Williamson argues for the contention that substructural logics are ‘ill-suited to acting as background logics for science’. That contention, if true, would be very important, but it is refutable, given what is already known about certain substructural logics. Classical Core Logic is a substructural logic, for it eschews the structural rules of Thinning and Cut and has Reflexivity as its only structural rule. Yet it suffices for classical mathematics, and it furnishes all the proofs and disproofs one needs for the (...) hypothetico-deductive method in science. We explain exactly what Classical Core Logic is, why it is a substructural logic par excellence, and what the basic requirements would be for a logic to be ‘suited to acting as [a] background logic for science’. We also explain how Classical Core Logic meets all these requirements. We end by examining Williamson’s argument in order to expose where its error lies. (shrink)

Non‐classical logicians do not typically reject classically valid logical principles across the board. In fact, they sometimes suggest that their preferred logic recovers classical reasoning in most circumstances. This idea has come to be known in the literature as ‘classical recapture’. Recently, classical logicians have raised various doubts about it. The main problem is said to be that no rigorous explanation has been given of how is it exactly that classical logic can be recovered. The goal of the paper is (...) to address this problem. First, I discuss a number of different ways to characterize the idea of classical recapture and I examine how far it can be pushed. Secondly, I argue that a palatable form of classical recapture is given by the thought that classical logic can be retained for statements that are grounded (in Kripke’s sense). To substantiate this view I provide a formal fixed‐point semantics for a language containing a predicate standing for the concept of groundedness and I address a couple of objections that have been deployed against the claim that classical recapture can aid the non‐classical logician’s cause. (shrink)

It is not uncommon among theorists favoring a deviant logic on account of the semantic paradoxes to subscribe to an idea that has come to be known as ‘classical recapture’. The main thought underpinning it is that non-classical logicians are justified in endorsing many instances of the classically valid principles that they reject. Classical recapture promises to yield an appealing pair of views: one can attain naivety for semantic concepts while retaining classicality in ordinary domains such as mathematics. However, Julien (...) Murzi and Lorenzo Rossi have recently suggested that revisionary approaches to truth breed revenge paradoxes when they are coupled with the thought that classical reasoning can be recaptured in certain circumstances. What’s novel about the paradoxes they put forward is that they cannot be dismissed so easily. The concepts used to generate these paradoxes—those of paradoxicality and unparadoxicality—are concepts that non-classical theorists need in order to offer a diagnosis of the truth-theoretic paradoxes. My goal in this paper is to argue that non-classical theorists can represent the concept of paradoxicality without falling prey to revenge paradoxes. In particular, I will show how to provide a formal fixed-point semantics for a language extended with a paradoxicality predicate that adequately expresses the non-classical logician’s notion of paradoxicality. (shrink)

Mathematical pluralism notes that there are many different kinds of pure mathematical structures—notably those based on different logics—and that, qua pieces of pure mathematics, they are all equally good. Logical pluralism is the view that there are different logics, which are, in an appropriate sense, equally good. Some, such as Shapiro, have argued that mathematical pluralism entails logical pluralism. In this brief note I argue that this does not follow. There is a crucial distinction to be drawn between the preservation (...) of truth and the preservation of truth-in-a-structure; and once this distinction is drawn, this suffices to block the argument. The paper starts by clarifying the relevant notions of mathematical and logical pluralism. It then explains why the argument from the first to the second does not follow. A final section considers a few objections. (shrink)

As a response to the semantic and logical paradoxes, theorists often reject some principles of classical logic. However, classical logic is entangled with mathematics, and giving up mathematics is too high a price to pay, even for nonclassical theorists. The so-called recapture theorems come to the rescue. When reasoning with concepts such as truth/class membership/property instantiation, (These are examples of concepts that are taken to satisfy naive rules such as the naive truth schema and naive comprehension, and that therefore are (...) compatible with a solution to paradox cast in the logics considered below. Other notions of similar kind can be added to the list.) if one is interested in consequences of the theory that only contain mathematical vocabulary, nothing is lost by reasoning in the nonclassical framework. This article shows that this claim is highly misleading, if not simply false. Under natural assumptions, some well-established approaches to recapture are incorrect. (shrink)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants of Kripke–Feferman truth. The first, \, features a consistent but partial truth predicate. The second, \, an inconsistent but complete truth predicate. It is known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability, under natural assumptions, coincides with (...) definitional equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of \ and \ raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (shrink)

Under the influence of Quine’s famous manifesto, many philosophers have thought that logical theories are scientific theories that can be ‘adopted’ and tested as scientific theories. Here we argue that this idea is untenable. We discuss it with special reference to Putnam’s proposal to ‘adopt’ a particular non-classical logic to solve the foundational problems of quantum mechanics in his famous paper ‘Is Logic Empirical?’ (1968), which we argue was not really coherent.

Timothy Williamson has maintained that the applicability of classical mathematics in science raises a problem for the endorsement, in non-mathematical domains, of a wide range of non-classical logics. We show that this is false.

The PhD thesis advances a new approach to vagueness as dispersion, comparing it with the main philosophical theories of vagueness in the analytic tradition.

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 semantics, 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 of rules. 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)

Logicians disagree about how validity—the very heart of logic—should be understood. Many different formal systems have been born due to this disagreement. This thesis examines how teachers explain the subject matter of logic to students in introductory logic textbooks, and demonstrates the different explanations teachers use. These differences help explain why logicians have different intuitions about validity.

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants Kripke-Feferman truth. The first, KF+CONS, features a consistent but partial truth predicate. The second, KF+COMP, an inconsistent but complete truth predicate. It is well-known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability under natural assumptions coincides with definitional (...) equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of KF+CONS and KF+COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (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)