We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, whereas Hilbert dismisses elucidation and consequently treats the primitives as schematic.

What is it to believe something might be the case? We develop a puzzle that creates difficulties for standard answers to this question. We go on to propose our own solution, which integrates a Bayesian approach to belief with a dynamic semantics for epistemic modals. After showing how our account solves the puzzle, we explore a surprising consequence: virtually all of our beliefs about what might be the case provide counterexamples to the view that rational belief is closed under logical implication.

The rules for Core Logic are stated, and various important results about the system are summarized. We describe its relationship to other systems, such as Classical Logic, Intuitionistic Logic, Minimal Logic, and the Anderson–Belnap relevance logic R. A precise, positive explication is offered of what it is for the premises of a proof to connect relevantly with its conclusion. This characterization exploits the notion of positive and negative occurrences of atoms in sentences. It is shown that all Core proofs are relevant in this precisely defined sense. We survey extant results about variable-sharing in rival systems of relevance logic, and find that the variable-sharing conditions established for them are weaker than the one established here for Core Logic (and for its classical extension). Proponents of other systems of relevance logic (such as R and its subsystems) are challenged to formulate a stronger variable-sharing condition, and to prove that R or any of its subsystems satisfies it, but that Core Logic does not. We give reasons for pessimism about the prospects for meeting this challenge.

Are counterfactuals with true antecedents and consequents automatically true? That is, is Conjunction Conditionalization: (X ∧ Y) ⊃ (X > Y) valid? Stalnaker and Lewis think so, but many others disagree. We note here that the extant arguments for Conjunction Conditionalization are unpersuasive, before presenting a family of more compelling arguments. These arguments rely on some standard theorems of the logic of counterfactuals as well as a plausible and popular semantic claim about certain semifactuals. Denying Conjunction Conditionalization, then, requires rejecting other aspects of the standard logic of counterfactuals or else our intuitive picture of semifactuals.

Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic that norms belief as classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterize degrees of determinacy. It applies the methodology above to focus on degree logic. This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allow degrees of consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail.

We describe a logic of preference in which modal connectives reflect reasons to desire that a sentence be true. Various conditions on models are introduced and analyzed.

I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language.

We analyze a class of modal logics rendered insensitive to reflexivity by way of a modification to the semantic definition of the modal operator. We explore the extent to which these logics can be characterized, and prove a general completeness theorem on the basis of a translation between normal modal logics and their reflexive-insensitive counterparts. Lastly, we provide a sufficient semantic condition describing when a similarly general soundness result is also available.

Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteriors are equal” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s proof but are not explicit in the key conditional: (1) that agents commonly know, of some prior μ, that it is the common prior; (2) that agents commonly know that each of them updates on the prior by conditionalization; (3) that agents commonly know that if an agent knows a proposition, she knows that she knows that proposition (the “KK” principle); (4) that agents commonly know that they each update only on true propositions. It is shown that natural weakenings of any one of these strong assumptions can lead to countermodels to Aumann’s key conditional. Examples are given in which agents who have a common prior and commonly know what probability they each assign to a proposition nevertheless assign that proposition unequal probabilities. To alter Aumann’s famous slogan: people can “agree to disagree”, even if they share a common prior. The epistemological significance of these examples is presented in terms of their role in a defense of the Uniqueness Thesis: If an agent whose total evidence is E is fully rational in taking doxastic attitude D to P, then necessarily, any subject with total evidence E who takes a different attitude to P is less than fully rational.

We construct four binary consequence systems axiomatizing entailment relations between formulas of classical, intuitionistic, dual-intuitionistic and modal (S4) logics, respectively. It is shown that the intuitionistic consequence system is embeddable in the modal (S4) one by the usual modal translation prefixing □ to every subformula of the translated formula. An analogous modal translation of dual-intuitionistic formulas then consists of prefixing ◊ to every subformula of the translated formula. The philosophical importance of this result is briefly discussed.

We introduce the logic LRC, designed to describe and reason about agents’ abilities and capabilities in using resources. The proposed framework bridges two—up to now—mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC (soundness, completeness, canonicity, and disjunction property) and its associated analytic calculus (conservativity, cut elimination, and subformula property). These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multitype calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations.

This paper explores how a semantics for Prior’s infamous connective tonk should be, a connective defined by inference rules that trivialize the logic of a deductive system if that logic is supposed to be transitive. To avoid triviality, one must reject transitivity and in a relatively recent paper, Roy Cook develops a semantics for tonk with non-transitive entailment. However, I show in this paper that a cut-free sequent calculus for tonk - the arguably most natural and simplest deductive system for a non-transitive logic - can neither be complete with respect to Cook’s semantics nor with respect to a semantics with non-transitive entailment based on a semantics for vagueness and transparent truth developed by Cobreros et al. It is argued that the failure to adequately represent tonk is connected with the fact that tonk is not uniquely defined in a cut-free sequent calculus system unless the logic is in addition non-reflexive. To remedy this, the paper develops a semantics with non-transitive and non-reflexive entailment based on the idea that complex formulae are true or false relative to them being assessed as premise or as conclusion.

The paper argues that the view that the particular quantifier is ‘existentially loaded’ is a relatively new one historically and that it has become entrenched in modern philosophical logic for less than happy reasons.

Certain commentators on Russell’s “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions.” These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show that Russell did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extralinguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language.

Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”—making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an operator allowing it, against the opinion of many critics, to rule things out and manifest disagreement: for unlike other proposed exclusion-expressing-devices (for instance, the entailment of triviality), the operator used to formulate the notion of absolute contradiction appears to be immune both from crippling expressive limitations and from revenge paradoxes—pending a rigorous nontriviality proof for a formal dialetheic theory including it.

The operations of expansion and refinement on nondeterministic matrices (Nmatrices) are composed to form a new operation called rexpansion. Properties of this operation are investigated, together with their effects on the induced consequence relations. Using rexpansions, a semantic method for obtaining conservative extensions of (N)matrix-defined logics is introduced and applied to fragments of the classical two-valued matrix, as well as to other many-valued matrices and Nmatrices. The main application of this method is the construction and investigation of truth-preserving ¬-paraconsistent conservative extensions of Gödel fuzzy logic, in which ¬ has several desired properties. This is followed by some results regarding the relations between the constructed logics.

This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted by PC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to present PC as a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, in PC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection between PC and the alpha system.

This paper develops a new framework for combining propositional logics, called “juxtaposition.” Several general metalogical theorems are proved concerning the combination of logics by juxtaposition. In particular, it is shown that under reasonable conditions, juxtaposition preserves strong soundness. Under reasonable conditions, the juxtaposition of two consequence relations is a conservative extension of each of them. A general strong completeness result is proved. The paper then examines the philosophically important case of the combination of classical and intuitionist logics. Particular attention is paid to the phenomenon of collapse. It is shown that there are logics with two stocks of classical or intuitionist connectives that do not collapse. Finally, the paper briefly investigates the question of which rules, when added to these logics, lead to collapse.

This article discusses how to update one’s credences based on evidence that has initial probability 0. I advance a diachronic norm, Kolmogorov Conditionalization, that governs credal reallocation in many such learning scenarios. The norm is based upon Kolmogorov’s theory of conditional probability. I prove a Dutch book theorem and converse Dutch book theorem for Kolmogorov Conditionalization. The two theorems establish Kolmogorov Conditionalization as the unique credal reallocation rule that avoids a sure loss in the relevant learning scenarios.

In an earlier paper in this journal I provided detailed motivation for the constructive and relevant system of Core Logic; explained its main features; and established that, even though Cut is not a rule of the system, nevertheless Cut is admissible for it, and indeed with potential epistemic gain. If Π is a proof of the sequent Δ : A, and Σ is a proof of the sequent Г, A : θ, then there is a computable reduct [Π, Σ] of some subsequent of the ‘overall target sequent’ Δ, Г : θ. In the constructive case the potential epistemic gain consists in the possibility that the subsequent in question be a proper subsequent of Δ, Г : θ, and indeed even logically stronger than Δ, Г : θ.

In this paper it is established that Cut is likewise admissible for Classical Core Logic, which is obtained from Core Logic by adding a suitably relevantized form of the rule of Classical Dilemma. In the classical case there is an additional feature of potential epistemic gain: the proof [Π, Σ] might have a lower degree of non-constructivity than do Π and Σ.