Vann McGee has presented a putative counterexample to modus ponens. I show that (a slightly modified version of) McGee’s election scenario has the same structure as a famous lottery scenario by Kyburg. More specifically, McGee’s election story can be taken to show that, if the Lockean Thesis holds, rational belief is not closed under classical logic, including classical-logic modus ponens. This conclusion defies the existing accounts of McGee’s puzzle.

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)

In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of Preservation for conditionals, which guarantees that one’s prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one’s non-conditional beliefs. We first note that Bradley’s original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of Commutativity, then in a doxastic analogue of the rule of modus ponens (...) —are problematic in a significant number of respects. We then turn to a recent U-turn on his part, in which he winds up rescinding his commitment to modus ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: assuming that one gives up on full-blown modus ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of modus ponens in McGee’s famous counterexamples. (shrink)

Chancy modus ponens is the following inference scheme: ‘probably φ’, ‘if φ, then ψ’, therefore, ‘probably ψ’. I argue that Chancy modus ponens is invalid in general. I further argue that the invalidity of Chancy modus ponens sheds new light on the alleged counterexample to modus ponens presented by McGee. I close by observing that, although Chancy modus ponens is invalid in general, we can recover a restricted sense in (...) which this scheme of inference is valid. (shrink)

Recently, there has been a shift away from traditional truth-conditional accounts of meaning towards non-truth-conditional ones, e.g., expressivism, relativism and certain forms of dynamic semantics. Fueling this trend is some puzzling behavior of modal discourse. One particularly surprising manifestation of such behavior is the alleged failure of some of the most entrenched classical rules of inference; viz., modus ponens and modus tollens. These revisionary, non-truth-conditional accounts tout these failures, and the alleged tension between the behavior of modal (...) vocabulary and classical logic, as data in support of their departure from tradition, since the revisionary semantics invalidate some of these patterns. I, instead, offer a semantics for modality with the resources to accommodate the puzzling data while preserving classical logic, thus affirming the tradition that modals express ordinary truth-conditional content. My account shows that the real lesson of the apparent counterexamples is not the one the critics draw, but rather one they missed: namely, that there are linguistic mechanisms, reflected in the logical form, that affect the interpretation of modal language in a context in a systematic and precise way, which have to be captured by any adequate semantic account of the interaction between discourse context and modal vocabulary. The semantic theory I develop specifies these mechanisms and captures precisely how they affect the interpretation of modals in a context, and do so in a way that both explains the appearance of the putative counterexamples and preserves classical logic. (shrink)

I argue that we should solve the Lottery Paradox by denying that rational belief is closed under classical logic. To reach this conclusion, I build on my previous result that (a slight variant of) McGee’s election scenario is a lottery scenario (see Lissia 2019). Indeed, this result implies that the sensible ways to deal with McGee’s scenario are the same as the sensible ways to deal with the lottery scenario: we should either reject the Lockean Thesis or Belief Closure. After (...) recalling my argument to this conclusion, I demonstrate that a McGee-like example (which is just, in fact, Carroll’s barbershop paradox) can be provided in which the Lockean Thesis plays no role: this proves that denying Belief Closure is the right way to deal with both McGee’s scenario and the Lottery Paradox. A straightforward consequence of my approach is that Carroll’s puzzle is solved, too. (shrink)

ABSTRACT: This paper traces the earliest development of the most basic principle of deduction, i.e. modus ponens (or Law of Detachment). ‘Aristotelian logic’, as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as ‘hypothetical syllogisms’. However, Aristotle did not (...) discuss such arguments, nor did he call any arguments ‘hypothetical syllogisms’. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them ‘hypothetical syllogisms’; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle’s logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called ‘hypothetical syllogisms’? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle’s dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle’s logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories. (shrink)

There are two primary arguments against scientific realism, one pertaining to underdetermination, the other to the history of science. While these arguments are usually treated as altogether distinct, P. Kyle Stanford's ‘problem of unconceived alternatives’ constitutes one kind of synthesis: I propose that Stanford's argument is best understood as a broad modus ponens underdetermination argument, into which he has inserted a unique variant of the historical pessimistic induction. After articulating three criticisms against Stanford's argument and the evidence that (...) he offers, I contend that, as it stands, Stanford's argument poses no threat to contemporary scientific realism. Nonetheless, upon identifying two useful insights present in Stanford's general strategy, I offer an alternative variant of the modus ponens underdetermination argument, one that, although historically informed by science, requires no inductive premises. I contend that this non-inductive but historically informed variant of the modus ponens clarifies and considerably strengthens the case against scientific realism. (shrink)

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it (...) introduces a novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality. Similar results for paraconsistent set theories were obtained in author papers [13]-[16].

The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion of Tarski’s axiom (...) which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier ZF (aa). In this paper we look at a set theory NC# ∞# , based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory NC# ∞# based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed. (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)

Epistemic modals in consequent place of indicative conditionals give rise to apparent counterexamples to Modus Ponens and Modus Tollens. Familiar assumptions of fa- miliar truth conditional theories of modality facilitate a prima facie explanation—viz., that the target cases harbor epistemic modal equivocations. However, these explana- tions go too far. For they foster other predictions of equivocation in places where in fact there are no equivocations. It is argued here that the key to the solution is to drop (...) the assumption that modal claims are inherently relational (i.e., that they ex- press a logical relation between a prejacent and a premise-set) in favor of a view that treats them as inherently quantificational. In particular it is suggested that modals are mass noun descriptions of information. We demonstrate how this approach unlocks the equivocation problem. (shrink)

Many philosophers hold that if an agent acts intentionally, she must know what she is doing. Although the scholarly consensus for many years was to reject the thesis in light of presumed counterexamples by Donald Davidson, several scholars have recently argued that attention to aspectual distinctions and the practical nature of this knowledge shows that these counterexamples fail. In this paper I defend a new objection against the thesis, one modelled after Timothy Williamson’s anti-luminosity argument. Since this argument relies on (...) general principles about the nature of knowledge rather than on intuitions about fringe cases, the recent responses that have been given to defuse the force of Davidson’s objection are silent against it. Moreover, the argument suggests that even weaker theses connecting practical entities with knowledge are also false. Recent defenders of the thesis that there is a necessary connection between knowledge and intentional action are motivated by the insight that this connection is non-accidental. I close with a positive proposal to account for the non-accidentality of this link without appeal to necessary connections by drawing an extended analogy between practical and perceptual knowledge. (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)

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)

There are three questions associated with Simpson’s Paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP?, and (iii) What should be done about SP? By developing a logic-based account of SP, it is argued that (i) and (ii) must be divorced from (iii). This account shows that (i) and (ii) have nothing to do with causality, which plays a role only in addressing (iii). A counterexample is also presented against the causal account. Finally, the causal and (...) logic-based approaches are compared by means of an experiment to show that SP is not basically causal. (shrink)

There are three questions associated with Simpson’s paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP? and (iii) How to proceed when confronted with SP? An adequate analysis of the paradox starts by distinguishing these three questions. Then, by developing a formal account of SP, and substantiating it with a counterexample to causal accounts, we argue that there are no causal factors at play in answering questions (i) and (ii). Causality enters only in connection with action.

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.

This article argues, against contemporary experimentalist criticism, that conceptual analysis has epistemic value, with a structure that encourages the development of interesting hypotheses which are of the right form to be valuable in diverse areas of philosophy. The article shows, by analysis of the Gettier programme, that conceptual analysis shares the proofs and refutations form Lakatos identified in mathematics. Upon discovery of a counterexample, this structure aids the search for a replacement hypothesis. The search is guided by heuristics. The (...) heuristics of conceptual analysis are similar to those in other interesting areas of scholarship, and so hypotheses generated by it are of the right form to be applicable to diverse areas. The article shows that the explanationist criterion in epistemology was developed and applied in this way. The epistemic value of conceptual analysis is oblique because it contributes not towards the main purpose of conceptual analysis but towards the reliable development of epistemically valuable hypotheses in philosophy and scholarship. (shrink)

Basic Argument forms Modus Ponens , Modus Tollens , Hypothetical Syllogism and Dilemma contains ‘If –then’ conditions. Conclusions from the Arguments containing ‘If –then’ conditions can be deduced very easily without any significant memorization by applying Raval’s method. Method: In Raval’s method If P then Q is written as P (2$) – Q (1$) and viewed numerically, in currency form i.e. P is viewed as 2$ and Q is viewed as 1$ and implications from this notations are (...) valid conclusions. If one has 2$ then he definitely have 1$. If one do not have 2$, he may not have 1$. If one is having 1$, he may not have 2$. If one do not have 1$, he definitely doesn’t have 2$. (shrink)

The Principle of Identity of Indiscernibles (PII) is the focus of much controversy in the history of Metaphysics and in contemporary Physics. Many questions rover the debate about its truth or falsehood, for example, to which objects the principle applies? Which properties can be counted as discerning properties? Is the principle necessary? In other words, which version of the principle is the correct and is this version true? This thesis aims to answer this questions in order to show that PII (...) is a necessarily true principle of metaphysics. To accomplish this task, the reader will find, in this thesis, an encyclopaedical introduction to the history of PII and to the reasons it matters so much, followed by a presentation of the most famous arguments against it and the defences used against these arguments. Then, the reader finds in-depth discussion of the minutiae involved in postulating the principle as to make clear what is in fact being attacked and defended. With these preliminaries solved, a deeper analysis of these defences is presented aiming to discover which is the most appropriate example to use against the attacks to the principle. This analysis allowed a classification of these defences in four families with different strategies within them. Finally, with these defensive strategies at hand we are able to confront alleged counterexamples to PII in Mathematics with the intention to test these defences. (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)

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 virtue of what are we justified in employing the rule of inference Modus Ponens? One tempting approach to answering this question is to claim that we are justified in employing Modus Ponens purely in virtue of facts concerning meaning or concept-possession. In this paper, we argue that such meaning-based accounts cannot be accepted as the fundamental account of our justification.

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.

In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.

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.

A consistent finding in research on conditional reasoning is that individuals are more likely to endorse the valid modus ponens (MP) inference than the equally valid modus tollens (MT) inference. This pattern holds for both abstract task and probabilistic task. The existing explanation for this phenomenon within a Bayesian framework (e.g., Oaksford & Chater, 2008) accounts for this asymmetry by assuming separate probability distributions for both MP and MT. We propose a novel explanation within a computational-level Bayesian (...) account of reasoning according to which “argumentation is learning”. We show that the asymmetry must appear for certain prior probability distributions, under the assumption that the conditional inference provides the agent with new information that is integrated into the existing knowledge by minimizing the Kullback-Leibler divergence between the posterior and prior probability distribution. We also show under which conditions we would expect the opposite pattern, an MT-MP asymmetry. (shrink)

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.

In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality.

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)

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)

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)

The idea that knowledge can be extended by inference from what is known seems highly plausible. Yet, as shown by familiar preface paradox and lottery-type cases, the possibility of aggregating uncertainty casts doubt on its tenability. We show that these considerations go much further than previously recognized and significantly restrict the kinds of closure ordinary theories of knowledge can endorse. Meeting the challenge of uncertainty aggregation requires either the restriction of knowledge-extending inferences to single premises, or eliminating epistemic uncertainty in (...) known premises. The first strategy, while effective, retains little of the original idea—conclusions even of modus ponens inferences from known premises are not always known. We then look at the second strategy, inspecting the most elaborate and promising attempt to secure the epistemic role of basic inferences, namely Timothy Williamson’s safety theory of knowledge. We argue that while it indeed has the merit of allowing basic inferences such as modus ponens to extend knowledge, Williamson’s theory faces formidable difficulties. These difficulties, moreover, arise from the very feature responsible for its virtue- the infallibilism of knowledge. (shrink)

In contemporary discussions of the Ramsey Test for conditionals, it is commonly held that (i) supposing the antecedent of a conditional is adopting a potential state of full belief, and (ii) Modus Ponens is a valid rule of inference. I argue on the basis of Thomason Conditionals (such as ' If Sally is deceiving, I do not believe it') and Moore's Paradox that both claims are wrong. I then develop a double-indexed Update Semantics for conditionals which takes these (...) two results into account while doing justice to the key intuitions underlying the Ramsey Test. The semantics is extended to cover some further phenomena, including the recent observation that epistemic modal operators give rise to something very like, but also very unlike, Moore's Paradox. (shrink)

Expressing a widely-held view, David Hitchcock claims that "an enthymematic argument ... assumes at least the truth of the argument's associated conditional ... whose antecedent is the conjunction of the argument's explicit premises and whose consequent is the argument's conclusion." But even definitionally, this view is problematic, since an argument's being "enthymematic" or incomplete with respect to its explicit premises means that the conclusion is not implied by these premises alone. The paper attempts to specify the ways in which the (...) view is incorrect, as well as seemingly correct (e.g., the case of a Modus Ponens wherein the major premise is implicit). -/- . (shrink)

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle. Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will lose”… “ticket (...) n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”). As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises. In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions. One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox. -/- . (shrink)

Four intuitions are recurrent and influential in theories about conditionals: the Ramsey’s test, the Adams’ Thesis, the Equation, and the robustness requirement. For simplicity’s sake, I call these intuitions ‘the big four’. My aim is to show that: (1) the big four are interdependent; (2) they express our inferential dispositions to employ a conditional on a modus ponens; (3) the disposition to employ conditionals on a modus ponens doesn’t have the epistemic significance that is usually attributed (...) to it, since the acceptability or truth conditions of a conditional is not necessarily associated with its employability on a modus ponens. (shrink)

This paper assesses the prospects of a pragmatist theory of content. I begin by criticising the theory presented in D.H. Mellor’s essay ‘Successful Semantics’. I then identify problems and lacunae in the pragmatist theory of meaning sketched in Chapter 13 of Dummett’s The Logical Basis of Metaphysics. The prospects are brighter, I contend, for a tempered pragmatism, in which the theory of content is permitted to draw upon irreducible notions of truth and falsity. I sketch the shape of such a (...) theory and illustrate the role of its pragmatist elements by showing how they point towards a promising account of the truth conditions of indicative conditionals. A feature of the account is that it validates Modus Ponens whilst invalidating Modus Tollens. (shrink)

In this work, we propose a definition of logical consequence based on the relation between the quantity of information present in a particular set of formulae and a particular formula. As a starting point, we use Shannon‟s quantitative notion of information, founded on the concepts of logarithmic function and probability value. We first consider some of the basic elements of an axiomatic probability theory, and then construct a probabilistic semantics for languages of classical propositional logic. We define the quantity of (...) information for the formulae of these languages and introduce the concept of informational logical consequence, identifying some important results, among them: certain arguments that have traditionally been considered valid, such as modus ponens, are not valid from the informational perspective; the logic underlying informational logical consequence is not classical, and is at the least paraconsistent sensu lato; informational logical consequence is not a Tarskian logical consequence. (shrink)

In “Naming and Necessity” Saul Kripke describes some cases which, he claims, provide counterexamples both to cluster theories and, more generally, to description theories of proper names. My view of these cases is that while they do not provide counterexamples to cluster theories, they can be used to provide evidence against single-description theories. In this paper I shall defend both of the claims involved in my view.

The Ontology for Biomedical Investigations (OBI) is an ontology that provides terms with precisely defined meanings to describe all aspects of how investigations in the biological and medical domains are conducted. OBI re-uses ontologies that provide a representation of biomedical knowledge from the Open Biological and Biomedical Ontologies (OBO) project and adds the ability to describe how this knowledge was derived. We here describe the state of OBI and several applications that are using it, such as adding semantic expressivity to (...) existing databases, building data entry forms, and enabling interoperability between knowledge resources. OBI covers all phases of the investigation process, such as planning, execution and reporting. It represents information and material entities that participate in these processes, as well as roles and functions. Prior to OBI, it was not possible to use a single internally consistent resource that could be applied to multiple types of experiments for these applications. OBI has made this possible by creating terms for entities involved in biological and medical investigations and by importing parts of other biomedical ontologies such as GO, Chemical Entities of Biological Interest (ChEBI) and Phenotype Attribute and Trait Ontology (PATO) without altering their meaning. OBI is being used in a wide range of projects covering genomics, multi-omics, immunology, and catalogs of services. OBI has also spawned other ontologies (Information Artifact Ontology) and methods for importing parts of ontologies (Minimum information to reference an external ontology term (MIREOT)). The OBI project is an open cross-disciplinary collaborative effort, encompassing multiple research communities from around the globe. To date, OBI has created 2366 classes and 40 relations along with textual and formal definitions. The OBI Consortium maintains a web resource providing details on the people, policies, and issues being addressed in association with OBI. (shrink)

This chapter assesses John Gray’s modus vivendi-based justification for liberalism. I argue that his approach is preferable to the more orthodox deontological or teleological justificatory strategies, at least because of the way it can deal with the problem of diversity. But then I show how that is not good news for liberalism, for grounding liberal political authority in a modus vivendi undermines liberalism’s aspiration to occupy a privileged normative position vis-à-vis other kinds of regimes. So modus vivendi (...) can save liberalism from moralism, but at cost many liberals will not be prepared to pay. (shrink)

Neil Sinhababu and I presented Backward Clock, an original counterexample to Robert Nozick’s truth-tracking analysis of propositional knowledge. Fred Adams, John Barker and Murray Clarke argue that Backward Clock is no such counterexample. Their argument fails to nullify Backward Clock which also shows that other tracking analyses, such as Dretske’s and one that Adams et al. may well have in mind, are inadequate.

This paper describes a cubic water tank equipped with a movable partition receiving various amounts of liquid used to represent joint probability distributions. This device is applied to the investigation of deductive inferences under uncertainty. The analogy is exploited to determine by qualitative reasoning the limits in probability of the conclusion of twenty basic deductive arguments (such as Modus Ponens, And-introduction, Contraposition, etc.) often used as benchmark problems by the various theoretical approaches to reasoning under uncertainty. The probability (...) bounds imposed by the premises on the conclusion are derived on the basis of a few trivial principles such as "a part of the tank cannot contain more liquid than its capacity allows", or "if a part is empty, the other part contains all the liquid". This stems from the equivalence between the physical constraints imposed by the capacity of the tank and its subdivisions on the volumes of liquid, and the axioms and rules of probability. The device materializes de Finetti's coherence approach to probability. It also suggests a physical counterpart of Dutch book arguments to assess individuals' rationality in probability judgments in the sense that individuals whose degrees of belief in a conclusion are out of the bounds of coherence intervals would commit themselves to executing physically impossible tasks. (shrink)

According to Viktor Frankl, although people are not always free to choose the conditions in which they find themselves, they are always free to choose their attitude towards these conditions and, thus, are always free to find their lives meaningful. This basic tenet of Frankl’s theory is also often repeated approvingly in the secondary literature. I argue that the claim is wrong; not all people are free to find their lives meaningful. Counterexamples include people who suffer from severe depression or (...) people who, due to lack of sufficient intelligence, ability to focus, or determination cannot profit from psychological counseling (including logotherapeutic counseling). I also criticize Frankl’s oft-repeated argument that some people’s success in finding their lives meaningful in the concentration camps shows that all people are free to find their lives meaningful. Frankl’s discussion of the noetic dimension and its relation to other dimensions of the human personality is also insufficient for defending his claim about all people’s freedom to find their lives meaningful. Frankl’s theory of the noos suggests that all people’s lives are meaningful. But since not all people’s lives are meaningful, Frankl’s claims about the noos seem incorrect: either some people do not have (or are not also) a noetic dimension or the noetic dimension does not always endow life with meaning. Further, the claim that, thanks to all people’s noetic dimension, all people’s lives are already meaningful is in tension with the claim that all people can wrest meaning from life. I suggest that understanding Frankl as only claiming that all people have a potential for meaningful lives is also unhelpful. Finally, I discuss the implications of my criticism for Frankl’s theory at large. I argue that much in this very helpful theory can be retained, but identify those aspects of the theory that need to be modified. (shrink)

True beliefs and truth-preserving inferences are, in some sense, good beliefs and good inferences. When an inference is valid though, it is not merely truth-preserving, but truth-preserving in all cases. This motivates my question: I consider a Modus Ponens inference, and I ask what its validity in particular contributes to the explanation of why the inference is, in any sense, a good inference. I consider the question under three different definitions of ‘case’, and hence of ‘validity’: the orthodox (...) definition given in terms of interpretations or models, a metaphysical definition given in terms of possible worlds, and a substitutional definition defended by Quine. I argue that the orthodox notion is poorly suited to explain what's good about a Modus Ponens inference. I argue that there is something good that is explained by a certain kind of truth across possible worlds, but the explanation is not provided by metaphysical validity in particular; nothing of value is explained by truth across all possible worlds. Finally, I argue that the substitutional notion of validity allows us to correctly explain what is good about a valid inference. (shrink)

“Probably Peirce’s best-known works are the first two articles in a series of six that originally were collectively entitled Illustrations of the Logic of Science and published in Popular Science Monthly from November 1877 through August 1878. The first is entitled ‘The Fixation of Belief’ and the second is entitled ‘How to Make Our Ideas Clear.’ In the first of these papers Peirce defended, in a manner consistent with not accepting naive realism, the superiority of the scientific method over other (...) methods of overcoming doubt and ‘fixing belief.’” — Robert Burch, “Charles Sanders Peirce,” entry in the Stanford Encyclopedia of Philosophy (2021 revision) -/- “Pragmatist epistemologies often explore how we can carry out inquiries in a self-controlled and fruitful way. (Where much analytic epistemology centres around the concept of knowledge, considered as an idealised end-point of human thought, pragmatist epistemology centres around the concept of inquiry, considered as the process of knowledge-seeking and how we can improve it.) So pragmatists often provide rich accounts of the capacities or virtues that we must possess in order to inquire well, and the rules or guiding principles that we should adopt. A canonical account is Peirce’s classic early paper ‘The Fixation of Belief‘. Here Peirce states that inquiry is a struggle to replace doubt with “settled belief“, and that the only method of inquiry that can make sense of the fact that at least some of us are disturbed by inconsistent beliefs, and will subsequently reflect upon which methods of fixing belief are correct is the Method of Science, which draws on the Pragmatic Maxim described above. This contrasts with three other methods of fixing belief: i) refusing to consider evidence contrary to one’s favored beliefs (the Method of Tenacity), ii) accepting an institution’s dictates (the Method of Authority), iii) developing the most rationally coherent or elegant-seeming belief-set (the A Priori Method).“ — Catherine Legg, “Pragmatism,” entry in the Stanford Encyclopedia of Philosophy (2021 revision). (shrink)