Generalized Quantifier Theory defines superlative quantifiers at most n and at least n as truth-conditionally equivalent to comparative quantifiers fewer than n+1 and more than n \1. It has been demonstrated, however, that this standard theory cannot account for various linguistic differences between these two types of quantifiers. In this paper I discuss how the distinction between assertibility and truth-conditions can be applied to explain this phenomenon. I draw a parallel between the assertibility of disjunctions and superlative quantifiers, and argue (...) that those assertibility conditions are essentially modal. I use epistemic logic to formalize the assertibility conditions and revisit some of the linguistic puzzles related to superlative quantification. (shrink)

In _Truth and Objectivity_, Crispin Wright argues that the notion of superassertibility affords the antirealist (with respect to a given range of discourse) a viable alternative to the realist’s more robust notion of truth. Toward this end, he endeavors to prove that a superassertibility predicate can satisfy the traditional equivalence schema: it is true that P iff P. (Wright takes satisfaction of this schema to be a criterion of adequacy for any viable truth predicate.) In this paper, I will argue (...) that, not only does Wright’s attempt to prove that a superassertibility predicate satisfies the equivalence schema fail, an examination of its failure reveals a dilemma for the antirealist who would adopt superassertibility as a model of truth. (shrink)

Timothy Smiley’s wonderful paper ‘Rejection’ (1996) is still perhaps not as well known or well understood as it should be. This note first gives a quick presentation of themes from that paper, though done in our own way, and then considers a putative line of objection – recently advanced by Julien Murzi and Ole Hjortland (2009) – to one of Smiley’s key claims. Along the way, we consider the prospects for an intuitionistic approach to some of the issues discussed in (...) Smiley’s paper. (shrink)

Given a few assumptions, the probability of a conjunction is raised, and the probability of its negation is lowered, by conditionalising upon one of the conjuncts. This simple result appears to bring Bayesian confirmation theory into tension with the prominent dogmatist view of perceptual justification – a tension often portrayed as a kind of ‘Bayesian objection’ to dogmatism. In a recent paper, David Jehle and Brian Weatherson observe that, while this crucial result holds within classical probability theory, it fails within (...) intuitionistic probability theory. They conclude that the dogmatist who is willing to take intuitionistic logic seriously can make a convincing reply to the Bayesian objection. In this paper, I argue that this conclusion is premature – the Bayesian objection can survive the transition from classical to intuitionistic probability, albeit in a slightly altered form. I shall conclude with some general thoughts about what the Bayesian objection to dogmatism does and doesn’t show. (shrink)

Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can (...) extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented. (shrink)

The view that dominates the literature on intentional attitudes holds that beliefs and desires both have propositional content. A commitment to what I call “content uniformity” underlies this view. According to content uniformity, beliefs and desires are but different psychological modes having a uniform kind of content. Prima facie, the modes don’t place any constraint on the kinds of content the attitude can have. I challenge this consensus by pointing out an asymmetry between belief contents and desire contents which shows (...) content uniformity to be mistaken. I do this by revisiting the arguments of Richard (Philosophical Studies, 39(1): 1–13, 1981), and show that arguments which purport to show the temporal specificity of belief contents yield the opposite results for desire contents. I defend this preliminary conclusion from various strategies to neutralize the asymmetry claim. My defense provides occasions to respond to objections by Brogaard (2012) and Recanati (2007) to the Richard argument, and to get clearer on the role of temporal adjuncts in desire ascriptions. Finally, I consider whether the construal of attitude content as centered propositions (as in Lewis Philosophical Review, 88(4): 513–543, 1979) can be invoked to vindicate content uniformity. My conclusion is that while the framework itself doesn’t vindicate content uniformity, it could, but only if it availed itself of a further, substantive thesis about desire, which itself is in need of defense. (shrink)

Substantivism is a general philosophical methodology advocating a substantive approach to philosophical theorizing. In this article, I present an overview of this methodology with a special emphasis on the field of truth. I begin with a framework for understanding what is at stake in the substantivist–deflationist debate and describe the substantivist critique of deflationism. I then proceed to discuss contemporary substantivism as a positive methodology, present examples of recent substantivist theories of truth, delineate several principles of philosophical substantivism, and connect (...) it to contemporary thought about the nature and methods of philosophy. Due to limitations of space, I am unable to discuss all the forms contemporary substantivism has taken. But I try to give a clear sense of the central principles, challenges, and promise of this methodology. (shrink)

Philosophers are divided on whether the proof- or truth-theoretic approach to logic is more fruitful. The paper demonstrates the considerable explanatory power of a truth-based approach to logic by showing that and how it can provide (i) an explanatory characterization —both semantic and proof-theoretical—of logical inference, (ii) an explanatory criterion for logical constants and operators, (iii) an explanatory account of logic’s role (function) in knowledge, as well as explanations of (iv) the characteristic features of logic —formality, strong modal force, generality, (...) topic neutrality, basicness, and (quasi-)apriority, (v) the veridicality of logic and its applicability to science, (v) the normativity of logic, (vi) error, revision, and expansion in/of logic, and (vii) the relation between logic and mathematics. The high explanatory power of the truth-theoretic approach does not rule out an equal or even higher explanatory power of the proof-theoretic approach. But to the extent that the truth-theoretic approach is shown to be highly explanatory, it sets a standard for other approaches to logic, including the proof-theoretic approach. (shrink)

Famously, Michael Dummett argues that considerations concerning the role of language in communication lead to the rejection of classical logic in favor of intuitionistic logic. Potentially, this results in massive revisions of established mathematics. Recently, Neil Tennant (“The law of excluded middle is synthetic a priori, if valid”, Philosophical Topics 24 (1996), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle grounded on a metaphysical principle of determinacy. This article shows (...) that the for the anti-realist, the law of excluded middle entails that humans have wildly implausible abilities. The proposed synthesis between anti-realism and classical mathematics thus fails. (shrink)

The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the notion of “same meaning” (...) is itself context-sensitive, depending on the purposes of a given conversation. (shrink)

In this paper, I use the cases of intuitionistic arithmetic with Church’s thesis, intuitionistic analysis, and smooth infinitesimal analysis to argue for a sort of pluralism or relativism about logic. The thesis is that logic is relative to a structure. There are classical structures, intuitionistic structures, and (possibly) paraconsistent structures. Each such structure is a legitimate branch of mathematics, and there does not seem to be an interesting logic that is common to all of them. One main theme of my (...) ante rem structuralism is that any coherent axiomatization describes a structure, or a class of structures. If one weakens the logic, then more axiomatizations become coherent. (shrink)

The answers to the questions in the title depend on the kind of pluralism one is talking about. We will focus here on our own views. The purpose of this article is to trace out some possible connections between these kinds of pluralism. We show how each of them might bear on the other, depending on how certain open questions are resolved.

Performing an inference involves irreducibly dynamic cognitive procedures. The article proposes that a non-associative information frame, corresponding to a residuated pogroupoid, underpins the information structure involved. The argument proceeds by expounding the informational turn in logic, before outlining the cognitive actions at work in deductive inference. The structural rules of Weakening, Contraction, Commutation, and Association are rejected on the grounds that they cause us to lose track of the information flow in inferential procedures. By taking the operation of information application (...) as the primary operation, the fusion connective is retained, with commutative failure generating a double implication. The other connectives are rejected. (shrink)

The author suggests that educational philosophy should benefit from addressing questions traditionally asked within discourse in the philosophy of mind, namely: the relation between the mind and world and the problems of intentionality , meaning, and representation. Peirce's semiotics and his category of creative abduction provide a novel conceptual framework for exploring these questions. A model of reasoning and learning, based on Peirce's triadic logic of relations, is analysed. This model, it is argued, is fruitful for overcoming the paradox of (...) new knowledge that was first debated by Socrates in his dialogue with Meno. (shrink)

The author suggests that educational philosophy should benefit from addressing questions traditionally asked within discourse in the philosophy of mind, namely: the relation between the mind and world and the problems of intentionality (or aboutness), meaning, and representation. Peirce's semiotics and his category of creative abduction provide a novel conceptual framework for exploring these questions. A model of reasoning and learning, based on Peirce's triadic logic of relations, is analysed. This model, it is argued, is fruitful for overcoming the paradox (...) of new knowledge that was first debated by Socrates in his dialogue with Meno. (shrink)

This study reviews some of the principal themes in contemporary work on religious language. Unlike other recent surveys, the most pressing issues about religious language are addressed from the perspective of the philosophy of language; different positions taken on these issues by philosophers of religion and theologians are considered. Topics that are covered include: the subject matter of religious discourse, reductionism and subjectivism, expressivism, the nature of religious metaphor, religious fictionalism and truth in religious discourse. The study also looks at (...) the relationship between questions about religious language and cognate areas of philosophy of religion such as epistemology and metaphysics, and potential future directions of research. (shrink)

In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, intuitionistic, paraconsistent and quantum (...) logics. Its purpose is to show that there is no particular domain or reason that demands the use of a non-classical logic; the particular reasons given for the non-classical logic can also be handled within classical logic. The result of Sect. 2 is substantiated in Sect. 3, where it is shown (referring to other work) that all four kinds of non-classical logics can be translated into classical logic in a meaning-preserving way. Based on this fact a justification of classical logic is developed in Sect. 4 that is based on its representational optimality. It is pointed out that not many but a few non-classical logics can be likewise representationally optimal. However, the situation is not symmetric: classical logic has ceteris paribus advantages as a unifying metalogic, while non-classical logics can have local simplicity advantages. (shrink)

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It (...) is argued that these two sorts of concepts must be kept strictly apart. (shrink)

We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of proof-theoretic harmony. With every set of introduction rules a canonical elimination (...) rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework. (shrink)

From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a crucial (...) advantage over natural deduction, where substitution is built into the general framework. (shrink)

In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value. Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, (...) can be translated into classical logic by introducing additional intensional operators into the language. Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic. (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.

This paper discusses the philosophical and logical motivations for rejectivism, primarily by considering a dialogical approach to logic, which is formalized in a Question–Answer Semantics. We develop a generalized account of rejectivism through close consideration of Mark Textor's arguments against rejectivism that the negative expression ‘No’ is never used as an act of rejection and is equivalent with a negative sentence. In doing so, we also shed light upon well-known issues regarding the supposed non-embeddability and non-iterability of force indicators.

How should we understand validity? A standard way to characterize validity is in terms of the preservation of truth (or truth in a model). But there are several problems facing such characterizations. An alternative approach is to characterize validity epistemically, for instance in terms of the preservation of an epistemic status. In this paper, I raise a problem for such views. First, I argue that if the relevant epistemic status is factive, such as being in a position to know or (...) having conclusive evidence for, then the account runs into trouble if we endorse certain familiar logical principles. Second, I argue that if the relevant epistemic status is non-factive, such as is rationally committed to or has justification for believing, then a similar problem arises if we endorse the logical principles as well as a sufficiently strong epistemic “level-bridging” principle. Finally, I argue that an analogous problem arises for the most natural characterization of validity in terms of rational credence. (shrink)

Anti-realism is plagued by Fitch’s paradox: the remarkable result that if one accepts that all truths are knowable, minimal assumptions about the nature of knowledge entail that every truth is known. Dorothy Edgington suggests to address this problem by understanding p is knowable to be a counterfactual claim, but her proposal must contend with a forceful objection by Timothy Williamson. I revisit Edgington’s basic idea and find that Williamson’s objection is obviated by a refined understanding of counterfactual knowability that is (...) grounded in possible courses of inquiry. I arrive at a precise definition of knowability that is not just a technical avoidance of paradox, but is metaphysically sound and does justice to the anti-realist idea. (shrink)

We examine the proof-theoretic verificationist justification procedure proposed by Dummett. After some scrutiny, two distinct interpretations with respect to bases are advanced: the independent and the dependent interpretation. We argue that both are unacceptable as a semantics for propositional intuitionistic logic.

Semantic justifications of the classical rules of logical inference typically make use of a notion of bivalent truth, understood as a property guaranteed to attach to a sentence or its negation regardless of the prospects for speakers to determine it as so doing. For want of a convincing alternative account of classical logic, some philosophers suspicious of such recognition-transcending bivalence have seen no choice but to declare classical deduction unwarranted and settle for a weaker system; intuitionistic logic in particular, buttressed (...) by assertion-conditional semantics, is often considered to enjoy a degree of meaning-theoretical respectability unattainable by classical logic.The decision to forgo the classical inference rules is not always made lightly. Thus, Dummett: " In the resolution of the conflict between [the view that generally accepted classical modes of inference ought to be theoretically accommodated, and the demand that any such accommodation be achieved without recourse to bivalence] lies, as I see it, one of the most fundamental and intractable problems in the theory of meaning; indeed, in all philosophy. "The present article ventures to suggest that the conflict need not be irresoluble. Helping ourselves only to such conceptual resources as are standardly invoked by proponents of intuitionistic logic, we shall formulate a rudimentary meaning theory for a selection of logical constants, define a relation of valid inferability, and prove that the latter includes all of classical first-order logic. The result is essentially that reported in Sandqvist as Theorem 2.20.2. TheoryWe will be considering a standard-syntax first-order language with ⊃, ⊥ and ∀ as its only primitive logical constants. We assume countable supplies of individual constants, individual functors , and predicates , as well as a denumerably infinite set of individual variables. By a basic 1 formula we will mean one that …[Full Text of this Article]. (shrink)

Allocating the burden of proof and intuitions in philosophical disputes.– This paper criticises the view that in philosophical disputes the onus probandi rests on those who advance a position that contradicts our basic intuitions. Such a rule for allocating the burden of proof may be an adequate reconstruction of everyday justification, but is unreasonable in the area of philosophy. In philosophy it is not only difficult to determine the plausibility of a proposition, at the same time contradictory claims may be (...) equally plausible. – In contrast to such common sense proposals I try to show that in philosophical disputes the burden of proof does not depend on the material content of speech acts. A speaker simply bears the burden of proof for a proposition p if he has asserted that p and has agreed to justify it. (shrink)

According to judgment internalism, there is a conceptual connection between moral judgment and motivation. This paper offers an argument against that kind of internalism that does not involve counterexamples of the amoralist sort. Instead, it is argued that these forms of judgment internalism fall prey to a Frege-Geach type argument.

Scott Soames (2002) has recently developed and defended strategies for (i) accounting for the meaning of Millian terms, and (ii) extending Kripke's insights from proper names to natural kind terms. In this paper I argue that if we accept these strategies, and their implausible assumptions and consequences, then we can present a novel defence of descriptivism for at least some natural kind terms – those for substances – on that basis. The conclusion, then, will be that there is just no (...) motivation for being a Soamesian Kripkean for an important class of natural kind terms. (shrink)

ABSTRACTIn reply to Linnebo, I defend my analysis of Tait's argument against the use of classical logic in set theory, and make some preliminary comments on Linnebo's new argument for the same conclusion. I then turn to Shapiro's discussion of intuitionistic analysis and of Smooth Infinitesimal Analysis. I contend that we can make sense of intuitionistic analysis, but only by attaching deviant meanings to the connectives. Whether anyone can make sense of SIA is open to doubt: doing so would involve (...) making sense of mathematical quantities whose relationship to zero and to one another is inherently indeterminate. (shrink)

According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute (...) between classicists and intuitionists. I argue for this by presenting an account of consequence, and a cognate semantic theory for the language of the propositional calculus, which respects the meanings of the connectives as embodied in the familiar classical truth-tables, does not presuppose Bivalence, and with respect to which the rules of the intuitionist propositional calculus are sound and complete. Thus the disagreement between classicists and intuitionists, at least, need not stem from their attaching different senses to the connectives; one may deny the doctrine without changing the subject. The basic notion of my semantic theory is truth at a possibility, where a possibility is a way that things might be, but which differs from a possible world in that the way in question need not be fully specific or determinate. I compare my approach with a previous theory of truth at a possibility due to Lloyd Humberstone, and with a previous attempt to refute Quine’s thesis due to John McDowell. (shrink)

I explain why Fernando Ferreira's interesting formal result does not threaten the bilateralist account of the sense of the connectives.

All through the literatura, the question about what is a logical constant has recieved many answers, from model-theoretic aproaches,, to answers that focus in the inferential practice as meaning,,. Detractors of the second tradition presented many ineludible incovenients, in particular, the logical constant named ‘tonk’. Inferentialist tryed many solutions, in particular they presented the concept of ‘harmony’. The goal of this paper is to show that the different criteria of ‘harmony’ used in the proof-theoretic semantics to determine what is and (...) what is not a logical constant fail to be necessary or sufficient. I will show the philosophical reasons that make this concept appear and then i will describe the different ways in wich the literatura understads the concept of ‘harmony’. Then I will show that they subgenerate or overgenerate connectives with some counterexamples. Finaly, I will explain some philosophical reasons that should delimitate where to go towards a satisfactory definition of ‘harmony’. (shrink)

The aim of this paper is to provide a philosophical interpretation of bilateralism in terms of probabilism. In particular, to interpret the main concepts of bilateralism –acceptance, rejection and incoherence– in terms of the probabilistic notions of degree of belief and coherence. According to bilateralism, the meaning of logical connectives is determined by the acceptance and rejection conditions of the sentences in which they are involved, where acceptance and rejection cannot be reduced to one another. I will focus on a (...) variant of bilateralism that understands logical consequence as the statement that it is incoherent to accept all the premises of a valid argument while rejecting all its conclusions. On the other hand, probabilism states that it is possible to interpret our degrees of belief in terms of probabilities. The aim of this work is then to interpret the concept of incoherence in terms of probability functions and determine when it is coherent to accept or to reject a proposition according to some threshold defined in terms of degrees of belief. To achieve this goal, we need both an interpretation of the concept of incoherence coined by the bilateralists as well as an interpretation of acceptance and rejection. I will show that a good interpretation of coherence in probabilistic terms can already be found in the literature. Then, I will give an interpretation of acceptance and rejection in terms of degrees of belief. In particular, I will show that it is possible to interpret these concepts in accordance with Locke’s thesis, the thesis that states that there is some threshold r such that if you believe some sentence in degree equal or higher than r you should accept it, without falling into epistemic paradoxes. (shrink)

Boolos has suggested a plural interpretation of second-order logic for two purposes: to escape Quine’s allegation that second-order logic is set theory in disguise, and to avoid the paradoxes arising if the second-order variables are given a set-theoretic interpretation in second-order set theory. Since the plural interpretation accounts only for monadic second-order logic, Rayo and Yablo suggest an new interpretation for polyadic second-order logic in a Boolosian spirit. The present paper argues that Rayo and Yablo’s interpretation does not achieve the (...) goal. (shrink)

Realists claim that, amongst the statements weunderstand, there are some which are true, yetwhose truth potentially transcends the limits ofwhat we can recognize. Dummett and othershave argued that this realist thesis is incompatiblewith an account of understanding in termsof recognitional capacities. But careful analysis revealsthat this contention is mistaken. Thealleged incompatibility presupposes the truth of ametaphysical principle which cannot bevindicated on the basis of an account of understanding.Realists have independent reason toreject this metaphysical principle, as it leads to a collapseof (...) more modest forms of realisminto more contentious ones. According to moderate realism,we neither know that there areactually recognition-transcendent truthsnor do we know taht there are none. Pending apositive case for the metaphysical principle, thisagnosticism is sufficient to warrant therealists' key thesis, viz. that there arepotentially recognition-transcendent truths. Thusrealism, in its more moderate guise, turns outto be the default position. (shrink)

We conceive of many general terms we use as having satisfaction conditions that are objective in that the thought that something meets them neither entails nor is entailed by the thought that we are currently in a position in which we are ready, or warranted, to apply those terms to it. How do we manage to use a given term in such a way that it is thereby endowed, and conceived to be endowed, with satisfaction conditions that are objective in (...) this sense? In the first half of the paper, I present a number of interrelated problems for some extant metasemantical accounts of how use determines objective satisfaction conditions. In the second half, I then propose a novel account that avoids all of these problems. (shrink)

In this paper, I present an argument for the revision of classical logic. The argument is based on the coherence of a metaethical position which is a species of agnosticism. According to this view, the debate between cognitivists and noncognitivists about moral discourse is unresolved. I argue that there is something at stake in this debate and so something one can coherently be agnostic about. The revisionary argument also draws on principles of epistemic closure. I make these principles explicit and (...) indicate to what extent they can plausibly be assumed. The proposal to revise classical logic is likely to meet with some resistance: classical logic is too deeply entrenched in our reasoning. Before suggesting what to put in its place, I address and defuse four objections that might be levelled against the argument for its revision. I close with some general remarks on the force of arguments for logical reform. (shrink)

In light of the paradox of knowability anti‐realists ought to revise their wholesale equation of truth and knowability, lest they be committed to the absurd conclusion that there are no truths that will never be known. The task accordingly becomes to identify the problematic statements the knowability of whose truth would force that conclusion and to restrict the equation in appropriate ways to all but the problematic statements. This restriction strategy was first implemented by Tennant. However, recently Williamson and Brogaard (...) and Salerno have argued that the restriction strategy, and in particular Tennant's implementation of it, fail to avert the paradoxical conclusion. Here I argue, first, that the arguments devised by Brogaard and Salerno are ineffective because they rely on an invalid closure principle and, second, that while Williamson's argument may suffice to undermine Tennant's specific proposal, it fails to discredit the restriction strategy as such. To this end, I give a better characterisation of the problematic cases, which is immune to Williamson's criticism, and then show how the restricted anti‐realist thesis fares in light of the meaning‐theoretical arguments anti‐realists typically advance in support of their view. (shrink)

This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning---one which takes meaning to be constituted by assertibility and deniability conditions---and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system---ST---that conservatively extends classical (...) logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be non-transitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for non-classical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favourably to more familiar related systems, and some objections are considered. (shrink)

This paper consider Prior's connective Tonk from a particular bilateralist perspective. I show that there is a natural perspective from which we can see Tonk and its ilk as perfectly well-defined pieces of vocabulary; there is no need for restrictions to bar things like Tonk.

Michael Dummett has interpreted and expounded upon intuitionism under the influence of Wittgensteinian views on language, meaning and cognition. I argue against the application of some of these views to intuitionism and point to shortcomings in Dummett's approach. The alternative I propose makes use of recent, post-Wittgensteinian views in the philosophy of mind, meaning and language. These views are associated with the claim that human cognition exhibits intentionality and with related ideas in philosophical psychology. Intuitionism holds that mathematical constructions are (...) mental processes or objects. Constructions are, in the first instance, forms of consciousness or possible experience of a particular type. As such, they must be understood in terms of the concept of intentionality. This view has a historical basis in the literature on intuitionism. In a famous 1931 lecture Heyting in fact identifies constructions with fulfilled or fulfillable mathematical intentions. I consider some of the consequences of this identification and contrast them with Dummett's views on intuitionism. (shrink)

ABSTRACT In our response Field's ‘Properties, Propositions and Conditionals’, we explore the methodology of Field's program. We begin by contrasting it with a proof-theoretic approach and then commenting on some of the particular choices made in the development of Field's theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field's response to Restall’s [2010] ‘What Are We to Accept, and What Are We to Reject, (...) While Saving Truth from Paradox?’. (shrink)

In the middle of the 20th century, it was a common Wittgenstein-inspired idea in philosophy that for a linguistic expression to have a meaning is for it to be governed by a rule of use. In other words, it was widely believed that meanings are to be identified with use-conditions. However, as things stand, this idea is widely taken to be vague and mysterious, inconsistent with “truth-conditional semantics”, and subject to the Frege-Geach problem. In this paper I reinvigorate the ideas (...) that meaningfulness is a matter of being governed by rules of use and that meanings are best thought of in terms of use-conditions. I will do this by sketching the Rule-Governance view of the nature of linguistic meaningfulness, showing that the view isn’t by itself subject to the two problems, and explain why the idea has had a lasting appeal to philosophers from Strawson to Kaplan and why we should find it continually attractive. (shrink)

In order to explicate Gentzen’s famous remark that the introduction-rules for logical constants give their meaning, the elimination-rules being simply consequences of the meaning so given, we develop natural deduction rules for Sheffer’s stroke, alternative denial. The first system turns out to lack Double Negation. Strengthening the introduction-rules by allowing the introduction of Sheffer’s stroke into a disjunctive context produces a complete system of classical logic, one which preserves the harmony between the rules which Gentzen wanted: all indirect proof reduces (...) to direct proof. (shrink)

Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may (...) be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions. (shrink)

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett gives a mistaken elaboration (...) of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen's. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic. (shrink)