As we can see, analytic philosophy is primarily a philosophical style.
It is customary to oppose analytic philosophy and continental philosophy. Continental philosophy refers to the philosophical writings of French and German authors of the nineteenth and twentieth centuries, among whom are, without being exhaustive: Friedrich Hegel, Sören Kierkegaard, Friedrich Nietzsche, Karl Marx, Herbert Marcuse, Martin Heidegger, Jean-Paul Sartre, Maurice Merleau-Ponty and Michel Foucault. The writings of these philosophers are characterized by a greater literary involvement and often a stronger political commitment.
Analytic philosophy is sometimes associated with Anglo-Saxon countries and continental philosophy with the European continent. Such a viewpoint is, however, somewhat simplistic. Indeed, it is true that analytic philosophy is currently the dominant style in the United Kingdom, the United States, Canada, Australia and New Zealand, for example. However, it is also represented in France, Italy, Germany, Spain, Portugal, Greece, Belgium, etc. Moreover, if one takes into account ancient and classical philosophers, it is clear that such a viewpoint proves to be incorrect, because one can find a particularly pure analytical style on the banks of the Mediterranean, in the writings of several philosophers of antiquity. The classical Greek philosophers, inventors of famous unresolved paradoxes such as the Liar, the sorites paradox, but also the paradoxes of Zeno of Elea, constitute outstanding examples. With Plato in particular, one can also find the clarity of the argument in the famous Allegory of the cave.
Moreover, one will find in Pascal, with the Wager argument, all the criteria of a detailed, accurate and clear argumentation. And most importantly, we can notice that Descartes was practicing an astonishingly pure and avant-gardist analytical style. Many of the arguments of Descartes could have been included without change in the contemporary analytic philosophical literature. In the present work, one will thus find the famous cogito argument, the evil demon argument, the ontological argument of Descartes, as well as an argument in favor of mind/body dualism.
It would be rather awkward and Manichean to oppose the two styles—analytical and continental—by considering that one is better than the other. So less overtly subjectively, we can estimate that these are two different styles of practicing philosophy, which each have their own advantages and disadvantages. It is certainly necessary to preserve both styles, given their respective merits and complementarity. Finally, it turns out that the coexistence of the two styles essentially constitutes an expression of cultural diversity that proves to be synonymous with wealth.
1. The Liar Paradox
The Liar paradox is one of the oldest and deepest of the known paradoxes. It is attributed to the Greek philosopher Eubulides of Miletus, who lived in the fourth century BC. The Liar paradox can be expressed very simply, since it arises directly from the consideration of the following statement: “This sentence is false.” The paradox stems from the fact that if this last sentence is true, then it follows that it is false, but if the sentence is false, then it is false that it is false and therefore it is true. Thus “This sentence is false” is false if it is true, and true if it is false. In conclusion, “This sentence is false” is true if and only if it is false, and the latter conclusion turns out to be paradoxical.
One often denotes “This sentence is false” by (λ). At this stage, it is worth describing in detail the various steps of reasoning that lead to the Liar paradox (the symbol λ denotes the conclusion):
|(λ)||(λ) is false|
|(1)||(λ) is either true or false||bivalence|
|(2)||if (λ) is true||hypothesis 1|
|(3)||then it is true that (λ) is false||from (λ),(2)|
|(4)||then (λ) is false||from (3)|
|(5)||if (λ) is false||hypothesis 2|
|(6)||then it is false that (λ) is false||from (λ),(5)|
|(7)||then (λ) is true||from (6)|
|(8)||∴ (λ) is neither true nor false||from (4),(7)|
The conclusion (8) is paradoxical here, since it follows that (λ) is neither true nor false, in contradiction of the principle (1) of bivalence. The problem with the Liar is thus the following: what is the truth value of the proposition (λ), given that we can not assign it, without contradiction, the truth value true or false?
A first attempt at a solution to the Liar is to consider that the truth value of (λ) is neither true nor false, but a third truth value: indeterminate. One can then consider a three-valued logic, which thus includes three truth values: true, false, indeterminate. The Liar is then reintroduced under the following form:
|(λ3)||(λ3) is false or indeterminate|
In this new context, a proposition may now be assigned three different truth values: true, false or indeterminate. The principle of trivalence then states that (λ3) is either true or false, or indeterminate. However, the fact of considering in turn that (λ3) is true, false, or indeterminate still does not lead to a satisfactory solution, since by the same reasoning as with the simple Liar, the conclusion follows that (λ3) is neither true nor false nor indeterminate. This results in the impossibility of properly assigning a truth value to the proposition (λ3).
Moreover, it turns out that the problem resurfaces in the same way if we consider not three, but four truth values: true, false, indeterminate1 and indeterminate2. We must then make use of a four-valued logic. However, the following variation of the Liar emerges:
|(λ4)||(λ4) is false or indeterminate1 or indeterminate2|
which leads, as previously, to the impossibility of assigning a truth value to (λ4).
Another attempt at a solution is to reject the principle of bivalence, of trivalence, and more generally of n-valence on which the reasoning that leads to the Liar is based. However, such a line of solution also fails, since it faces a more powerful variation of the Liar, namely the Strengthened Liar, which does not require us to appeal to any principle of bivalence, of 3-valence or of n-valence:
|(λs)||(λs) is non-true|
This is because the Strengthened Liar leads to the following reasoning:
|(λs)||(λs) is non-true|
|(9)||(λs) is either true or non-true||dichotomy|
|(10)||if (λs) is true||hypothesis 1|
|(11)||then it is true that (λs) is non-true||from (λs),(10)|
|(12)||then (λs) is non-true||from (11)|
|(13)||if (λs) est non- true||hypothesis 2|
|(14)||then it is non-true that (λs) is non-true||from (λs),(13)|
|(15)||then (λs) is true||from (14)|
|(16)||∴ (λs) is neither true nor non-true||from (12),(15)|
Lastly, another attempt at a solution to the Liar paradox is to consider that the structure of the Liar is self-referential, since this very proposition makes direct reference to itself. According to this type of solution, it would suffice to prohibit the formation of self-referential propositions to prevent the emergence of the paradox. However, such a solution turns out to be too restrictive, since there exist many propositions whose structure is self-referential, but for which the attribution of a truth value does not pose any problem. It suffices to consider then the Contingent Liar:
|(λc)||either this proposition is false or 0 = 0|
But it turns out that one can validly assign the truth value true to the Contingent Liar. Thus, although the Contingent Liar presents a self-referential structure, we can successfully and without contradiction—unlike the Liar—assign it a truth value. In this context, it turns out that simply prohibiting all self-referential propositions is too high a price to pay for solving the Liar paradox, and therefore does not constitute a satisfactory solution.
2. The Sorites Paradox
The sorites paradox is one of the oldest and most important of the known paradoxes. Its origin is usually attributed to Eubulides of Miletus, the Greek philosopher from antiquity to whom we also owe the Liar paradox. The paradox can be described informally as follows. Firstly, it is commonly accepted that a set comprising 100,000 grains of sand is a heap. In addition, it proves that if a set comprising a given number of grains of sand is a heap, then a set with a grain of sand less is also a heap. Given these premises, the conclusion follows that a set comprising a single grain of sand is also a heap. In effect, if a set comprising 100,000 grains of sand is a heap, it follows that a set with 99,999 grains of sand is a heap, and the same goes for a set comprising 99,998 grains of sand and 99,997, 99996, 99995, …, and so on, up to a single grain of sand. The paradox stems from the fact that the corresponding reasoning seems quite valid, whereas the resulting conclusion proves to be unacceptable.
The various steps leading to the sorites paradox can be detailed as follows:
|(1)||a set comprising 100000 grains of sand is a heap|
|(2)||if a set comprising n grains of sand is a heap, then a set comprising n – 1 grains of sand is a heap|
|(3)||if a set comprising 100000 grains of sand is a heap, then a set comprising 99999 grains of sand is a heap|
|(4)||∴ a set comprising 99999 grains of sand is a heap|
|(5)||if a set comprising 99999 grains of sand is a heap, then a set comprising 99998 grains of sand is a heap|
|(6)||∴ a set comprising 99998 grains of sand is a heap|
|(7)||if a set comprising 99998 grains of sand is a heap, then a set comprising 99997 grains of sand is a heap|
|(8)||∴ a set comprising 99997 grains of sand is a heap|
|(10)||∴ a set comprising 1 grain of sand is a heap|
The conclusion of the paradox results from the repeated use of a logical and widely accepted principle, termed modus ponens, which has the form: p, if p then q, then q (where p and q denote two propositions).
Many variations of the sorites paradox can be found in the literature. Thus, another version of the paradox with the predicate tall is as follows:
|(11)||a man who measures 200 cm is tall|
|(12)||if a man who measures n cm is tall, then a man who measures n–1 cm is tall|
|(14)||∴ a man who measures 140 cm is tall|
Likewise, we can also construct some variations of the paradox with other vague concepts such as: rich, old, red, etc. This highlights the structure of the paradox (where P denotes a vague predicate):
|(16)||if P(n) then P(n – 1)||induction step|
It should be pointed out here that the structure of the paradox is reversible. Indeed, the previous versions of the paradox proceed decrementally. But the paradox can also operate incrementally, in the following way:
|(19)||a man who has 1 hair is bald||basis step|
|(20)||if a man who has n hairs is bald, then a man who has n + 1 hairs is bald||induction step|
|(22)||∴ a man who has 100000 hairs is bald|
The structure of the paradox is then as follows (P denotes a vague predicate):
|(24)||if P(n) then P(n + 1)||induction step|
Numerous solutions have been proposed to solve the sorites paradox. However, none of them has proved satisfactory so far. Thus, the sorites paradox remains one of the most widely studied contemporary paradoxes.
A solution that calls into question the induction step has been proposed to solve the paradox. This type of solution is based on an approach by degrees and argues thus that the induction step is only true for some instances—the proper instances—of the notion of heap. Such analysis is based on the fact that the notion of heap is a vague concept. Such a notion is thus characterized by the existence of proper instances (e.g. a value of n equal to 1 million), of proper counter-instances (e.g. a value of n equal to 2), but also of borderline cases (e.g. a value of n equal to 100), which constitute a penumbra zone between the notions of heap and non-heap. According to the approach by degrees, the truth value of the induction step is 1 when one is in the presence of proper instances. But when it comes to borderline cases, its truth value is less than 1. It follows finally that the truth value of the induction step, when considering all possible values of n, is slightly less than 1. And this is sufficient to partially block the deductive process and finally prevent us from reaching the final conclusion.
The induction step is also referred to in another type of solution, which considers that the induction step is not necessarily true. It suffices, for example, to consider a stack consisting of cubes stacked on top of each other. Such a stack may comprise, for example, up to twenty stacked cubes. Now, the reasoning that leads to the sorites paradox can also be applied to this stack, because intuitively, if we remove the cubes one by one from the top, we are still in the presence of a stack. Yet in fact, we cannot remove some cubes of strategic importance without all the others falling simultaneously, thus destroying the entire stack. Conversely, some cubes—especially those from the top—are less critical, so they can be removed without jeopardizing the very existence of the stack. Such an analysis of the sorites paradox suggests that there are other factors that should be taken into account, such as the position of each cube, their alignment, etc.. However, this type of solution also fails because it faces a purely numerical variation of the same problem, which consists of Wang’s paradox:
|(27)||100000000 is large||basis step|
|(28)||if n is large then n – 1 is large||induction step|
|(30)||∴ 1 is large|
Indeed, such a problem is an instance of the sorites paradox, to which the preceding type of solution fails to apply.
Lastly, according to another approach, of an epistemological nature, there exists a precise boundary in the number of grains that allows us to differentiate a heap from a non-heap, but we are unable to determine accurately where such boundary is located. The cause of the paradox lies in a deficiency in our knowledge, which is a kind of blind spot. Such a precise boundary also exists, according to this type of approach, at the level of notions such as young/non-young, small/non-small, bald/non-bald, etc., thus allowing us to distinguish them. As we can see, this type of solution tends to reject the induction step as false. However, such a solution does not prove satisfactory, since the existence, for each vague notion, of a precise numerical cut-off allowing us to distinguish the instances from the proper counter-instances seems rather counter-intuitive. And this type of solution does not do justice to the intuition that there exists, for each vague concept, a penumbra zone corresponding to borderline cases.
3. Russell’s Paradox
Russell’s paradox is one of the most famous paradoxes of mathematical set theory. The paradox, stated by Bertrand Russell, results informally from the fact of taking into consideration the set of all sets that do not contain themselves. The very existence of this set leads straightforwardly to a contradiction. Indeed, it follows, on the one hand, that if this set belongs to itself, then it does not belong to itself. And if it does not belong to itself, on the other hand, then it belongs to itself. Thus, such a set both does not belong to itself and does belong to itself.
A classical variation of Russell’s paradox is the barber problem. Such a barber shaves all men who do not shave themselves, and only those ones. The following question then ensues: does the barber shave himself? If the barber shaves himself, then by definition, he belongs to the class of the men who shave themselves, and therefore he does not shave himself. On the other hand, if the barber does not shave himself, then by definition, he belongs to the class of the men who do not shave themselves, and therefore he shaves himself. In conclusion, if the barber shaves himself, then he does not shave himself, and if he does not shave himself, then he shaves himself. Thus, whatever assumption we consider, a contradiction ensues.
Another version of Russell’s paradox arises under the following form: let us consider the catalog of all catalogs that do not mention themselves. The following question then ensues: does this catalog make mention of itself? If it mentions itself, then it is not part of this catalog and does not mention itself, and if it does not mention itself then it is part of the catalog and then it does mention itself. In both cases, one is faced with a contradiction.
Russell’s paradox can be stated more formally as follows. Let R be the set of all sets that do not contain themselves. We then have the following definition of R (where ∈ denotes set membership and ∉ denotes set non-membership):
|(1)||x ∈ R | x ∉ x|
Now, given this general definition, let us consider the specific case of the set R. Two cases are now possible: either R belongs to itself, or R does not belong to itself. On the assumption that R does not belong to itself, the reasoning can be stated as follows:
|(2)||R ∈ R||hypothesis 1|
|(3)||R ∉ R||from (2)|
And likewise, under the assumption that R does not belong to itself, it follows by definition that:
|(4)||R ∉ R||hypothesis 2|
|(5)||R ∈ R||from (4)|
The resulting conclusion is that the set R belongs to itself if and only if it does not belong to itself. The different steps of the reasoning can be detailed as follows:
|(6)||x ∈ R | x ∉ x||definition|
|(7)||R ∈ R||hypothesis 1|
|(8)||R ∉ R||from (6),(7)|
|(9)||∴ if (R ∈ R) then (R ∉ R)||from (7),(8)|
|(10)||R ∉ R||hypothesis 2|
|(11)||R ∈ R||from (6),(10)|
|(12)||∴ if (R ∉ R) then (R ∈ R)||from (10),(11)|
|(13)||∴ R ∉ R and R ∈ R||from (9),(12)|
Thus, the fact of taking into account the very existence of the set R of all sets that do not contain themselves leads straightforwardly to a contradiction.
The paradox stems from naive set theory, into which it is possible to define a set unrestrictedly. Naive set theory is proved thus to be too liberal, by allowing the construction of some sets the nature of which proves finally to be contradictory, such that the set R. In particular, it turns out that the axiom of comprehension of naive set theory was responsible for the emergence of Russell’s paradox. The axiom of comprehension, in fact, allowed the construction of any set that conformed to the following schema:
|(14)||x ∈ E | P(x)|
where P(x) denotes any property of an object x, such that any x with property P belongs to the set E. Thus, the solution to Russell’s paradox consisted in restricting the expressive power of set theory. The axioms of set theory were thus modified to prohibit the construction of the set R of all sets that do not contain themselves. In 1908, Ernst Zermelo proposed a set theory with an axiom of comprehension thus amended, which did not allow for the construction of the set R. This resulted in Zermelo-Fraenkel’s set theory, which is still in use nowadays, and whose axioms render impossible the construction of the set R, thus avoiding the ensuing contradiction.
4. The Surprise Examination Paradox
The surprise examination paradox originated, it is said, in an announcement by the Swedish authorities during the Second World War. According to this announcement, a civil defense exercise was scheduled for the following week, but the specific day was not revealed, so that the exercise would truly take place by surprise. Professor Lennart Elkbom understood the subtle problem that resulted from the announcement and told his students. Subsequently, the problem spread in academic circles and then gave rise to many discussions.
The surprise examination paradox is classically described as follows. A professor tells his students that an examination will take place on the following week. However, the professor adds that it will not be possible to the students to know in advance the date of the examination, because it will occur surprisingly. A clever student then reasons as follows: the examination can not take place on the last day of the week—Friday—because otherwise he would know, with certainty, that the examination would take place on Friday. So Friday can be eliminated. Similarly the student reasons, the examination cannot take place on the penultimate day of the week—Thursday—because otherwise he would know that the examination would take place on Thursday. Thus Thursday is also eliminated. By the same reasoning, the student concludes that the examination cannot take place either on Wednesday or Tuesday or Monday. Finally, the student concludes that the examination cannot take place on any day of the week. However, this does not prevent the examination from occurring surprisingly, for example on Wednesday. The paradox arises here because the student’s reasoning seems valid, but turns out to be ultimately in contradiction with the facts, since the examination finally occurs totally unexpectedly.
The student’s reasoning that leads to the surprise examination paradox can be detailed as follows:
|(1)||if the examination takes place on Friday||hypothesis 1|
|(2)||then I will know that the examination will take place on Friday||from (1)|
|(3)||then the examination will not occur surprisingly||from (2)|
|(4)||∴the examination cannot take place on Friday||from (1),(3)|
|(5)||if the examination takes place on Tuesday||hypothesis 2|
|(6)||then I will know that the examination will take place on Tuesday||from (5)|
|(7)||then the examination will not occur surprisingly||from (6)|
|(8)||∴the examination cannot take place on Tuesday||from (5),(7)|
|(9)||if the examination takes place on Wednesday||hypothesis 3|
|(10)||then I will know that the examination will take place on Wednesday||from (9)|
|(11)||then the examination will not occur surprisingly||from (10)|
|(12)||∴the examination cannot take place on Wednesday||from (9),(11)|
|(13)||if the examination takes place on Tuesday||hypothesis 4|
|(14)||then I will know that the examination will take place on Tuesday||from (13)|
|(15)||then the examination will not occur surprisingly||from (14)|
|(16)||∴the examination cannot take place on Tuesday||from (13),(15)|
|(17)||if the examination takes place on Monday||hypothesis 5|
|(18)||then I will know that the examination will take place on Monday||from (17)|
|(19)||then the examination will not occur surprisingly||from (18)|
|(20)||∴the examination cannot take place on Monday||from (17),(19)|
|(21)||∴ the examination can take place on no day of the week||from (4),(8),(12),(16),(20)|
Several solutions have been proposed to solve the surprise examination paradox. None of them is currently, however, the subject of a consensus. A first attempt at a solution was put forward by O ‘Connor, in an article published in the Mind journal in 1948. According to O ‘Connor, the paradox is due to the contradictory nature that results from the professor’s announcement and its implementation. For O’Connor, the professor’s announcement, according to which the examination should occur unexpectedly, stands in contradiction with the known data from the implementation of the examination. Thus, the statement of the surprise examination paradox is, according to O’Connor, self-refuting. However, such an analysis has not proven satisfactory because it became apparent that the examination could finally occur by surprise, without contradiction, for example on Wednesday. And the fact that the examination could finally occur surprisingly clearly confirmed the professor’s announcement, without refuting it.
A second type of solution has also been proposed by Quine, who highlighted the fact that four possibilities exist (by denoting the last day of the week by n):
|(i)||the examination will take place on day n and the student will know that the examination will take place on day n|
|(ii)||the examination will take place on day n and the student will know that the examination will not take place on day n|
|(iii)||the examination will not take place on day n and the student will know that the examination will take place on day n|
|(iv)||the examination will not take place on day n and the student will know that the examination will not take place on day n|
According to Quine, the problem is that the student, when he develops his reasoning, only considers the cases (i) and (iv), disregarding the possibilities (ii) and (iii). In particular, he does not consider the case (ii), which is the actual situation in which he eventually finds himself, thus allowing the examination to take place finally as a surprise. But if the student had considered this possibility from the beginning, Quine concludes, he would not have reached an erroneous conclusion.
As part of the solutions, it was also proposed that the surprise examination paradox reduces to the sorites paradox. Such a view has been particularly endorsed, though with different nuances, by P. Dietl in 1973 and by J. W. Smith in 1984. Both authors argue that the two paradoxes exhibit a common structure, so that the surprise examination paradox ultimately proves equivalent to the sorites paradox. According to this analysis, the various stages of the two paradoxes are equivalent and the surprise examination paradox thus finds its origin in the fact that the notion of surprise is a vague concept. But such an analysis has been criticized by Roy Sorensen, in his book Blindspots, published in 1988, where he argued that the two problems are not really similar. Sorensen argues first, that the version of the sorites paradox that is equivalent to the surprise examination paradox would be too fast. And second, Sorensen argues, the basic premises of the two paradoxes cannot truly be considered equivalent.
5. Goodman’s Paradox
Goodman’s paradox was introduced by Nelson Goodman in an article published in 1946 in the Journal of Philosophy. Goodman exposes his paradox as follows (with some slight adaptations). Consider an urn containing 100 balls. Every day, a ball is drawn from the urn; this is repeated for 99 days, until today. At each draw, it turns out that the ball taken from the urn is red. At this stage, one expects intuitively that the 100th ball drawn will also be red. This prediction is based on the generalization according to which all balls present in the urn are red. The reasoning on which the latter conclusion is based consists in an enumerative induction.
We can translate the previous inductive reasoning more formally as follows. Let R be the predicate red. Also, let b1, b2, b3, …, b100 be the 100 balls in the urn (∧ denoting the logical connector and).
|(1)||Rb1 ∧ Rb2 ∧ Rb3 ∧ … ∧ Rb99||enumeration|
|(2)||Rb1∧ Rb2 ∧ Rb3 ∧ … ∧ Rb99 ∧ Rb100||from (1), induction|
|(3)||∴ Rb100||from (2)|
At this stage, if we consider the property S, “drawn before today and red, or drawn after today and non-red,” we can see that this property is also verified by the 99 instances already observed. But this time the resulting prediction, based on the generalization that all balls are S, is that the 100th ball will not be red. This contradicts the previous conclusion, which is itself, however, consistent with our intuition. The corresponding reasoning can be detailed thus:
|(4)||Sb1 ∧ Sb2 ∧ Sb3 ∧ … ∧ Sb99||enumeration|
|(5)||Sb1∧ Sb2 ∧ Sb3 ∧ … ∧ Sb99 ∧ Sb100||from (4), induction|
|(6)||∴ Sb100||from (5)|
But here, the conclusion that the 100th ball is S is equivalent to the fact that the latter will be non-red. Now this is at odds with the conclusion resulting from the previous inductive reasoning that the 100th ball will be red. The paradox arises here because of the fact that the two conclusions (3) and (6) are contradictory. Intuitively, the application of the inductive enumeration to (4) seems erroneous. But the difficulty lies here in the fact of diagnosing accurately the flaw in the reasoning at the origin of this false conclusion.
Goodman also gives, in his book Facts, fictions and predictions, published in its original version in 1954, a slightly different version of his paradox, this time applied to emeralds:
Suppose that all emeralds examined before a certain time t are green. At time t, then, our observations support the hypothesis that all emeralds are green; and this is in accord with our definition of confirmation. […] Now let me introduce another predicate less familiar than “green”. It is the predicate “grue” and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at time t we have, for each evidence statement asserting that a given emerald is green, a parallel evidence statement asserting that that emerald is “grue”.
This version of Goodman’s paradox is well known and based on the predicate “grue”. The definition of “grue” is the following: green and observed before T or non-green and observed after T. This results in two types of competing lines of reasoning. A first reasoning implements a classical inductive enumeration: from the observation that all emeralds observed before T were green, we conclude that the next observed emerald will also be green (V denoting green, and e1, e2, e3, …, e100 denoting the emeralds):
|(7)||Ve1 ∧ Ve2 ∧ Ve3 ∧ … ∧ Ve99||enumeration|
|(8)||Ve1 ∧ Ve2 ∧ Ve3 ∧ … ∧ Ve99 ∧ Ve100||from (7), induction|
|(9)||∴ Ve100||from (8)|
The alternative reasoning is based on the same type of inductive enumeration applied to the predicate “grue.” From the fact that all emeralds observed before T were “grue,” we conclude this time that the next observed emerald will be “grue” (“grue” being denoted by G):
|(10)||Ge1 ∧ Ge2 ∧ Ge3 ∧ … ∧ Ge99||enumeration|
|(11)||Ge1∧ Ge2 ∧ Ge3 ∧ … ∧ Ge99 ∧ Ge100||from (10), induction|
|(12)||∴ Ge100||from (11)|
A contradiction then follows, since according to (9), the 100th emerald will be green, whereas it follows from (11) that the 100th emerald will be non-green. The two problems described by Goodman are two variations of the same paradox, since the predicate S used by Goodman in his article of 1946 shares with “grue” a common structure. P and Q being two predicates, the latter structure corresponds to the definition: (P and Q) or (not-P and not-Q).
Goodman’s paradox has generated a huge literature and many different types of solution have been proposed to solve it. Goodman himself has proposed a solution based on the notion of entrenchment. In Fact, Fiction and Forecast, he considers thus that the problem reduces to that of drawing a distinction between predicates that are projectible and those that do not. Projectible predicates can validly be used to support an enumerative induction, while others, among which is “grue,” are not suitable for this. According to Goodman, projectible predicates are those that are integrated, embedded in our current inductive practice. They consist then of an inductive use, which is thus validated by practice. Projectible predicates are those that are somehow validated by current usage, present and past. Conversely, non-projectible predicates such as “grue” are not suitable for inductive use. However, Goodman’s solution based on entrenchment into the language and common usage did not prove satisfactory, as it turns out that new predicates appear every day. Many neologisms are indeed created, that quickly integrate into everyday language and practice. Even the predicate “grue,” which was originally much criticized, has become somewhat familiar.
Another notable solution that has been proposed to solve Goodman’s paradox is based on the fact that the predicate “grue” has a time reference, unlike the predicate “green.” According to this type of solution, predicates such as “grue”, which include such temporal clauses, should not be used for induction. Nevertheless, this type of solution has proved too restrictive, as there are predicates that have a time reference but whose inductive projection is not a problem. Consider then a tomato: it is green when immature and red after. This property applies to the 99 tomatoes that I have just found in my garden, but also to the 100th tomato that is located in my neighbor’s garden. Second, it is quite possible to build a version of Goodman’s paradox that is devoid of such a temporal clause. It suffices then to build a predicate G based, for example, on a color-space combination, replacing the color-time association, to create a variation of Goodman’s paradox that overcomes a time reference. Finally, the response of Nelson Goodman himself against this type of objection is that the predicate “green” can also be defined with a time reference when using “grue” as a primitive concept. It suffices thus to draw a parallel between, on the one hand, the predicates “green” and “blue” and on the other hand, “grue” (green before T and blue after T) and “bleen” (blue before T and green after T). In this case, it is quite possible to define “green” and “blue” with the primitive notions of “grue” and “bleen”. A “green” object is then defined as “grue” before T and “bleen” after T, and similarly, a “blue” object is defined as “bleen” before T and “grue” after T. Thus, the definitions of “green” and “blue,” and on the other hand, of “grue” and “bleen,” turn out to be perfectly symmetrical and present identically a time reference.
6. Newcomb’s Problem
Newcomb’s Problem was described in 1960 by physicist William Newcomb, and was then introduced into the philosophical literature in an essay published in 1969 by Robert Nozick. We can describe the problem as follows. Two boxes, A and B, are placed in front of you. One of them—box A—is transparent and contains 1000 dollars. You will be faced with a choice: either take only the contents of box B or take the contents of both boxes A and B. You also know that a diviner, whose predictions have been extremely reliable so far, will put one million dollars in box B if he predicts that you will only take the latter. However, if he predicts that you will take both boxes A and B, the diviner will leave box B empty. Now, do you choose to take only box B, or to take boxes A and B? By virtue of a first argument (I), it turns out that the predictions made in the past by the diviner have proved very reliable and there is no reason that the prediction that he will make for you will not hold true once again. Therefore, it seems prudent to take only box B in order to collect one million dollars, which is already a very nice amount of money. At this point, however, it turns out that an alternative argument (II) can also be held, for at the moment when you prepare to open box B or both boxes, the diviner has already made his choice. Therefore, if the diviner predicted that you will only open box B, he has then placed one million dollars in the box. Wouldn’t it then be absurd to leave the 1000 dollars that are in box A, for the latter box is transparent, and you can observe its content. You reason, and you find that it can no longer affect the choice of the diviner. Therefore, it is better to open both boxes, and thus collect 1001000 dollars. At this point, it turns out that each of the two arguments (I) and (II) seems to be valid. However, both lead to conflicting conclusions. And the riddle posed by Newcomb’s problem is precisely that of knowing which of the arguments (I) and (II) is valid.
It is worth somewhat formalizing some aspects of Newcomb’s problem, in order to highlight some elements of its internal structure. It thus appears that the structure of the statement is that of a double conditional:
|(1)||if < the diviner predicts that the subject will open box B> then < the diviner will put 1000000 dollars in box B>|
|(2)||if < the diviner predicts that the subject will open both boxes A and B> then < the diviner will put 0 dollars in box B >|
Similarly, the reasoning (I) can be described in detail in the following way:
|(3)||the predictions made in the past by the diviner proved very reliable||premise|
|(4)||the predictions made by the diviner are very reliable||generalization|
|(5)||this time also, the diviner should predict my choice||from (4), induction|
|(6)||if the diviner predicted that I would only open box B, then he placed 1000000 dollars in box B||from (1)|
|(7)||if the diviner predicted that I would open boxes A and B, then he has placed 0 dollar in the box B||from (2)|
|(8)||∴ I have interest in opening up box B||from (6),(7)|
And we can also formalize thus the reasoning (II):
|(9)||by the time I make my choice, the amounts of money are already placed within the boxes, and they will not be affected by my choice||premise|
|(10)||if the diviner has put 1000000 dollars in box A, then by also taking box B, I shall win 1001000 dollars in place of 1000000 dollars||from (9)|
|(11)||if the diviner has put 0 dollars in box A, then by also taking box B, I shall win 1000 dollars||from (9)|
|(12)||in both cases, I get a higher gain by also taking box A||from (10),(11)|
|(13)||∴ I have interest in opening up boxes A and B||from (12)|
Newcomb’s paradox has gained tremendous popularity and has engendered a huge literature. Among the solutions that have been proposed to solve the paradox, one of them focuses on the fact that the situation corresponding to the paradox is actually impossible and is such that we cannot meet it in practice. According to this analysis, the part of the statement according to which the diviner can accurately predict the choice of the agent is unlikely. Under this analysis, such a clause relies on extravagant properties that are not those of our physical world, such as retroactive causality (the fact that an effect can act on its own cause) or the lack of free will in individuals. Such a solution, however, has not proved satisfactory, for if it is allowed to question the existence of retroactive causation or the lack of free will, we can nevertheless highlight some other variations of the paradox that do not make use of such singular properties. It suffices for this to consider a probabilistic version of the paradox where the diviner’s prediction is most often accurate, for the diviner may well rely on merely psychological considerations. A study of Newcomb’s paradox showed in effect that 70% of people choose to take only box B. The diviner may thus have a computer program simulating human behavior in a very powerful way when faced with this type of situation and make its forecasts accordingly. In this context, the clause of the statement according to which the predictions of the diviner are very often accurate would be entirely respected.
7. The Prisoner’s Dilemma
The Prisoner’s Dilemma was described by Merrill Flood and Melvin Dresher in 1950. It can be formulated as follows. Two prisoners, John and Peter, are questioned by a judge who suspects them of having committed a crime. The judge offers to each of them the following options: “You have two options: either confess or not confess. But beware, the choice you make will have a very important consequence on the sentence you will be imposed. Thus, if one of you confesses but the other does not confess, the one who has confessed will be free, while the one who has refused to confess will be sentenced to ten years in prison. However, if you both confess, each of you will have only five years in prison. Finally, if neither of you confess, I would inflict on you both one year in prison. Now, think and then decide. I will then let you know my sentence.”
At this point, it is worth describing in more detail the structure of the prisoner’s dilemma. It turns out then that the four following cases are possible:
|(i)||John confesses and Peter confesses|
|(ii)||John confesses and Peter doesn’t confess|
|(iii)||John doesn’t confess and Peter confesses|
|(iv)||John doesn’t confess and Peter doesn’t confess|
In addition, the judge’s announcement can be described with the help of the following matrix, which defines the penalties assigned to each of the two prisoners depending on their attitude:
|(i)||John confesses and Peter confesses||John (5 years) and Peter (5 years)|
|(ii)||John confesses and Peter doesn’t confess||John (0 year) and Peter (10 years)|
|(iii)||John doesn’t confess and Peter confesses||John (10 years) and Peter (0 year)|
|(iv)||John doesn’t confess and Peter doesn’t confess||John (1 year) and Peter (1 year)|
The problem inherent to the prisoner’s dilemma stems from the fact that two different types of reasoning both seem valid. Indeed, by virtue of the first type (I) of reasoning, it turns out that the fact of not confessing is what gives to each the best chance of being free. Indeed, if one of the prisoners confesses, there ensues a sentence of 5 years (if the other confesses as well) or null (if the other does not confess); thus, the resulting sentence is an average of 2.5 years (5+0)/2. However, if the prisoner does not confess, there follows a sentence of 10 years (if the other confesses) or 1 year (if the other does not confess as well), so the result is a sentence which is on average 5.5 years (10+1)/2. It therefore seems much more rational to confess. However, another type of reasoning is also possible. In effect, according to another viewpoint (II), it turns out that the fact of not confessing proves to be very interesting for both prisoners, for it only results in a one-year sentence for each of them. Finally, one is faced with a dilemma, since each of the options resulting from the two competing arguments (I) and (II) proves to be, from a certain standpoint, optimal.
The prisoner’s dilemma corresponds to a concrete, practical situation, which has implications in the fields of game theory, economics, political science, biology, etc. In terms of game theory, one distinguishes thus classically between zero-sum games and non-zero sum games. For zero-sum games, there is a winner and a loser, but no intermediate situation (this is the case, for example, for tennis). In contrast, for non-zero sum games, there is a winner, a loser, and one or more intermediate situations (chess, where the possibility of a draw exists, is one example). In this context, the prisoner’s dilemma proves to be a non-zero-sum game, since there are two cases where both prisoners receive the same sentence: (1) if both confess, and (2) if both do not confess.
One can observe that the prisoner’s dilemma leads to a significant variation when the dilemma is repeated. This is termed the iterated prisoner’s dilemma. In this context, several strategies then turn out to be possible. This results in the following basic strategies: always confess or never confess. But other more complex strategies are possible, based notably on the option chosen by the other prisoner in the previous moves. In this case, the iterations then lead us to analyze the sequence of moves by the prisoner as a type of behavior. At this point, the possibilities become numerous. A strategy that has proven very successful has been called tit-for-tat. The strategy on which it is based is the following: confess at first move, then play at move n+1 what the other prisoner has played at move n. For the iterated prisoner’s dilemma, there is no strategy that we can say with certainty is better than any other.
8. Cantor’s Paradox
Cantor’s paradox was discovered by Georg Cantor in 1899, but was not published until 1932. The main idea of the paradox resides in the fact that the consideration of the set of all sets leads to a contradiction. In effect, if we call C the set of all sets, then it follows that there exists a set C*, which is itself defined as the set composed of the parts of the set C. By definition, the set C, which is the set of all sets, therefore also includes the set C*. This implies that the cardinal—that is to say the number of elements—of the set C is greater than or equal to the cardinal of the set C*. Now a theorem established by Cantor states that given a set E, the cardinal of E is less than the cardinal of the set E*, which consists of all parts of E. Thus, by virtue of Cantor’s theorem, it follows that the cardinal of the set C*, which includes all the parts of C, is necessarily greater than the cardinal of the set C. This results in a contradiction.
The reasoning corresponding to Cantor’s paradox can be detailed thus more formally (card denotes here the cardinal of a given set):
|(1)||C is the set of all sets||definition|
|(2)||C* is the set of all parts of the set C||premise|
|(3)||card (C) ≥ card (C*)||from (1)|
|(4)||for any set E, the set E* of all parts of E is such that card (E) < card (E*)||Cantor’s theorem|
|(5)||for the set C, the set C* of all parts of C is such that card (C) < card (C*)||from (4)|
|(6)||∴ card (C) ≥ card (C*) and card (C) < card (C*)||from (3),(5)|
Cantor’s paradox, like Russell’s paradox, belongs to the category of the paradoxes of set theory. Just like Russell’s paradox, it emerges within the naive set theory, where the construction of the set C of all sets is allowed. In the present set theory, that of Zermelo-Fraenkel, the paradox is avoided because one cannot construct the set C. In effect, one of the axioms of Zermelo-Fraenkel theory, the axiom of comprehension, was designed more restrictively than in naive set theory in order to prohibit the construction of the set C of all sets. But such an approach may seem ad hoc, in the sense that it consists of a restriction of set theory which has the sole purpose of avoiding the paradoxes and the resulting contradiction. In this context, as with Russell’s paradox, one cannot truly consider that we now have a genuine solution to Cantor’s paradox.
9. Grelling’s Paradox
This paradox was invented by Kurt Grelling. It is also termed the paradox of heterological words. Grelling’s paradox can be stated as follows: some adjectives describe properties that apply to themselves, such as “polysyllabic”, “French.” Such adjectives can be referred to as autological. Some other adjectives, in contrast, describe properties that do not apply to themselves. For example, “long”, “Italian”, “monosyllabic”. Such words can be termed heterological. This leads us to classify words into two categories: (i) autological and (ii) heterological. Such a distinction, however, leads to a paradox. Given the above definitions, the paradox emerges, in fact, when one questions the status of the predicate heterological itself. Is “heterological” autological or heterological? For if “heterological” is heterological, then by definition, “heterological” is autological. And conversely, if “heterological” is autological, it follows that it is heterological. The conclusion is paradoxical, since it follows that “heterological” is heterological if and only if it is autological.
The definitions and the reasoning that lead to Grelling’s paradox can be described in more detail in the following way (H and ~H denote respectively heterological and non-heterological—that is to say autological—and denotes a given property):
|(1)||H(“Φ”) if and only if ~Φ(“Φ”)||definition 1|
|(2)||~H(“Φ”) if and only if Φ(“Φ”)||definition 2|
|(3)||if H(“H”)||hypothesis 1|
|(4)||then ~H(“H”)||from (1)|
|(5)||if ~H(“H”)||hypothesis 2|
|(6)||then H(“H”)||from (2)|
|(7)||∴ H(“H”) if and only if ~H(“H”)||from (3),(4),(5),(6)|
It turns out that one cannot validly assign the predicate “heterological” either to the property heterological or to the property autological.
At this stage, it is also interesting to examine the status of the word “autological” itself. Is “autological” heterological or autological? The rationale relating to “autological” goes as follows:
|(1)||H(“Φ”) if and only if ~Φ(“Φ”)||definition 1|
|(2)||~H(“Φ”) if and only if Φ(“Φ”)||definition 2|
|(8)||if H(“~H”)||hypothesis 1|
|(9)||then ~~H(“~H”)||from (1)|
|(10)||then H(“~H”)||from (9)|
|(11)||if ~H(“~H”)||hypothesis 2|
|(12)||then ~H (“~H”)||from (2)|
Here, the specific stage (10) is justified by the elimination of the double negation. In this case, it turns out that if “autological” is heterological then it is heterological, and in the same way, if “autological” is autological then it is autological. Hence, it turns out that one cannot properly determine whether “autological” is heterological or not.
Among the solutions that have been proposed to solve the paradox of heterological words, one of them leads us to notice that the structure of the paradox is very similar to that of Russell’s paradox. Thus, the two paradoxes would present a common structure and would then lead to a solution of the same nature.
Another solution leads us, as with the Liar paradox, to reject the definitions of all predicates that exhibit self-referential structure. However, such a solution does not prove to be satisfactory. In effect, it is far too restrictive, since it turns out that one succeeds quite safely to determine the status of numerous self-referential predicates such as polysyllabic. To merely outlaw all predicates of which the structure is self-referential would be to pay much too high a price for merely eliminating the paradox.
10. The Two-Envelope Paradox
The two-envelope paradox can be enunciated as follows: in front of you are two envelopes that each contain a given amount of money and you know with certainty that one of them contains twice as much as the other. You take one of the two envelopes at random. Now you have the choice of keeping the envelope in your hand or switching with the other envelope. What do you decide to do? A first type of reasoning (I) comes immediately to your mind: the situation with regard to each of the two envelopes is quite similar. By choosing only one of the two envelopes, you do not get any new information. Therefore, the choice of either is equivalent, so you decide to keep the envelope that you initially took. However, it turns out that another type of reasoning (II) also proves to be possible: let x be the amount contained in the envelope that you have in your hands. The other envelope then contains an amount that is equal to either 2x or 1/2x. Both situations are equally probable and each can be assigned a probability of 1/2. Therefore, the general probability can be calculated as follows: 2x x 1/2 + 1/2x x 1/2 = 5/4x. It follows that in the general case, the other envelope contains a sum equal to 5/4x: that is to say, 1.25 x. Thus, it turns out that the other envelope contains an amount that is a quarter greater than the one that you have in your hands. Therefore, it is in your interest to switch with the other envelope. However, once the envelope has been exchanged, a similar reasoning leads you to switch the envelope again, and so on ad infinitum.
In the two-envelope paradox, it is clearly the reasoning (II) that is at issue, since it leads to the absurd conclusion that one should exchange the envelopes ad infinitum. Yet the task of determining accurately the fallacious step in the reasoning (II) proves to be very difficult. To this end, it is helpful to further formalize the various steps involved in reasoning (II):
|(1)||the other envelope contains either (i) the amount 2x or (ii) the amount 1/2x||premise|
|(2)||the probability of each situation (i) and (ii) is 1/2||premise|
|(3)||the general probability is that the other envelope contains: 2x x 1/2 + 1/2x x 1/2||from (1),(2)|
|(4)||the general probability is that the other envelope contains 1,25x||from (3)|
|(5)||∴ I have interest to switch with the other envelope||from (4)|
Among the solutions that have been proposed to solve the paradox, one of them argued that the assertion (2), according to which the second envelope contains 2x or 1/2x with a probability equal to 1/2, is not true in all cases. Thus, Frank Jackson and his co-authors argued in an article published in 1994 that in reality, the values of x and the resulting pairs of values do not all have the same probability of being in the envelopes. In effect, there are certain limit values—either very small or very large—that one has very few chances to encounter for practical reasons. Thus, the two values that can be in one envelope are not equally probable and therefore the premise (2) is not exact. However, such a solution did not prove to be satisfactory. Indeed, as pointed out by McGrew and his co-authors in an article published in 1997, one succeeds in making the paradox reappear by considering a variant of the latter, wherein the envelopes do not contain money, but merely pieces of paper on which are inscribed numbers.
11. Moore’s Paradox
|(1)||It’s raining and I do not believe that it’s raining|
It follows that such a proposition is in principle absurd. Intuitively, such a proposition presents a contradictory nature. However, it turns out that there are some situations where an assertion such as (1) may be validly expressed. Thus, that person may firmly believe that it is not raining today on the basis of the weather forecast that he/she heard the day before, while it is raining in reality. In this context, the assertion (1) will seem plausible again. Such a situation corresponds, for example, to a situation where a person has a justified belief that a given event will not occur, but this event eventually occurs, thus ultimately rendering the initial belief false. Hence, this person may strongly believe that it is not raining today, based on weather forecasts heard the day before, whereas it is raining in reality. In this context, the assertion (1) regains plausibility.
It is worth analyzing here in greater detail the structure of (1). If we consider then any proposition P, it follows that the assertion (1) has the following structure:
|(2)||P and I do not believe that P|
The logical structure of (2) is the following (Q denoting “I believe” and ~ negation):
|(3)||P ∧ ~Q(P)|
One classically distinguishes two variations of Moore’s paradox: Moore’s paradox of Hintikka and Moore’s paradox of Wittgenstein. Moore’s paradox of Hintikka presents a structure which is that of (2) and corresponds to the original version of Moore’s paradox. In contrast, Moore’s paradox deals with Wittgenstein’s proposition
|(4)||P and I do believe non-P|
which presents the logical structure:
|(5)||P ∧ Q(~P)|
According to some authors, the surprise examination paradox assimilates itself to Moore’s paradox. This was notably the view expressed by Robert Binkley, in an article published in 1968, where he argued that if the period in which the examination can take place is only one day, then the professor’s announcement presents the structure of Moore’s paradox, as the professor’s announcement made to the students is the following: “There will be an examination tomorrow, but you will not know that this examination will take place tomorrow.” Once the students conclude that the examination can take place, they are then on the very day of the examination, in a situation that permits the professor’s announcement to be validated. The result is then a real situation which corresponds, without contradiction, to proposition (1).
12. Löb’s Paradox
Löb’s paradox is mentioned in Jon Barwise and John Etchemendy’s book The Liar, published in 1987. The authors point out that the paradox was brought to their attention by Dag Westerstahl. Löb’s paradox, starting from one proposition that seems harmless, leads to the devastating conclusion that any proposition is true. The proposition that constitutes the starting point of the reasoning is as follows:
|(1)||if proposition (1) is true, then 0 = 1||premise|
Such a proposition presents the structure of a conditional proposition (that is to say, it takes the form: if <antecedent> then <consequent>) whose antecedent is “proposition (1) is true” and the consequent is “0 = 1.” The paradox emerges when one considers the hypothesis that the antecedent of (1), that is to say “proposition (1) is true”, is true. If the antecedent of (1) is true, then it follows that 0 = 1. But the latter proposition is nothing but (1) itself. The result is, by application of modus ponens (a logical principle in virtue of which if P, P Q, then Q), that proposition (1) itself is true. Consequently, the proposition (1) has just been proven. It consists here of a case of application of conditional proof. However, if (1) is true, another application of modus ponens leads finally to the fact that 0 = 1.
One can describe in more detail the various steps of reasoning leading to Löb’s paradox:
|(1)||if proposition (1) is true, then 0 = 1||premise|
|(2)||if proposition (1) is true||hypothesis|
|(3)||then 0 = 1||from (1),(2)|
|(4)||if proposition (1) is true, then 0 = 1||from (2),(3)|
|(5)||∴ (1) is true||from (4)|
|(6)||∴ 0 = 1||from (1),(5)|
Löb’s paradox thus goes on to prove, starting from one proposition that seems harmless, any proposition. As with other contemporary paradoxes, the task of diagnosing the exact cause of the paradox proves to be very difficult.
An attempted solution leads us to notice that the structure of (1) is self-referential. It consists here of a common feature with other paradoxes, especially the Liar paradox. But the solution that consists in prohibiting the propositions with a self-referential structure is not appropriate here either. Indeed, this is far too drastic and restrictive, thus leading to the elimination of the propositions whose structure is self-referential, but which do not, however, present a problem in being assigned a truth value. Here again, the problem arises of defining the criteria for distinguishing between: (i) self-referential propositions that admit a valid truth value, and (ii) self-referential propositions that cannot be validly assigned a truth value.
13. The Race Course Paradox
[The race course paradox] asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.
Informally, the paradox—which is also termed the Dichotomy paradox—can be described as follows. A runner wants to travel the distance from point A to point B. To get to B, the runner must first traverse half the distance that separates point A from point B. But once he has traveled half the distance, the runner still has to go through half the distance that separates him from arrival at B. Once at this point, the runner will have traveled three quarters of the distance that separates him from B. But from there, he will still have to go through half the distance to the arrival, and so on ad infinitum. Thus, the runner must have to go through an infinite number of times the distances that are themselves finite. Now this should take an infinite time. Therefore the runner will never succeed in reaching B, and it follows that all motion is impossible.
The paradox can be described somewhat more formally. Let d be the distance from A to B. In this case, the runner must first have to go through half of d, then 1/4 d, then 1/8 d and 1/16 d, and so on ad infinitum. The reasoning that leads to the race course paradox can be described as follows:
|(1)||to travel from one point to another, a runner must first go through half the distance between two points||premise|
|(2)||the runner wants to travel the distance d which separates point A from point B||premise|
|(3)||to go from A to B, the runner must first travel 1/2 d||from (1),(2)|
|(4)||once reached 1/2 d, the runner must then travel 1/4 d||from (1),(2),(3)|
|(5)||once reached 3/4 d, the runner must then travel 1/8 d||from (1),(2),…,(4)|
|(7)||the runner shall have to travel an infinite number of times a fraction of d||from (3),(4),…,(6)|
|(8)||it is impossible to traverse an infinite number of distances in a finite time||premise|
|(9)||∴ the runner will never reach point B||from (7),(8)|
A first type of response that can be made with respect to the paradox was formulated by Aristotle via Simplicius: everyone knows from personal experience that one can move from one point to another. Therefore, one can also move from point A to point B in the case corresponding to the statement of the paradox. So the runner will reach point B, in the same way that we are able to reach a point to which we want to move in real life. This objection, however, did not prove to be convincing. In effect, the empirical finding that it highlights turns out indeed to be true. However, it consists precisely of one component of the paradox, for what represents here the heart of the paradox is that the reasoning inherent to the race course paradox leads to a conclusion that contradicts the current data that result from the experiment. Thus, this objection only mentions one inner element of the paradox. What is required, however, is to determine accurately the fallacious step in the reasoning described by Zeno.
Another answer, regarded by many as a compelling resolution of the race course paradox, results directly from the work of Cauchy and his theory of infinite series. In effect, Cauchy has shown that the sum of an infinite series was sometimes finite. In this case, it turns out that the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 +… + 1/2n equals 1. Under these circumstances, each intermediate distance is thus covered in a finite time. The distance d is therefore traveled in a finite time, which is equal to the sum of the intermediate laps.
14. The Stone Paradox
The stone paradox was described by W. Savage in 1967 in an article published in the Philosophical Review journal. It can be stated as follows: let be first the definition that God is an omnipotent being. Then consider a stone that has the following characteristic: it is so heavy that God cannot lift it. At this stage, there are two possibilities: either God can create it, or God cannot create it. Let us consider first the first hypothesis. If God can create such a stone, it follows that God cannot lift it. Therefore, if God can create such a stone, then there exists a task that God cannot accomplish. Now consider the second hypothesis, by which God cannot create such a stone. In this case, it also follows that there is a task that God cannot accomplish. Thus, the fact of taking into account each of the two assumptions leads to the conclusion that in each case there exists a task that God cannot accomplish. And this proves to be inconsistent with the fact that God is omnipotent. It follows then that God does not exist.
The steps of the argument can be decomposed as follows:
|(1)||God is an omnipotent being||definition|
|(2)||either God can create a stone that he cannot lift, or God cannot create it||dichotomy|
|(3)||God can create a stone that he cannot lift||hypothesis 1|
|(4)||God cannot lift a stone||from (3)|
|(5)||there exists a task that God cannot do||from (4)|
|(6)||God cannot create a stone that he cannot lift||hypothesis 2|
|(7)||there exists a task that God cannot do||from (6)|
|(8)||there exists a task that God cannot do||from (5),(7)|
|(9)||∴ God is not an omnipotent being||from (8)|
A solution that has been formulated to solve the stone paradox rests on the fact that the concept of a stone that God can not lift has an inherently contradictory nature. The status of such a stone, if it existed, would be contradictory in essence. It is therefore not surprising that the use of a contradictory notion in an argument yields illogical consequences. The concept of a stone that God can not lift can then be compared with a “square circle” or a “single married,” for one can in effect have the exact same type of argument with a “square circle,” thus leading in the same way to an inconsistent consequence.
According to another point of view, resulting from the writings of St. Thomas Aquinas, the concept of omnipotence cannot be used without restriction, for the notion of divine omnipotence should only be considered in relation to things that are actually possible. In no case does the notion of omnipotence entail the ability to do impossible things. Such a perspective can be applied directly to the stone paradox. It follows then that the fact of lifting a stone that no one can lift is precisely an impossible task.
15. The Doomsday Argument
The Doomsday Argument is an argument that was enunciated by astrophysicist Brandon Carter in the early 1990s. This type of reasoning was also discovered independently by Richard Gott and H. Nielsen. The Doomsday Argument was then developed in detail and defended by the Canadian philosopher John Leslie in a series of publications. The main feature of the argument is that the premises of the corresponding reasoning seem quite acceptable, whereas the corresponding conclusion proves to be unacceptable to most people.
The reasoning on which the argument is based is as follows. One considers first an urn that contains either 10 or 1000 balls. The balls are numbered 1, 2, 3, 4, 5,…. The competing hypotheses are thus as follows:
|(H1)||the urn contains 10 numbered balls|
|(H2)||the urn contains 1000 numbered balls|
Let us consider that the initial probability that the urn contains 10 balls or 1000 balls is 1/2. Now, you pull a ball at random from the urn and you find out that it has the number 5. Does this draw make more likely the hypothesis that the urn contains 10 balls, or the one according to which it contains 1000 balls? Given the novel information that the ball retrieved from the urn is numbered #5, it turns out that an upward revision of the initial probability of the hypothesis that the urn contains only ten balls must be carried out. In effect, the random draw of the ball # 5 renders much more likely this last hypothesis. For if the urn contains only 10 balls, there is one chance out of 10 that you will pick ball # 5. In contrast, if the urn contains 1000 balls, there is one chance in 1000 of picking ball # 5. An accurate calculation by using Bayes’ theorem leads to a revision to 0.99 of the initial probability that the urn contains 10 balls. Such reasoning, based on the contents of an urn, proves to be consensual.
At this stage, we can now draw a parallel with the human situation. One considers two assumptions regarding the evolution of humankind. It can be envisaged that the total population of humans that have ever existed will reach 100 billion or 10 trillion. We thus formulate the following two assumptions regarding the future of humanity:
|(H3)||humankind will have a total of 100 billion humans|
|(H4)||humankind will have a total of 10000 billion humans|
The first hypothesis corresponds to a fast and imminent extinction of humanity, whereas the second corresponds to a very long lifetime of humanity, which could then colonize other planets and spread across the galaxy. We assign, for simplicity, a probability of 1/2 to each of these two hypotheses. At this stage I am led to consider my position since the birth of humanity. Considering then that I am the 70 billionth human, I am led to reason in the same way as I did before with the urn. This leads to an upward revision of the initial probability that the total population of humans who ever lived will only reach 100 billion. Finally, this argues for the probability—much greater than we could have imagined at first glance—of a relatively imminent extinction of humanity. But unlike the previous case concerning the urn, the latter conclusion turns out this time to be quite unacceptable and contrary to intuition. In the reasoning that led to the conclusion that humankind should face imminent extinction, a step proves to be defective. But the task of assessing accurately the weakness in the Doomsday argument proves to be a very difficult one, about which opinions differ widely.
A first approach to try to solve the problem raised by the argument is simply to accept its conclusion. According to some authors, especially John Leslie, the argument is correct and the conclusion that results should be accepted (with an important caveat, however, regarding the case where our universe is not completely deterministic). Leslie bases this position on the fact that in two articles published in 1992 in the journal Mind and in his book The End of the World published in 1996, he has refuted, often convincingly, an impressive number of objections to the Doomsday argument. However, accepting the conclusion of the argument remains quite counterintuitive. On the other hand, the acceptance that the mere knowledge of our birth rank leads to an upward shift in the probability of an imminent extinction of humanity leads to a similar conclusion in many common analog situations. It follows, for example, an upward revision in the probability of an imminent disappearance of the association that I have just joined, and so on.
Another type of solution that I developed in an article published in 1999 by the Canadian Journal of Philosophy is to consider that the reference class which bears on the Doomsday argument – that is to say, the human species – is not precisely defined, for must we assimilate it to the subspecies homo sapiens sapiens, to the species homo sapiens, to the homo gender, etc.? We can choose the reference class differently by operating by restriction or extension. In the statement of the Doomsday argument, no objective criteria allowing us to choose the reference class are present. It therefore follows an arbitrary choice of the latter. Suppose then that I assimilate, in an arbitrary way, the reference class to the subspecies homo sapiens sapiens. There then follows, by applying the Doomsday argument, a bayesian shift in favor of the hypothesis that the subspecies homo sapiens sapiens is promised future extinction. However, the extinction of the subspecies homo sapiens sapiens could also be accompanied by the appearance of one or more new sub-species, such as homo sapiens supersapiens. In this case, the disappearance of the reference class that identifies, by restriction, to the subspecies homo sapiens sapiens is accompanied by the survival of a broader reference class, which assimilates itself with the species homo sapiens. Such reasoning has the effect of rendering the Doomsday argument harmless and of neutralizing its initially devastating conclusion. It may be objected, however, that such a solution always admits the validity of the argument with regard to a limited reference class such as homo sapiens sapiens, even though such a conclusion—although inoffensive—seems counterintuitive.
Another solution that has been proposed recently by George Sowers, in an article published in 2002 in the Mind journal, is as follows. According to the author, the analogy with the urn behind the Doomsday argument is not valid, because our individual birth rank is not obtained randomly as are the numbers of the balls extracted from the urn. Indeed, our birth rank is indexed on the temporal position which corresponds to our birth. Therefore, Sowers concludes, the reasoning underlying the Doomsday argument is misleading, since it is based on a false analogy. However, Sowers’ analysis is not fully convincing. Indeed, one can easily imagine an analogy with a slightly different urn, where the drawing of the ball occurs randomly, but where the number of the ball is indexed on the corresponding time position. It suffices for this to consider a device with an urn from which ball #n for example is extracted at random. Then the mechanism expels the ball #1 at time T1, the ball #2 at time T2, the ball #3 at time T3, the ball #4 time T4, … and finally the ball #n at time Tn. The device then stops. In this case, it seems clear that the draw of the ball was made randomly, even though the number of the ball is indexed on the corresponding time position.
16. The Ship of Theseus Problem
In the literature, one finds the first trace of the problem of the ship of Theseus in the works of Plutarch. The problem can be described as follows. Theseus has a ship with which he goes overseas on a given day, accompanied by several of his companions. Let A be this ship, which is thus the “ship of Theseus.” During the trip, multiple damages necessitate many repairs and thus quite often, parts of the ship must be replaced with new ones. Many years go by, and as the return time approaches, it transpires that all parts of the ship have finally been replaced. Thus, upon the return of Theseus to Greece, the ship does not include any of its original parts. Let us call B the ship of Theseus upon his return to Greece. Now the question arises: Is ship A identical to ship B? In other words, is ship B still the ship of Theseus?
It is interesting to model this problem more accurately. Let us consider thus that ship A possesses n parts (boards, metal, rope, etc.), which are so many parts that can be denoted by a1, a2, a3,…, an-1, an. Similarly, the parts of ship B are b1, b2, b3,…, bn-1, bn. We can then denote ship A by a1a2a3… an-1an and ship B by b1b2b3… bn-1bn. Over the years, i.e. from time T0 to time Tn, the replacement process of the n parts comprises the following successive steps:
|(1)||a1a2a3… an-1an||in T0|
|(2)||b1a2a3… an-1an||in T1|
|(3)||b1b2a3… an-1an||in T2|
|(4)||b1b2b3… an-1an||in T3|
|(5)||b1b2b3… bn-1an||in Tn-1|
|(6)||b1b2b3… bn-1bn||in Tn|
It turns out at this stage that two hypotheses can be made:
|(7)||ship B is identical to ship A|
|(8)||ship B is not identical to ship A|
Intuitively, what justifies the fact that ships A and B are identical is that in everyday life, the mere fact of changing a piece of a device does not entail that this device is different. In the same way, intuitively, the ship’s identity remains identical each time a board or a metal part is replaced. On this basis, it can be concluded that ship B is identical to ship A.
However, another strong argument favors, conversely, the hypothesis that ships A and B are not identical. In effect, all parts of the ship have been changed over the years. Thus, ship B has no original parts of ship A. How, under these conditions, can we still consider that ships A and B are identical? Under the principle that two objects that have no part in common are different, the conclusion follows that the two vessels are different.
The statement of the problem of the ship of Theseus is often associated with a second part, which is as follows. As the vessel moves away from Greece at the time of departure, it is accompanied by a second ship, in charge of assistance. Each time a repair is carried out on the ship of Theseus, the assistance ship gets the old piece that has been changed. And the master of the assistance ship decides, with the help of his crew, to rebuild identically the original ship of Theseus. This way, when it reaches Greece upon his return, the second ship has all the planks of the original ship. Let C be the assistance ship. The question is then: Is ship C identical to ship A? Now it turns out even more clearly than before that ship C is identical to ship A, since both are made up of exactly the same boards. In the latter variation of the problem of the ship of Theseus, we now face four hypotheses:
|(i)||ship B is identical to ship A and ship C is identical to ship A|
|(ii)||ship B is identical to ship A and ship C is not identical to ship A|
|(iii)||ship B is not identical to ship A and ship C is identical to ship A|
|(iv)||ship B is not identical to ship A and ship C is not identical to ship A|
A first type of solution that was proposed to solve the problem of the ship of Theseus is the idea that it is but a variation of the sorites paradox. Yet a closer examination reveals that the problem of the ship of Theseus is based on defining the criteria of identity between two objects. The crucial question that arises here is the following: under which conditions is an object A identical to an object B, and in particular under which conditions does the identity of an object persist over time? Given the lack of a consensual response to the latter question, we can consider that we do not have a satisfactory solution to the problem of the ship of Theseus.
Another type of solution has been put forth by Derek Parfit, in his book Reasons and Persons, published in 1984. According to Parfit, it is the fact of formulating the two hypotheses in terms of an identity relationship that is at the origin of the problem, because it would require us to reformulate the problem with regard to a different type of relationship, which can be denoted by R. The conclusion thus results that the original ship of Theseus is in relation R with the two ships, A and B. However, such an analysis does not prove entirely convincing, because while the fact of replacing the identity relation with another relation eliminates the problem, such a solution does not veritably handle the pressing issue raised by the problem of the ship of Theseus, which deals specifically with our intuitive notion of identity and the conditions of its persistence through time.
17. Hempel’s Problem
Hempel’s problem was described by Carl Hempel in an article published in 1945 in the journal Mind, in the context of the study of confirmation theory. The starting point is the following assertion: “All ravens are black.” Clearly, the discovery of a black raven confirms this hypothesis. Similarly, this hypothesis would also be invalidated by the discovery of a blue raven. However, it turns out that the assertion that “All ravens are black” is equivalent to the assertion that “All non-black objects are non-ravens.” Similarly, we can consider validly that everything which confirms a given proposition P also confirms a proposition P* which is equivalent to it. But this then entails that the discovery of a pink flamingo or a blue umbrella, which confirms the proposition that “All non-black objects are non-ravens,” also confirms the assertion that “All ravens are black.” And the latter conclusion proves to be paradoxical.
The reasoning on which Hempel’s problem is based can thus be described in detail:
|(1)||All ravens are black||hypothesis 1|
|(2)||All non-black objects are non-ravens||hypothesis 2|
|(3)||(2) is equivalent to (1)||contraposition|
|(4)||the instances that confirm a proposition P also confirm a proposition P* which is equivalent||premise|
|(5)||the discovery of a pink flamingo confirms (2)||from (3),(4)|
|(6)||∴ the discovery of a pink flamingo confirms (1)||from (4),(5)|
One can observe here that the logical structure of the proposition (1) according to which “All ravens are black” has the form:
|(7)||All X are Y|
whereas that of (2) according to which “All non-black objects are non-ravens” is the following:
|(8)||All non-Y are non-X|
In fact, the structure of the contrapositive form (8) is clearly equivalent to (7). As we see it, the propositions (1) and (2) are based on four properties, which correspond respectively to: raven, non-raven, black, and non-black. These four properties determine four categories of objects: black ravens, non-black ravens, black non-ravens and non-black non-ravens.
It should be noticed here that Hempel’s problem is not, strictly speaking, a paradox, because it does not entail a genuine contradiction. However, the conclusion resulting from the reasoning inherent to Hempel’s problem proves highly counter-intuitive. However, one solution that has been proposed to solve Hempel’s problem is based on the acceptance of its conclusion (6). According to this type of solution, the discovery of a pink flamingo actually confirms that all ravens are black but only to an infinitesimal degree, for the class of non-ravens contains an extremely high number of objects. Thus, under this type of solution, the discovery of a non-raven confirms the proposition (1) according to which “All ravens are black,” but only at an infinitesimal degree.
Paul Feyerabend, in an article published in 1968 in the British Journal for the Philosophy of Science, believes that Hempel’s paradox and Goodman’s paradox admit the same type of solution. According to Feyerabend, one should only consider valid, from a scientific perspective, the negative instances (those that refute a given hypothesis), thus leading one to purely and simply ignore the positive instances (those that confirm a hypothesis). Once one ignores the latter, the step that leads one to put on the same level the instances confirming (2) and those confirming (1) is then blocked, and therefore, the paradox disappears. Nevertheless, Feyerabend’s approach proved too radical, because it turns out that to confirm a hypothesis H1 is also to refute the opposite hypothesis H2, and conversely, to refute the hypothesis H1 is also to confirm the opposite hypothesis H2. Thus, a given instance constitutes a positive instance for a given hypothesis as well as a negative instance for the opposite hypothesis. For this reason, Feyerabend’s approach did not prove to be truly convincing.
Another type of solution that has been proposed to solve Hempel’s problem is that a predicate such as “black” should not be used unrestrictedly in inductive practice. According to this type of solution, one should restrict oneself to those predicates that are projectible, since every predicate is likely to give rise to many variations built on the model of “grue.” According to this type of analysis, Hempel’s problem and Goodman’s paradox are the result of the unrestricted application of all predicates for inductive processes. However, such an analysis does not prove convincing. Indeed, “black”, unlike “grue,” does not include any temporal clause. It is “black” that is projected here, not “black before T,” for the projection of a predicate such as “black” is carried out regularly in common practice in an entirely satisfactory way, so that it cannot reasonably be considered to renounce any logical inference relating to the predicate “black”.
18. McTaggart’s Argument
In an article that remains famous, published in 1908 in the Mind journal, John Ellis McTaggart described an argument intended to prove that time is not real. McTaggart begins by distinguishing two types of properties of time positions:
Positions in time, as time appears to us prima facie, are distinguished in two ways. Each position is earlier than some, and later than some, of the other positions. And each position is either past, present, or future. The distinctions of the former class are permanent, while those of the latter are not. If M is ever earlier than N, it is always earlier. But an event, which is now present, was future and will be past.
McTaggart terms B series the first distinction, by virtue of which any temporal position M is placed before but also after some other time positions. He also points out a constant property of the B Series: when an event M is earlier than an event N at a given time, it turns out to be earlier than N permanently. McTaggart also terms A Series the second distinction, by virtue of which any temporal position M belongs either to the past or the present or the future. McTaggart notices that the A series are such that each event M is in turn past, present and future. Thus, an event that is present was future and will be past. Similarly, an event that is past was present and future. Finally, an event that is future will be present and past. Hence, the second distinction highlights a non-permanent feature of time.
McTaggart then continues his argument by showing how time must necessarily have all properties of the A series. Suppose, McTaggart says, that time is defined only by means of the B series. In this case, we are not in a position to account for an essential element of time, namely change. Thus, McTaggart continues, it proves to be necessary to resort to the A series to account for the essential properties of time.
Finally, McTaggart sets out to demonstrate how the properties of the A series lead to a contradiction, for the A series are mutually exclusive: an event cannot be past, present and future. The intuition which governs our notion of time is that a given event cannot be past, present and future simultaneously. However, McTaggart considers a given time position M: the latter is present, will be past and was future. But “will be past” is tantamount to “is past at a future time position” and similarly, “was future” is tantamount to “is future at a past time position.” Thus, we define past with regard to future, and future with regard to past. This results in a circular definition. This shows the inconsistency of the A series. Therefore, no event can have all properties of the A series. It follows that time cannot present all the properties of the A series. Thus, concludes McTaggart, time has no reality.
The structure of McTaggart’s argument can thus be described in detail as follows:
|(1)||any time position has two distinct properties: the A series and the B Series||premise|
|(2)||the B Series cannot account for change||premise|
|(3)||change is an essential element of time||premise|
|(4)||the B series cannot account for an essential element of time||from (2),(3)|
|(5)||time must have the properties of the A Series in order to account for an essential element: change||from (1),(4)|
|(6)||time possesses the properties of the A Series||hypothesis|
|(7)||in the A Series, a future event is defined with respect to the past|
|(8)||in the A Series, a present event is defined with respect to the present|
|(9)||in the A Series, a past event is defined with respect to the future|
|(10)||∴ in the A Series, the definitions are circular||from (7),(8),(9)|
|(11)||time cannot have the properties of the A series||from (10)|
|(12)||∴ time is unreal||from (5),(11)|
An objection that can be raised against McTaggart’s argument is that the fact that the B series are not sufficient to account for the essential properties of time does not prove that it is indispensable to have recourse to the A series. For perhaps could one find another series—let us call it the D series—which would account for the properties of time, in combination with the B series, but without exhibiting the disadvantages of the A series. In other words, there may be other alternatives to the A series, which would adequately account for the intrinsic properties of time.
Another objection that was made against McTaggart’s argument, especially by Bertrand Russell, is that the A series may be obtained logically starting from the B series. Thus, according to Russell, the concepts of past, present and future can be defined starting from the relationships before, during and after, which are then the primitive terms. Thus, past, present and future are respectively defined as: before T, during T, after T. Russell’s objection is intended to show how the A series are ultimately not necessary to describe the properties of time. However, Russel’s definition has the disadvantage of including a reference to time T. And it is allowed to think that this implicit reference to T assimilates itself to the “present time”. This leads us finally to define the present as “during present time” in a manner which proves, however, also circular.
19. The Ontological Argument
An ontological argument is an argument that concludes as to the existence of God from a priori considerations only: that is to say, from premises that are not based on empirical data or physical evidence. An ontological argument is intended to constitute a proof of the existence of God. However, unlike classical proofs resulting from mere observation of the reality, such evidence is based only on reasoning. There exist, then, several types of ontological argument. The earliest is due to St. Anselm of Canterbury (1077). The starting point is the consideration of a being of which we cannot conceive of a greater being. If it does not exist, we can therefore conceive of a being of which one cannot conceive of a greater being and which moreover exists. But this implies that we can conceive of a greater being that the being of which we cannot conceive of a greater being, and this latter conclusion turns out to be contradictory. Thus, taking into account the assumption according to which the being of which we cannot conceive of a greater being does not exist, leads to a contradiction. Therefore the being of which we cannot conceive of a greater being exists. The ontological argument of St. Anselm can thus be described in detail as follows:
|(1)||I can conceive of a being of which one cannot conceive a greater being||premise|
|(2)||either a being of which one cannot conceive a greater being exists or he does not exist||dichotomy|
|(3)||if a being of which one cannot conceive a greater being does not exist||hypothesis 1|
|(4)||then I can conceive of a being of which one cannot conceive a greater being but who exists||from (3)|
|(5)||I can conceive a being greater than the being of which one cannot conceive a greater being||from (3),(4)|
|(6)||∴ a being of which one cannot conceive a greater being exists||from (2),(4)|
A slightly different ontological argument is due to Descartes, who describes it in his Meditations. According to Descartes, God, by definition, is a perfect being. It therefore has all qualities. Hence, it also has that of existing. Therefore God exists. Descartes’ ontological argument emphasizes the definition of God as a perfect being. The original excerpt of the Meditations that contains Descartes’ ontological argument is the following:
But now, if just because I can draw the idea of something from my thought, it follows that all which I know clearly and distinctly as pertaining to this object does really belong to it, may I not derive from this an argument demonstrating the existence of God? It is certain that I no less find the idea of God, that is to say, the idea of a supremely perfect Being, in me, than that of any figure or number whatever it is; and I do not know any less clearly and distinctly that an [actual and] eternal existence pertains to this nature than I know that all that which I am able to demonstrate of some figure or number truly pertains to the nature of this figure or number, and therefore, although all that I concluded in the preceding Meditations were found to be false, the existence of God would pass with me as at least as certain as I have ever held the truths of mathematics (which concern only numbers and figures) to be.
This indeed is not at first manifest, since it would seem to present some appearance of being a sophism. For being accustomed in all other things to make a distinction between existence and essence, I easily persuade myself that the existence can be separated from the essence of God, and that we can thus conceive God as not actually existing. But, nevertheless, when I think of it with more attention, I clearly see that existence can no more be separated from the essence of God than can its having its three angles equal to two right angles be separated from the essence of a [rectilinear] triangle, or the idea of a mountain from the idea of a valley; and so there is not any less repugnance to our conceiving a God (that is, a Being supremely perfect) to whom existence is lacking (that is to say, to whom a certain perfection is lacking), than to conceive of a mountain which has no valley.
More precisely, the structure of Descartes’ ontological argument can be thus defined:
|(1)||God is a perfect being||definition|
|(2)||God is a being who possesses all qualities||from (1)|
|(3)||existence constitutes a quality||premise|
|(4)||∴ God exists||from (2),(3)|
Ontological arguments have been the subject of multiple objections in the literature. Famous criticism notably emanates in particular from Kant, in his Critique of Pure Reason, who considers that existence does not constitute a genuine property. This has the effect of blocking premise (3) of Descartes’ ontological argument, thereby neutralizing the reasoning that leads to the conclusion that God exists. According to Kant, we cannot consider that the mere fact of existing is a property, in the same way as red is the property of a tomato or hard is the property of a stone. For Kant, it is the very existence of a thing x which is a necessary condition for the attribution of its properties (color, size, density, roughness, hardness, etc.).
In a general way, ontological arguments are usually not regarded as truly convincing evidence of God’s existence, and they usually prove insufficient to convince non-theists of God’s existence.
20. The Fine-Tuning Argument
The fine-tuning argument belongs to the category of arguments which aim to prove the existence of God. The argument rests on the fact that a large number of cosmological constants governing our universe are such that if they had been slightly different, the emergence of intelligent life based on carbon chemistry as we observe it on Earth would not have been possible. Among these constants are: the ratio of the respective masses of the electron and the proton, the age of the universe, the neutrino mass, the average distance between the stars, the speed of light, the universal cosmological constant, Planck’s constant, etc. The argument is underpinned by the fact that each of these parameters could have had a slightly different value, which then would not have allowed the emergence of life. Consider, for example, the speed of light in a vacuum (v = 299,792.458 km/s): if it had been even slightly higher, then the stars would have emitted too much light to allow for the emergence of life. And even if the speed of light had been fractionally lower, light emission by the stars would have been insufficient to permit the emergence of life. The same goes for the gravitational constant (G = 6,672.10-11 Nm2kg-2): if the latter had had a slightly higher value, the stars would have had too high a temperature and would have been consumed far too quickly to allow the emergence of life based on carbon chemistry. Similarly, if the gravitational constant had been slightly lower, the stars’ temperature would have been too low to allow for the formation of many of the chemical elements required for the appearance of life. One can also consider the ratio of the mass of the electron relative to the proton (me/mp = 5.446170232x10-4): if it had been slightly different, the chemical bonds that would have resulted would have been insufficient to permit the emergence of life. Finally, if the rate of expansion of the universe had been slightly higher, no galaxy would have formed, and likewise, if it had been slightly lower, the universe would have collapsed even before the formation of stars, etc.
Thus, these different parameters, in virtue of the fine-tuning argument, were not determined randomly, but according to a specific purpose: the emergence of intelligent life in the universe. This particular purpose reflects the presence of a divine plan and thus finally the existence of God.
We can detail as follows the different steps of the fine-tuning argument:
|(1)||several cosmological constants governing our universe have values such that they permit the emergence of intelligent carbon-based life||premise|
|(2)||the cosmological constants that govern our universe could have had many different values||hypothesis|
|(3)||if the values of these cosmological constants were slightly different, then the emergence of intelligent carbon-based life would not have been possible||from (2)|
|(4)||if the cosmological constants had been obtained at random, then the probability that their setting was optimal would have been extremely low||from (2)|
|(5)||the optimal setting of the cosmological constants is not due to chance||from (4)|
|(6)||the optimal setting of the cosmological constants was done purposely in order to allow the emergence of intelligent life||from (3),(5)|
|(7)||the optimal setting of the cosmological constants was made by God||from (6)|
|(8)||∴ God exists||from (7)|
Several objections have been raised against the fine-tuning argument. One of them in particular relies on the speculative idea, defended by a number of cosmologists, that the universe we observe is not the only one, but is only one universe among many others, in a system composed of multiple causally independent universes. In this context, there exist many other universes, entirely different from ours, which have quite distinct cosmological parameters. As we can see, this objection targets directly step (5) of the reasoning underlying the fine-tuning argument, according to which the optimal setting of cosmological constants is not due to chance, for the hypothesis of multiple universes is quite consistent with the fact that the parameters of our universe may have been obtained randomly.
21. The Dreaming Argument
The Dreaming argument is due to Descartes. It can be formulated very simply. It consists of an argument which leads to the conclusion that our current perceptions may well be illusory and misleading because they are in all respects similar to those that we have when we dream. When we are indeed in a dream state, our perceptions are in effect realistic enough that they are able to create the illusion of reality. The Dreaming argument is described in the following passage (First Meditation) of the Metaphysical Meditations:
But it may be that although the senses sometimes deceive us concerning things which are hardly perceptible, or very far away, there are yet many others to be met with as to which we cannot reasonably have any doubt, although we recognize them by their means. For example, there is the fact that I am here, seated by the fire, attired in a dressing gown, having this paper in my hands and other similar matters. And how could I deny that these hands and this body are mine, were it not perhaps that I compare myself to certain persons, devoid of sense, whose cerebella are so troubled and clouded by the violent vapors of black bile, that they constantly assure us that they think they are kings when they are really quite poor, or that they are clothed in purple when they are really without covering, or who imagine that they have an earthenware head or are nothing but pumpkins or are made of glass. But they are mad, and I should not be any the less insane were I to follow examples so extravagant.
At the same time I must remember that I am a man, and that consequently I am in the habit of sleeping, and in my dreams representing to myself the same things or sometimes even less probable things, than do those who are insane in their waking moments. How often has it happened to me that in the night I dreamt that I found myself in this particular place, that I was dressed and seated near the fire, whilst in reality I was lying undressed in bed! At this moment it does indeed seem to me that it is with eyes awake that I am looking at this paper; that this head which I move is not asleep, that it is deliberately and of set purpose that I extend my hand and perceive it; what happens in sleep does not appear so clear nor so distinct as does all this. But in thinking over this I remind myself that on many occasions I have in sleep been deceived by similar illusions, and in dwelling carefully on this reflection I see so manifestly that there are no certain indications by which we may clearly distinguish wakefulness from sleep that I am lost in astonishment. And my astonishment is such that it is almost capable of persuading me that I now dream.
The Dreaming argument can be detailed as follows:
|(1)||when I am awake, I have some perceptions||premise|
|(2)||when I dream, I also have some perceptions||premise|
|(3)||the perceptions that I have when I am awake are identical in all respects to those that I have when I dream||premise|
|(4)||I do not have a criterion that allows me to distinguish my perceptions when I am awake or when I dream||from (3)|
|(5)||I have no proof that I am not currently in a state of dream||from (4)|
|(6)||∴ it is possible that I am currently in a state of dream||from (5)|
|(7)||when I dream, my perceptions are false||premise|
|(8)||∴ it is possible that all my current perceptions are false||from (6),(7)|
Descartes’ Dreaming argument has given rise to several contemporary variations. One of these modern variations rests on the idea that we are “brains in a vat.” The film Matrix, by Larry and Andy Wachowski, also develops a variation of this idea.
An objection to the Dreaming argument was raised by Barry Stroud, in a book published in 1989. According to this objection, premise (4) is false, because it is quite possible to carry out a test to determine whether each of us is or is not in a dream state. By using sensors that determine whether brain waves characteristic of the dream state are produced by the brain, we can determine whether or not a person dreams, and thus provide a definitive and reliable answer to this question. However, this objection has failed to convince many authors, who have argued that such a response presupposes that you are not dreaming when you perform the test. Under this assumption, the fact of carrying out a test proves indeed successful. But suppose on the contrary that we are in a dream state when we perform the test. In this case, the test is part of our dream and we cannot validly rely on it trustfully. Thus, the idea underlying this objection presupposes ultimately that we are not dreaming, whereas it is this very question which is at hand here.
Another type of objection may also be raised against the Dreaming argument. Suppose that this last argument is perfectly valid and that its conclusion is irrefutable. In this case, we then have compelling evidence that we are in a dream state. But if this were the case, would it not follow that the Dreaming argument itself is a pure product of our dreams, and therefore something illusory? Thus, in no case could it be an argument on which we could base our knowledge. As we see it, such a property has the effect of rendering the Dreaming argument self-refuting.
22. The “Brains in a Vat” Experiment
The “brains in a vat” experiment was enunciated by Hilary Putnam in his book Reason, Truth and History, published in 1982. The argument begins with the following question: am I not a brain in a vat? In other words, am I certain that some mad scientist did not abduct me, and did not then remove my brain and place it in a nutrient liquid, and did not finally simulate all information usually reaching my brain with an especially sophisticated device. In this way, my feelings, my perceptions, my thoughts, etc. would be the effect of stimulations that the mad scientist sends to all of my neurons using his device. Am I quite sure that I do not find myself in a situation like this? If this were the case, stimulations sent to my brain would produce exactly the impressions that are mine as they occur when I have sensations, perceptions, emotions or thoughts, in normal conditions. How then can I be absolutely sure that I am I not a brain in a vat?
However, Putnam’s argument does not aim to suggest that we are actually brains in vats. For Putnam, it is indeed clear, however, that we are not “brains in vats.” For him, this results from the simple consideration of the assertion that “we are brains in vats.” Putnam proposes to prove that the latter assertion is always false. He distinguishes two hypotheses: if (i) we are not brains in vats, then it is false that we are brains in vats, and if (ii) we are brains in vats, then the concepts and words that we use on a daily basis do not refer to real objects, but to virtual objects, which are the result of a simulation. Such is the case when we use concepts such as “table”, “chair”, “umbrella”, etc. In this case, our concepts of “table” or “umbrella” refer not to a table or an umbrella, but to a simulation of table or of umbrella that comes from electrical impulses sent to our brains by a sophisticated electronic device. And this is also the case when we make use of words like “brain” or “vat”. In this case, we are then referring ourselves to a simulated brain or vat. Thus, when we affirm that “we are brains in vats”, we state that “we are simulated brains in simulated vats.” But this does not correspond to reality. Thus, if we consider the hypothesis that we are brains in vats, it is also true that it is false that we are brains in vats. In conclusion, whatever the hypothesis, it is false that we are brains in vats.
Putnam’s argument can be described more precisely as follows:
|(1)||there does not exist any internal criterion to determine whether or not our sensations, our perceptions, our emotions and our thoughts are stimulated by a device||premise|
|(2)||if we are brains in vats||hypothesis 1|
|(3)||then our sensations, our perceptions, our emotions and our thoughts are stimulated by a device||from (2)|
|(4)||then “we are brains in vats” means “we are simulated brains in simulated vats”||from (3)|
|(5)||if we are not brains in vats||hypothesis 2|
|(6)||then our sensations, our perceptions, our emotions and thoughts are not stimulated by a device||from (5)|
|(7)||then it is false that “we are brains in vats”||from (5)|
|(8)||∴ it is necessary to resort to an external criterion to determine whether or not we are brains in vats||from (1),(4),(7)|
Putnam’s thought experiment aims to emphasize that the internal states resulting from stimulation by, on the one hand, an external device of a brain in a vat, and on the other hand, the thoughts and perceptions of a normal person, cannot be distinguished, for the resulting internal mental states are identical in both cases. Therefore, it is necessary to resort to external criteria to differentiate them. Thus, the point of view expressed by Putnam turns out to be fundamentally externalist. Putnam’s argument stresses that the meaning of words or phrases depends not only on internal content: that is to say, our thoughts, our emotions, etc. According to Putnam’s famous formula, “Meanings just are not in the head.”
23. The Teleological Argument
The teleological argument or argument from design belongs, like the ontological argument, to a family of arguments that aim to prove the existence of God. The teleological argument is based on the simple idea that our universe is so complex and so well-arranged that this can only be the manifestation of the design of an intelligent entity. The complex scheduling of our universe thus demonstrates that it has a Creator.
The argument from design can be described in more detail as follows:
|(1)||our universe is very complex and well organized||premise|
|(2)||the complexity and settlement of our universe can only be the manifestation of the design of an intelligent being||from (1)|
|(3)||an intelligent being is the Creator of our universe||from (2)|
|(4)||∴ God is the Creator of our universe||from (3)|
A famous formulation of the argument from design is notably due to William Paley (1743-1805), in his book Natural Theology, published in 1802. Paley describes the argument in the following terms:
In crossing a heath, suppose I pitched my foot against a stone, and were asked how the stone came to be there; I might possibly answer, that, for anything I knew to the contrary, it had lain there forever: nor would it perhaps be very easy to show the absurdity of this answer. But suppose I had found a watch upon the ground, and it should be inquired how the watch happened to be in that place; I should hardly think of the answer which I had before given, that, for anything I knew, the watch might have always been there.
According to Paley, the reason why we cannot conceive that the watch has been there forever is that its different parts have been assembled purposely, and that this purpose can only be the work of an intelligent being. Paley’s argument is based on an analogy between the watch and the universe, and leads to the conclusion that the universe was created purposely, and that design is that of God.
An objection that has been raised against the design argument is targeted against premise (1), that our universe is very well arranged. But to this, it can be retorted that this is only the expression of a specific viewpoint regarding our universe, because from another standpoint, our universe could seem poorly arranged. It would suffice for that to consider that disorder is present everywhere in the world, as it could be observed that our world is troubled by frequent earthquakes, devastating tidal waves, destructive cyclones, etc. and undergoes, in general, many natural disasters. From this point of view, we cannot really consider the universe as well-arranged.
Another objection targets directly step (2), according to which the scheduling of our beautiful universe can only be the work of an intelligent being. Under this objection, the complexity of our universe and its sophisticated layout are well proven, but that does not mean that it results from the work of a creator, for one could also imagine that many universes coexist, some being very simple and basic, while others are complex and sophisticated. As observers, we obviously can only find ourselves in a complex and well-ordered universe, allowing in particular the emergence of life based on carbon chemistry. However, there could well exist many universes very different from ours, some of which are very crude and rudimentary, and devoid of any observers.
24. Pascal’s Wager
Pascal’s wager is an argument contained in paragraph 233 of the Pensées. It is one of the most famous arguments in the philosophy of religion, and aims to provide the reader with solid reasons to believe in the existence of God. Pascal exposes therein the alternative in front of which we find ourselves placed: either God exists or he does not exist. Faced with such a situation, we can bet in favor of the existence of God, or in favor of his non-existence. Pascal then analyzes the consequences that follow from a bet in favor of either option. He then considers the four cases that are thus determined. If I bet in favor of the existence of God and God exists (i), then I receive an infinite gain. If I bet in favor of the existence of God and God does not exist (ii), then this results in a zero loss. If I bet for the non-existence of God and God exists (iii), there follows an infinite loss. Lastly, if I bet for the non-existence of God and God does not exist (iv) then I receive no gain or loss. Thus, it turns out that if I bet for the non-existence of God, I am exposed to an infinite loss. Therefore, concludes Pascal, it is wiser to bet in favor of the existence of God, because there follows either an infinite gain or a zero loss.
The excerpt from Pascal’s Pensées that contains the Wager’s argument is the following:
If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is. This being so, who will dare to undertake the decision of the question? Not we, who have no affinity to Him.
Who then will blame Christians for not being able to give a reason for their belief, since they profess a religion for which they cannot give a reason? They declare, in expounding it to the world, that it is a foolishness, stultitiam; and then you complain that they do not prove it! If they proved it, they would not keep their word; it is in lacking proofs, that they are not lacking in sense. “Yes, but although this excuses those who offer it as such, and takes away from them the blame of putting it forward without reason, it does not excuse those who receive it.” Let us then examine this point, and say, “God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.
Do not then reprove for error those who have made a choice; for you know nothing about it. “No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all.”
Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.—“That is very fine. Yes, I must wager; but I may perhaps wager too much.”—Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. It is all divided; wherever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all. And thus, when one is forced to play, he must renounce reason to preserve his life, rather than risk it for infinite gain, as likely to happen as the loss of nothingness.
For it is no use to say it is uncertain if we will gain, and it is certain that we risk, and that the infinite distance between the certainty of what is staked and the uncertainty of what will be gained, equals the finite good which is certainly staked against the uncertain infinite. It is not so, as every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason. There is not an infinite distance between the certainty staked and the uncertainty of the gain; that is untrue. In truth, there is an infinity between the certainty of gain and the certainty of loss. But the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss. Hence it comes that, if there are as many risks on one side as on the other, the course is to play even; and then the certainty of the stake is equal to the uncertainty of the gain, so far is it from fact that there is an infinite distance between them. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain. This is demonstrable; and if men are capable of any truths, this is one.
“I confess it, I admit it. But, still, is there no means of seeing the faces of the cards?”—Yes, Scripture and the rest, etc. “Yes, but I have my hands tied and my mouth closed; I am forced to wager, and am not free. I am not released, and am so made that I cannot believe. What, then, would you have me do?”
True. But at least learn your inability to believe, since reason brings you to this, and yet you cannot believe. Endeavor then to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith, and do not know the way; you would like to cure yourself of unbelief, and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make you believe, and deaden your acuteness.—“But this is what I am afraid of.”—And why? What have you to lose?
But to show you that this leads you there, it is this which will lessen the passions, which are your stumbling-blocks.
The end of this discourse.—Now, what harm will befall you in taking this side? You will be faithful, honest, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others? I will tell you that you will thereby gain in this life, and that, at each step you take on this road, you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognize that you have wagered for something certain and infinite, for which you have given nothing.
“Ah! This discourse transports me, charms me,” etc.
If this discourse pleases you and seems impressive, know that it is made by a man who has knelt, both before and after it, in prayer to that Being, infinite and without parts, before whom he lays all he has, for you also to lay before Him all you have for your own good and for His glory, that so strength may be given to lowliness.
The Wager’s argument can be described more precisely as follows:
|(1)||either God exists or God does not exist||dichotomy 1|
|(2)||I can bet either for the existence of God or for its non-existence||dichotomy 2|
|(3)||if I bet if in favor of the existence of God and God exists||case 1|
|(4)||then I get an infinite gain||from (3)|
|(5)||if I bet if in favor of the existence of God and God does not exist||case 2|
|(6)||then it follows a zero loss||from (5)|
|(7)||if I bet if in favor of the non-existence of God and God exists||case 3|
|(8)||then it results in an infinite loss||from (7)|
|(9)||if I bet if in favor of the non-existence of God and God does not exist||case 4|
|(10)||then it does not follow any gain or loss||from (9)|
|(11)||it is rational to make a choice in order to maximize the expected gain and loss||premise|
|(12)||if I bet if in favor of the existence of God||from (3),(5)|
|(13)||then the maximum gain is infinite and the maximum loss is null||from (4),(6)|
|(14)||if I bet if in favor of the non-existence of God||from (7),(9)|
|(15)||then the maximum gain is null and the maximum loss is infinite||from (8),(10)|
|(16)||∴ it is rational to bet in favor of the existence of God||from (11),(13),(15)|
Pascal’s wager argument has led to a number of objections. Some critics, such as Jeffrey in his book The Logic of Decision, published in 1983, or McClennen in an essay published in 1994, blamed steps (3)-(4) and argued that the infinite utility that results from the expected gain when betting in favor of the existence of God is not a realistic gain and therefore has no real practical interest.
In addition, some other authors have pointed out that the intrinsic attitude which underlies the wager is itself questionable. Voltaire particularly found this attitude unbecoming, as it consists of deciding on a matter as serious as the existence of God according exclusively to considerations of public interest. In the situation of the wager, Voltaire considers that one may well have some rational elements for deciding on the existence of God, but that one lacks, however, the moral elements.
25. The Argument from Evil
The argument from Evil is an argument that tends to prove the non-existence of God. Its formulation is very simple. The argument from Evil is based on the fact that evil is present in the world. The presence of suffering and pain is an inherent feature of our world today. Even worse, atrocities and horrendous crimes unfortunately occur every day in the world. The argument from Evil takes into account these undeniable facts and concludes that this demonstrates that God does not exist. There are different formulations of the argument from Evil. According to one of them, God is by definition a perfect being. God, moreover, is the creator of all things. Yet the obvious evil that exists in the world is one of those things. Therefore, according to this variation of the argument, God is the creator of evil. If this is the case, the claim that God is perfect is thus contradicted. The latter contradiction leads to the conclusion that God does not exist. The different steps of the argument from Evil can thus be described as follows:
|(1)||God is perfect||definition|
|(2)||God is the creator of all that exists||definition|
|(3)||evil exists in the world||premise|
|(4)||God is the creator of evil that exists in the world||from (2),(3)|
|(5)||God is not perfect||from (4)|
|(6)||∴ God does not exist||from (1),(5)|
Another formulation of the argument from Evil emphasizes the all-powerfulness of God, and especially the notion of omnipotence. The argument considers that if God exists, then God is all-powerful and as such has the power to eliminate evil. But it turns out that there is evil in the world, in contradiction with the assumption that God exists. This results in the conclusion that God does not exist. This variant of the argument from evil can be described as follows:
|(7)||if God exists||hypothesis|
|(8)||then God is all-powerful||definition|
|(9)||then God has the power to remove evil||from (8)|
|(10)||if God exists then God has the power to remove evil||from (7),(9)|
|(11)||evil exists in the world||premise|
|(12)||God has not the power to remove evil||from (10),(11)|
|(13)||∴ God does not exist||from (10),(12)|
The argument from Evil has been the subject of both old and recent objections. According to a recent objection raised by Alvin Plantinga in his book God and Other Minds, published in 1967, the argument is not valid because it is based on the false premise that God creates evil, or has the power to remove evil. Plantinga considers in contrast that free will is a necessary virtue, and therefore God aims to enable the development of free will in humans. According to Plantinga, God is not responsible for evil (by creating it or making it possible) because evil results directly from the exercise of human choice. And those choices—whether good or bad—made by men themselves are essential to the development of free will.
26. Descartes’ Cogito
The Cogito argument is due to Descartes. It can be formulated very briefly and simply: “I think, therefore I am.” However, in order to understand the exact scope of the Cartesian cogito, it is necessary to delve more deeply into its structure and context.
The original formulation of the Cogito can be found in the Discourse on the Method (Part IV):
I am in doubt as to the propriety of making my first meditations in the place above mentioned matter of discourse; for these are so metaphysical, and so uncommon, as not, perhaps, to be acceptable to every one. And yet, that it may be determined whether the foundations that I have laid are sufficiently secure, I find myself in a measure constrained to advert to them. I had long before remarked that, in relation to practice, it is sometimes necessary to adopt, as if above doubt, opinions which we discern to be highly uncertain, as has been already said; but as I then desired to give my attention solely to the search after truth, I thought that a procedure exactly the opposite was called for, and that I ought to reject as absolutely false all opinions in regard to which I could suppose the least ground for doubt, in order to ascertain whether after that there remained aught in my belief that was wholly indubitable. Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (presentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all the objects (presentations) that had ever entered into my mind when awake, had in them no more truth than the illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth, I think, therefore I am (COGITO ERGO SUM), was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the skeptics capable of shaking it, I concluded that I might, without scruple, accept it as the first principle of the philosophy of which I was in search.
It is tempting at this stage to consider that the Cogito argument can be formulated very briefly: “I think, therefore I am” and that its structure can be described as follows:
|(2)||if I think therefore I am||from (1)|
|(3)||∴ I am||from (1),(2)|
However, this is an interpretation of Descartes’ argument that proves to be restrictive. It appears preferable to describe the Cartesian cogito in a way that better captures its essence, by better taking into account the context of doubt in which occurs the Cogito argument itself, for the Cogito is an argument which aims at demonstrating the existence of self, by taking into account the possibility of being self-deceived about one’s thoughts or perceptions. Descartes goes so far as to consider the case where the object of his own thoughts is wrong, that is to say he is deceived about the existence of sensible things that surround him, for example because he is dreaming. But even in this case, the conclusion that he exists also imposes itself to Descartes. The strength of the argument lies in the fact that even if I admit that I am currently deceived by my own thoughts because their object is false, it follows that I exist by the mere fact that my thoughts are erroneous. Therefore, what the Cogito argument ultimately shows is that I cannot be deceived about the mere fact that I exist, whether my thoughts are misleading or not. Thus, the Cogito argument can be rendered more accurately as follows:
|(4)||the object of my thoughts is either true or false||dichotomy|
|(5)||if the object of my thoughts is true||hypothesis 1|
|(6)||then I think||consequence 1|
|(7)||if the object of my thoughts is false||hypothesis 2|
|(8)||then I think||consequence 2|
|(9)||I think||from (4),(6),(8)|
|(10)||if I think then I do exist||from (9)|
|(11)||∴ I do exist||from (9),(10)|
The Cogito argument is an application of the methodological doubt implemented by Descartes. He undertakes then to doubt the reality of all the knowledge he had previously acquired and that he had always considered as certain, not because he casts serious doubt on its existence, but because such a method allows him to achieve, in an optimum manner, some very certain and better assured knowledge. The Cogito argument is thus an illustration of this methodological doubt, which allows Descartes, in this context, to obtain a firm and steady knowledge, which corresponds to the certainty of his own existence.
27. Lewis Caroll’s Argument
Lewis Carroll’s argument was published in 1895 in the Mind journal. The argument is presented there under the form of a dialogue between Achilles and the tortoise. The problem resulting from this argument can be stated as follows. Let us consider the following steps of reasoning:
|(1)||two things which are equal to a third one are themselves equal||premise|
|(2)||the sides AB and AC of a triangle ABC are both equal to the length DE||premise|
|(Z)||∴ the sides AB and AC of the triangle ABC are equal||from (1),(2)|
At this stage, such reasoning seems to be quite valid. But now consider the following argument, which comprises an additional step (3):
|(1)||two things which are equal to a third one are themselves equal||premise|
|(2)||the sides AB and AC of a triangle ABC are both equal to the length DE||premise|
|(3)||if (1) and (2) are true then (Z) is true||from (1),(2)|
|(Z)||∴ the sides AB and AC of the triangle ABC are equal||from (1),(2),(3)|
Before asserting the conclusion (Z), would it not be better to first acknowledge step (3) as true? Step (3) considers that the reasoning leading to (Z) is valid. It consists here of a necessary step to establish that (Z) is true, for if step (3) turned out to be false, we could not legitimately conclude that (Z) is true. Therefore, it is legitimate to include this step in the reasoning that leads to (Z). At this stage, however, it turns out that if we restore step (3), we must also take into account a new additional step (4), which leads us to consider the whole reasoning that follows:
|(1)||two things which are equal to a third one are themselves equal||premise|
|(2)||the sides AB and AC of a triangle ABC are both equal to the length DE||premise|
|(3)||if (1) and (2) are true then (Z) is true||from (1),(2)|
|(4)||if (1), (2) and (3) are true then (Z) is true||from (1),(2),(3)|
|(Z)||∴ the sides AB and AC of the triangle ABC are equal||from (1),(2),(3),(4)|
But again, it turns out that the above reasoning can be extended by incorporating a new further step:
|(1)||two things which are equal to a third one are themselves equal||premise|
|(2)||the sides AB and AC of a triangle ABC are both equal to the length DE||premise|
|(3)||if (1) and (2) are true then (Z) is true||from (1),(2)|
|(4)||if (1), (2) and (3) are true then (Z) is true||from (1),(2),(3)|
|(5)||if (1), (2), (3) and (4) are true then (Z) is true||from (1),(2),(3),(4)|
|(Z)||∴ the sides AB and AC of the triangle ABC are equal||from (1),(2),(3),(4),(5)|
Such reasoning can be extended ad infinitum and thus results in an infinite regress. Therefore, it follows that we never reach the conclusion (Z).
Lewis Carroll’s argument rests on the fact that, before reaching the conclusion (Z), it should be recognized that the reasoning leading to this conclusion is valid. In a general way, the argument—just like Zeno of Elea’s paradox of the race, but also the paradox of Achilles and the tortoise, which is another paradox from Zeno—emphasizes that before reaching the conclusion (Z), we must go through an infinite series of steps and in these conditions, we never succeed in formulating the conclusion (Z).
Lewis Carroll’s argument notably emphasizes the importance of modus ponens. This inference rule allows the reasoning which has the following structure (P and Q being two propositions):
|(6)||P is true||premise|
|(7)||if P is true then Q is true||premise|
|(8)||∴ Q is true||from (6),(7)|
The argument points to the fact that before applying a rule of inference such as modus ponens, it is necessary to have available a second rule describing how to apply modus ponens, then a third rule describing how we should apply the rule that describes how to apply modus ponens, and so on. An infinite regress ensues.
An objection that has been traditionally raised against Caroll’s argument is that such problem does not occur within formal logic, where each rule is formalized. In this case, the deductive mechanism reduces to a manipulation of symbols. However, such a formal system has the disadvantage of not taking into account the semantic aspect of things, yet this is essential, for the latter aspect proves totally absent from what only reduces then to a manipulation of meaningless symbolic characters.
28. The Twin Earth Thought Experiment
The Twin Earth thought experiment was introduced by Hilary Putnam in an essay published in 1975. Putnam presents three thought experiments, and one of them—the H2O-XYZ thought experiment—introduces the problem of the Twin Earth. Putnam features a planet, the Twin Earth, which proves to be identical in all respects to the Earth, with one difference. This difference relates to the compound body that is referred to as “water” on Earth and whose atomic structure is H2O. On the Twin Earth, there exists indeed a compound body that possesses all the properties of our water, such as being liquid, transparent, odorless, etc. but whose chemical composition is XYZ. Let us term such a compound body water*. On the Twin Earth, people also call the latter compound body “water”. At this stage, according to Putnam, it turns out that water refers to the compound body H2O and water* refers to the compound body XYZ. Thus, water and water* are respectively used in a quite identical manner by Earth people and by the inhabitants of Twin Earth. Moreover, the content of thoughts of an inhabitant of the Earth or of the Twin Earth is quite identical when they each think of water or of water*. Therefore, it follows that the semantic content of their respective thoughts cannot be determined in a purely internal way, and therefore can only be elucidated by resorting to external data. It is here, according to Putnam, that the lesson of the problem resulting from the Twin Earth experiment situates itself. One can indeed wonder whether or not the meaning, the semantic content of a word or concept, finds itself exclusively in our brain. According to Putnam, what the Twin Earth experiment demonstrates is that a negative answer should be given to this question, for only the use of external data in the Twin Earth thought experiment allows us to determine the semantic content of the thoughts of a human being and of an inhabitant of the Twin Earth when they think or speak respectively of water or of water*. Thus, Putnam concludes, it is necessary to adopt an externalist conception for the determination of mental content.
The reasoning to which the Twin Earth experiment leads can be thus detailed:
|(1)||on Earth, there exists a liquid, transparent, odorless compound body, etc. whose composition is H2O||premise|
|(2)||the compound body whose composition is H2O is water||definition|
|(3)||on Twin Earth, there exists a liquid, transparent, odorless compound body, etc. whose composition is XYZ||premise|
|(4)||the compound body whose composition is XYZ is water*||definition|
|(5)||the inhabitants of Earth call “water” the compound body whose composition is H2O||premise|
|(6)||the inhabitants of Twin Earth call “water” the compound body whose composition is XYZ||premise|
|(7)||the content of the thoughts of an inhabitant of Earth when he/she thinks to water is x||from (1),(5)|
|(8)||the content of the thoughts of an inhabitant of Twin Earth when he/she thinks to water* is x||from (2),(6)|
|(9)||the content of the thoughts of an inhabitant of Earth or of Twin Earth when they think respectively of water or to water* is identical||from (7),(8)|
|(10)||∴ we must resort to external data in order to distinguish the semantic content of the thoughts of an inhabitant of Earth who thinks of water from the content of the thoughts of an inhabitant of Twin Earth who thinks of water*||from (9)|
At this stage, it turns out that the scope of the problem raised by Putnam extends beyond the sole Twin Earth thought experiment and our concept of water, for a similar reasoning can be applied to all categories of objects referred to by our ordinary language, such as a cloud, a mountain, a chair, etc. For each of our everyday and familiar objects, it proves necessary, by virtue of the Twin Earth experiment, to resort to an external criterion to understand the corresponding semantic content.
It has been objected that the corresponding situation is unrealistic. Indeed, if a compound body should possess properties very similar to those of our water, shouldn’t then its composition be the same as that of water, that is to say H2O? According to this objection, proposition (3), that there exists another planet with a compound body with identical properties to those of water and whose chemical composition is different, proves unrealistic and even contradictory.
29. The Argument against the Principle of Verifiability
The argument against the principle of verifiability results from the work of a group of philosophers belonging to the school of thought of logical positivism. This line of thinking is part of the ideas put forward in 1920-1930 by the Vienna Circle, which notably included Rudolf Carnap and Kurt Gödel. Logical positivism distinguishes two types of meaningful propositions: some propositions (i) are analytical, while some others (ii) can be verified experimentally. Analytical propositions include, for example, mathematical propositions, such as “a triangle has two right angles” (which is analytically false) or “a dog is a mammal” (which is analytically true), of which one can determine the truth or falsity by deduction only. In contrast, experimentally verifiable propositions can be confirmed or disconfirmed empirically. Thus, “I measure 1.73 meters” or “Proxima Centauri is 4.23 light-years from Earth” are propositions that can be tested experimentally. Any other type of proposition, i.e. one that is neither analytically nor experimentally verifiable, is meaningless. Logical positivism, influenced by ideas put forth by Ludwig Wittgenstein, thus leads to the rejection of metaphysical propositions, considered insignificant because they do not meet one of the two above-mentioned criteria. According to this viewpoint, the metaphysical assertions do not have a logical basis, because they do not meet the criterion of verifiability, under which any claim must be experimentally verified. In this respect, a metaphysical statement should be subject to the possibility of confirmation or disconfirmation. Such is not the case, and therefore, metaphysical assertions must be discarded.
However, such an argument based on the principle of verifiability has been the subject of the following objection, notably due to Ewing, in his book The Fundamental Questions of Philosophy published in 1962: the principle of verifiability itself is not experimentally verifiable. Thus, the principle of verifiability itself does not satisfy the criterion of verifiability, for one does not have at one’s disposal a process allowing one to verify it experimentally. Thus, the principle of verifiability itself falls victim to the very principle that it claims to promote. This shows how this principle actually proves too restrictive. The argument against the principle of verifiability can be described step by step as follows:
|(1)||either the principle of verifiability prevails or it does not prevail||dichotomy|
|(2)||in virtue of the principle of verifiability, any claim must be verifiable||premise|
|(3)||to be verifiable, for a given proposition, consists in the fact that it is possible to confirm or to disconfirm it||definition|
|(4)||the principle of verifiability cannot be confirmed experimentally||premise|
|(5)||the principle of verifiability cannot be disconfirmed experimentally||premise|
|(6)||the principle of verifiability cannot be confirmed nor disconfirmed experimentally||from (4),(5)|
|(7)||the principle of verifiability is not verifiable||from (3),(6)|
|(8)||∴ the principle of verifiability does not prevail||from (7)|
30. The Allegory of the Cave
The celebrated Allegory of the Cave was described by Plato in the Republic (Book VII). Plato depicts humans who have been chained, since their childhood, to the walls of a cave. These prisoners are chained in such a way that they cannot move their heads and cannot see each other. However, the cave communicates with the outside via an opening. All that these prisoners can observe are merely reflections of people and animals that pass outside the cave and shadows of flowers, rocks, etc. as they appear on the walls of the cave. For the prisoners, reality itself is restricted to shadows and reflections that they observe on these walls. But one day, one of the prisoners manages to break his chains and escape the cave. He goes out of the cave for the first time, and in the light of day he discovers real people, authentic animals, genuine flowers, etc. in their original shapes and colors. His one idea is then to return to the cave and inform his former companions that what they see on the walls of the cave are mere reflections, shadows and images of another level of reality, which would appear if they also broke their ties and went into the light of day. Returning to the cave, he begins to explain to his chained companions that what they see is only a reflection of the true reality. But his former companions do not believe him, and eventually kill him. The allegory clearly has the structure of an analogy, since for Plato, the shadows that appear on the walls of the cave represent the world of appearances. In contrast, real objects such that one can observe in the daylight belong to the world of Ideas.
The extract of the Republic that includes the Allegory of the cave depicts the following dialogue between Socrates and Glaucon:
[Socrates:] And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: — Behold! human beings living in a underground cave, which has a mouth open towards the light and reaching all along the cave; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets.
[Glaucon:] I see.
— And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent.
— You have shown me a strange image, and they are strange prisoners.
— Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave?
— True, he said; how could they see anything but the shadows if they were never allowed to move their heads?
— And of the objects which are being carried in like manner they would only see the shadows?
— Yes, he said.
— And if they were able to converse with one another, would they not suppose that they were naming what was actually before them?
— Very true.
— And suppose further that the prison had an echo which came from the other side, would they not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow?
— No question, he replied.
— To them, I said, the truth would be literally nothing but the shadows of the images.
— That is certain.
— And now look again, and see what will naturally follow if the prisoners are released and disabused of their error. At first, when any of them is liberated and compelled suddenly to stand up and turn his neck round and walk and look towards the light, he will suffer sharp pains; the glare will distress him, and he will be unable to see the realities of which in his former state he had seen the shadows; and then conceive someone saying to him, that what he saw before was an illusion, but that now, when he is approaching nearer to being and his eye is turned towards more real existence, he has a clearer vision, —what will be his reply? And you may further imagine that his instructor is pointing to the objects as they pass and requiring him to name them, —will he not be perplexed? Will he not fancy that the shadows which he formerly saw are truer than the objects which are now shown to him?
— Far truer.
— And if he is compelled to look straight at the light, will he not have a pain in his eyes which will make him turn away to take and take in the objects of vision which he can see, and which he will conceive to be in reality clearer than the things which are now being shown to him?
— True, he now.
— And suppose once more, that he is reluctantly dragged up a steep and rugged ascent, and held fast until he is forced into the presence of the sun himself, is he not likely to be pained and irritated? When he approaches the light his eyes will be dazzled, and he will not be able to see anything at all of what are now called realities.
— Not all in a moment, he said.
— He will require to grow accustomed to the sight of the upper world. And first he will see the shadows best, next the reflections of men and other objects in the water, and then the objects themselves; then he will gaze upon the light of the moon and the stars and the spangled heaven; and he will see the sky and the stars by night better than the sun or the light of the sun by day?
— Last of he will be able to see the sun, and not mere reflections of him in the water, but he will see him in his own proper place, and not in another; and he will contemplate him as he is.
— He will then proceed to argue that this is he who gives the season and the years, and is the guardian of all that is in the visible world, and in a certain way the cause of all things which he and his fellows have been accustomed to behold?
— Clearly, he said, he would first see the sun and then reason about him.
— And when he remembered his old habitation, and the wisdom of the cave and his fellow-prisoners, do you not suppose that he would felicitate himself on the change, and pity them?
— Certainly, he would.
— And if they were in the habit of conferring honors among themselves on those who were quickest to observe the passing shadows and to remark which of them went before, and which followed after, and which were together; and who were therefore best able to draw conclusions as to the future, do you think that he would care for such honors and glories, or envy the possessors of them? Would he not say with Homer, “Better to be the poor servant of a poor master,” and to endure anything, rather than think as they do and live after their manner?
— Yes, he said, I think that he would rather suffer anything than entertain these false notions and live in this miserable manner.
— Imagine once more, I said, such a one coming suddenly out of the sun to be replaced in his old situation; would he not be certain to have his eyes full of darkness?
— To be sure, he said.
— And if there were a contest, and he had to compete in measuring the shadows with the prisoners who had never moved out of the cave, while his sight was still weak, and before his eyes had become steady (and the time which would be needed to acquire this new habit of sight might be very considerable) would he not be ridiculous? Men would say of him that up he went and down he came without his eyes; and that it was better not even to think of ascending; and if any one tried to loose another and lead him up to the light, let them only catch the offender, and they would put him to death.
— No question, he said.
We can detail, at this stage, the different steps underlying the Allegory of the cave:
|(1)||the prisoners of the cave are convinced that the objects they observe daily are the actual objects||premise|
|(2)||the prisoners of the cave observe in reality on the walls shadows and reflections of the actual objects||from (1)|
|(3)||the situation of the prisoners of the cave is similar to our present situation||analogy|
|(4)||we are convinced that the objects we see every day are the actual objects||premise|
|(5)||∴ the objects that we see are in reality the shadows and reflections of the actual objects||from (2),(3),(4)|
Plato’s conclusion is that the human situation is analogous to that of the prisoners of the cave. In this sense, the Allegory of the cave is clearly an argument by analogy. However, at this stage, the resulting conclusion can be diversely interpreted. We can thus distinguish two main interpretations. According to the first interpretation, the prisoners of the cave are men, and the objects they see are merely a pale reflection of real objects, which are the ideas or archetypes. There are therefore archetypes of the number “7”, of courage, of tolerance, of a lion and of the sun, etc. in the world of ideas. In this sense, humans believe that ultimate reality is the one which corresponds to their perceptions, while this is illusory, and that the true reality is located at the level of the Archetypes. Thus, we live every day in what constitutes only the second plane corresponding to the projection of the real objects which situate themselves in the foreground: that is to say, at the level of the Archetypes. In this sense, the Allegory of the cave proves to be close to the “brains in a vat” experiment and its modern illustration through the Matrix movie.
A second type of interpretation, however, can be applied to the Allegory of the cave. Such an interpretation is directly related to Plato’s theory of knowledge, for Plato distinguishes between the knowledge gained from opinion and genuine knowledge. Thus, the knowledge of the beings and objects that the prisoners of the cave possess is merely knowledge gained from opinion. It does not consist of true knowledge, because it is shaped, transformed and distorted by the education that was received. Basic knowledge that we possess is, according to Plato, perverted by the turmoil of human passions, ambition, competition, received ideas, etc. By contrast, authentic and true knowledge situates itself beyond passions, hatred, honors and established ideas. According to Plato, each human must rise well above the passions that bind him in chains in order to reach true knowledge.
31. The Simulation Argument
The Simulation Argument has been described recently by Nick Bostrom, in an article published in 2003 in the Philosophical Quarterly journal. The argument is basically grounded on the fact that it appears quite likely that post-human civilization will carry out simulations of humans. Indeed, it seems likely that far advanced post-human civilizations will have both the ability and the willingness to carry out extremely realistic simulations of humans. If this were the case, the number of simulated humans would then greatly exceed the number of genuine humans. In such a case, it follows that taking into account the fact that each of us exists leads us to consider it more probable that we belong to the simulated humans rather than to the authentic humans. According to Bostrom, the conclusion that follows from the Simulation argument is that the probability of each of the three following statements is approximately 1/3:
|(1)||humanity will face a relatively imminent extinction|
|(2)||a post-human civilization will not carry out human simulations|
|(3)||we currently live within a simulation|
These probabilities are not surprising regarding the assertions (1) and (2), but the inherent likelihood of the assertion (3) under which we now live in a simulation, turns out to be quite counter-intuitive.
The Simulation argument is also described more succinctly by Brian Weatherson, in a response to Bostrom’s original article, published in 2004. According to him, the real core of the Simulation argument can be described as follows. Firstly, it is very likely that post-human civilizations will be able to produce realistic simulations of human beings. Similarly, it is very likely that the number of simulated humans will greatly exceed the number of real humans. Thus, at a post-human age, the ratio between simulated humans and genuine human beings should be largely in favor of simulated humans. At this step, it turns out that the simple fact of taking into account our present existence leads to the conclusion that it is likely that we are simulated humans. This invites us to think that the probability that our thoughts, feelings, sensations, etc. are the result of a simulation is high.
The conclusion of the Simulation argument—in a somewhat similar way to the Doomsday argument—proves to be counter-intuitive and contrary to common sense. However, in the same way as for the Doomsday argument, the task that consists in determining accurately the fallacious step at the level of the Simulation argument proves to be very difficult.
A first objection that could be raised against the Simulation argument focuses on the need to appeal to a principle of indifference (in virtue of which there is no reason to favor a priori any of the assumptions here). For are we really human beings randomly chosen within the reference class that includes both humans and simulated human beings? It seems in fact that the Simulation argument is worthwhile only if we are chosen randomly within the reference class. Is there not here the same problem that emerges when faced with the Doomsday argument? Bostrom, however, responds to this objection by arguing that the principle of indifference used in the Simulation argument is not of the same nature as that referred to by the Doomsday argument. Indeed, in the Doomsday argument, an important premise is that every human being, given its birth rank, should be considered as randomly chosen within the reference class. The principle of indifference used in the Simulation argument proves to be weaker, since it is applied regardless of the birth rank (or any other criterion of a similar nature), but proceeds from the mere observation of our existence as members of the reference class.
Another objection that might be raised is that the simulation argument itself is self-refuting. Indeed, if its conclusion is true, it follows that the argument itself is the product of a simulation and that all our logic is in turn simulated. In this case, we cannot therefore accept as valid the conclusions resulting from the argument. However, we can see that such an objection also applies to the Dreaming argument, to the “brains in a vat” experiment, etc. Thus, such an objection turns out to be too general, and it seems that it does not respond accurately to the specific problem posed by the Simulation argument.
32. The Dualist Argument from Divisibility
In the course of the Metaphysical Meditations (Sixth Meditation), Descartes develops an argument that aims to prove the existence of the mind/body duality. He proposes to show how body and mind are two vital components of human nature, which prove, however, fundamentally different in nature. This argument takes place within the debate opposing materialism to idealism. Materialism is the doctrine according to which only material and physical things exist. In this context, the phenomena of mental nature can be reduced to mere phenomena of material origin. Thus, according to materialism, everything that exists is material and can be characterized in purely physical terms. In contrast, idealism is the view that only things of a mental nature exist. In this context, material things only gain their existence through our own perceptions. According to the idealist standpoint, all that exists reduces thus to a purely mental existence. In essence, materialism and idealism are monistic views. In contrast, dualism is a pluralist perspective that considers that things of physical and mental nature both exist. According to this view, the mental and the physical, whose very nature is fundamentally different, coexist. The dualistic point of view was famously defended by Descartes. There exists, according to Descartes, a mind/body duality, which constitutes the counterpart relevant to human beings of the mental/physical dualism. Descartes based his reasoning on the respective properties of the body and the mind, which are fundamentally different. He thus considers that physical matter that constitutes our body has an extension in space and proves therefore divisible. Conversely, the mind, according to Descartes, has no spatial extension and therefore does not have the same property of divisibility. Thus, the body and the mind have at least one different property and are therefore, in virtue of Leibniz’s law—according to which two objects are identical if and only if all their properties are identical—fundamentally different.
The Dualist argument from Divisibility comes from the following passage of the Metaphysical Meditations:
In order to begin this examination, then, I here say, in the first place, that there is a great difference between mind and body, inasmuch as body is by nature always divisible, and the mind is entirely indivisible. For, as a matter of fact, when I consider the mind, that is to say, myself inasmuch as I am only a thinking thing, I cannot distinguish in myself any parts, but apprehend myself to be clearly one and entire; and although the whole mind seems to be united to the whole body, yet if a foot, or an arm, or some other part, is separated from my body, I am aware that nothing has been taken away from my mind. And the faculties of willing, feeling, conceiving, etc. cannot be properly speaking said to be its parts, for it is one and the same mind which employs itself in willing and in feeling and understanding. But it is quite otherwise with corporeal or extended objects, for there is not one of these imaginable by me which my mind cannot easily divide into parts, and which consequently I do not recognize as being divisible; this would be sufficient to teach me that the mind or soul of man is entirely different from the body, if I had not already learned it from other sources.
The different steps of Descartes’ Dualist Argument from Divisibility can be detailed as follows:
|(1)||my body has a space extension||premise|
|(2)||everything that has a space extension is divisible||premise|
|(3)||my body is divisible||from (1),(2)|
|(4)||my mind has no space extension||premise|
|(5)||my mind is not divisible||premise|
|(6)||my body and my mind have at least one different property||from (3),(5)|
|(7)||two things are identical if and only if they have identical properties||Leibniz law|
|(8)||if two things have different properties then these two things are distinct||from (7)|
|(9)||∴ my body and my mind are two distinct things||from (6),(8)|
Descartes’ dualistic point of view gave rise to an important objection, which is the following: if there exists a mind/body duality, how do these two fundamentally different components of the same human being interact? The nature of the interaction resulting from the doctrine of mind/body duality has so far not been elucidated. This is an important gap in the dualist approach, since a comprehensive dualistic theory needs to explicitly describe the modalities of the interaction between the body and the mind.
33. The Sleeping Beauty Problem
The Sleeping Beauty Problem has been the subject of a number of recent discussions, notably by Adam Elga and David Lewis in articles published respectively in 2000 and 2001 in the Analysis journal. The Sleeping Beauty problem was described as follows by Elga. Some researchers have planned an experiment in which they propose to put to sleep Sleeping Beauty. She will be asleep for two days, namely Monday and Tuesday. However, during her sleep, she will be awakened once or twice. The number of times she will be awakened will depend on the outcome of the throw of a perfectly balanced coin. If the coin lands on heads, Beauty will be awakened once on Monday. However, if the coin lands on tails, she will be awakened twice, on Monday and on Tuesday. In both cases, after being awakened on Monday, Beauty will be put to sleep again and will forget that she has been awakened. Given these elements, when Beauty awakens, to what extent should she believe that the coin fell heads?
At this stage, a first type (I) of reasoning suggests that the probability that the coin fell equals 1/2. Indeed, the coin is balanced, and therefore, if the experiment is repeated, it will result in a roughly equal number of tail draws. The initial likelihood of heads or tails is thus 1/2. But when Beauty awakens, she receives no new information. Therefore, she has no reason to change her initial belief. It would have been rational to modify the initial probabilities if new information had been provided, but such is not the case, and therefore, Beauty has no justification to change her initial probabilities. Such reasoning corresponds, in a simplified way, to that carried out by David Lewis.
It turns out, however, that a second type (II) of response is possible. The corresponding reasoning leads to the conclusion that the probability that the coin fell heads is 1/3. Let us suppose that the experiment is repeated many times. In this case, it will turn out that about 1/3 of the awakenings will occur when the coin fell heads. Similarly, about 2/3 of the awakenings will occur when the coin fell tails. So when Beauty awakes, she can legitimately consider that it is a heads-awakening with a probability of 1/3. Therefore, Beauty must conclude that the probability that the coin fell heads is 1/3.
It is worth formalizing the elements of the Sleeping Beauty problem, in order to cast light on its internal structure. The problem is based on the following two competing hypotheses:
|(H1)||Sleeping Beauty will be awakened once (HEADS)|
|(H2)||Sleeping Beauty will be awakened twice (TAILS)|
Similarly, it turns out that three cases are possible:
|(i)||the coin fell HEADS and Sleeping Beauty will be awakened on Monday|
|(ii)||the coin fell TAILS and Sleeping Beauty will be awakened on Monday|
|(iii)||the coin fell TAILS and Sleeping Beauty will be awakened on Tuesday|
The problem resulting from the situation corresponding to the Sleeping Beauty problem is that both rationales (I) and (II) seem a priori correct, whereas they lead to contradictory conclusions. Thus, one of the two arguments must be false. But which one? And why? In the contemporary literature on the Sleeping Beauty problem, the two competing arguments have their advocates and detractors, and there is currently no consensual solution.
34. The Evil Demon Argument
The Evil Demon argument is a famous argument described by Descartes in the Metaphysical Meditations. The evil demon argument is an argument in favor of skepticism. The argument itself is based on a thought experiment. Descartes considers thus the hypothesis that there exists an evil genius who is capable of deceiving him not only at the level of all his sensory perceptions, but also at the level of the whole of his knowledge, including that relating to mathematics. Considering that he does not have absolute certainty that this evil demon does not exist, Descartes concludes that it is possible that all his knowledge is false and that he is therefore entitled to question his overall knowledge.
The evil demon argument is mentioned in the following passage from the Meditations on First Philosophy:
I shall then suppose, not that God who is supremely good and the fountain of truth, but some evil genius not less powerful than deceitful, has employed his whole energies in deceiving me; I shall consider that the heavens, the earth, colors, figures, sound, and all other external things are nought but the illusions and dreams of which this genius has availed himself in order to lay traps for my credulity; I shall consider myself as having no hands, no eyes, no flesh, no blood, nor any senses, yet falsely believing myself to possess all these things; I shall remain obstinately attached to this idea, and if by this means it is not in my power to arrive at the knowledge of any truth, I may at least do what is in my power [i.e. suspend my judgment], and with firm purpose avoid giving credence to any false thing, or being imposed upon by this arch deceiver, however powerful and deceptive he may be.
The evil demon argument can be detailed as follows:
|(1)||it is possible that there exists an evil genius, capable of deceiving me on the whole of my sensory perceptions and my mathematical knowledge||hypothesis|
|(2)||if I am deceived at the level of the whole of my sensory perceptions and my mathematical knowledge (i.e. the fact that I am now in front of the fire in the fireplace or that the sum of the angles of a triangle is equal to a flat angle) then the whole of my beliefs are false||premise|
|(3)||it is possible that the whole of my beliefs are false||from (1),(2)|
|(4)||if I do not have the certainty that such evil genius does not exist, then I can not consider that the whole of my beliefs are true||from (1),(3)|
|(5)||I do not have the certainty that such an evil genius does not exist||premise|
|(6)||I cannot consider that the whole of my beliefs are true||from (4),(5)|
|(7)||∴ I have good reason to doubt the whole of my beliefs||from (6)|
The argument clearly targets a posteriori knowledge related to physical objects (i.e. a table, a horse or the planet Saturn), but also a priori knowledge such as that resulting from mathematics (i.e. the fact that the sum of the angles of a triangle is equal to a flat angle, or 1 + 3 = 4).
It is doubtful, however, whether the evil demon argument allows generalized doubt: that is to say, if it applies to our overall knowledge. Indeed, as demonstrated by Descartes himself, it seems that a proposition such as “I think, therefore I am” escapes such a doubt of universal scope. In this sense, the conclusion of the evil demon argument proves too strong. However, it turns out that even if we restrict the scope of the argument’s conclusion, much of it remains and still permits us to conclude in favor of skepticism.
Another objection that can be raised against the argument is that the argument is self-refuting, for the latter applies to both a posteriori and a priori knowledge. But the conclusion that follows from the evil demon argument itself constitutes a priori knowledge. I am also entitled to doubt the latter conclusion. Thus, the argument itself is undermined by its own conclusion. Given that I am justified in doubting all my a priori knowledge, I am well founded to doubt that I can doubt the whole of my beliefs.
35. Searle’s Chinese Room Argument
The Chinese Room argument was described by John Searle in an article published in 1980 in the Behavioral and Brain Sciences journal. This argument is based on a thought experiment, which can be described as follows. Suppose you have no knowledge of the Chinese language and you are locked alone in a room that contains the following items: (i) a set of typed text in Chinese, entitled “script”; (ii) a second set of documents in Chinese, entitled “the story”, accompanied by a set of rules in English to relate the initial documents with the latter; (iii) a third set of documents, entitled “questions”, with symbols in Chinese and instructions in English to relate the Chinese symbols with the first two sets of documents. At this point, a Chinese text is sent to you under the door. Consulting your four sets of documents, you then write another text in Chinese, entitled “answers” that you pass in your turn under the door of the room.
Searle’s thought experiment is based on an analogy. It draws a parallel between the situation of the person who is in the room and the situation corresponding to a computer program performing a translation. The person who is in the room receives a text written in Chinese, and then, after consulting a series of documents, writes in turn a new document in Chinese, which is a response to the first document received. Such a response is no different from the one that a person with a thorough understanding of the Chinese language would have made. And this stresses how the true understanding of the Chinese text that was submitted to him/her in fact completely escapes him/her, for the person who is in the room is able to respond competently to the question that is asked, but completely ignores the content of this answer. The experiment thus aims to highlight how the semantic content of the text escapes the machine, even though it possesses control of its syntactic content.
Searle’s argument is intended to constitute an objection to the view that a computer program is capable of thinking. This viewpoint represents the thesis called “strong AI” (strong artificial intelligence). According to this latter view, the computers have the ability to actually think in the same way that humans do. In this sense, a computer program can have a true understanding of a given situation. Strong AI is thus opposed to the thesis of weak AI, according to which computer programs are only simulations of the human mind. In this sense, the result of a computer program is not a genuine process of thought, but a mere simulation, however successful it may be, of the latter.
Searle’s argument itself, illustrated by the Chinese room experiment, can be detailed as follows:
|(1)||either strong AI prevails or weak AI prevails||dichotomy|
|(2)||computer programs make use of symbols||premise|
|(3)||the symbols correspond to the syntactic content of a text||premise|
|(4)||the human mind makes use of the semantic content of a text||premise|
|(5)||the Chinese room experiment shows that the syntactic content of a text does not suffice to determine its semantic content||from (3),(4)|
|(6)||the situation of the person within the Chinese room is analogous to that of a computer program performing a translation||analogy|
|(7)||∴ computer programs do not reach to determine the semantic content of a text||from (5),(6)|
|(8)||strong AI does not prevail||from (7)|
|(9)||∴ it is weak AI that prevails||from (1),(8)|
The Chinese room argument has generated a huge controversy. Although Searle responds in advance in his original article to a number of objections, the argument has left many authors unconvinced. However, to date, none of them has been able to indicate in a consensual way the accurate step in Searle’s reasoning that proves defective.
36. The Turing Test
Alan Turing, in a famous paper published in 1950 in the journal Mind, proposes to clarify the question, “Can machines think?” Instead of trying to answer that in a conventional manner by defining the concepts of “machine” and “think”, Turing chooses a different path. He seeks then to describe the following game, which he calls the Imitation game:
The imitation game This game is played with three people, a man (A), a woman (B) and an interrogator (C) of either sex. The interrogator is located in a room separate from the other two. The goal of the game for the interrogator is to be able to determine which one among the two others is the man or the woman. The interrogator knows each of them by the names X and Y, and at the end of the game, he must say either “X is A and Y is B” or “X is B and Y is A”. For this purpose, the interrogator is allowed to ask questions to A and B.
Nowadays, the original version of the Imitation game described by Turing is usually replaced by a simplified experiment which is as follows:
The Imitation Game (modern version) This game is played with two people and one machine: a man (A), a machine (M) and an interrogator (C). A and C are of one or the other sex. The interrogator is connected to A and M with a terminal, through which they can communicate. However, the interrogator can see neither man nor machine and does not know who is human and which is the machine. Its mission is to endeavor to determine who is human and which is the machine by asking questions. The interrogator is located in a room separate from that where the other two are. The machine and the human are trying to convince the interrogator that each of them is human. The goal of the game for the interrogator is to be able to determine who is truly human. If the interrogator cannot distinguish the human from the machine, it is then considered that the machine is intelligent.
It should be noted that an earlier version of the Turing test can be traced back to Descartes, who, in his Discourse on the Method, imagines a situation of a similar nature in the following passage:
And here I specially stayed to show that, were there such machines exactly resembling organs and outward form an ape or any other irrational animal, we could have no means of knowing that they were in any respect of a different nature from these animals; but if there were machines bearing the image of our bodies, and capable of imitating our actions as far as it is morally possible, there would still remain two most certain tests whereby to know that they were not therefore really men. Of these the first is that they could never use words or other signs arranged in such a manner as is competent to us in order to declare our thoughts to others: for we may easily conceive a machine to be so constructed that it emits vocables, and even that it emits some correspondent to the action upon it of external objects which cause a change in its organs; for example, if touched in a particular place it may demand what we wish to say to it; if in another it may cry out that it is hurt, and such like; but not that it should arrange them variously so as appositely to reply to what is said in its presence, as men of the lowest grade of intellect can do. The second test is, that although such machines might execute many things with equal or perhaps greater perfection than any of us, they would, without doubt, fail in certain others from which it could be discovered that they did not act from knowledge, but solely from the disposition of their organs: for while reason is a universal instrument that is alike available on every occasion, these organs, on the contrary, need a particular arrangement for each particular action; whence it must be morally impossible that there should exist in any machine a diversity of organs sufficient to enable it to act in all the occurrences of life, in the way in which our reason enables us to act.
Secondly, on the basis of the Imitation game, Turing made the following prediction. He considered that by the year 2000, it would be quite possible to program a computer so that the average human interrogator would have not more than a 70/100 chance at the imitation game to correctly identify the human and the machine, after asking a series of questions for five minutes. In a general way, the Turing test aims to demonstrate that the time is not far off when it will be impossible to distinguish man from machine. According to Turing, this is a demonstration that human intelligence can be fully simulated by computer.
The argument underlying the Turing test can thus be presented in detail as follows:
|(1)||if a test is performed to distinguish human intelligence from the simulated intelligence of the machine||hypothesis|
|(2)||then one cannot define a criterion to make such a distinction||from (1)|
|(3)||it is almost impossible to discern human intelligence from the simulated intelligence of the machine||from (2)|
|(4)||∴ human intelligence can be fully simulated||from (3)|
In this context, the argument based on the Turing test appears closely related to the Simulation argument recently described by Nick Bostrom.
It may be objected to the Turing experiment that the potentialities of the human brain and intelligence are barely known. Thus, new capabilities of human intelligence may well be discovered, which would then entirely escape the Turing test. In the same vein, we can also consider that what allows us to conclude the Turing test is that at present and in the near future, it will be difficult to distinguish a machine from a human being. However, this does not allow us to conclude that such differentiation will never be possible. Isn’t it too strong a conclusion? To conclude validly that human intelligence can be fully computer simulated, it would be necessary to have an absolute certainty that the differentiation between human and machine, under the conditions of the test, cannot be performed.
37. Gettier’s Problem
The Gettier problem was exposed by Edmund Gettier, in an article published in 1963 in the Analysis journal. Traditionally, one considers that a given person S knows a given proposition P provided that three conditions are simultaneously met: (i) the proposition P is true, (ii) S believes that P is true, (iii) S is justified in his/her belief that P is true. Hence, S knows that P if S has a justified true belief that P. This threefold condition of knowledge is widely accepted. However, Gettier undertakes to demonstrate that this threefold condition of knowledge is ungrounded and that these three criteria do not constitute a sufficient condition.
The first concrete case described by Gettier is as follows. Two protagonists, Smith and Jones, both apply for a job. Smith has decisive elements that allow him to think that the following proposition, whose structure is that of a conjunction, is true:
|(1)||Jones is the one who will get the job and Jones has ten coins in his pocket|
The key elements that are at the disposal of Smith are, on the one hand, the fact that the president of the company assured him that it would be Jones who would get the job, and on the other hand, the fact that Smith had previously counted the number of coins—that amount to ten—that were in Jones’ pocket. Hence, (1) has the following consequence:
|(2)||the one who will get the job has ten coins in his pocket|
In this case, we can consider that Smith knows that (2), since the above-mentioned three conditions are satisfied: proposition (2) is true, Smith believes that (2) is true and Smith is justified by (1) in his belief that (2) is true. Now let us imagine that, unknown to Smith, it is Smith himself who has actually got the job and he also has ten coins in his pocket. In this case, (1) turns out to be false. In addition, it turns out that Smith did not really know that (2), even though the threefold condition of knowledge was, however, satisfied. Thus, it turns out in this case that Smith does not know P, although the three aforementioned conditions are met.
The Gettier’s second practical case is the following. Consider the following proposition:
|(3)||Jones owns a Ford|
Moreover, Smith knows that Jones has always had a Ford and that the latter has recently made a trip with him. Smith thus possesses crucial elements in favor of (3). In addition, it turns out that Smith has another friend, Brown, about whom he ignores, however, a number of things. Let us consider now the three propositions:
|(4)||either Jones owns a Ford, or Brown is in Boston|
|(5)||either Jones owns a Ford, or Brown is in Barcelona|
|(6)||either Jones owns a Ford, or Brown is in Brest-Litovsk|
At this stage, it proves that each of these three propositions represents a logical consequence of (3). However, we can consider that Smith knows that (4), (5) and (6), because each of these propositions is true, and secondly Smith has for each of them a justified belief. But let us suppose now that Brown does not have a Ford, but uses a rental Chrysler, and that Brown, unknown to Smith, is located, in Barcelona. In this case, it proves that Smith does not really know that (5) is true, even though the three conditions of knowledge relating to (5) are once again satisfied.
The two preceding examples, Gettier concludes, demonstrate that the aforementioned threefold condition does not constitute a sufficient condition to ensure that S knows that P. However, a number of answers have been given to Gettier’s problem. One of these responses stresses that the justification that is present in the two cases mentioned by Gettier proves insufficient. For shouldn’t knowledge be motivated by actual evidence, not by what merely constitutes a weak justification? Smith, in effect, grounds his belief on the mere fact that the president of the company assured him that Jones would get the job. However, at this step, Smith has the certainty of the President’s declarations but lacks evidence of the corresponding facts, for couldn’t the President change his mind later? Therefore, we can assume that the justification step proves insufficient. In this sense, the two examples described by Gettier are characterized by a weak justification, whereas a strong justification is clearly needed. According to this type of objection, as we can see, the threefold condition of knowledge remains acceptable, but the condition of justification must be replaced by a stronger condition, which corresponds to evidence. In this context, genuine knowledge corresponds to a true and proved belief. However, such type of response to the Gettier problem proves insufficient to dissipate its consequences, because this type of response has the disadvantage of being too radical. Its application leads us, then, to consider that many situations of everyday life where we lack such definitive and absolute proof do not lead to genuine knowledge.
Several proposed solutions to the Gettier problem are intended to prevent the emergence of the cases described by Gettier by adding a supplementary condition. Solutions of this type are based on the fact that knowledge is the result of a true and justified belief, but also that this threefold condition cannot be obtained accidentally. The latter condition is intended to prevent the cases described by Gettier from occurring. But such an approach has not proved entirely satisfactory, because the very definition of accidental conditions has proved problematic. In effect, in some cases, the accidental occurrence of the threefold aforementioned condition does not lead to true knowledge, whereas in other circumstances, the accidental occurrence of this threefold condition generates genuine knowledge.
38. Frege’s Puzzle about Identity Statements
The puzzle about identity statements was described by Gottlob Frege in his essay On Sense and Reference, published in 1892. This puzzle is as follows. Let us consider first a statement such as “the morning star is the evening star.” In this case, it turns out that the expressions “the morning star” and “the evening star” refer to one and the same object: the planet Venus. We can see that the structure of the proposition “the morning star is the evening star” has the form “A” = “B”. In a general way, propositions that have such a structure are true if and only if “A” and “B” refer to the same object. This can also be formulated in terms of numbers. If we consider the terms “160 + 10” and “153 + 17”, it proves that these two expressions refer to the same natural number, which is 170. Frege endeavored thus to describe a theory of truth for propositions with the structure “A” = “B” by defining the conditions under which such propositions turn out to be true. However, Frege noticed that a problem emerged with this type of analysis. It occurred indeed that the conditions under which a proposition of the form “A” = “B” proved true (the truth conditions) were identical to those under which a proposition of the form “A” = “A” was also true. But a proposition of the form “A” = “A” as “the morning star is the morning star” proves to be, from a semantic point of view, very different from a proposition such as “the morning star is the evening star.” The conclusion follows that the truth conditions are identical for semantically very different propositions of the form “A” = “B” or “A” = “A”.
The reasoning that leads to Frege’s puzzle about identity statements can be formalized as follows:
|(1)||the Morning Star is the Evening Star||premise|
|(2)||“the Morning Star” and “the Evening Star” refer to the planet Venus||definition|
|(3)||(1) is true||from (1),(2)|
|(4)||the Morning Star is the Morning Star||identity|
|(5)||“the Morning Star” refer to the planet Venus||definition|
|(6)||(4) is true||from (4),(5)|
|(7)||(1) has the structure “A” = “B”||from (1)|
|(8)||(4) has the structure “A” = “A”||from (4)|
|(9)||a proposition which has the structure “A” = “B” is true if and only if “A” and “B” refer to the same object||generalization|
|(10)||a proposition which has the structure “A” = “A” is true if and only if “A” and “A” refer to the same object||from (9)|
|(11)||the truth conditions of a proposition which has the structure “A” = “B” and of a proposition which has the structure “A” = “A” are identical||from (9),(10)|
|(12)||from a semantic viewpoint, a proposition which has the structure “A” = “B” is very different from a proposition which has the structure “A” = “A”||from (1),(4)|
|(13)||∴ the truth conditions of two propositions which are semantically very different are identical||from (11),(12)|
39. The Paradox of Analysis
The paradox of analysis results from the work of George Edward Moore. The paradox is based on a methodological approach which consists in analyzing a given concept. Let us call α such a concept. The analysis of this concept, then, presents the form: α = E. Here, α is the concept which is analyzed (the analysandum) whereas E is an expression (the analysans)—more or less complex—that defines and describes the semantic content of α. The paradox emerges when we consider the two possibilities that arise: (i) either the analysans accurately describes the content of the concept α or (ii) the analysans does not accurately describe the content of the concept α. In the first case, it follows that the analysis is trivial, and therefore has no interest. In the second case, it turns out that the analysans does not accurately describe the content of the concept α and therefore, the analysis is false. Thus, the analysans is either trivial or false. In both cases, the analysis is useless. However, this is in contradiction with the data resulting from our pretheoretical intuition, according to which the analysis of a given concept proves most often useful.
The reasoning corresponding to the paradox of analysis can be detailed as follows:
|(1)||either the analysans describes accurately the content of concept α, or the analysans does not describe accurately its content||dichotomy|
|(2)||if the analysans describes accurately the content of concept α||hypothesis 1|
|(3)||then the analysis is trivial||from (2)|
|(4)||if the analysans does not describe accurately the content of concept α||hypothesis 2|
|(5)||then the analysis is inexact||from (4)|
|(6)||the analysis of concept α is either trivial or inexact||from (3),(5)|
|(7)||∴ the analysis of concept α is useless||from (6)|
A notable solution to the paradox of analysis that results from the ideas put forward by Gottlob Frege in his essay On Sense and Reference is as follows. This solution undermines the transition from step (2) to step (3), which leads to the conclusion that the analysis is trivial if the analysans accurately describes the content of the concept α. Frege distinguished two types of semantic content: first, the meaning, and secondly, the reference. In this context, it turns out that if the concept α and the analysans have the same reference, then the resulting analysis is exact. However, this does not preclude the analysans from having a different meaning of the concept α. And under such conditions, the analysis proves not trivial but useful, since it provides new information.
40. Heraclitus’ river puzzle
The Problem of Heraclitus’ River originates from the Fragments of the work of Heraclitus that have reached us. Heraclitus affirms that it is not possible to cross the same river twice, because the waters that constitute the latter are constantly renewed. The underlying idea in this last problem is that between two crossings, the river has undergone changes such that it is no longer exactly the same river.
We can formulate more accurately the problem of Heraclitus’ river as follows:
|(1)||I cross the river r at time T1||premise|
|(2)||I cross the river r at time T2 (with T1 < T2)||premise|
|(3)||the river r has undergone changes between T1 and T2||premise|
|(4)||∴ the river r at time T1 is different from the river at time T2||from (3)|
|(5)||∴ at time T2 I cross a river which is different from the river r that I crossed at time T1||from (1),(2),(4)|
An objection that has been raised with regard to the problem of Heraclitus’ river is that the changes undergone by the river between T1 and T2 are not significant enough to transform the river in T1 into a different river in T2. According to this standpoint, the changes undergone by the river are minor and do not affect its identity as a river. This type of objection, as we can see, has the effect of blocking the passage from step (3) to step (4). It thus emphasizes the persistence of the identity of an object o through time, despite the changes of a secondary nature that are undergone by this object, for according to this standpoint, steps (3) and (4) should be replaced by:
|(3*)||the river r has undergone minor changes between T1 and in T2||premise|
|(4*)||∴ the river r at time T1 is no different from the river r at time T2||from (3*)|
However, such an objection is insufficient to permanently resolve the problem of Heraclitus’ river. Indeed, the underlying distinction between substantive and non-substantive changes that can affect a given object proves difficult to apply. Thus, between two given time positions, the river water is completely new, so that the elements that compose the latter have been entirely changed. It is therefore difficult to consider that the totality of elements that compose an object at a given time are not essential to it.
The aforementioned paradoxes, philosophical problems and arguments represent only a selection of the many issues addressed in the rich literature constituted by contemporary analytical philosophy, for it consists of a lively and evolving field, where every year, new arguments are created, and then exposed and discussed. As we have seen, millennial unresolved paradoxes coexist with philosophical arguments that have just been described.
On the other hand, the presentation of these contemporary problems of analytic philosophy is primarily designed to allow a better understanding of the analytic style with regard to the so-called continental philosophy, as both styles of philosophizing, as we have seen, are deserving of respect. The goal here was simply to introduce an often overlooked facet of contemporary philosophy. Some will immediately feel a natural affinity with the analytic style. Others will prefer the “continental” style. All, however, I hope, will benefit from a better understanding of the diversity of philosophical styles.
Of the statement of the foregoing paradoxes and arguments, it is also apparent, I believe, that human reasoning is perfectible and surprisingly vulnerable to error. For the pitfalls of reasoning that have been described, the contradictions to which paradoxes easily lead us indicate that the thinking of all of us proves to be vulnerable. It is quite fascinating to see how we are all prone to think in a way that leads to paradoxical conclusions, leaving us with the contradictions that result from reasoning that seemed quite valid. The reasoning that leads to error is common to us, and again, if a solution had to be provided to a particular problem or paradox, it should prove to be consensual to be validated. As we can see, such a field has a significant practical importance. It consists here of improving and enhancing the way of thinking that we share in common. In this context, the discovery of a consensual solution to a specific argument or an unresolved paradox should, then, be of benefit to everyone.
I thank Francis Antona, Christian Carayon and Eric Odin for very useful comments on earlier drafts.
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