Twisted Logic: Navigating Life's Puzzles, Paradoxes, and Big Questions draws upon an array of popular and novel puzzles and paradoxes to help us understand and navigate our everyday world, as well as engaging with the big questions beyond. It will appeal to all those interested in learning about twisted logic and the ways in which intuition and common sense can sometimes lead us astray. The book is designed for everyone and is accessible to the layman and (...) student alike. No prior mathematical skill or technical knowledge is required. (shrink)

A solitaire version of the classic paper and pencil game! A grid shows an "ocean," in which a flotilla of vessels--that you have to find--is concealed. Most of the puzzles have helpful hints revealing the contents of a few squares, and indicate how many squares in the corresponding rows and columns that contain ships. Sample puzzles show strategies for eliminating possibilities and clueing into likely arrangements. Puzzles are ranked according to difficulty.

"The Hardest Logic Puzzle Ever" was first described by the late George Boolos in the Spring 1996 issue of the Harvard Review of Philosophy. Although not dissimilar in appearance from many other simpler puzzles involving gods (or tribesmen) who always tell the truth or always lie, this puzzle has several features that make the solution far from trivial. This paper examines the puzzle and describes a simpler solution than that originally proposed by Boolos.

Rabern and Rabern (Analysis 68:105–112 2 ) and Uzquiano (Analysis 70:39–44 4 ) have each presented increasingly harder versions of ‘the hardest logic puzzle ever’ (Boolos The Harvard Review of Philosophy 6:62–65 1 ), and each has provided a two-question solution to his predecessor’s puzzle. But Uzquiano’s puzzle is different from the original and different from Rabern and Rabern’s in at least one important respect: it cannot be solved in less than three questions. In this paper we solve Uzquiano’s (...) puzzle in three questions and show why there is no solution in two. Finally, to cement a tradition, we introduce a puzzle of our own. (shrink)

Puzzles of logic involve knights, knaves, gods, demons, and mortals, and Inspector Craig conducts a summer-long adventure in combinatory logic, basic to computer science and artificial intelligence.

Seventy-five fun puzzles to help improve mathematical and thinking skills. Decipher various clues by adding, subtracting, multiplying, dividing, reading carefully, and using powers of reason. Transfer the information onto grids that are provided, and find the solution to a given set of circumstances. "...the book is a delightful collection of problem situations that teachers and students will enjoy using..."--Mathematics.

Rabern and Rabern (2008) have noted the need to modify `the hardest logic puzzle ever’ as presented in Boolos 1996 in order to avoid trivialization. Their paper ends with a two-question solution to the original puzzle, which does not carry over to the amended puzzle. The purpose of this note is to offer a two-question solution to the latter puzzle, which is, after all, the one with a claim to being the hardest logic puzzle ever.

In this entertaining and challenging collection of logic puzzles, Raymond Smullyan-author of Forever Undecided-continues to delight and astonish us with his gift for making available, in the thoroughly pleasurable form of puzzles, some of the most important mathematical thinking of our time.

We present the simplest solution ever to 'the hardest logic puzzle ever'. We then modify the puzzle to make it even harder and give a simple solution to the modified puzzle. The final sections investigate exploding god-heads and a two-question solution to the original puzzle.

This piece is intended to explicate - by providing a precising definition of - the common cinematic figure which I term “the failure of the cinematic imagination,“ while presenting a logical puzzle and its solution within a simple Gricean framework. -/- It should be noted that this is neither fully accurate nor fully precise, because of the audience; one should examine the remaining articles in the issue to understand what I mean.

In this paper, we first propose a simple formal language to specify types of agents in terms of necessary conditions for their announcements. Based on this language, types of agents are treated as ‘first-class citizens’ and studied extensively in various dynamic epistemic frameworks which are suitable for reasoning about knowledge and agent types via announcements and questions. To demonstrate our approach, we discuss various versions of Smullyan’s Knights and Knaves puzzles, including the Hardest Logic Puzzle Ever (HLPE) proposed (...) by Boolos (in Harv Rev Philos 6:62–65, 1996). In particular, we formalize HLPE and verify a classic solution to it. Moreover, we propose a spectrum of new puzzles based on HLPE by considering subjective (knowledge-based) agent types and relaxing the implicit epistemic assumptions in the original puzzle. The new puzzles are harder than the previously proposed ones in the literature, in the sense that they require deeper epistemic reasoning. Surprisingly, we also show that a version of HLPE in which the agents do not know the others’ types does not have a solution at all. Our formalism paves the way for studying these new puzzles using automatic model checking techniques. (shrink)

An entertaining series of logic problems and puzzles of increasing difficulty, and all relating important mathematical and logical concepts, includes mind-benders, paradoxes, metapuzzles, number exercises, and a mathematical novel.

Puzzle-solving, like several other everyday activities, appears in a more sophisticated and ramified form in the realm of natural science. Improving on Thomas Kuhn's rudimentary account of puzzles in science, this paper formulates logical and functional criteria for the occurrence of scientific puzzles, and examines the two-fold nature of their solutions. Then, with the aid of erotetic logic, puzzle-posing questions are identified, their presuppositional relations to scientific theory and explanations are explored, and a new tool for history (...) of science research is developed. (shrink)

In this paper, I will present a puzzle for logical analyses, such as Russell’s analysis of definite descriptions and Recanati’s analysis of ‘that’-clauses. I will argue that together with Kripke’s disquotational principles connecting sincere assent and belief such non-trivial logical analyses lead to contradictions. Following this, I will compare the puzzle about logical analysis with Frege’s puzzle about belief ascriptions. We will see that although the two puzzles do have similarities, the solutions to Frege’s puzzle cannot be applied mutatis (...) mutandis to the puzzle about logical analysis. Hence, to say it with Kripke, the main thesis of this paper is that the puzzle is a puzzle. A complete solution to the puzzle promises a better understanding of both logical analyses and belief ascriptions. (shrink)

Paradox Lost covers ten of philosophy’s most fascinating paradoxes, in which seemingly compelling reasoning leads to absurd conclusions. The following paradoxes are included: The Liar Paradox, in which a sentence says of itself that it is false. Is the sentence true or false? The Sorites Paradox, in which we imagine removing grains of sand one at a time from a heap of sand. Is there a particular grain whose removal converts the heap to a non-heap? The Puzzle of the Self-Torturer, (...) in which a series of seemingly rational choices has us accepting a life of excruciating pain, in exchange for millions of dollars. Newcomb’s Problem, in which we seemingly maximize our expected profit by taking an unknown sum of money, rather than taking the same sum plus $1000. The Surprise Quiz Paradox, in which a professor finds that it is impossible to give a surprise quiz on any particular day of the week... but also that if this is so, then a surprise quiz can be given on any day. The Two Envelope Paradox, in which we are asked to choose between two indistinguishable envelopes, and it is seemingly shown that each envelope is preferable to the other. The Ravens Paradox, in which observing a purple shoe provides evidence that all ravens are black. The Shooting Room Paradox, in which a deadly game kills 90% of all who play, yet each individual’s survival turns on the flip of a fair coin. Each paradox is clearly described, common mistakes are explored, and a clear, logical solution offered. Paradox Lost will appeal to professional philosophers, students of philosophy, and all who love intellectual puzzles. (shrink)

Lampert and Waldrop have recently presented a puzzle about moral responsibility and logical truth, in which they derive a contradiction from three apparently plausible principles: (A) no one is responsible for any logical truth; (B) if no one is responsible for something, then no one is responsible for what it strictly implies; and (C) someone is responsible for something. They argue that, in response, we must give up (B)—a principle that plays a key role in arguments for incompatibilism. In this (...) paper, I argue that this response is mistaken, and that the lesson to be drawn lies elsewhere. For the puzzle also rests on a conception of logical truth according to which there are contingent logical truths. And, given this conception, (A) is false—for reasons independent of (B). So the import here concerns, not a connection between responsibility and the debate over incompatibilism, but a connection between responsibility and the debate over the correct conception of logical truth: one can hold on to (A), or hold on to the relevant conception of logical truth, but not both. (shrink)

I argue that plausible claims in the logic of partial grounding, when combined with a plausible analysis of that concept, entail the falsity of plausible grounding claims. As our account of the concept of partial grounding and its logic should be consistent with plausible grounding claims, this is problematic. The argument hinges on the idea that some facts about what grounds what are grounded in others, which is an idea the paper aims to motivate.

In the Posterior Analytics (I 6, 75a18–27) Aristotle discusses a puzzle which endangers the possibility of inferring a non-necessary conclusion. His solution relies on the distinction between the necessity of the conclusion's being the case and the necessity of admitting the conclusion once one has admitted the premisses. The former is a factual necessity, whereas the latter is meant to be a normative or deontic necessity that is independent of the facts stated by the premisses and the conclusion. This paper (...) maintains that Aristotle resorts to this distinction because he thinks that, as long as it is conceived as a factual relation, logical consequence cannot exist independently of the facts expressed by the premisses and the conclusion. As a corollary, the necessity of such a consequence relation always requires the necessity of these facts. Aristotle holds this factual conception of logical consequence responsible for the puzzle, since it cannot account for valid syllogisms with contingent or false premisses. The alternative conception of necessity is then introduced by him in order to make good this deficiency. The distinction between the necessity of being and the necessity of saying was revived by the Oxford logician E. W. B. Joseph, and taken over by Frank Ramsey in his seminal Truth and Probability, but has not received attention from recent interpreters of Aristotle's logic. This paper, however, argues that, in spite of its intrinsic interest, the distinction bore no significant fruit in Aristotle's logical doctrine. (shrink)

Logical puzzles like the doctrinal paradox raise the problem of how to aggregate individual judgements into a collective judgement, or alternatively, how to merge collectively inconsistent knowledge bases. In this paper, we view judgement aggregation as a function on propositional logic valuations, and we investigate how logic constrains judgement aggregation. In particular, we show that there is no non-dictatorial decision method for aggregating sets of judgements in a logically consistent way if the decision method is local, i.e., (...) only depends on the individual judgements on the proposition under consideration. (shrink)

Logical puzzles like the doctrinal paradox raise the problem of how to aggregate individual judgements into a collective judgement, or alternatively, how to merge collectively inconsistent knowledge bases. In this paper, we view judgement aggregation as a function on propositional logic valuations, and we investigate how logic constrains judgement aggregation. In particular, we show that there is no non-dictatorial decision method for aggregating sets of judgements in a logically consistent way if the decision method is local, i.e., (...) only depends on the individual judgements on the proposition under consideration. (shrink)

ABSTRACT In this paper, I argue that a number of influential Millian responses to Frege’s puzzle, which consist in denying that Frege’s data apply to natural languages, are not viable if logic is to play its role in legitimizing the logical appraisal of rational subjects. A notion of validity which does justice to the normativity of logic must make room for a distinction between valid inferences and enthymemes. I discuss the prospects of formal, relevant and manifest validity as (...) candidates for a notion which complies with this desideratum. Their success, or failure is argued to hang on the viability of a semantical account of de jure co-reference, which is in tension with standard Millian tenets. I conclude that these Millian theories face the following dilemma: either accept that there is no notion of logical validity which makes logic normative for reasoning, thus jeopardizing our well entrenched practices of rational appraisal; or accept that de jure co-reference is a real semantical relation. (shrink)

Natural language meaning has properties of both (embodied) cognitive representations and formal/mathematical structures. But it is not clear how they actually relate to one another. This article argues that how properties of cognitive representations and formal/mathematical structures of natural language meaning can be united remains one of the puzzles in cognitive science. That is primarily because formal/mathematical structures of natural language meaning are abstract, logical, and truth‐conditional properties, whereas cognitive/conceptual representations are embodied and grounded in sensory‐motor systems. After reviewing (...) the current progress, this work offers, in outline, the general formulations that show how these two different kinds of representations for semantic structures can (potentially) be unified and also proposes three desiderata for testing, in brain dynamics, the mathematical equivalence between formal symbolic representations (and their transitions), and neuronal population codes (and their transitions). (shrink)

Our puzzle about moral motivation can be seen as a tension that we encounter when we try to reconcile intellectual and practical aspects of morality. Cognitivists interpret moral judgments as expressing cognitive attitudes, such as belief. Moral judgments ascribe properties – axiological, deontic, and aretaic – to persons, actions, institutions, and policies. Internalists believe that moral judgments necessarily engage the will and motivate. We expect people to be motivated to act in accord with their moral judgments and would find it (...) odd for people to be systematically indifferent to what they judge morally significant. It is also a common view that motivation involves pro-attitudes, such as desires. We explain intentional action as the product of informational states, such as beliefs, and practical states, such as desires. But beliefs and desires are logically independent states; no belief entails any particular desire. These assumptions are in tension. (shrink)

The philosophy of science has produced numerous accounts of how scientific facts are generated, from very specific facilitators of belief, such as neo-Kantian constitutive principles, to global frameworks, such as Kuhnian paradigms. I consider a recent addition to this canon: van Fraassen’s notion of an epistemic stance—a collection of attitudes and policies governing the generation of factual beliefs—and his commitment to voluntarism in this context: the idea that contrary stances and sets of beliefs are rationally permissible. I argue that while (...) scientific inquiry inevitably favours a high degree of consensus in our choices of stance, there is no parallel constraint in the case of philosophical inquiry, such as that concerned with how scientific knowledge should be interpreted. This leads, in the latter case, to a fundamental and apparently irresolvable mystery at the heart of stance voluntarism, regarding the grounds for choosing basic epistemic stances. (shrink)

Contemporary Complex Cinema. Complex conditions: the resurgence of narrative complexity ; Complex cinema as brain-candy for the empowered viewer ; Narrative taxonomies: simple, complex, puzzle plots -- Cognitive Approach to Contemporary Complex Cinema. Why an (embodied-)cognitive approach? ; Various forms of complexity and their effects on sense making ; Problematizing narrative linearity ; Complicating narrative structures and ontologies ; Under-stimulation and cognitive overload ; Contradictions and unreliabilities ; A cognitive approach to classifying complexity ; Deceptive unreliability and the twist film (...) ; Disorienting but solvable puzzle films ; Impossible puzzle films -- Narrative Complexity and Dissonant Cognitions. The concept of cognitive dissonance ; Cognitions in dissonance: from social psychology to narrative engagement ; Types of dissonance in narrative comprehension ; Narrative incongruities ; Narrative impossibilities ; Cognitive access to impossible storyworlds: immersed and reflected operations ; 'Impossibilities' and embodied cognition ; Disrupting viewers' reliance on image schemas by perceptual impossibility ; Disrupting viewers' reliance on image schemas by formal impossibility -- Taming Dissonance: Cognitive Operations and Interpretive Strategies. Cognitive dissonance versus narrative coherence ; Reducing dissonance: interpretation and naturalization ; Foregrounding ; Narrating agency and authorship ; Artefact emotions and metareflexive appreciation ; Interpretation as dissonance reduction ; Coping with dissonance: frame-switches and poetic and aesthetic readings ; Frame-switching as hermeneutic play in impossible puzzle films ; Switching between narrative and symbolical readings: Enemy and Mulholland Drive ; Cognitive hesitation and the fantastic -- Impossible Puzzle Films: Between Art-Cinema and (Post-) Classical Narration. From art-cinema to puzzle films ; Art-cinema as a narrative mode ; Dissonance in modernist art-cinema ; Art-cinema as a cognitive reception frame ; Narrative complexity and meaning-making in art-cinema ; Impossible puzzle films and (post-)classical narration ; High degree of tellability ; Identification with goal-oriented characters ; Strong reliance on classical genre elements ; Adherence to classical narrative cohesion devices ; Inclusion of quasi-rational frames of naturalization -- Wallowing in Dissonance: The Attractiveness of Impossible Puzzles. Hermeneutic play and interpretive multiplicity ; Orientation, navigation, mapping ; Game logic and the fascination in failure ; Effort justification ; Diegetization of decoupling ; Fascination in infinity ; Destabilized ontological certainty ; Eudaimonic motivations and intrinsic needs. (shrink)

As scholars have commonly observed, a central difference between epistemic and practical normativity is the fact that the reasons of the former kind balance prohibitively, while reasons of the latter kind do so permissively. To explain the prohibition to believe or disbelieve in the face of tied evidence, several scholars have appealed to normative reasons in favour of a third doxastic option, the suspension of judgement. However, the question remains as to what happens if these latter reasons are defeated. In (...) such cases, one seems to be stuck in an impasse: either one is forced into an epistemic dilemma– where neither of one’s doxastic options seems permissible– or one must relax the prohibitive balancing principle– where there might be, after all, situations in which believing in the face of tied evidence is rationally permissible. In this paper, I discuss different ways to derive this puzzle of defeated suspension, as well as the most common strategy in responding to it. I argue that an appeal to undefeated reasons to suspend judgement comes with significant downsides and, thus, one should seek a different response: in extreme forms of the puzzle of defeated suspension, a form of higher-order suspension becomes rational. If one treats higher-order suspension as an additional doxastic option, one can avoid the puzzle in its entirety. Furthermore, an analogue puzzle of higher-order doubt merits the same treatment. (shrink)

Frege’s puzzling remarks on the beginning of On Sense and Reference challenge us to explain how true identity sentences of the form a = a can differ in cognitive value from sentences of the form a = b when they are made true by the same object’s self-identity. Some philosophers (e.g. Almog, Glezakos and Paganini) suggest that the puzzle cannot be set up in the context of natural languages since natural sentences, unlike those of regimented formal ones, do not wear (...) their logical properties on their sleeves. In this paper we argue that, on the contrary, there exists a notion of coordination between names which is apt to track the relevant logical properties of natural sentences and therefore to set up the puzzle in natural languages. Frege’s puzzle is here to stay. (shrink)

Logic is the study of the validity of arguments, which is to say the study of when a conclusion follows or does not follow from a set of premises. Logic is an ancient discipline pioneered by Aristotle and developed by some of the greatest thinkers in the Middle Ages. However, in the nineteenth century logic underwent a remarkable transformation into a precise branch of mathematics that changed the nature of logic, and the study of religion, forever. (...) Both religious adherents and critics of religion have sometimes claimed that logic and religion do not mix, but this is a puzzling claim since logic itself is not a set of substantive premises about the world which may come into conflict with religious claims, but a means of testing mathematically which premises come into conflict with which other premises. In fact, sense can be made of whether logic and religion mix, but the question, and the answer, turn out to be more complex than many have thought. (shrink)

In this manuscript, we define and discuss a new type of logical puzzles. These puzzles are based on the simplest truth-teller and liar puzzles. Graphs are used to represent graphically the puzzles. these logical puzzles contain three types of people. Strong Truth-tellers who can say only true statements, Strong Liars who can make only false statements and Weak Crazy people who must make at least one self-contradicting statement if he/she says anything. Self-contradicting statements are related (...) to the Liar paradox, such that, there is no Truth-teller or a Liar could say “I am a Liar”. In any good puzzle there is a unique solution, while the puzzle is clear if only the people of the puzzle and their statements are given to solve the puzzle. It is well-known that there is no good and clear SS-puzzle. However, in this paper, we show that there are clear and good SSW-puzzles. Characteristics of the newly investigated type of people, the ‘Weak Crazy’ people, has also been studied. Some statistical results about the new type of puzzles and a comparison with other types of puzzles are also shown: the number of solvable and also the number of good puzzles is much larger than in the previously known SS-puzzles. (shrink)

The author tends to emphasize that there are at least three reasons to analyze Callimachus\' epigram about Diodorus : First of all, the date of this epigram shows us that it represents the earliest information about Diodorus doctrine. Second, another support of its authenticity could be found in fact that this epigram expressing part of the atmosphere following, and also remaining after, discussing the Diodorian topics. Third, its philosophical relevance, usually minimized in classical literature, could be found in those facts (...) that it could show the way out in many today dilemmas about his philosophical claims and support some of our contemporary assumptions about its logical conception, as well as that of space, time, and meaning of statements. The author defends a position that it is necessary to develop well-grounded and methodologically relevant base covering the historical reconstruction and the interpretation of ancient logical theories. (shrink)

Summary“There are some types of reasoning which are acceptable in a given situation but not justifiable according to the rules of formal logic. This sort of reasoning seems to depend on a judgment about what the speaker knows along with an Assumption of Maximum Information, that if the speaker is serious he is making the logically strongest statement he knows to be true. Because such reasoning can be informally correct, formal logic should be understood as establishing rules not (...) for all acts of inference but only for those which do not depend on contingent judgments about the context of an assertion.”RésuméIl y a des types de raisonnement qui sont acceptables dans une situation donnée, mais qui ne sont pas justifiables selon les règles de la logique formelle. Cette sorte de raisonnement semble dépendre d'un jugement sur ce que sait celui qui parle, ainsi que d'un Postulat d'Information Maximale selon lequel, si le locuteur est sérieux, il choisit l'énoncé le plus fort logiquement parmi ceux qu'il sait être vrais. Un tel raisonnement étant informellement correct, la logique formelle devrait être conçue comme édictant des règles non pas pour tous les actes d'inférence, mais seulement pour ceux qui ne dépendent pas de jugements contingents quant au contexte d'une assertion.Zusammenfassung〈Es gibt Argumentationsweisen, die in einer bestimmten Situation annehmbar sind, obschon sie sich auf Grund der Regeln der formalen Logik nicht rechtfertigen lassen. Sie hängen ab von einem Urteil über das, was der Redende weiss, wie auch von der Annahme maximaler Information, die besagt, dass er, wenn er im Ernst spricht, auch die starkste Aussage macht, von der er weiss, dass sie wahr ist. Weil es korrekte informelle Folgerungen dieser Art gibt, kann man sagen, dass die formate Logik nicht Regeln fiir aile Schlussfolgerungen liefert, sondern nur für solche, die nicht von kontingenten Urteilen über den Aeusserungskontext abhängen.〉. (shrink)

Collects a variety of mathematics and logic puzzles, some based on the theorems of the mathematician Kurt Godel.

This paper presents a puzzle about moral responsibility. The problem is based upon the indeterminacy of relevant reference classes as applied to action. After discussing and rejecting a very tempting response I propose moral contextualism instead, that is, the idea that the truth value of judgments of the form S is morally responsible for x depends on and varies with the context of the attributor who makes that judgment. Even if this reply should not do all the expected work it (...) is a first step. (shrink)

The two works reprinted in this volume are a unique fusion of logical thought and inimitable whimsy. Written by the 19th-century mathematician who also gave us "Alive in Wonderland", they are among the most entertaining logical works ever written, and contain some of the most thought-provoking puzzles ever devised.

Several logical puzzles, riddles and problems are defined based on the notion of games in informative contexts. Hintikka argues that epistemology or the theory of knowledge must be considered from the notion of information. So, knowledge cannot just be based on the notions of belief and justification. The present proposal will focus on the logical structure of information, and not only on the quantification of information as suggested by Claude A. Shannon. In many cases, the information bits, although seemingly (...) or factually contradictory, are quite relevant. The paraconsistent systems of logic offer a formalization of reasoning that can support certain contradictions. The well-known “Bar-Hillel–Carnap Paradox” causes embarrassment when it concludes that the informational content of a contradiction would be maximum, opposing the traditional notion that the semantic information must be true, and that contradictions are necessarily false. (shrink)

This chapter examines puzzle about the word ‘or’ and some proposed solutions. The theorems of propositional logic are interpreted in such a way as to be provided with an analogous account. The contrast lies in the absence of any such compositional account of ‘or’ and the vocabulary of preference. The distributive puzzle has merely been displaced. We should have to justify the assumption that every list formed with ‘or’ is a disjunctive list. A grammaticological account of distribution over or-lists (...) depends upon our being able to understand the list-containing sentence as a sentence-embedding sentence. Readers have to add a parallel observation about ‘or’. Disjunctive occurrences of ‘or’ as a sentential connective are, if not quite rare, at least not among the commonest uses of ‘or’ in spoken or written English. (shrink)

We present a new puzzle about logical truth, necessity, and moral responsibility. We defend one solution to the puzzle. A corollary of our preferred solution is that prominent arguments for the incompatibility of determinism and moral responsibility are invalid.