In the paper “A Dichotomic Analysis of the Surprise Examination Paradox” (originally published in French under the title Une analyse dichotomique du paradoxe de l’examen-surprise, I offer a comprehensive and novel approach to resolving the Surprise Examination Paradox (SEP). The proposed solution is grounded in a detailed analysis of the paradox, incorporating previous literature and introducing new conceptual distinctions. The paper is structured into five main sections, each contributing to the development of the proposed solution.
The Paradox and Its Background
I begin by tracing the origins of the Surprise Examination Paradox (SEP) to a real-life event in Sweden during 1943-1944 and its subsequent academic discussion, starting with D. O’Connor’s 1948 article. The paradox arises when a professor announces a surprise examination, leading students to a logical conundrum where they conclude that a surprise exam is impossible, yet it occurs. The students reason as follows. The examination cannot take place on Saturday, they think, for otherwise they would know in advance that the examination would take place on Saturday and thus it could not occur surprisingly. Thus, Saturday is ruled out. Moreover, the examination cannot take place on Friday, for otherwise the students would know in advance that the examination would take place on Friday and thus it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, all days of the week are then ruled out. However, this does not prevent the examination from finally occurring surprisingly, say, on Wednesday.
Review of the literature
I review various attempts to solve the paradox, including O’Connor’s self-refuting statement theory, Quine’s analysis of student error, Shaw’s self-referential structure argument, Montague and Kaplan’s reduction to the Liar paradox, Binkley’s reduction to Moore’s paradox, Dietl and Smith’s sorites paradox analogy, and Wright and Sudbury’s temporal retention principle. Each solution is shown to have limitations or fail to address certain variations of the paradox.
Monist or Dichotomic Analysis of the Paradox
I introduce then a crucial distinction between monist and dichotomic analyses of SEP. Monist solutions attempt a single, overarching resolution applicable to all forms of SEP, while dichotomic solutions differentiate between two distinct scenarios or structures within the paradox, offering separate solutions for each. I advocate then for a dichotomic approach, proposing that SEP can be understood in terms of two structurally different versions: one with a conjoint structure (where surprise and non-surprise can coexist) and another with a disjoint structure (where surprise and non-surprise are mutually exclusive). This dichotomy is further elucidated through matrix notation, illustrating how different days of the week can be categorized as surprise or non-surprise under various conditions.
The Surprise Notion Corresponding to the Conjoint Structure
In this section, I examine the version of SEP where the concept of surprise is based on a conjoint structure. This version aligns with Quine’s solution, where the student’s error is in failing to consider the possibility of the exam occurring surprisingly on the last day. I also discuss Hall’s reduction, which addresses objections to Quine’s solution by transforming an n-day scenario into an (n-1) day scenario, thereby making Quine’s solution applicable.
The Surprise Notion Corresponding to the Disjoint Structure
Here, I shift focus to the version of SEP with a disjoint structure of surprise and non-surprise. In this scenario, Quine’s solution does not apply as it requires the possibility of surprise and non-surprise coexisting. I argue that the student’s error in this version is not recognizing the vague nature of the concept of surprise, which includes a penumbral zone of borderline cases. This understanding prevents the successive elimination of days and acknowledges the possibility of surprise on any given day.
The Solution to the Paradox
I conclude by synthesizing the analysis into a comprehensive solution to SEP. The student, upon hearing the professor’s announcement, should recognize two distinct conceptions of surprise (conjoint and disjoint) and consider them separately. In both cases, the traditional reasoning of eliminating each day is flawed. With the conjoint structure, the error lies in not considering the possibility of a surprise exam on the last day. In the disjoint structure, the mistake is in not recognizing the vagueness of the surprise concept, which includes borderline cases between surprise and non-surprise. This dichotomic approach, considering both conjoint and disjoint structures of surprise, leads to the rejection of the classical backward-induction reasoning and provides a resolution to the SEP.
This analysis addresses the traditional SEP but also extends to various modifications and versions of the paradox. The paper’s contribution lies in its comprehensive approach, which not only critiques and builds upon previous solutions but also introduces a novel framework that can be applied to understand and resolve different forms of the SEP. This dichotomic analysis, with its clear distinction between conjoint and disjoint structures of surprise, provides a more versatile and robust solution to the paradox, accommodating a wider range of scenarios than previous monist approaches.