This paper defends a counterexample to Modus Tollens, and uses it to draw some conclusions about the logic and semantics of indicative conditionals and probability operators in natural language. Along the way we investigate some of the interactions of these expressions with 'knows', and we call into question the thesis that all knowledge ascriptions have truth-conditions.

Ethicists and economists commonly assume that if A is all things considered better than B, and B is all things considered better than C, then A is all things considered better than C. Call this principle Transitivity. Although it has great conceptual, intuitive, and empirical appeal, I argue against it. Larry S. Temkin explains how three types of ethical principle, which cannot be dismissed a priori, threaten Transitivity: (a) principles implying that in some cases different factors are relevant to comparing (...) A to C than to comparing A to B or B to C; (b) principles of limited scope; (c) principles implying that morally relevant differences in degree can amount to differences in kind. My counterexamples employ a principle of type (c): pleasures and pains enormously different in intensity differ in kind. Temkin has also endorsed this type of counterexample, using arguments based on earlier drafts of this paper. (shrink)

In this essay, I present a straightforward counterexample to the Uniqueness Thesis, which holds, roughly speaking, that there is a unique rational response to any particular body of evidence.

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged (...) an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object‐wise, could underlie diagram‐based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions. (shrink)

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)

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.

The theory of causal Bayes nets [15, 19] is, from an empirical point of view, currently one of the most promising approaches to causation on the market. There are, however, counterexamples to its core axiom, the causal Markov condition. Probably the most serious of these counterexamples are EPR/B experiments in quantum mechanics (cf. [13, 23]). However, these are also the only counterexamples yet known from the quantum realm. One might therefore wonder whether they are the only phenomena in the quantum (...) realm that create problems for causal Bayes nets. The aim of this paper is to demonstrate that not only the phenomenon of quantum correlations in EPR/B experiments create problems for causal Bayes nets, but also the temporal evolution of quantum systems, which is described as dualistic by quantum mechanics. For this purpose, it is shown that single photon experiments in a Mach-Zehnder interferometer (MZI) violate the causal Markov condition as well. It is then argued, however, that the Markov violation does not occur under the de Broglie-Bohm interpretation of Bohmian mechanics. (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).

Frankfurt sets out to refute the principle according to which ‘a person is morally responsible for what he has done only if he could have done otherwise’ . Frankfurt devises a counterexample in which an agent is intuitively responsible even though she could not have done otherwise: Suppose someone – Black, let us say – wants Jones to perform a certain action. Black is prepared to go to considerable lengths to get his way, but he prefers to avoid showing (...) his hand unnecessarily. So he waits until Jones is about to make up his mind what to do, and he does nothing unless it is clear to him that Jones is going to decide to do something other than what he wants him to do. If it does become clear that Jones is going to decide to do something else, Black takes effective steps to ensure that Jones decides to do, and that he does do, what he wants him to do. Whatever Jones's initial preferences and inclinations, then, Black will have his way … Now suppose that Black never has to show his hand because Jones, for reasons of his own, decides to perform and does perform the very action Black wants him to perform. In that case, it seems clear, Jones will bear precisely …. (shrink)

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.

Many philosophers (myself included) have been converted to causal decision theory by something like the following line of argument: Evidential decision theory endorses irrational courses of action in a range of examples, and endorses “an irrational policy of managing the news”. These are fatal problems for evidential decision theory. Causal decision theory delivers the right results in the troublesome examples, and does not endorse this kind of irrational news-managing. So we should give up evidential decision theory, and be causal decision (...) theorists instead. Unfortunately, causal decision theory has its own family of problematic examples for which it endorses irrational courses of action, and its own irrational policy that it is committed to endorsing. These are, I think, fatal problems for causal decision theory. I wish that I had another theory to offer in its place. (shrink)

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)

Manley and Wasserman (2008) join the chorus of opposition to the possibility of conditional analysis of dispositions. But that score cannot be settled without more careful attention to the implicit philosophical methodology. Some of the opposition to such an analysis badly overestimates the effect of counterexamples, as if the Gettier example were sufficient to refute the possibility of conjunctive analysis of knowledge. A general objection to a form of analysis must satisfy a number of constraints, and Manley and Wasserman join (...) the chorus too in failing to satisfy them. Most significant is the optional presupposition that the conditional used in analysis will exhibit some sort of centring. We show that even a careful effort to repair and reform Manley and Wasserman's objection to provide a satisfactory argument requires, ultimately, appeal to centring. Worse, the particular positive proposal they offer is vulnerable to a minor variant of their own counterexample. (shrink)

In recent years there has been a growing consensus that ordinary reasoning does not conform to the laws of classical logic, but is rather nonmonotonic in the sense that conclusions previously drawn may well be removed upon acquiring further information. Even so, rational belief formation has up to now been modelled as conforming to some fundamental principles that are classically valid. The counterexample described in this paper suggests that a number of the most cherished of these principles should not (...) be regarded as valid for commonsense reasoning. An explanation of this puzzling failure is given, arguing that a problem in the theory of rational choice transfers to the realm of belief formation. (shrink)

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)

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)

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.

Frankfurt sets out to refute the principle according to which ‘a person is morally responsible for what he has done only if he could have done otherwise’. Frankfurt devises a counterexample in which an agent is intuitively responsible even though she could not have done otherwise: Suppose someone – Black, let us say – wants Jones to perform a certain action. Black is prepared to go to considerable lengths to get his way, but he prefers to avoid showing his (...) hand unnecessarily. So he waits until Jones is about to make up his mind what to do, and he does nothing unless it is clear to him that Jones is going to decide to do something other than what he wants him to do. If it does become clear that Jones is going to decide to do something else, Black takes effective steps to ensure that Jones decides to do, and that he does do, what he wants him to do. Whatever Jones's initial preferences and inclinations, then, Black will have his way … Now suppose that Black never has to show his hand because Jones, for reasons of his own, decides to perform and does perform the very action Black wants him to perform. In that case, it seems clear, Jones will bear precisely …. (shrink)

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)

Plantinga’s Free Will Defense is an argument that, possibly, God cannot actualize a world containing significant creaturely free will and no wrongdoings. I will argue that if standard Molinism is true, there is a pair of worlds w1 and w2 each of which contains a significantly free creature who never chooses wrongly, and that are such that, necessarily, at least one of these worlds is a world that God can actualize.

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.

The likelihood theory of evidence (LTE) says, roughly, that all the information relevant to the bearing of data on hypotheses (or models) is contained in the likelihoods. There exist counterexamples in which one can tell which of two hypotheses is true from the full data, but not from the likelihoods alone. These examples suggest that some forms of scientific reasoning, such as the consilience of inductions (Whewell, 1858. In Novum organon renovatum (Part II of the 3rd ed.). The philosophy of (...) the inductive sciences. London: Cass, 1967), cannot be represented within Bayesian and Likelihoodist philosophies of science. (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.

There is currently much discussion about how decision making should proceed when an agent's degrees of belief are imprecise; represented by a set of probability functions. I show that decision rules recently discussed by Sarah Moss, Susanna Rinard and Rohan Sud all suffer from the same defect: they all struggle to rationalize diachronic ambiguity aversion. Since ambiguity aversion is among the motivations for imprecise credence, this suggests that the search for an adequate imprecise decision rule is not yet over.

David Makinson has argued that the compelling character of counterexamples to the Recovery Condition on contraction is due to an appeal to justificational structure. In “naked theories” where such structure is ignored or is not present, Recovery does apply. This note attempts to show that Makinson is mistaken on both counts. Recovery fails when no appeal is made to justificational structure.

According to Jody Azzouni’s “deflationary nominalism,” the singular terms of mathematical language applied or unapplied to science refer to nothing at all. What does exist, Azzouni claims, must satisfy the quaternary condition he calls “thick epistemic access” (TEA). In this paper I argue that TEA surreptitiously reifies some mathematical entities. The mathematical entity that I take TEA to reify is the Fourier harmonic, an infinite-duration monochromatic sinusoid applied throughout engineering and physics. I defend the reality of the harmonic, in Azzouni’s (...) account, not by satisfying all four TEA conditions with respect to it, but by showing that the harmonic renders satisfiable the TEA condition called “grounding,” specifically in Azzouni’s (Talking about nothing: numbers, hallucinations, and fictions. New York: Oxford University Press, 2010) example of a human visually perceiving a vase. The harmonic thereby plays what Azzouni calls an “epistemic role,” and merits inclusion in the deflationary nominalist ontology. Against the “coding” objection of Azzouni and Bueno (Br J Philos Sci 67:781–816, 2016), which would nominalize the harmonic to nonexistent status, I reply that in the context of grounding, the coding objection begs the question. (shrink)

Two alternative accounts have emerged as viable competitors to the forerunning counterfactual comparative account in the recent debate concerning the nature of harm. These are the “non-comparative statebased account of harm ” defended by Elizabeth Harman, the “event-based account of harm ” defended by Matthew Hanser. I raise one simple but serious counterexample involving “non-regrettable disabilities” that applies to both of these alternative accounts but that is avoided by the counterfactual comparative account. I point out that my counterexample (...) is one instance of a broader problem for alternatives to the counterfactual comparative account. The problem is that each of them divorces the concept of harm from the intuitive idea that we have moral and prudential reasons to avoid it. (shrink)

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).

In his book God and Necessity and in four subsequent papers, Brian Leftow argues against metaphysical theories which hold that “God’s nature makes necessary truths true or gives rise to their truthmakers,” asserting that all such “deity theories commit us to the claim that God’s existence depends on there being truthmakers for particular necessary truths about creatures.” Leftow supports this by arguing that all deity theories entail that if it is untrue that water = H2O, then God does not exist. (...) This paper presents a counterexample deity theory along with a synopsis of its correlative theory of truth and truthmaking. (shrink)

Many epistemologists treat knowledge as a binary relation that holds between a subject and a proposition. The contrastive account of knowledge developed by Jonathan Schaffer maintains that knowledge is a ternary, contrastive relation that holds between a subject, a proposition, and a set of contextually salient alternative propositions the subject’s evidence must eliminate. For the contrastivist, it is never simply the case that S knows that p; in every case of knowledge S knows that p rather than q. This paper (...) offers a counterexample to the contrastive account of knowledge. Part 1 summarizes the contrastive theory developed by Schaffer in a series of recent papers. Part 2 presents an example from a class of cases characterized by compatibility between the proposition p and each of the alternative propositions that occupy q. In such cases the alternative propositions that partially constitute the ternary contrastive relation play no role in the acquisition of knowledge. Part 3 considers and rejects potential responses to the counterexample. The paper concludes that the contrastive theory is not a general account of knowledge. (shrink)

I discuss three recent counterexamples to the transitivity of grounding due to Jonathan Schaffer. I argue that the counterexamples don’t work and draw some conclusions about the relationship between grounding and explanation.

The Direct Argument is an important argument for demonstrating that moral responsibility is incompatible with determinism because it makes no presuppositions about the nature of free will. One of the inference rules employed in the Direct Argument is rule A: If a proposition is broadly logically necessary, then it is true and no one is, nor ever has been, even partially morally responsible for the fact that the proposition is true. While inference rule A is assumed by all parties to (...) the debate, I demonstrate that the rule is invalid. (shrink)

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)

ABSTRACT Agents in Frankfurt-style counterexamples only appear to be responsible insofar as they act willingly in the actual sequence, but would need to be manipulated against their will into forming the relevant intention in the alternative sequence. This difference appears ineliminable and unavoidably morally significant. ‘Neo-Frankfurtians’ concede that the sequences must be physically differentiated, but deny their moral differentiation. In contrast, I explore whether the alternatives could be physically undifferentiated, despite their moral difference. The reason there is an ineliminable moral (...) difference is because there is an ineliminable causal one: The agent’s choice must be caused by his own inclinations in one sequence, but by Black’s manipulation in the other. Whether this causal difference entails a physical one depends on whether we suppose causal relations conform to Humean supervenience. The denial of the latter requires us to either reject any broadly Humean view of causation, or else to suppose that we can individuate events or their qualities independently of the physical facts. If these are coherent possibilities, such examples may show that physically differentiated alternatives are not required for moral responsibility. This could support compatibilism, but only if such examples adequately parallel determinism, and I remain sceptical on this point. (shrink)

A criticism of Frankfurt-counterexamples presented by David Widerker and known as the W-defense has been resilient for years and has been considered one of the strongest challenges these counterexamples have to face. In this paper I intend to offer an explanation of one of the appeals on the W-Defense, mainly, that it allows us to pass over the intricate debate on whether a successful Frankfurt counterexample can be presented or not. I defend this debate, although interesting and fruitful, misses (...) the main point Frankfurt counterexamples intend to make. Next I offer a defense of Frankfurt counterexamples from Widerker’s attack by presenting a dilemma for the Principle of Alternative Expectations, it’s main premise: either on the one hand, this principle rests on the Kantian maxim ought implies can, which makes the principle, and Widerker’s whole argument, redundant and unnecessary; or the principle does not rest on such maxim, but then there is no good reason to accept it a valid. (shrink)

We give a counterexample to the claim that every provably total function of Basic Arithmetic is a polynomially bounded primitive recursive function.

We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal.

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.

The well‐known “Consequence Argument” for the incompatibility of freedom and determinism relies on a certain rule of inference; “Principle Beta”. Thomas Crisp and Ted Warfield have recently argued that all hitherto suggested counterexamples to Beta can be easily circumvented by proponents of the Consequence Argument. I present a new counterexample which, I argue, is free from the flaws Crisp and Warfield detect in earlier examples.

We exhibit several counterexamples showing that the famous Serrin's symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for such partially overdetermined systems and justify their assumptions as well.

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)

This paper reveals and corrects a flaw in Nozick’s account of knowledge via inference. First, two counterexamples are provided by considering cases which would not typically be regarded as instances of knowledge although they are counted as such by Nozick’s theory. Then the general form of these counterexamples is given. From this it is apparent that the counterexamples show that Nozick’s theory fails to take account of cases in which the subject infers q from p, but in counterfactual situations some (...) proposition other than p would entail q. In view of this, the theory is then revised to eliminate the counterexamples. (shrink)