Corcoran, J. 2005. Counterexamples and proexamples. Bulletin of Symbolic Logic 11(2005) 460. -/- John Corcoran, Counterexamples and Proexamples. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 E-mail: [email protected] Every perfect number that is not even is a counterexample for the universal proposition that every perfect number is even. Conversely, every counterexample for the proposition “every perfect number is even” is a perfect number that is not even. Every perfect number that is odd is a proexample for the existential proposition (...) that some perfect number is odd. Conversely, every proexample for the proposition “some perfect number is odd” is a perfect number that is odd. As trivial these remarks may seem, they can not be taken for granted, even in mathematical and logical texts designed to introduce their respective subjects. One well-reviewed book on counterexamples in analysis says that in order to demonstrate that a universal proposition is false it is necessary and sufficient to construct a counterexample. It is easy to see that it is not necessary to construct a counterexample to demonstrate that the proposition “every true proposition is known to be true” is false–necessity fails. Moreover the mere construction of an object that happens to be a counterexample does not by itself demonstrate that it is a counterexample–sufficiency fails. In order to demonstrate that a universal proposition is false it is neither necessary nor sufficient to construct a counterexample. Likewise, of course, in order to demonstrate that an existential proposition is true it is neither necessary nor sufficient to construct a proexample. This article defines the above relational concepts of counterexample and of proexample, it discusses their surprising history and philosophy, it gives many examples of uses of these and related concepts in the literature and it discusses some of the many errors that have been made as a result of overlooking the challenging subtlety of the proper use of these two basic and indispensable concepts. (shrink)

According to a bedrock assumption in the current methodology of armchair philosophy, we may refute a theory aiming at analyzing a concept by providing a counterexample in which it intuitively seems that a hypothetical or real situation does not fit with what the theory implies. In this paper, we shall argue that this assumption is at most either untenable or otherwise useless in bringing about what is commonly expected from it.

A symposium on my *Objects: Nothing Out of the Ordinary* (2015). In response to Wallace, I attempt to clarify the dialectical and epistemic role that my arguments from counterexamples were meant to play, I provide a limited defense of the comparison to the Gettier examples, and I embrace the comparison to Moorean anti-skeptical arguments. In response to deRosset, I provide a clearer formulation of conservatism, explain how a conservative should think about the interaction between intuition and science, and discuss (...) what conservatives should say about scattered territories, clonal colonies, and arbitrary systems. In response to Tillman and Spencer, I fortify my original presentation of the debunking arguments by clarifying why, even while trees (if they exist) are paradigmatically causal, conservatives are meant to be rationally obstructed from believing that it is trees that are causing our tree beliefs. (shrink)

We provide counterexamples to some purported characterizations of dilation due to Pedersen and Wheeler :1305–1342, 2014, ISIPTA ’15: Proceedings of the 9th international symposium on imprecise probability: theories and applications, 2015).

Kopec and Titelbaum collect five alleged counterexamples to Uniqueness, the thesis that it is impossible for agents who have the same total evidence to be ideally rational in having different doxastic attitudes toward the same proposition. I argue that four of the alleged counterexamples fail, and that Uniqueness should be slightly modified to accommodate the fifth example.

The essay presents a novel counterexample to Causal Decision Theory (CDT). Its interest is that it generates a case in which CDT violates the very principles that motivated it in the first place. The essay argues that the objection applies to all extant formulations of CDT and that the only way out for that theory is a modification of it that entails incompatibilism. The essay invites the reader to find this consequence of CDT a reason to reject it.

I present two counterexamples to the recently back-in-favour truth-tracking account of knowledge: one involving a true belief resting on a counterfactually robust delusion, one involving a true belief acquired alongside a bunch of false beliefs. These counterexamples carry over to a recent modification of the theory due to Briggs and Nolan (2012), and seem invulnerable to a recent defence of the theory against known counterexamples, by Adams and Clarke (2005).

One strategy in recent discussions of theological fatalism is to draw on Harry Frankfurt’s famous counterexamples to the principle of alternate possibilities (PAP) to defend human freedom from divine foreknowledge. For those who endorse this line, “Frankfurt counterexamples” are supposed to show that PAP is false, and this conclusion is then extended to the foreknowledge case. This makes it critical to determine whether Frankfurt counterexamples perform as advertised, an issue recently debated in this journal via a pair (...) of articles by David Widerker and John Martin Fischer. I suggest that this debate can be avoided: divine foreknowledge is itself aparadigmatic counterexample to PAP, requiring no support from suspect Frankfurt counterexamples. (shrink)

Derek Parfit argues that everyone ought to follow the principles whose universal acceptance would make things go best. I present a counterexample: a world in which no one's moral beliefs have any motivating force. I explain how Parfit's metaethical commitments imply that such a world is possible, and why this possibility is a problem for Parfit's project of reconciling Kantianism, contractualism, and consequentialism. I consider two of Parfit's responses to my counterexample.

Most ordinary folks think that there are ordinary objects such as trees and frogs. They do not think there are extraordinary objects such as the mereological sum of trees and frogs, as the permissivist does. Nor do they deny the existence of ordinary composite objects such as tables, as the eliminativist does. In his recent book, Objects: Nothing Out of the Ordinary, Korman positions himself alongside ordinary folk. He deftly defends the common sense view of ordinary objects, and argues against (...) the permissivist and eliminativist. One of the ways he does this is by launching what he calls ‘arguments from counterexample’. In this paper, I focus on the dialectical effectiveness of such arguments generally. I discuss whether and how we might sometimes be justified in using arguments from counterexample and commonsense in philosophical debate, even if they appear to be (or are) question-begging. I conclude that, given some of Korman’s other commitments, there is little reason for his opponents to take his arguments from counterexample seriously. (shrink)

This paper considers five counterexamples to Mary Douglas's definition of dirt, one of which is extracted from a scene from George Eliot's novel Middlemarch and another from Marilyn Strathern's essay on anthropology at home.

In a recent paper, Breckenridge and Magidor argue for an interesting and counterintuitive account of instantial reasoning. According to this account, in arguments such as one beginning with 'There is some x such that x is mortal. Let O be such an x. ...', the 'O' refers to a particular object, although we cannot know which. I give and defend a simple counterexample involving the notion of an unreferred-to object.

In a recent paper, Tristan Haze offers two examples that, he claims, are counterexamples to Nozick's Theory of Knowledge. Haze claims his examples work against Nozick's theory understood as relativized to belief forming methods M. We believe that they fail to be counterexamples to Nozick's theory. Since he aims the examples at tracking theories generally, we will also explain why they are not counterexamples to Dretske's Conclusive Reasons Theory of Knowledge.

Years ago, when I was an undergraduate math major at the University of Wyoming, I came across an interesting book in our library. It was a book of counterexamples t o propositions in real analysis (the mathematics of the real numbers). Mathematicians work more or less like the rest of us. They consider propositions. If one seems to them to be plausibly true, then they set about to prove it, to establish the proposition as a theorem. Instead o f (...) setting out to prove propositions, the psychologists, neuroscientists, and other empirical types among us, set out to show that a proposition is supported by the data, and that it is the best such proposition so supported. The philosophers among us, when they are not causing trouble by arguing that AI is a dead end or that cognitive science can get along without representations, work pretty much like the mathematicians: we set out to prove certain propositions true on the basis of logic, first principles, plausible assumptions, and others' data. But, back to the book of real analysis counterexamples. If some mathematician happened t o think that some proposition about continuity, say, was plausibly true, he or she would then set out to prove it. If the proposition was in fact not a theorem, then a lot of precious time would be wasted trying to prove it. Wouldn't it be great to have a book that listed plausibly true propositions that were in fact not true, and listed with each such proposition a counterexample to it? Of course it would. (shrink)

In this paper I will offer a comprehensive defense of the safety account of knowledge against counterexamples that have been recently put forward. In Sect. 2, I will discuss different versions of safety, arguing that a specific variant of method-relativized safety is the most plausible. I will then use this specific version of safety to respond to counterexamples in the recent literature. In Sect. 3, I will address alleged examples of safe beliefs that still constitute Gettier cases. In (...) Sect. 4, I will discuss alleged examples of unsafe knowledge. In Sect. 5, I will address alleged cases of safe belief that do not constitute knowledge for non-Gettier reasons. My overall goal is to show that there are no successful counterexamples to robust anti-luck epistemology and to highlight some major presuppositions of my reply. (shrink)

Commonsense holds that testimony transfers knowledge from a speaker to the hearer. If the speaker has knowledge, then the hearer acquires it. Call that sufficiency. And a hearer acquires knowledge only if the speaker has it to transfer. Call that necessity. This article reviews counterexamples--and some replies to those counterexamples--to both claims.

McGee argued that modus ponens was invalid for the natural language conditional ‘If…then…’. Many subsequent responses have argued that, while McGee’s examples show that modus ponens fails to preserve truth, they do not show that modus ponens fails to preserve rational full acceptance, and thus modus ponens may still be valid in the latter informational sense. I show that when we turn our attention from indicative conditionals to subjunctive conditionals, we find that modus ponens does not preserve either truth or (...) rational full acceptance, and thus is not valid in either sense. In concluding I briefly consider how we can account for these facts. (shrink)

English translation of an entry on pages 137–42 of the Spanish-language dictionary of logic: Luis Vega, Ed. Compendio de Lógica, Argumentación, y Retórica. Madrid: Trotta. -/- DEDICATION: To my friend and collaborator Kevin Tracy. -/- This short essay—containing careful definitions of ‘counterargument’ and ‘counterexample’—is not an easy read but it is one you’ll be glad you struggled through. It contains some carefully chosen examples suitable for classroom discussion. -/- Using the word ‘counterexample’ instead of ‘counterargument’ in connection with Aristotle’s invalidity (...) proofs leads to a misinterpretation of Aristotle’s intentions. The two relations expressed by these two words are drastically different—as explained in this essay. There is however a subtle overlap: Every argument that is a counterargument for a given argument is a counterexample to the universal proposition that every argument in the same form as the given argument is valid. The word ‘counterexample’ was coined in the 1800s. The word ‘counterargument’ was given this sense in the 1960s. Saying that one argument is a counterexample for another argument is bad English—taking the word ‘counterexample’ in its Standard English sense. The word ‘counterinstance’ is an exact synonym for ‘counterexample’. Which logic books define ‘counterinstance’ or ‘counterexample’? Who started using ‘counterinstance’ or ‘counterexample’ in the sense of ‘counterargument’? Did they note that they were not using the word in the Standard sense? Why did it catch on? -/- Also see: https://www.academia.edu/8840909/Argumentations_and_logic_1989_ . (shrink)

In a recent article, David Hunt has proposed a theological counterexample to the principle of alternative possibilities involving divine foreknowledge (G-scenario). Hunt claims that this example is immune to my criticism of regular Frankfurt-type counterexamples to that principle, as God’s foreknowing an agent’s act does not causally determine that act. Furthermore, he claims that the considerations which support the claim that the agent is morally responsible for his act in a Frankfurt-type scenario also hold in a G-scenario. In reply, (...) Icontest Hunt’s symmetry claim and also raise a worry whether, given theological fatalism, the agent’s act in a G-scenario can be deemed a free act in the libertarian sense. Finally, I offer an independent argument why in a G-scenario the agent should not regarded morally blameworthy for his act. (shrink)

Externalists argue that motivation is external to moral judgments on the grounds that people can be unmoved by their moral judgments. I reply that people sometimes act indifferently to their moral considerations not because their moral judgments lack motivation but because their moral judgments are obstructed by rival desires. It appears that the moral motivation wanes while the moral judgments linger. In reality, however, the moral motivation is only made inconspicuous by the motivation of the opposing desires. A moral judgment (...) is subject to obstruction just like an emotive judgment and a gustatory judgment. (shrink)

The consequence argument is a salient argument in favor of incompatibilism which is the thesis that if determinism is true, then it is not the case that we have free will. In a nutshell, the consequence argument has it that if determinism is true, then our acts are determined by the laws of nature and events of the past. But we are neither able to change the past nor the laws of nature. Therefore, we are not able to change the (...) consequences of them. This short paper consists of two main sections. In the first section, I will explain the consequence argument and rule β. In addition, in the second section of the paper, I will introduce the no matter what interpretation which was suggested by Huemer (2000). Moreover, I will offer a counterexample to rule β∗ and α∗. (shrink)

Recent literature contains influential arguments for variabilism, the view that we should understand proper names as analogues not of constants but of variables. In particular, proper names are said to sometimes take semantic values that are not referential but purely general. I present a counter-example to this view.

Intuitively, Gettier cases are instances of justified true beliefs that are not cases of knowledge. Should we therefore conclude that knowledge is not justified true belief? Only if we have reason to trust intuition here. But intuitions are unreliable in a wide range of cases. And it can be argued that the Gettier intuitions have a greater resemblance to unreliable intuitions than to reliable intuitions. Whats distinctive about the faulty intuitions, I argue, is that respecting them would mean abandoning a (...) simple, systematic and largely successful theory in favour of a complicated, disjunctive and idiosyncratic theory. So maybe respecting the Gettier intuitions was the wrong reaction, we should instead have been explaining why we are all so easily misled by these kinds of cases. (shrink)

The author argues that Plato’s “proof” that happiness follows justice has a fatal flaw – because the philosopher king in Plato’s Republic is itself a counter example.

I argue that dreams can contain perceptual elements in multifarious, heretofore unthought-of ways. I also explain the difference between dreams that contain perceptual elements, perceptual experiences that contain dream elements, and having a dream and a perceptual experience simultaneously. I then discuss two applications of the resulting view. First, I explain how my taxonomy of perception in dreams will allow “dream engineers”—who try to alter the content of people’s dreams—to accurately classify different dreams and explore creating new forms of perception (...) in dreams. Second, I consider the consequences of the view for the role of memory in dreaming and imagination. I argue that not every element of dreams or sensory imaginations must rely on memory. The resultant view of sensory imagination provides a counterexample to Hume’s account of sensory imagination, according to which sensory imagination must be built up from faint copies of sensory impressions stored in memory. (shrink)

Must rational thinkers have consistent sets of beliefs? I shall argue that it can be rational for a thinker to believe a set of propositions known to be inconsistent. If this is right, an important test for a theory of rational belief is that it allows for the right kinds of inconsistency. One problem we face in trying to resolve disagreements about putative rational requirements is that parties to the disagreement might be working with different conceptions of the relevant attitudes. (...) My aim is modest. I hope to show that there is at least one important notion of belief such that a thinker might rationally hold a collection of beliefs (so understood) even when the thinker knows their contents entail a contradiction. (shrink)

Sungho Choi has criticised Michael Strevens's counterexample to DavidLewis's final theory of "token" causation, causation as "influence." Iargue that, even if Choi's points are correct, Strevens's counterexampleremains useful in revealing a shortcoming of Lewis's theory. Thisshortcoming is that Lewis's theory does not properly account for*degrees* of causation. That is, even if Choi's points are correct,Lewis's theory does not capture an intuition we have about the*comparative* causal statuses of those events involved in Strevens'scounterexample (we might, for example, intuit that Sylvie's ball-firingis (...) *as much*/*more*/*less* a cause of the jar's shattering as/than isBruno's ball-firing). (shrink)

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as (...) the set of numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching! (shrink)

Legal anti-positivism is widely believed to be a general theory of law that generates far too many false negatives. If anti-positivism is true, certain rules bearing all the hallmarks of legality are not in fact legal. This impression, fostered by both positivists and anti-positivists, stems from an overly narrow conception of the kinds of moral facts that ground legal facts: roughly, facts about what is morally optimific—morally best or morally justified or morally obligatory given our social practices. A less restrictive (...) view of the kinds of moral properties that ground legality results in a form of anti-positivism that can accommodate any legal rule consistent with positivism, including the alleged counter-examples. This form of “inclusive” anti-positivism is not just invulnerable to extensional challenges from the positivist. It is the only account that withstands extensional objections, while incorporating, on purely conceptual grounds, a large part of the content of morality into law. (shrink)

Aristotle’s doctrine of the mean states that each moral virtue stands opposed to two types of vice: one of excess and one of deficiency, respectively. Critics claim that some virtues—like honesty, fair-mindedness, and patience—are counterexamples to Aristotle’s doctrine. Here, I develop a generalizable strategy to defend the doctrine of the mean against such counterexamples. I argue that not only is the doctrine of the mean defensible, but taking it seriously also allows us to gain substantial insight into particular (...) virtues. Failure to take the doctrine seriously, moreover, exposes us to the risk of mistaking certain vices for virtues. (shrink)

In debates concerning the consequence argument, it has long been claimed that [McKay, T. J., and D. Johnson. 1996. “A Reconsideration of an Argument Against Compatibilism.” Philosophical Topics 24 (2): 113–122] demonstrated the invalidity of rule (β). Here, I argue that their result is not as robust as we might like to think. First, I argue that McKay and Johnson's counterexample is successful if one adopts a certain interpretation of ‘no choice about’ and if one is willing to deny the (...) conditional excluded middle principle. In order to make this point I demonstrate that (β) is valid on Stalnaker's theory of counterfactuals. This result is important and should not be neglected, I argue, because there is a particular line of objection to the revised formulations of the consequence argument that does not succeed against the original version. (shrink)

In this paper, I examine a new line of response to Frankfurt’s challenge to the traditional association of moral responsibility with the ability to do otherwise. According to this response, Frankfurt’s counterexample strategy fails, not in light of the conditions for moral responsibility per se, but in view of the conditions for action. Specifically, it is claimed, a piece of behavior counts as an action only if it is within the agent’s power to avoid performing it. In so far as (...) Frankfurt’s challenge presupposes that actions can be unavoidable, this view of action seems to bring his challenge up short. Helen Steward and Maria Alvarez have independently proposed versions of this response. Here I argue that this response is unavailable to Frankfurt’s incompatibilist opponents. This becomes evident when we put this question to its proponents: “Are actions that originate deterministically ipso facto unavoidable?” If they answer “yes,” they encounter one horn of a dilemma. If they answer “no,” they encounter the other horn. Since no one has a clearer stake in meeting Frankfurt’s challenge than these theorists do, it is significant that the Steward-Alvarez response is unavailable to them. (shrink)

This paper aims to defend deliberation-compatibilism against several objections, including a recent counterexample by Yishai Cohen that involves a deliberator who believes that whichever action she performs will be the result of deterministic manipulation. It begins by offering a Moorean-style proof of deliberation-compatibilism. It then turns to the leading argument for deliberation-incompatibilism, which is based on the presumed incompatibility of causal determinism and the ‘openness’ required for rational deliberation. The paper explains why this argument fails and develops a coherent account (...) of how one can rationally deliberate and believe in causal determinism without inconsistency. The second half of the paper then takes up Cohen's proposed counterexample and his Four-Case Deliberation Argument against deliberation-compatibilism, which is meant to mirror Derk Pereboom's famous Four-Case Manipulation Argument. In response, the author defends a hard-line reply to FCDA but also argues that the notion of ‘sourcehood’ relevant to rational deliberation differs from that involved in free will. (shrink)

Epistemic closure under known implication is the principle that knowledge of "p" and knowledge of "p implies q", together, imply knowledge of "q". This principle is intuitive, yet several putative counterexamples have been formulated against it. This paper addresses the question, why is epistemic closure both intuitive and prone to counterexamples? In particular, the paper examines whether probability theory can offer an answer to this question based on four strategies. The first probability-based strategy rests on the accumulation of (...) risks. The problem with this strategy is that risk accumulation cannot accommodate certain counterexamples to epistemic closure. The second strategy is based on the idea of evidential support, that is, a piece of evidence supports a proposition whenever it increases the probability of the proposition. This strategy makes progress and can accommodate certain putative counterexamples to closure. However, this strategy also gives rise to a number of counterintuitive results. Finally, there are two broadly probabilistic strategies, one based on the idea of resilient probability and the other on the idea of assumptions that are taken for granted. These strategies are promising but are prone to some of the shortcomings of the second strategy. All in all, I conclude that each strategy fails. Probability theory, then, is unlikely to offer the account we need. (shrink)

This self-contained one page paper produces one valid two-premise premise-conclusion argument that is a counterexample to the entire three traditional rules of distribution. These three rules were previously thought to be generally applicable criteria for invalidity of premise-conclusion arguments. No longer can a three-term argument be dismissed as invalid simply on the ground that its middle is undistributed, for example. The following question seems never to have been raised: how does having an undistributed middle show that an argument's conclusion does (...) not follow from its premises? This result does nothing to vitiate the theories of distribution developed over the period beginning in medieval times. What it does vitiate is many if not all attempts to use distribution in tests of invalidity outside of the standard two-premise categorical arguments—where they were verified on a case-by-case basis without further theoretical grounding. In addition it shows that there was no theoretical basis for many if not all claims of fundamental status of rules of distribution. These results are further support for approaching historical texts using mathematical archeology. (shrink)

The economist David K. Levine claims that if a government of a country makes torture legal, the inevitable result will be torture that is out of control. I point out an inconsistency in his approach to torture. I then argue that we should be open to rare counterexamples to his claim and describe a kind of counterexample.

This document is a brief summary of the literature. It presents two arguments for thinking that it is irrational to have cyclic preferences — to prefer option A over B, B over C, but C over A. It then presents four counterexamples, where this seems rational.

Unadorned process reliabilism (hereafter UPR) takes any true belief produced by a reliable process (undefeated by any other reliable process) to count as knowledge. Consequently, according to UPR, to know p, you need not know that you know it. In particular, you need not know that the process by which you formed your belief was reliable; its simply being reliable is enough to make the true belief knowledge. -/- Defenders of UPR are often presented with purported counterexamples describing subjects (...) who have true beliefs resulting from reliable (and undefeated) processes, but whom we do not intuitively take to know the propositions that they believe (call this “the internalist objection”). Mark McEvoy has recently challenged such arguments claiming (1) that the internalist objection against UPR simply begs the question against it, and (2) our intuitions about cases structurally similar to the standard examples characteristic of the internalist objection are actually often in line with UPR. In what follows I’ll argue that the plausibility of (1) depends on McEvoy’s success in establishing (2), but with the level of description provided, (2) seems undermotivated. (shrink)

An important objection to signaling approaches to representation is that, if signaling behavior is driven by the maximization of usefulness, then signals will typically carry much more information about agent-dependent usefulness than about objective features of the world. This sort of considerations are sometimes taken to provide support for an anti-realist stance on representation itself. The author examines the game-theoretic version of this skeptical line of argument developed by Donald Hoffman and his colleagues. It is shown that their argument only (...) works under an extremely impoverished picture of the informational connections that hold between agent and world. In particular, it only works for cue-driven agents, in Kim Sterelny’s sense. In cases in which the agents’ understanding of what is useful results from combining pieces of information that reach them in different ways, and that complement one another, maximizing usefulness involves construing first a picture of agent-independent, objective matters of fact. (shrink)

In this paper, I argue that, contrary to common opinion, a counterexample against a philosophical theory does not amount to conclusive evidence against that theory. Instead, the method of counterexamples allows for the derivation of a disjunction, i.e., ‘either the theory is false or an auxiliary assumption is false’, not a negation of the target theory. This is so because, whenever the method of counterexamples is used in an attempt to refute a philosophical theory, there is a crucial (...) auxiliary assumption that needs to be taken into account. The auxiliary assumption is that making intuitive judgments in response to hypothetical cases about the subject matter in question (e.g., knowledge or proper names) is a good method for finding out truths about that subject matter. Without good reasons to think that this assumption is warranted, the negation of a philosophical theory whose content is alleged to be in conflict with the content of an intuition cannot be justifiably derived using an argument that employs the method of counterexamples. (shrink)

In this article, I present a schema for generating counterexamples to the argument form known as Hypothetical Syllogism with indicative conditionals. If my schema for generating counterexamples to HS works as I think it does, then HS is invalid for indicative conditionals.

First, ‘Is Justified True Belief Knowledge’ is imprecise but Gettier is explicit that ‘know’ is analysed as the definiendum is ‘S knows that P’. Second, Gettier does not misrepresent as Plato’s definition as the expressions used are ‘Plato considers’ and ‘seems to accept’. Third, Gettier is not mistaken to apply Plato’s definition to propositions since propositional knowledge is a species of Plato’s definition. Fourth, for Plato true belief temporally precedes an account. ‘Jones owns a Ford’ is never a true opinion, (...) hence no account for it can be given. The counterexample is reconstructed with temporality built into it. Fifth, Gettier does not fail to establish the equivalence of ‘believe’, ‘accepts’ and ‘sure’ in the three versions as this is implicitly established in the shifts made in the paper. Sixth, ‘entails’ logically is used only when the entailing proposition is true, but in the counterexamples a false proposition is taken to imply a true one. ‘Entail’ is to be taken in the ordinary sense of implies. Seventh, in Case I, the implication is preserved with the proper representation: → [Gy ↔ ], which implies { → [Gy ↔ ]}’. Eighth, the counterexample is reworked to avoid the objection that justification for p and justification for q may not be sufficient justification for ‘p & q’. (shrink)

Moti Mizrahi (2013) presents some novel counterexamples to Hypothetical Syllogism (HS) for indicative conditionals. I show that they are not compelling as they neglect the complicated ways in which conditionals and modals interact. I then briefly outline why HS should nevertheless be rejected.

The author considers the model-theoretic character of proofs and disproofs by means of attempted counterexample constructions, distinguishes this proof format from formal derivations, then contrasts two approaches to semantic tableaux proposed by Beth and Lambert-van Fraassen. It is noted that Beth's original approach has not as yet been provided with a precisely formulated rule of closure for detecting tableau sequences terminating in contradiction. To remedy this deficiency, a technique is proposed to clarify tableau operations.

Several counterexamples are adduced against the view that surprise is an essential ingredient of humor.

The standard counterexamples to David Lewis’s account of intrinsicality involve two sorts of properties: identity properties and necessary properties. Proponents of the account have attempted to deflect these counterexamples in a number of ways. This paper argues that none of these moves are legitimate. Furthermore, this paper argues that no account along the lines of Lewis’s can succeed, for an adequate account of intrinsicality must be sensitive to hyperintensional distinctions among properties.