Logical Consequence and Entailment

The material account claims that indicative conditionals are material. However, the conventional wisdom even among material account enthusiasts is that the material account cannot be extended to subjunctive conditionals. There are mainly three reasons that motivate this consensus: (1) the belief that if subjunctives were material, most subjunctive conditionals would be vacuously true, which is implausible; (2) its inconsistency with Adam pairs, which suggest that indicative and subjunctive conditionals have different truth conditions; and (3) the belief that it is an (...) inferior hypothesis compared to the possible world theories. I will argue against (1) that the counterintuitive aspects of vacuously true conditionals can be explained away in a uniform fashion, regardless of whether they are indicatives or subjunctives. I reinforce this assumption by showing that the positive arguments for the material account of indicatives are also intuitively valid for subjunctives. The point mentioned in (2) is resisted by explaining Adam pairs as logically equivalent conditionals that can be appropriate at different times, depending of the speaker’s epistemic situation. Finally, (3) is criticised by making the case that the possible world account faces insurmountable problems and that a full-blown material account of indicatives and subjunctives is overall a more elegant solution. (shrink)

Semantic relativism is the view that the truth-value of some types of statements can vary depending on factors besides possible worlds and times, without any change in their propositional content. It has grown increasingly popular as a semantic theory of several types of statements, including statements that attribute knowledge of a proposition to a subject. The ways of knowing claim is the view that perception logically implies knowledge. In my “Semantic Relativism and Ways of Knowing” I argued that a relativist (...) semantics for knowledge attributions is incompatible with the ways of knowing claim. I suggested that this incompatibility depends on some basic features of the logic of relativist semantics, and therefore can be shown to generalise beyond the discussion of knowledge attributions to semantic relativism more broadly. Here I make this generalisation. I demonstrate that for any proposition p expressed by a statement that does not have a relativist semantics, and for any proposition q expressed by a statement that does have a relativist semantics, p fails to logically imply q. I explain why this happens, discuss some of its philosophical consequences, and consider a way to modify relativist semantics to avoid it. I conclude that semantic relativism raises interesting philosophical questions that have gone largely unnoticed in discussions of this view until now. (shrink)

The material account of indicative conditionals states that indicative conditional sentences and the material implication have the same truth conditions. Many conditional logics are motivated by attempts to fix the counter-intuitive aspects associated with the material account. Some counter-intuitive instances of classical argumentative forms, e.g., strengthening of the antecedent, contraposition, and conditional negation, are regarded as evidence that the material account is wrong and that classical logic should be rejected in favour of a new logic system in which these argumentative (...) forms are invalid. It is argued that these logical revisions are ad hoc, because those controversial argumentative forms are implied by other argumentative forms we want to keep. It is impossible to remove a counter-intuitive argumentative form from a logical system without getting entangled in an intricate logical web, since these revisions imply the removal of other parts of logic we want to maintain. Consequently, these revisions are incoherent and unwarranted. At the very least, the usual approach in the analysis of putative counter-examples of argumentative forms must be seriously reconsidered. (shrink)

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by (...) his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

Substructural logics and their application to logical and semantic paradoxes have been extensively studied, but non-reexive systems have been somewhat neglected. Here, we aim to (at least partly) ll this lacuna, by presenting a non-reexive logic and theory of naïve consequence (and truth). We also investigate the semantics and the proof-theory of the system. Finally, we develop a compositional theory of truth (and consequence) in our non-reexive framework.

Philosophers will promise you the world. They advertise that we can learn with logic textbooks how to spot a valid argument by their logical form alone and that the conclusion of any valid argument is already contained in its premises. They also advertise that their textbooks contain classifications with the most basic valid argumentative forms people of flesh and blood can use in deductive matters. Think about this for a moment. These are bold statements. If these statements were true, philosophers (...) would be in higher demand than software engineers and no one would be able to make contributions to theoretical physics without ever taking a logic class. Of course, none of that is true. Their basic argumentative forms are nowhere to be seen in real-life argumentation and people will carry on their cognitive business without ever putting their hands on a logic textbook. I believe philosophers overpromise and underdeliver due to a subtle, but profound confusion they make about their own object of study. They take metalogical principles such as hypothetical syllogism for real forms of the arguments used in natural language. I try to connect these principles with real-life argumentation by interpreting their ‘premises’ as background assumptions and their ‘conclusions’ as either arguments or categorical assertions. This is not enough to save face for two reasons. First, the arguer’s assumptions should count as additional premises, but they will involve numerous arguments and the resulting formal interpretation will be too cumbersome to be practical. Second, we can’t determine the validity of an argument by formal means except when the conclusion repeats the premise, the premise is contradictory or the conclusion is tautological. These limitations are usually ignored due to the prevalent but questionable assumption that conditionals are connectives and not claims to implication. But conditionals can be elegantly explained as claims to implication in a variety of modal ranges that are all classical in nature. Consequently, the substantial validity of a deductive argument can only be determined in an informal manner and requires an extensive investigation of the nature of the subject. I conclude the article with some observations about the need to rethink the role of logic, its cognitive importance, and its teaching. Logic is important because it allows us to understand the nature of logical consequence and not because it is supposed to be useful for other deductive endeavours. I also make some observations about the need to rethink informal logic. Similarly to baby logic conventions involving formal logic, the textbook fallacies have no uncontroversial corresponding examples in real-life argumentation and are better interpreted as a list of epistemic vices that any honest epistemic agent should avoid. In this sense, informal logic has been misunderstood as a poor cousin of formal logic for decades, when it should be viewed as a speciality of virtue epistemology. (shrink)

Goodman and Lederman (2020) argue that the traditional Fregean strategy for preserving the validity of Leibniz’s Law of substitution fails when confronted with apparent counterexamples involving proper names embedded under propositional attitude verbs. We argue, on the contrary, that the Fregean strategy succeeds and that Goodman and Lederman’s argument misfires.

This paper presents a new semantics for the weak relevant logic DW that makes the role of the infamous Routley star more explicable. Central to this rewriting is combining aspects of both the American and Australian plan for understanding negations in relevance logics.

Our question is: can we embed minimal negation in implicative logics weaker than I→? Previous results show how to define minimal negation in the positive fragment of the logic of relevance R and in contractionless intuitionistic logic. Is it possible to endow weaker positive logics with minimal negation? This paper prooves that minimal negation can be embedded in even such a weak system as Anderson and Belnap’s minimal positive logic.

Etude de l'extension par la negation semi-intuitionniste de la logique positive des propositions appelee logique C, developpee par A. Urquhart afin de definir une semantique relationnelle valable pour la logique des valeurs infinies de Lukasiewicz (Lw). Evitant les axiomes de contraction et de reduction propres a la logique classique de Dummett, l'A. propose une semantique de type Routley-Meyer pour le systeme d'Urquhart (CI) en tant que celle-la ne fournit que des theories consistantes pour la completude de celui-ci.

The aim of this article is to discuss the extent to which certain substructural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collect...

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry's logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.

In a typical modus ponens the reasoner will (a) assert that a premise materially implies a conclusion in a given world; (b) assert this premise and (c) infer the conclusion. But this restricted modus ponens has little in common with the unrestricted textbook modus ponens, since the latter claims that there are no possible worlds where: (a’) a premise materially implies a conclusion, (b’) that premise is true and (c’) the conclusion is false. It is clear that this textbook modus (...) ponens cannot be a real argumentative form since the same inference cannot be repeated across all possible worlds by one reasoner. What is wrong? (shrink)

In a material implication the relation of logical consequence is restricted to a world parameter, whereas in a formal implication the relation of logical consequence extends over many worlds. One could infer from this pattern that the material implication should be reduced to formal implication since it is just a restricted version of it, or, inversely, that formal implication should be reduced to material implication since it is an unrestricted version of it. But both reductionist claims would betray a superficial (...) understanding of this connection. The obvious kinship between material and formal implication just means that they are versions of the same type of implication. The only difference is that in each case the same type of implication is presented in two degrees. (shrink)

Russell’s role in the controversy about the paradoxes of material implication is usually presented as a tale of how even the greatest minds can fall prey of basic conceptual confusions. Quine accused him of making a silly mistake in Principia Mathematica. He interpreted ‘if-then’ as a version of ‘implies’ and called it material implication. Quine’s accusation is that this decision involved a use-mention fallacy because the antecedent and consequent of ‘if-then’ are used instead of being mentioned as the premise and (...) the conclusion of an implication relation. It was his opinion that the criticisms and alternatives to the material implication presented by C. I. Lewis and others would never be made in the first place if Russell simply called the Philonian construction ‘material conditional’ instead of ‘material implication’. Quine’s interpretation on the topic became hugely influential, if not universally accepted. This paper will present the following criticisms against this interpretation: (1) the notion of material implication does not involve a use-mention fallacy, since the components of ‘if-then’ are mentioned and not used; (2) Quine’s belief that the components of ‘if-then’ are used was motivated by a conditional-assertion view of conditionals that is widely controversial and faces numerous difficulties; (3) the Philonian construction remains paradoxical even if it is called ‘material conditional’; (4) the Philonian construction is more plausible when it is interpreted as a material implication. (shrink)

The material account of indicative conditionals faces a legion of counterexamples that are the bread and butter in any entry about the subject. For this reason, the material account is widely unpopular among conditional experts. I will argue that this consensus was not built on solid foundations, since these counterexamples are contextual fallacies. They ignore a basic tenet of semantics according to which when evaluating arguments for validity we need to maintain the context constant, otherwise any argumentative form can be (...) rendered invalid. If we maintain the context fixed, the counterexamples to the material account are disarmed. Throughout the paper I also consider the ramifications of this defence, make suggestions to prevent contextual fallacies, and anticipate some possible misunderstandings and objections. (shrink)

Lee Archie argued that if any truth values are consistently assigned to a natural language conditional to which Modus Ponens and Modus Tollens are valid argumentative forms, and affirming the consequent is an invalid argumentative form, this conditional would have the same truth conditions than a material implication. This argument is simple and it requires few assumptions that are relatively uncontroversial. We show that it is possible to extend Archie’s argument to three-valued logics and five-valued logics and vindicate a slightly (...) weaker conclusion, but that is still important: even if we would not believe in bivalence and in the classical negation operator, we would still have good reasons to accept that natural language conditionals and the material implication share truth conditions. (shrink)

It is usually accepted that conditional sentences are sui generis and enigmatic. In this paper I try to make them more accessible by interpreting them as claims to relations of implication restricted to a parameter world. This interpretation revives an old idea that fell into disuse, but in its improved version leads to refreshing solutions to known problems in conditional theory. The many benefits of this approach are evidenced by its insightful explanation of some apparent counter-examples to classical argumentative forms (...) (the paradoxes of material implication, antecedent strengthening, contraposition, hypothetical syllogism, conditional negation), conditional standoffs, the problem of counterfactuals, the referential/inferential nature of conditionals, the Apartheid thesis, the triviality results and conditionals embedding. (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)

Intensional evidence is any reason to accept a proposition that is not the truth values of the proposition accepted or, if it is a complex proposition, its propositional contents. Extensional evidence is non-intensional evidence. Someone can accept a complex proposition, but deny its logical consequences in two circumstances: (1) when her acceptance is based on intensional evidence, while the logical consequences of the proposition presuppose the acceptance of extensional evidence, e.g., she can refuse the logical consequence of a proposition she (...) accepts because she doesn’t know what the truth-values of its propositional contents; (2) when she accepts a proposition based on extensional evidence, but thinks that this evidence is insufficient to establish its logical consequences, which would require intensional evidence. It is argued that this tension is responsible for the counter-intuitive aspects of the material account of conditionals involving the negation of conditionals, hypothetical syllogism, contraposition and the inferential passage from disjunctions to conditionals (or-to-if). This tension is also behind some known puzzles involving conditionals, namely, conditional stand-offs, Adam pairs, the cheating partner example, jump-out conditionals, the problem of counterfactuals and the burglar’s puzzle. It is shown that this tension is always dissolved in favour of extensional evidence, since intensional evidence is defeasible, while extensional evidence is not. Thus, it is irrational to deny the logical consequences of an accepted proposition due to its reliance on intensional evidence and ignorance of its extensional evidence. (shrink)

Moti Mizrahi (2013) presented a putative counterexample to Hypothetical Syllogism (HS) for indicative conditionals aiming to succeed where previous attempts to refute HS have failed. Lee Walters (2014a) objected that Mizrahi’s putative counterexample results from an inadequate analysis of conditionals with embedded modals, but advanced new putative counterexamples to HS for subjunctive conditionals that are supposed to bypass this issue (Walters, 2014a; 2014b). It is argued that Walter’s analysis of embedded modals is unnecessary to prevent Mizrahi’s putative counterexample, since the (...) example can be disarmed either as a fallacy of equivocation or as a contextual fallacy. Walters’s putative counterexamples to HS are also criticized as contextual fallacies. The conclusion is that neither Mizrahi nor Walters provided compelling reasons to reject HS. (shrink)

There is a profound, but frequently ignored, relationship between the classical conception of logical consequence and the material implication. The first repeats the patterns of the latter, but with a wider modal reach. This relationship suggests that there should be also a connection between the notion of logical consequence and the conditional connective of any given logical system. This implies, among other things, that it is incoherent to propose alternatives to the material implication while maintaining the classical conception of logical (...) consequence. The other important implication is that we need to posit different conceptions of logical consequence that are consistent with different theories of conditionals in order to evaluate their relative merits from new and unexplored angles. As a pilot study of this research program, we evaluate two new notions of logical consequence motivated by conditional-assertion theory and possible world theories. Those alternatives are compared unfavourably with the classical conception of logical consequence. (shrink)

The material account of indicative conditionals states that indicative conditional sentences and the material implication have the same truth conditions. Recently, Adam Rieger has carried out a survey of arguments in favour of the material account. In this paper, I extend this survey by presenting yet more arguments for the material account. On top of presenting more arguments, I also want to argue that it is plausible to extend the material account to subjunctive conditionals. For that reason, the arguments here (...) presented contain principles that hold for both indicative and subjunctive conditionals. Towards the end, I defend why there are good reasons to accept these new arguments and try to explain why similar arguments have been somewhat unpopular. (shrink)

Brogaard and Salerno (2008) argued that counter-examples to contraposition, strengthening the antecedent, and hypothetical syllogism involving subjunctive conditionals only seem to work because they involve a contextual fallacy where the context assumed in the premise(s) is illicitly shifted in the conclusion. To avoid such counter-examples they have proposed that the context must remain fixed when evaluating an argument for validity. That is the Brogaard-Salerno Stricture. Tristan Haze (2016), however, has recently objected that intuitively valid argumentative forms such as conjunction introduction (...) do not satisfy this constraint. This paper has two goals. First, it argues that the Brogaard-Salerno Stricture is not violated in Haze’s putative counter-example. Second, it argues that since this stricture blocks the usual counter-examples to instances of classical argumentative forms that involve indicative or subjunctive conditionals, it is reasonable to infer that indicative and subjunctive conditionals are material. (shrink)

The truth-functional hypothesis states that indicative conditional sentences and the material implication have the same truth conditions. Haze (2011) has rejected this hypothesis. He claims that a self-referential conditional, coupled with a plausible assumption about its truth-values and the assumption that the truth-functional hypothesis is true, lead to a liar’s paradox. Given that neither the self-referential conditional nor the assumption about its truth-values are problematic, the culprit of the paradox must be the truth-functional hypothesis. Therefore, we should reject it. In (...) this paper I argue that, contrary to what Haze thinks, the truth-functional hypothesis is not to blame. In fact, no liar’s paradox emerges when the truth-functional hypothesis is true; it emerges only if it is false. (shrink)

It is usually accepted that one of the properties of classical logic is monotonicity, which states that the validity of implication is not affected by the addition of new premises. In this piece, I will argue that this common notion is unjustified since it is motivated by a category mistake. The notion of monotonicity is primarily epistemic in character and can’t be meaningfully attributed to a system. This is acutely clear in the contrast of monotonicity with non-monotonicity, which we tend (...) to associate with defeasible inferences, where reasoners can abandon a previous inference based on new information. So non-monotonicity is actually the reasoners’ willingness to abandon a previous inference based on new information. Therefore, it can be a property of a system only in a manner of speaking. But since non-monotonicity is the reasoners’ willingness to abandon a previous inference based on new information, monotonicity should be understood in a similar epistemic fashion, namely, as the reasoners’ willingness to maintain a previous inference after newly discovered information. By analyzing the problem from this perspective, a classical inference can be considered non-monotonic if the reasoner is willing to retract her previous conclusions based on new information. (shrink)

By the beginning of the 19th century Hegel's dialectic turn contradiction (conceived as unity of a concept with its determined negation) into distinguished inference. In the course of 20th century a family of systems known as "paraconsistent" formalized dialectical logic according to the contemporary paradigm of inference, oriented to truth-preserving, and not powered anymore solely by contradiction. In this way, nevertheless, Hegel's idea of logic as unfolding of concepts ordered by degree of "determination" reached at every step of the process, (...) was put aside. We attempt to recover the importance of that idea for the hegelian concept of "logical necessity", pointing out in addition that it may well play the role of an order relation for dialectical inference. (shrink)

Lycan (2010: 617) maintained that if we know about a deductive argument P1, P2…, ∴C that it is valid, this shows that a set containing the premises and the negation of conclusion is internally inconsistent {P1, P2,…, ~C}. Using a similar reasoning, we should conclude that an invalid deductive argument shows that a set containing the premises and the negation of conclusion is internally consistent. This implies that premises and conclusion of any deductive argument are truth-apt. Since it is uncontroversial (...) that the argument that motivates the liar paradox is deductive, it follows that the liar sentences are truth-apt. The reason is obvious: if a sentence can’t be true, it can’t be a part of a consistent or inconsistent set, by definition. Consequently, the liar paradox cannot be solved by arguing that liar sentences are truth-valueless. (shrink)

The modus ponens can be interpreted as an answer to a circularity charge, but this strategy is only feasible if the additional conditional premise is interpreted as a claim to a material implication. Here’s how it works.

The conclusion of a valid deductive argument is contained in its premises, but the deductive argument can still be informative if the arguer is trying to find the missing premises that together with the accepted premises will ensure the truth of the conclusion. Necessarily true conditionals have a deductive-like character so their consequents are contained in their antecedents. These conditionals can be informative, so we need to find the missing antecedent that makes the connection between the antecedent and the consequent. (...) So where is it? (shrink)

An argument is deductive when the arguer believes the truth of the premises necessitates the truth of the conclusion. A deductive argument is valid when the arguer’s claim is true, i.e., when there are no possible worlds whether the premises are true and the conclusion is false. But in order to evaluate this claim in an accurate manner we need to consider three qualifications that have been repeatedly ignored in the literature, namely, consistency in the attribution of truth values, world (...) consistency, and the inclusion of the reasons offered by the arguer in support of the premises. These apparently innocuous restrictions have significant and far-reaching implications. (shrink)

It is argued that contraposition is valid for a class of natural language conditionals, if some modifications are allowed to preserve the meaning of the original conditional. In many cases, implicit temporal indices must be considered, making a change in verb tense necessary. A suitable contrapositive for implicative counterfactual conditionals can also usually be found. In some cases, the addition of certain words is necessary to preserve meaning that is present in the original sentence and would be lost or changed (...) in the contrapositive without them. A distinction is made between adding new meaning and adding new words to preserve existing meaning. For concessive conditionals and relevance conditionals, however, no valid contrapositive can be found. They do not belong to the class of contraposable conditionals, which can be independently defined. Difficult cases are also discussed in which the contradictory of the consequent semantically entails the truth of the antecedent. In such cases the content of the antecedent is implicit in the meaning of the consequent. Contraposition becomes possible if what is implicit in the original consequent is made explicit in the contrapositive antecedent. (shrink)

The premise that it is logically necessary for a necessary condition of value to be valuable is sometimes used in metaethics in support of the claim that agency, or some constitutive condition of agency or action, has value for all agents. I focus on the most recent application of this premise by Caroline T. Arruda and argue that the premise is false. Despite this defect the relevant evaluative step could still work just in case of agency if an additional condition (...) were satisfied. (shrink)

I argue against abductivism about logic, which is the view that rational theory choice in logic happens by abduction. Abduction cannot serve as a neutral arbiter in many foundational disputes in logic because, in order to use abduction, one must first identify the relevant data. Which data one deems relevant depends on what I call one's conception of logic. One's conception of logic is, however, not independent of one's views regarding many of the foundational disputes that one may hope to (...) solve by abduction. (shrink)

In this paper, a critical discussion is made of the role of entailments in the so-called New Paradigm of psychology of reasoning based on Bayesian models of rationality (Elqayam & Over, 2013). It is argued that assessments of probabilistic coherence cannot stand on their own, but that they need to be integrated with empirical studies of intuitive entailment judgments. This need is motivated not just by the requirements of probability theory itself, but also by a need to enhance the interdisciplinary (...) integration of the psychology of reasoning with formal semantics in linguistics. The constructive goal of the paper is to introduce a new experimental paradigm, called the Dialogical Entailment task, to supplement current trends in the psychology of reasoning towards investigating knowledge-rich, social reasoning under uncertainty (Oaksford and Chater, 2019). As a case study, this experimental paradigm is applied to reasoning with conditionals and negation operators (e.g. CEM, wide and narrow negation). As part of the investigation, participants’ entailment judgments are evaluated against their probability evaluations to assess participants’ cross-task consistency over two experimental sessions. (shrink)

We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system. Since self-contradictory expressions are neither provable nor disprovable (...) only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete. (shrink)

The focus of this paper are Dummett's meaning-theoretical arguments against classical logic based on consideration about the meaning of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett's outlook on the theory of meaning. In particular, I (...) shall grant Dummett his general bias towards verificationism, encapsulated in the slogan 'meaning is use'. The second general assumption I see no need to question is Dummett's particular breed of molecularism. Some of Dummett's assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inference in the Dummettian framework. (shrink)

In 2016 Beziau, introduce a more restricted concept of paraconsistency, namely the genuine paraconsistency. He calls genuine paraconsistent logic those logic rejecting φ, ¬φ |- ψ and |- ¬(φ ∧ ¬φ). In that paper the author analyzes, among the three-valued logics, which of these logics satisfy this property. If we consider multiple-conclusion consequence relations, the dual properties of those above mentioned are: |- φ, ¬φ, and ¬(ψ ∨ ¬ψ) |- . We call genuine paracomplete logics those rejecting the mentioned properties. (...) We present here an analysis of the three-valued genuine paracomplete logics. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

This paper explains the importance of classifying argumentation schemes, and outlines how schemes are being used in current research in artificial intelligence and computational linguistics on argument mining. It provides a survey of the literature on scheme classification. What are so far generally taken to represent a set of the most widely useful defeasible argumentation schemes are surveyed and explained systematically, including some that are difficult to classify. A new classification system covering these centrally important schemes is built.

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic value of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide the valid deductive inference. Sound deductive conclusions are the result of these finite string transformation rules.

Logical pluralism is the view that there is more than one correct logic. This very general characterization gives rise to a whole family of positions. I argue that not all of them are stable. The main argument in the paper is inspired by considerations known as the “collapse problem”, and it aims at the most popular form of logical pluralism advocated by JC Beall and Greg Restall. I argue that there is a more general argument available that challenges all variants (...) of logical pluralism that meet the following three conditions: that there are at least two correct logical systems characterized in terms of different consequence relations, that there is some sort of rivalry among the correct logics, and that logical consequence is normative. The hypothesis I argue for amounts to conditional claim: If a position satisfies all these conditions, then that position is unstable in the sense that it collapses into competing positions. (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)

I address a type of circularity threat that arises for the view that we employ general basic logical principles in deductive reasoning. This type of threat has been used to argue that whatever knowing such principles is, it cannot be a fully cognitive or propositional state, otherwise deductive reasoning would not be possible. I look at two versions of the circularity threat and answer them in a way that both challenges the view that we need to apply general logical principles (...) in deductive reasoning and defuses the threat to a cognitivist account of knowing basic logical principles. (shrink)

For semantic inferentialists, the basic semantic concept is validity. An inferentialist theory of meaning should offer an account of the meaning of "valid." If one tries to add a validity predicate to one's object language, however, one runs into problems like the v-Curry paradox. In previous work, I presented a validity predicate for a non-transitive logic that can adequately capture its own meta-inferences. Unfortunately, in that system, one cannot show of any inference that it is invalid. Here I extend the (...) system so that it can capture invalidities. (shrink)

There is a natural story about what logic is that sees it as tied up with two operations: a ‘throw things into a bag’ operation and a ‘closure’ operation. In a pair of recent papers, Jc Beall has fleshed out the account of logic this leaves us with in more detail. Using Beall’s exposition as a guide, this paper points out some problems with taking the second operation to be closure in the usual sense. After pointing out these problems, I (...) then turn to fixing them in a restricted case and modulo a few simplifying assumptions. In a followup paper, the simplifications and restrictions will be removed. (shrink)

Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By defining a logical-term-free language with the (...) full expressive power of first-order logic with identity, I show that this is true of logic more generally. Furthermore, having, in a logical theory, non-trivial valid forms that do not involve logical terms is not merely a technical possibility. As the case of adverbs shows, issues about the range of argument forms logic should countenance can quite naturally arise in such a way that they do not turn on whether we countenance certain terms as logical. (shrink)

This article reviews John Wood’s Truth in Fiction: Rethinking its Logic.

The many-property problem has traditionally been taken to show that the adverbial theory of perception is untenable. This paper first shows that several widely accepted views concerning the nature of perception---including both representational and non-representational views---likewise face the many-property problem. It then presents a solution to the many-property problem for these views, but goes on to show how this solution can be adapted to provide a novel, fully compositional solution to the many-property problem for adverbialism. Thus, with respect to the (...) many-property problem, adverbialism and several widely accepted views in the philosophy of perception are on a par, and the problem is solved. (shrink)