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)

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)

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.

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.

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.

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.

"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.

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)

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.

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.

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.

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.

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)

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)

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)

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)

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)

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.

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)

In general, a given object could have been different in certain respects. For example, the Great Pyramid could have been somewhat shorter or taller; the Mona Lisa could have had a somewhat different pattern of colours; an ordinary table could have been made of a somewhat different quantity of wood. But there seem to be limits. It would be odd to suppose that the Great Pyramid could have been thimble-sized; that the Mona Lisa could have had the pattern of colours (...) that actually characterizes The Scream; or that the table could have been made of the very quantity of wood that in fact made some other table. However, there are puzzling arguments that purport to show that so long as an object is capable of being somewhat different in some respect, it is capable of being radically different in that respect. These arguments rely on two tempting thoughts: first, that an object’s capacity for moderate variation is a non-contingent matter, and second, that what is possibly possible is simply possible. This book systematically investigates competing strategies for resolving these puzzles, and defends one of them. Along the way it engages with foundational questions about the metaphysics of modality. (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 this paper, using Mark Schroeder’s (2008a) expressivist semantic framework for normative language as a case study, I will identify difficulties that even an expressivist semantic theory capable of addressing the Frege-Geach problem will encounter in handling the logical possibility of moral dilemmas. To this end, I will draw on a classical puzzle formulated by McConnell (1978) that the logical possibility of moral dilemmas conflicts with some of the prima facie plausible axioms of the standard deontic logic, which include (...) obligation implies permission. On the tentative assumption that proponents of ethical expressivism should be generally committed to securing the logical possibility of moral dilemmas in their semantic theories, I will explore whether and how expressivists can successfully invalidate obligation implies permission within the framework developed by Schroeder. The case study eventually reveals that this can indeed be a hard task for expressivists. Generalizing from the case study, I will suggest that the source of the difficulty ultimately lies in the mentalist assumption of the expressivist semantic project that the logico-semantic relations exhibited by normative sentences should be modeled in terms of the psychological attitudes that speakers express by uttering them. My final goal will be to show that the difficulty expressivists face in dealing with the logical possibility of moral dilemmas is a reflection of the more general problem that their commitment to the mentalist assumption prevents them from flexibly adopting or dropping axioms in their semantic theories to get the right technical results. (shrink)

How is it that a speaker _S_ can at once make it obvious to an audience _A_ that she intends to communicate some proposition _p_, and yet at the same time retain plausible deniability with respect to this intention? The answer is that _S_ can bring it about that _A_ has a high justified credence that ‘_S_ intended _p_’ without putting _A_ in a position to know that ‘_S_ intended _p_’. In order to achieve this _S_ has to exploit a (...) sense in which communication can be lottery-like. After defending this view of deniability I argue that it compares favorably to a rival account recently developed by Dinges and Zakkou (Mind, https://doi.org/10.1093/mind/fzac056, 2023 ). (shrink)

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

The first page of Frege’s classic “Uber Sinn und Bedeutung” sets for more than a hundred years now the agenda for much of semantics and the philosophy of mind. It presents a purported puzzle whose solution is said to call upon the “entities” of semantics (meanings) and psychological explanation (Psychological states, beliefs, concepts). The paper separates three separate alleged puzzles that can be read into Frege’s data. It then argues that none are genuine puzzles. In turn, much of (...) the Frege-driven theoretical development, motivated as an inevitable “solution”, is thrown into doubt. (shrink)

Leave logic at the door and prepare to test your creative thinking skills with these 100 brand new lateral thinking puzzles. Each puzzle describes a situation which at first sounds odd or unlikely, and you have to work your way to the answer using creativity, imagination, and intuition. To cater for all ability levels, the 100 puzzles are arranged into Easy, Moderate, and Difficult categories, and a series of hints is provided for each puzzle to help you (...) along if you get stuck. As lateral thinking puzzles are fun, engaging, and make good icebreakers, they are a perennial favourite at dinner parties, break times, and social evenings. (shrink)

Hempel's paradox of the ravens has to do with the question of what constitutes confirmation from a logical point of view; Wason 's selection task has been used extensively to investigate how people go about attempting to confirm or disconfirm conditional claims. This paper presents an argument that the paradox is resolved, and that people's typical performance in the selection task can be explained, by consideration of what constitutes an effective strategy for seeking evidence of the tenability of universal or (...) conditional claims in everyday life. (shrink)

According to a plausible and influential account of perceptual knowledge, the truth-makers of beliefs that constitute perceptual knowledge must feature in the causal explanation of how we acquire those beliefs. However, this account runs into difficulties when it tries to accommodate time perception – specifically perception of order and duration – since the features we are apparently tracking in such perception are not causal. The central aim of the paper is to solve this epistemological puzzle. Two strategies are examined. The (...) first strategy locates the causal truth-makers within the psychological mechanism underlying time perception, thus treating facts about time order and duration as mind-dependent. This strategy, however, is problematic. The second strategy modifies the causal account of perceptual knowledge to include a non-causal component in the explanation of belief-acquisition, namely chronometric explanation. Applying this much more satisfactory approach to perceptual knowledge of time, we can preserve the mind-independence of order and duration, but not that of time's flow. (shrink)

According to traditional Western theism, God is maximally great (or perfect). More explicitly, God is said to have the following divine attributes: omnipotence, omniscience, and omnibenevolence. In this paper, I present three puzzles about this conception of a maximally great (or perfect) being. The first puzzle about omniscience shows that this divine attribute is incoherent. The second puzzle about omnibenevolence and omnipotence shows that these divine attributes are logically incompatible. The third puzzle about perfect rationality and omnipotence shows that (...) these divine attributes are logically incompatible. (shrink)

_Puzzled?!_ seamlessly fuses two traditional approaches to the study of philosophy at the introductory level. It is thematic, examining fundamental issues in epistemology, metaphysics, philosophy of religion, and more. It is also historical, introducing major philosophical arguments that have arisen throughout the history of Western philosophy. But its real innovation lies elsewhere. Each of its twelve chapters begins with a traditional argument of a thoroughly puzzling kind: a valid philosophical argument with highly plausible premises but a surprising conclusion. The remainder (...) of the chapter shows how major innovations in the history of philosophy arise as logical responses to that argument. Written with a light touch, _Puzzled?!_ nevertheless offers a rigorous introduction to the ideas it explores and to the foundations of critical thinking itself. It will serve as effectively as a main or supplementary text in an introduction to critical thinking as it will in Philosophy 101. (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)