This paper provides both a solution and a problem for the account of compositionality in Christopher Peacocke’s modest inferentialism. The immediate issue facing Peacocke’s account is that it looks as if compositionality can only be understood at the level of semantics, which is difficult to reconcile with inferentialism. Here, following up a brief suggestion by Peacocke, I provide a formal framework wherein compositionality occurs the level of the determining relation between inference and semantics. This, in turn provides a “test” for (...) compositionality, which, problematically, Peacocke’s natural deduction framework for classical logic can not meet. To finish, I briefly outline an alternative, bilateralist, framework for modest inferentialism, for which compositionality holds. (shrink)

In [5], Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract “core” of logics in general, where logical syntax and semantics are “two sides of the same coin”. The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic. (...) However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics. (shrink)

Mark Schroeder has argued that all reasonable forms of inconsistency of attitude consist of having the same attitude type towards a pair of inconsistent contents (A-type inconsistency). We suggest that he is mistaken in this, offering a number of intuitive examples of pairs of distinct attitudes types with consistent contents which are intuitively inconsistent (B-type inconsistency). We further argue that, despite the virtues of Schroeder's elegant A-type expressivist semantics, B-type inconsistency is in many ways the more natural choice in developing (...) an expressivist account of moral discourse. We close by showing how to adapt ordinary formality-based accounts of logicality to define a B-type account of logical inconsistency and distinguish it from both semantic and pragmatic inconsistency. In sum, we provide a roadmap of how to develop a successful B-type expressivism. (shrink)

Rejecting structural contraction has been proposed as a strategy for escaping semantic paradoxes. The challenge for its advocates has been to make intuitive sense of how contraction might fail. I offer a way of doing so, based on a “naive” interpretation of the relation between structure and logical vocabulary in a sequent proof system. The naive interpretation of structure motivates the most common way of blaming Curry-style paradoxes on illicit contraction. By contrast, the naive interpretation will not as easily motivate (...) one recent noncontractive approach to the Liar paradox. (shrink)

Real-world agents do not know all consequences of what they know. But we are reluctant to say that a rational agent can fail to know some trivial consequence of what she knows. Since every consequence of what she knows can be reached via chains of trivial cot be dismissed easily, as some have attempted to do. Rather, a solution must give adequate weight to the normative requirements on rational agents’ epistemic states, without treating those agents as mathematically ideal reasoners. I’ll (...) argue that agents can fail to know trivial consequences of what they know, but never determinately. Such cases are epistemic oversights on behalf of the agent in question, and the facts about epistemic oversights are always indeterminate facts. As a result, we are never in a position to assert that such-and-such constitutes an epistemic oversight for agent i (for we may rationally assert only determinate truths). I then develop formal epistemic models according to which epistemic accessibility relations are vague. Given these models, we can show that epistemic oversights always concern indeterminate cases of knowledge. (shrink)

I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.

This paper offers an interpretation of multiple-conclusion sequents as a kind of meta-inference rule: just as single-conclusion sequents represent inferences from sentences to sentences, so multiple-conclusion sequents represent a certain kind of inference from single-conclusion sequents to single-conclusion sequents. The semantics renders sound and complete the standard structural rules of reflexivity, monotonicity (or thinning), and transitivity (or cut). The paper is not the first one to attempt to account for multiple-conclusion sequents without invoking notions of truth or falsity—but unlike earlier (...) such efforts, which have typically helped themselves to primitive notions of both acceptance and rejection, the present one makes do with the former alone. For technical reasons, the treatment is limited to sequents with non-empty succedents. (shrink)

Ontological pluralism is the doctrine that there are different ways or modes of being. In contemporary guise, it is the doctrine that a logically perspicuous description of reality will use multiple quantifiers which cannot be thought of as ranging over a single domain. Although thought defeated for some time, recent defenses have shown a number of arguments against the view unsound. However, another worry looms: that despite looking like an attractive alternative, ontological pluralism is really no different than its counterpart, (...) ontological monism. In this paper, after explaining the worry in detail, I argue that considerations dealing with the nature of the logic ontological pluralists ought to endorse, coupled with an attractive philosophical thesis about the relationship between logic and metaphysics, show this worry to be unfounded. (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)

I present an account of truth values for classical logic, intuitionistic logic, and the modal logic s5, in which truth values are not a fundamental category from which the logic is defined, but rather, an idealisation of more fundamental logical features in the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical consequence.

A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: 1. If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich (...) will enroll. There are many different things one can say about this argument, but many agree that if we do not equivocate (if the terms mean the same thing in the premises and the conclusion) then the argument is valid, that is, the conclusion follows deductively from the premises. This does not mean that the conclusion is true. Perhaps the premises are not true. However, if the premises are true, then the conclusion is also true, as a matter of logic. This entry is about the relation between premises and conclusions in valid arguments. (shrink)

This paper shows how to conservatively extend classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truth—involving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system allows for Cut—elimination, but the (...) other does not.). (shrink)

Is it possible to give a justification of our own practice of deductive inference? The purpose of this paper is to explain what such a justification might consist in and what its purpose could be. On the conception that we are going to pursue, to give a justification for a deductive practice means to explain in terms of an intuitively satisfactory notion of validity why the inferences that conform to the practice coincide with the valid ones. That is, a justification (...) should provide an analysis of the notion of validity and show that the inferences that conform to the practice are just the ones that are valid. Moreover, a complete justification should also explain the purpose, or point, of our inferential practice. We are first going to discuss the objection that any justification of our deductive practice must use deduction and therefore be circular. Then we will consider a particular model of justificatory explanation, building on Georg Kreisel’s concept of informal rigour. Finally, in the main part of the paper, we will discuss three ideas for defining the notion of validity: (i) the classical conception according to which the notion of (bivalent) truth is taken as basic and validity is defined in terms of the preservation of truth; (ii) the constructivist idea of starting instead with the notion of (a canonical) proof (or verification) and define validity in terms of this notion; (iii) the idea of taking the notions of rational acceptance and rejection as given and define an argument to be valid just in case it is irrational to simultaneously accept its premises and reject its conclusion (or conclusions, if we allow for multiple conclusions). Building on work by Dana Scott, we show that the last conception may be viewed as being, in a certain sense, equivalent to the first one. Finally, we discuss the so-called paradox of inference and the informativeness of deductive arguments. (shrink)

According to some dialetheists, we ought to reject the distinction between object and meta‐languages. Given that dialetheists advocate truth‐value gluts within their object‐language, whether in order to solve the liar paradox or for some other reason, this rejection of the object‐/meta‐language distinction comes with the commitment to use a glutty metatheory. While it has been pointed out that a glutty metatheory brings with it expressive deficiencies, we highlight here another complication arising from the use of a glutty metatheory, this time (...) evidential in nature. According to this countermodel problem, while the thoroughgoing dialetheist who embraces a glutty metatheory can justify their acceptance of a rule of inference's invalidity using countermodels, to justify their renunciation of an unwanted rule they actually require the means to warrant their rejection of the rule's validity—which cannot be supplied by countermodels based on a standard dialetheic semantics. We end by sketching out a possible solution for the thoroughgoing dialetheist using a bilaterialist semantics. (shrink)

Thomas Hofweber takes the semantic paradoxes to motivate a radical reconceptualization of logical validity, rejecting the idea that an inference rule is valid just in case every instance thereof is necessarily truth-preserving. Rather than this “strict validity”, we should identify validity with “generic validity”, where a rule is generically valid just in case its instances are truth preserving, and where this last sentence is a generic, like “Bears are dangerous”. While sympathetic to Hofweber’s view that strict validity should be replaced (...) by something allowing for exceptions, I argue that this should instead be truth-conduciveness, a matter of a large majority of instances being truth-preserving. The fact that inference rules have uncountably many instances, I argue, can be handled by defining truth-conduciveness as relative to finite sets of instances. Moreover, I argue that Hofweber’s position, while seemingly radical, may be less so under closer scrutiny. For the mere claim that classical rules and unrestricted truth rules are generically valid (or truth-conducive) is not in itself controversial. Also, if there is an answer to the question of which of these are also strictly valid, then the claim that generic validity is the central notion in logic is at most an interest-relative matter, since there is then also a fact of the matter as to which inference rules are (not) strictly valid. If no solution operating with strict validity to the paradoxes can be had, however, then the claim that validity should be identified with a weaker notion is better motivated. I argue that it is reasonable, in view of past failures, to conjecture that no ordinary solution will score high enough relative to standard (uncontroversial) desiderata to merit justified belief. Hence, it is unknowable which solution is correct. I further argue that this is best explained by its being metaphysically indeterminate which solution is correct. If this conjecture is true, then we ought instead to think of validity as allowing exceptions, and then take the “true logic” to be one that takes both the classical rules and the truth rules to be valid, i.e., truth-conducive. This logic scores very high on the desiderata on logics, and is therefore preferable to any logic operating with strict validity. (shrink)

Bilateral proof systems, which provide rules for both affirming and denying sentences, have been prominent in the development of proof-theoretic semantics for classical logic in recent years. However, such systems provide a substantial amount of freedom in the formulation of the rules, and, as a result, a number of different sets of rules have been put forward as definitive of the meanings of the classical connectives. In this paper, I argue that a single general schema for bilateral proof rules has (...) a reasonable claim to inferentially articulating the core meaning of all of the classical connectives. I propose this schema in the context of a bilateral sequent calculus in which each connective is given exactly two rules: a rule for affirmation and a rule for denial. Positive and negative rules for all of the classical connectives are given by a single rule schema, harmony between these positive and negative rules is established at the schematic level by a pair of elimination theorems, and the truth-conditions for all of the classical connectives are read off at once from the schema itself. (shrink)

This book concerns the metasemantics of quantum mechanics (QM). Roughly, it pursues an investigation at an intersection of the philosophy of physics and the philosophy of language, and it offers a critical analysis of rival explanations of the semantic facts of standard QM. Two problems for such explanations are discussed: categoricity and permanence. New results include 1) a reconstruction of Einstein's incompleteness argument, which concludes that a local, separable, and categorical QM cannot exist, 2) a reinterpretation of Bohr's principle of (...) correspondence, grounded in the principle of permanence, 3) a meaning-variance argument for quantum logic, which follows a line of critical reflections initiated by Weyl, and 4) an argument for semantic indeterminacy leveled against inferentialism about QM, inspired by Carnap's work in the philosophy of classical logic. (shrink)

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)

Why must incompatibility be symmetric? An odd question, but recent work in the semantics of non-classical logic, which appeals to the notion of incompatibility as a primitive and defines negation in terms of it, has brought this question to the fore. Francesco Berto proposes such a semantics for negation argues that, since incompatibility must be symmetric, double negation introduction must be a law of negation. However, he offers no argument for the claim that incompatibility really must be symmetric. Here, I (...) provide such an argument, showing that, insofar as we think of incompatibility in normative pragmatic terms, it can play its basic pragmatic function only if it is symmetric. The upshot is that we can vindicate Berto’s claim about the symmetry of incompatibility but only if we, pace Berto, think about incompatibility, in the first instance, as a pragmatic relation between acts rather than a semantic relation between contents. (shrink)

In bilateral logic formulas are signed by + and –, indicating the speech acts assertion and denial. I argue that making an assumption is also speech act. Speech acts cannot be embedded within other speech acts. Hence we cannot make sense of the notion of making an assumption in bilateral logic. Attempts to solve this problem are considered and rejected.

In this paper I will show the problems that are encountered when dealing with uniqueness of connectives in a bilateralist setting within the larger framework of proof-theoretic semantics and suggest a solution. Therefore, the logic 2Int is suitable, for which I introduce a sequent calculus system, displaying - just like the corresponding natural deduction system - a consequence relation for provability as well as one dual to provability. I will propose a modified characterization of uniqueness incorporating such a duality of (...) consequence relations, with which we can maintain uniqueness in a bilateralist setting. (shrink)

Deflationists about truth hold that the function of the truth predicate is to enable us to make certain assertions we could not otherwise make. Pragmatists claim that the utility of negation lies in its role in registering incompatibility. The pragmatist insight about negation has been successfully incorporated into bilateral theories of content, which take the meaning of negation to be inferentially explained in terms of the speech act of rejection. We implement the deflationist insight in a bilateral theory by taking (...) the meaning of the truth predicate to be explained by its inferential relation to assertion. We combine this account of the meaning of the truth predicate with a new diagnosis of the Liar Paradox: its derivation requires the truth rules to preserve evidence, but these rules only preserve commitment. The result is a novel inferential deflationist theory of truth, which solves the Liar Paradox in a principled manner. We end by showing that our theory and simple extensions thereof have the resources to axiomatize the internal logic of several supervaluational hierarchies, including Cantini’s. This solves open problems of Halbach (2011) and Horsten (2011). (shrink)

In the chapter “Information and Content” of their Impossible Worlds, Berto and Jago provide us with a semantic account of information in deductive reasoning such that we have an explanation for why some, but not all, logical deductions are informative. The framework Berto and Jago choose to make sense of the above-mentioned idea is a semantic interpretation of Sequent Calculus rules of inference for classical logic. I shall argue that although Berto and Jago’s idea and framework are hopeful, their definitions (...) do not capture what is intended. This is so because the definitions are solely based on the logical complexity of an argument and they fail to capture the richness of the non- logical content of that argument. Then I will suggest some amendments to address this problem. Finally, I will extend the application of the definitions to first-order logic. It shall be observed that in some classical deductions, applying contraction may lead to hiding some epistemic impossibilities. This happens when formulae which contribute different propositions to the proof get contracted at the quantified level. (shrink)

This paper has two related goals. First, we develop an expressivist account of negation which, in the spirit of Alan Gibbard, treats disagreement as semantically primitive. Our second goal is to make progress toward a unified expressivist treatment of modality. Metaethical expressivists must be expressivists about deontic modal claims. But then metaethical expressivists must either extend their expressivism to include epistemic and alethic modals, or else accept a semantics for modal expressions that is radically disjunctive. We propose that expressivists look (...) to Amie Thomasson’s work for a general strategy for offering a unified expressivist account of modality. Modals in general, we propose, are devices for expressing metalinguistic commitments within the object language, with deontic, epistemic, and metaphysical modals all expressing different kinds of metalinguistic commitments. (shrink)

This paper sharpens the distinction between inferential and logcon arguments. Inferential arguments represent possible inferences, logcon ones need not. This distinction clarifies the roles that arguments play in accounting for the normativity of validity for inferential reasoning and in establishing the theoretical connection between validity and logical consequence. There are two related takeaways. First, the normativity of validity for inferential reasoning is grounded on the notion of an inferential argument. This will account for the use of validity to judge inference (...) in the face of well-known skepticism that a theory of validity has any special relevance to inferential reasoning. Second, a valid logcon argument needn't be an inferential argument. Accordingly, the normativity of validity for inferential reasoning doesn't require that every valid logcon argument be an inferential argument. The takeaways harmonize the use of validity to evaluate inferential reasoning with the use of the concept of logical consequence to characterize validity. (shrink)

People reason not only in beliefs, but also in intentions, preferences, and other attitudes. They form preferences from existing preferences, or intentions from existing beliefs and intentions, and so on. This often involves choosing between rival conclusions. Building on Broome (Rationality through reasoning, Hoboken, Wiley. https://doi.org/10.1002/9781118609088, 2013) and Dietrich et al. (J Philos 116:585–614. https://doi.org/10.5840/jphil20191161138, 2019), we present a philosophical and formal analysis of reasoning in attitudes, with or without facing choices in reasoning. We give different accounts of choosing, in (...) terms of a conscious activity or a partly subconscious process. Reasoning in attitudes differs fundamentally from reasoning _about_ attitudes, a form of theoretical reasoning in which one discovers rather than forms attitudes. We show that reasoning in attitudes has standard formal properties (such as monotonicity), but is indeterministic, reflecting choice in reasoning. Like theoretical reasoning, it need not follow logical entailment, but for a more radical reason, namely indeterminism. This makes reasoning in attitudes harder to model logically than theoretical reasoning. But it can be studied abstractly, using indeterministic consequence operators. (shrink)

Some think that logic concerns the “laws of truth”; others that logic concerns the “laws of thought.” This paper presents a way to reconcile both views by building a bridge between truth-maker theory, à la Fine, and normative bilateralism, à la Restall and Ripley. The paper suggests a novel way of understanding consequence in truth-maker theory and shows that this allows us to identify a common structure shared by truth-maker theory and normative bilateralism. We can thus transfer ideas from normative (...) bilateralism to truth-maker theory, such as non-transitive solutions to paradox, and vice versa, such as notions of factual equivalence and containment. (shrink)

This paper provides a proof-theoretic account of imperative logical consequence by generalising Greg Restall’s multiple conclusion bilateralism for declarative logic. According to imperative bilateralism, a sequent Γ ⊢ Δ is valid iff jointly commanding all the imperatives Φ ∈ Γ and prohibiting all the imperatives Ψ ∈ Δ clashes. This account has three main virtues: (1) it provides a proof-theoretic account of imperatives; (2) it does not rely on the controversial notion of imperative inference; and (3) it is neutral regarding (...) cognitivism about imperatives. (shrink)

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material inference and Greg Restall’s (...) bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates. (shrink)

P.F. Strawson contributed to many philosophical domains, including the philosophy of language, the history of philosophy, metaphysics, moral philosophy and philosophical methodology. Most of his contributions in these areas have influenced contemporary debates, either because his views are still defended or because they are still considered worthy of detailed responses. His views on the philosophy of logic have been only rarely discussed, however. My aim in this paper is threefold. First, I provide a systematic account of Strawson’s philosophy of logic. (...) I argue that Strawson is an important predecessor of logical expressivism, a contemporary position in the philosophy of logic most notably defended by Robert Brandom. My main focus is on Strawson’s largely-neglected 1982 paper ‘Logical Form and Logical Constants’, which contains his most mature ideas on the topic. Second, while Strawson’s position is of historical and independent philosophical interest, I argue that he leaves many points unclear. Finally, I propose several clarifications of Strawson’s position. (shrink)

What is the relation of logic to thinking? My dissertation offers a new argument for the claim that logic is constitutive of thinking in the following sense: representational activity counts as thinking only if it manifests sensitivity to logical rules. In short, thinking has to be minimally logical. An account of thinking has to allow for our freedom to question or revise our commitments – even seemingly obvious conceptual connections – without loss of understanding. This freedom, I argue, requires that (...) thinkers have general abilities to respond to support and tension among their thoughts. And these abilities are constituted by following logical rules. So thinkers have to follow logical rules. But there isn’t just one correct logic for thinking. I show that my view is consistent with logical pluralism: there are a range of correct logics, any one of which a thinker might follow. A logic for thinking does, however, have to contain certain minimal principles: Modus Ponens and Non-Contradiction, and perhaps others. We follow logical rules by exercising logical capacities, which display a distinctive first-person/third-person asymmetry: a subject can find the instances of a rule compelling without seeing them as instances of a rule. As a result, there are two limits on illogical thinking. First, thinkers have to tend to find instances of logical rules compelling. Second, thinkers can’t think in obviously illogical ways. So thinking has to be logical – but not perfectly so. When we try to think, but fail, we produce nonsense. But our failures to think are often subjectively indistinguishable from thinking. To explain how this occurs, I offer an account of nonsense. To be under the illusion that some nonsense makes sense is to enter a pretence that the nonsense is meaningful. Our use of nonsense within the pretence relies on the role of logical form in understanding. Finally, while the normativity of logic doesn’t fall directly out of logical constitutivism, it’s possible to build an attractive account of logical normativity which has logical constitutivism as an integral part. I argue that thinking is necessary for human flourishing, and that this is the source of logical normativity. (shrink)

It’s commonly thought that, in conversation, speakers accept and reject propositions that have been asserted by others. Do speakers accept and reject questions as well? Intuitively, it seems that they do. But what does it mean to accept or reject a question? What is the relationship between these acts and those of asking and answering questions? Are there clear and distinct classes of reasons that speakers have for acceptance and rejection of questions? This chapter seeks to address these issues. Beyond (...) their intrinsic interest to those working on the nature of questions, solutions to these problems may aid the extension of inferentialist approaches to logic and language. Inferentialists who think that inferences should be conceived in terms of the norms we are subject to in virtue of both the sentences we accept or assert as well as those we reject or deny are known as bilateralists. A coherent account of accepting and rejecting questions raises the prospects for a bilateralism that interprets question-involving or erotetic inferences according to these additional primitives. While a full-blown bilateralism for erotetic inferences and ultimately for question-forming operators themselves far exceeds this chapter’s scope, the work presented here does sketch the first steps in that direction. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem (...) for second-order —these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

En este trabajo presentaré una forma de evitar los problemas más recurrentes en cierta versión del pluralismo lógico, aquella que defiende que incluso considerando un lenguaje fijo existen múltiples sistemas lógicos legítimos. Para ello, será necesario considerar los puntos de partida del programa pluralista y explicitar los problemas que de ellos surgen, principalmente el Desafío de Quine y el Problema del Colapso. Luego, propondré una modificación respecto de lo que se entiende por consecuencia lógica, para poder considerar una familia de (...) sistemas lógicos, las lógicas mixtas, que abarcan tanto a las lógicas puras como a las impuras. Finalmente, mostraré que con una interpretación razonable del formalismo se puede eludir aquellos problemas a la vez que se respeta el espíritu del programa pluralista. (shrink)

This thesis is about the metaphysics of logic. I argue against a view I refer to as ‘logical realism’. This is the view that the logical constants represent a particular kind of metaphysical structure, which I dub ‘logico-metaphysical structure’. I argue instead for a more metaphysically lightweight view of logic which I dub ‘logical expressivism’. -/- In the first part of this thesis (Chapters I and II) I argue against a number of arguments that Theodore Sider has given for logical (...) realism. In Chapter I, I present an argument of his to the effect that logico-metaphysical structure provides the only good explanation of the semantic determinacy of the logical constants. I argue that other explanations are possible. In Chapter II, I present another argument of his to the effect that logico-metaphysical structure is something that comes along with ontological realism: the view that there is a non-language-relative fact of the matter about what exists. I argue that the connection between logical and ontological realism is not as close as Sider makes it out to be. -/- In the second part of this thesis (Chapters III – V) I set out a positive view of the logical constants that can explain both why their meanings are semantically determinate, and why they form part of our vocabulary. On that view, the primary bearers of logical structure are propositional attitudes, and the logical constants are in our language as vehicles for the expression of logically complex propositional attitudes. In Chapter III, I set out an expressivist theory of propositional logic. In Chapter IV, I use this theory to explain how the logical connectives end up having determinate meanings. In Chapter V, I extend the expressivist theory to predicate logic. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (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)

We present epistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati and Schlöder 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic S5 modulo an appropriate translation. The logical framework developed provides the (...) basis for a novel, proof-theoretic approach to the study of epistemic modality. To demonstrate the fruitfulness of the approach, we show how the framework allows us to reconcile classical logic with the contradictoriness of so-called Yalcin sentences and to distinguish between various inference patterns on the basis of the epistemic properties they preserve. (shrink)

Whether there are philosophically relevant connections between the expressive role of first-personal vocabulary and self-knowledge is an on-going debate in analytical philosophy. We will take a look at this debate by considering Ludwig Wittgenstein’s distinction between the two uses of ‘I’ as object and as subject and work out a further distinction within the subject-use of ‘I’. This relates to a problem that is inherent in Robert Brandom’s inferentialist program regarding the role of first-personal vocabulary. It can be shown that (...) subject-related aspects of language are necessary elements of inferentially articulated discourses—and not, like Brandom assumes, merely contingent features. (shrink)

This paper presents a sound and complete five-sided sequent calculus for first-order weak Kleene valuations which permits not only elegant representations of four logics definable on first-order weak Kleene valuations, but also admissibility of five cut rules by proof analysis.

Real-world agents do not know all consequences of what they know. But we are reluctant to say that a rational agent can fail to know some trivial consequence of what she knows. Since every consequence of what she knows can be reached via chains of trivial cot be dismissed easily, as some have attempted to do. Rather, a solution must give adequate weight to the normative requirements on rational agents’ epistemic states, without treating those agents as mathematically ideal reasoners. I’ll (...) argue that agents can fail to know trivial consequences of what they know, but never determinately. Such cases are epistemic oversights on behalf of the agent in question, and the facts about epistemic oversights are always indeterminate facts. As a result, we are never in a position to assert that such-and-such constitutes an epistemic oversight for agent i (for we may rationally assert only determinate truths). I then develop formal epistemic models according to which epistemic accessibility relations are vague. Given these models, we can show that epistemic oversights always concern indeterminate cases of knowledge. (shrink)

Anti-exceptionalism about logic is the Quinean view that logical theories have no special epistemological status, in particular, they are not self-evident or justified a priori. Instead, logical theories are continuous with scientific theories, and knowledge about logic is as hard-earned as knowledge of physics, economics, and chemistry. Once we reject apriorism about logic, however, we need an alternative account of how logical theories are justified and revised. A number of authors have recently argued that logical theories are justified by abductive (...) argument (e.g. Gillian Russell, Graham Priest, Timothy Williamson). This paper explores one crucial question about the abductive strategy: what counts as evidence for a logical theory? I develop three accounts of evidential confirmation that an anti-exceptionalist can accept: (1) intuitions about validity, (2) the Quine-Williamson account, and (3) indispensability arguments. I argue, against the received view, that none of the evidential sources support classical logic. (shrink)

Many think that expressivists have a special problem with negation. I disagree. For if there is a problem with negation, I argue, it is a problem shared by those who accept some plausible claims about the nature of intentionality. Whether there is any special problem for expressivists turns, I will argue, on whether facts about what truth-conditions beliefs have can explain facts about basic inferential relations among those beliefs. And I will suggest that the answer to this last question is, (...) on most plausible attempts at solving the problem of intentionality, ‘no’. (shrink)

We develop a novel solution to the negation version of the Frege-Geach problem by taking up recent insights from the bilateral programme in logic. Bilateralists derive the meaning of negation from a primitive *B-type* inconsistency involving the attitudes of assent and dissent. Some may demand an explanation of this inconsistency in simpler terms, but we argue that bilateralism’s assumptions are no less explanatory than those of *A-type* semantics that only require a single primitive attitude, but must stipulate inconsistency elsewhere. Based (...) on these insights, we develop a version of B-type expressivism called *inferential expressivism*. This is a novel semantic framework that characterises meanings by inferential roles that define which *attitudes* one can *infer* from the use of terms. We apply this framework to normative vocabulary, thereby solving the Frege-Geach problem generally and comprehensively. Our account moreover includes a semantics for epistemic modals, thereby also explaining normative terms under epistemic modals. (shrink)

We present an inferentialist account of the epistemic modal operator might. Our starting point is the bilateralist programme. A bilateralist explains the operator not in terms of the speech act of rejection ; we explain the operator might in terms of weak assertion, a speech act whose existence we argue for on the basis of linguistic evidence. We show that our account of might provides a solution to certain well-known puzzles about the semantics of modal vocabulary whilst retaining classical logic. (...) This demonstrates that an inferentialist approach to meaning can be successfully extended beyond the core logical constants. (shrink)

Starting from a proof-theoretic perspective, where meaning is determined by the inference rules governing logical operators, in this paper we primarily aim at developing a proof-theoretic alternative to the model-theoretic meaning-invariant logical pluralism discussed in Beall and Restall. We will also outline how this framework can be easily extended to include a form of meaning-variant logical pluralism. In this respect, the framework developed in this paper—which we label two-level proof-theoretic pluralism—is much broader in scope than the one discussed in Beall (...) and Restall’s book. (shrink)

This paper considers whether incompatibilism, the view that negation is to be explained in terms of a primitive notion of incompatibility, and Fregeanism, the view that arithmetical truths are analytic according to Frege’s definition of that term in §3 of Foundations of Arithmetic, can both be upheld simultaneously. Both views are attractive on their own right, in particular for a certain empiricist mind-set. They promise to account for two philosophical puzzling phenomena: the problem of negative truth and the problem of (...) epistemic access to numbers. For an incompatibilist, proofs of numerical non-identities must appeal to primitive incompatibilities. I argue that no analytic primitive incompatibilities are forthcoming. Hence incompatibilists cannot be Fregeans. (shrink)

According to logical inferentialists, the meanings of logical expressions are fully determined by the rules for their correct use. Two key proof-theoretic requirements on admissible logical rules, harmony and separability, directly stem from this thesis—requirements, however, that standard single-conclusion and assertion-based formalizations of classical logic provably fail to satisfy :1035–1051, 2011). On the plausible assumption that our logical practice is both single-conclusion and assertion-based, it seemingly follows that classical logic, unlike intuitionistic logic, can’t be accounted for in inferentialist terms. In (...) this paper, I challenge orthodoxy and introduce an assertion-based and single-conclusion formalization of classical propositional logic that is both harmonious and separable. In the framework I propose, classicality emerges as a structural feature of the logic. (shrink)

This thesis engages with three topics and the relationships between them: (i) the phenomenon of disagreement (paradigmatically, where one person makes a claim and another denies it); (ii) the normative character of disagreements (the issue of whether, and in what sense, one of the parties is “at fault” for believing something that’s untrue); (iii) the issue of which theory of what truth is can best accommodate the norms relating belief and truth. People disagree about all sorts of things: about whether (...) climate is changing, death penalty is wrong, sushi is delicious, or Louis C.K. is funny. However, even focusing on disagreements in the evaluative domain (e.g., taste, moral and comedic), where people have the intuition that there is ‘no fact of the matter’ about who is right, there are significant differences that require explanation. For instance, disagreement about taste is generally perceived as shallow. People accept to disagree and live comfortably with that fact. By contrast, moral disagreement is perceived as deep and sometimes hard to tolerate. Comedic disagreement is similar to taste. However, it may involve an element of ‘intellectual snobbery’ that is absent in taste disagreement. The immediate questions are whether these contrasts allow of precise characterization and what is responsible for them. I argue that, once a case is made for the truth-aptness of judgments in these areas, the contrast can be explained in terms of variable normative function of truth – as exerting a lightweight normative constraint in the domain of taste and a stricter constraint in the moral domain. In particular I claim that while truth in the moral domain exerts a sui generis deontic control, this normative feature of truth is silent in both the taste and the comedic domains. This leads me to investigate how to conceive of truth in the light of normative variability. I argue that an amended version of deflationism – minimally inflated deflationism – can account for the normative variability of truth. (shrink)