In “Truth by Convention” W.V. Quine gave an influential argument against logical conventionalism. Even today his argument is often taken to decisively refute logical conventionalism. Here I break Quine’s arguments into two— the super-task argument and the regress argument—and argue that while these arguments together refute implausible explicit versions of conventionalism, they cannot be successfully mounted against a more plausible implicit version of conventionalism. Unlike some of his modern followers, Quine himself recognized this, but argued that implicit conventionalism was explanatorily (...) idle. Against this I show that pace Quine’s claim that implicit conventionalism has no content beyond the claim that logic is firmly accepted, implicit rules of inference can be used to distinguish the firmly accepted from the conventional. As part of my case, I argue that positing syntactic rules of inference as part of our linguistic competence follows from the same methodology that leads contemporary linguists and cognitive scientists to posit rules of phonology, morphology, and grammar. The upshot of my discussion is a diagnosis of the fallacy in Quine’s master critique of logical conventionalism and a re-opening of possibilities for an attractive conventionalist theory of logic. (shrink)

This paper argues that logical inferentialists should reject multiple-conclusion logics. Logical inferentialism is the position that the meanings of the logical constants are determined by the rules of inference they obey. As such, logical inferentialism requires a proof-theoretic framework within which to operate. However, in order to fulfil its semantic duties, a deductive system has to be suitably connected to our inferential practices. I argue that, contrary to an established tradition, multiple-conclusion systems are ill-suited for this purpose because they fail (...) to provide a 'natural' representation of our ordinary modes of inference. Moreover, the two most plausible attempts at bringing multiple conclusions into line with our ordinary forms of reasoning, the disjunctive reading and the bilateralist denial interpretation, are unacceptable by inferentialist standards. (shrink)

Carnap’s seminal ‘Empiricism, Semantics and Ontology’ makes important use of the notion of a framework and the related distinction between internal and external questions. But what exactly is a framework? And what role does the internal/external distinction play in Carnap’s metaontology? In an influential series of papers, Matti Eklund has recently defended a bracingly straightforward interpretation: A Carnapian framework, Eklund says, is just a natural language. To ask an internal question, then, is just to ask a question in, say, English. (...) To try to ask an external question is to try, absurdly, to ask a question in no language at all. Finding that so trivial an I/e distinction can’t help to explain Carnap’s deflationary metaontology, Eklund is led to attribute to Carnap a view he calls ontological pluralism. In this paper, I show that Eklund misreads Carnap, and I argue that this misreading obscures fundamental features of Carnap’s philosophy. I then defend an account of frameworks as what Carnap called semantical systems, and I place this account in the context of Carnap’s philosophical program of explication. Finally, I discuss the role that frameworks and the I/e distinction play in ESO, showing that ESO provides no reason to attribute the doctrine of ontological pluralism to Carnap. (shrink)

We ask, when is a property of a model a logical property? According to the so-called Tarski–Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim–Skolem theorem, Completeness theorem, and absoluteness, are relevant from the logicality (...) point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic. We also offer a refinement of the result of McGee that logical properties of models can be expressed in $L_{\infty \infty }$ if the expression is allowed to depend on the cardinality of the model, based on replacing $L_{\infty \infty }$ by a “tamer” logic. (shrink)

The paper argues against an intuitive reading of the Stone-von Neumann theorem as a categoricity result, thereby pointing out that this theorem does not entail any model-theoretical difference between the theories that validate it and those that don't.

In philosophical logic and proof theory, we often find multiple-conclusion systems that induce a conjunctive reading of premises and a disjunctive reading of conclusions. In mathematical logic, in contrast, we often find multiple-conclusion systems that induce a conjunctive reading of both premises and conclusions. This paper studies some technical and philosophical aspects of this latter approach to multiple-conclusion consequence. The takeaway is that, while the importance of disjunctive multiple conclusions is beyond doubt, conjunctive multiple conclusions also have philosophical interest. First, (...) because there is some evidence that there are arguments with conjunctive multiple conclusions in natural language. Second, because conjunctive multiple conclusions are compatible with the reflexivity and transitivity of logical consequence, and this allows them to cohere better with some of our best accounts of what logical consequence is. (shrink)

Our goal is to develop a syntactical apparatus for propositional logics in which the accepted and rejected propositions have the same status and obeying treated in the same way. The suggested approach is based on the ideas of Łukasiewicz used for the classical logic and in addition, it includes the use of multiple conclusion rules. More precisely, a consequence relation is defined on a set of statements of forms “proposition _A_ is accepted” and “proposition _A_ is rejected”, where _A_ is (...) a proposition,—a unified consequence relation. Accordingly, the rules defining a unified consequence relation,—the unified rules, have statements as premises and as conclusions. A special attention is paid to the logics in which each proposition is either accepted or rejected. If we express this property via unified rules and add them to a unified deductive system, such a unified deductive system defines a reversible unified consequence: a statement “proposition _B_ is accepted” is derived from the statement “proposition _A_ is accepted” if and only if a statement “proposition _A_ is rejected” is derived from the statement “proposition _B_ is rejected”. (shrink)

Some versions of empiricism have been accused of being neither empirically confirmable nor analytically true and therefore meaningless or unknowable by their own lights. Carnap, and more recently van Fraassen, have responded to this objection by construing empiricism as a stance containing non-cognitive attitudes. The resulting stance empiricism is not subject to the norms of knowledge, and so does not self-defeat as per the objection. In response to this proposal, several philosophers have argued that if empiricism is a stance, then (...) there can be no distinctively epistemic reasons in favor of adopting it, but only prudential or moral reasons. I defend stance empiricism against this objection by showing that stance empiricism furthers many plausibly epistemic goals, such as false belief avoidance, wisdom, and justification. I respond to three objections to my argument: that I assume a conception of epistemic reason that leads to problematic tradeoffs, that to have epistemic reason is just to be epistemically justified, and that my premise that experience is the only source of information has no empirical content. (shrink)

The problem of logical constants consists in finding a principled way to draw the line between those expressions of a language that are logical and those that are not. The criterion of invariance under permutation, attributed to Tarski, is probably the most common answer to this problem, at least within the semantic tradition. However, as the received view on the matter, it has recently come under heavy attack. Does this mean that the criterion should be amended, or maybe even that (...) it should be abandoned? I shall review the different types of objections that have been made against invariance as a logicality criterion and distinguish between three kinds of objections, skeptical worries against the very relevance of such a demarcation, intensional warnings against the level at which the criterion operates, and extensional quarrels against the results that are obtained. I shall argue that the first two kinds of objections are at least partly misguided and that the third kind of objection calls for amendment rather than abandonment. (shrink)

We investigate two different broad traditions in the abstract valuational model theory for nontransitive and nonreflexive logics. The first of these traditions makes heavy use of the natural Galois connection between sets of valuations and sets of arguments. The other, originating with work by Grzegorz Malinowski on nonreflexive logics, and best systematized in Blasio et al. : 233–262, 2017), lets sets of arguments determine a more restricted set of valuations. After giving a systematic discussion of these two different traditions in (...) the valuational model theory for substructural logics, we turn to looking at the ways in which we might try to compare two sets of valuations determining the same set of arguments. (shrink)

Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment (...) relation over a single-conclusion one. (shrink)

Our deepest commitments about history are reflected in how we break it down into periods. (Cf. Galison 1988) By drawing a break at a certain point we emphasize the novelty and importance of a new development. It is also how we contain and dismiss certain work as no longer relevant. Thus, in the history of physics we break the story with Newton, both to emphasize his roles in bringing previous developments to a close and in initiating new lines of work, (...) and also to suggest that the ongoing practice of physics thereafter can appropriately in large measure ignore what preceeds Newton. Periodizing history is essential to understanding it, including when we periodize the work of a given writer. In philosophy, anyone who did not see a gulf between Kant’s early work and his critical philosophy or between Wittgenstein’s Tractatus and his Philosophical Investigations would be missing something enormously important. (shrink)

In his Formalization of Logic (1943) Carnap pointed out that there are non-normal interpretations of classical logic: non-standard interpretations of the connectives and quantifiers that are consistent with the classical consequence relation of a language. Different ways around the problem have been proposed. In a recent paper, Bonnay and Westerståhl argue that the key to a solution is imposing restrictions on the type of interpretation we take into account. More precisely, they claim that if we restrict attention to interpretations that (...) are (a) compositional, (b) non-trivial and (c) in the case of the quantifiers, invariant under permutations of the domain, Carnap’s Problem is avoided. This paper has two goals. The first is to show that Bonnay and Westerståhl’s solution to Carnap’s Problem doesn’t work. The second is to argue that something similar to their proposal seems to do the job. The problems with Bonnay and Westerståhl’s approach trace back to issues concerning the (un)definability of subsets of the domain of first-order structures, as well as to the compositionality of first-order languages. After expanding on these problems, I’ll propose a way to modify Bonnay and Westerståhl’s account and solve Carnap’s Problem. (shrink)

The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy “one level up” into the meta-theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This paper argues (...) that the challenge can be met. We show how the quantifier “there are infinitely many” can be uniquely determined in a naturalistically acceptable fashion and thus be used in the formulation of a theory of arithmetic. We compare the approach pursued here with Field’s justification of the same device and the popular strategy of invoking a second-order formalism, and argue that it is more robust than either of the alternative proposals. (shrink)

Even though the strong relationship between proof-theoretic and model-theoretic notions in one’s logical theory can be shown by soundness and completeness proofs, whether we can define the model-theoretic notions by means of the inferences in a proof system is not at all trivial. For instance, provable inferences in a proof system of classical logic in the logical framework do not determine its intended models as shown by Carnap (Formalization of logic, Harvard University Press, Cambridge, 1943), i.e., there are non-Boolean models (...) that satisfy its provable inferences. In the literature, this is known as the Categoricity problem or Carnap’s problem. In this paper, we will discuss the Categoricity problem (or Carnap’s problem) for three-valued logics K3 and LP. We will provide three different restrictions on admissible models that will deliver us categoricity results, some of which draw from the solutions provided for the Categoricity problem for classical logic in Belnap and Massey (Stud Log 49(1):67–82, 1990) and Bonnay and Westerståhl (Erkenntis 81(4):721–739, 2016). We will then argue that two of those solutions are philosophically well-motivated: (1) restricting the admissible models where negation is interpreted as a Strong Kleene truth-function, and (2) restricting the admissible models where a complex formula is assigned the third value when its immediate subformulas are assigned the third value. (shrink)

In this paper, we evaluate Button’s claim that knot is a nasty connective. Knot’s nastiness is due to the fact that, when one extends the set \ with knot, the connective provides counterexamples to a number of classically valid operational rules in a sequent calculus proof system. We show that just as going non-transitive diminishes tonk’s nastiness, knot’s nastiness can also be reduced by dropping Reflexivity, a different structural rule. Since doing so restores all other rules in the system as (...) validity-preserving, we are inclined to conclude that there, knot is not that nasty. However, since motivating non-reflexivity is harder than motivating non-transitivity, we also acknowledge that disagreement with our conclusion is possible. (shrink)

This is the Editorial Introduction to “S.I.: Strong and Weak Kleene Logics”.

The connectives of classical propositional logic are given an analysis in terms of necessary and sufficient conditions of acceptance and rejection, i.e. the connectives are analyzed within an expressivist bilateral meaning-is-use framework. It is explained how such a framework differs from standard inferentialist frameworks and it is argued that it is better suited to address the particular issues raised by the expressivist thesis that the meaning of a sentence is determined by the mental state that it is conventionally used to (...) express. Furthermore, it is shown that the classical requirements governing the connectives completely characterize classical logic, are conservative and separable, are in bilateral harmony, are structurally preservative with respect to the classical coordination requirements and resolve the categoricity problem. These results are taken to show that one can give an expressivist bilateral meaning-is-use analysis of the connectives that confer on them a determinate coherent classical interpretation. (shrink)

Formal Axiomatic method as exemplified in Hilbert’s Grundlagen der Geometrie is based on a structuralist vision of mathematics and science according to which theories and objects of these theories are to be construed “up to isomorphism”. This structuralist approach is tightly linked with the idea of making Set theory into foundations of mathematics. Category theory suggests a generalisation of Formal Axiomatic method, which amounts to construing objects and theories “up to general morphism” rather than up to isomorphism. It is shown (...) that this category-theoretic method of theorybuilding better fits mathematical and scientific practice. Moreover so since the requirement of being determined up to isomorphism (i.e. categoricity in the usual model-theoretic sense) turns to be unrealistic in many important cases. The category-theoretic approach advocated in this paper suggests an essential revision of the structuralist philosophy of mathematics and science. It is argued that a category should be viewed as a far-reaching generalisation of the notion of structure rather than a particular kind of structure. Finally, I compare formalisation and categorification as two alternative epistemic strategies. (shrink)

The long-standing dispute between absolutists and relativists traditionally focuses on whether there are absolute truths, absolute epistemic norms, and absolute.