Infinitary Action Logic: Complexity, Models and Grammars

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the –completeness of the equational theories of action lattices of subsets of a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics. Presented by Jacek Malinowski     

Truth Values and Proof Theory

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.     

Constructive Logic, Truth and Warranted Assertability

Shapiro and Taschek have argued that simply using intuitionistic logic and its Heyting semantics, one can show that there are no gaps in warranted assertability. That is, given that a discourse is faithfully modelled using Heyting's semantics for the logical constants, then if a statement S is not warrantedly assertable, its negation ^∼ S is. Tennant has argued for this conclusion on similar grounds. I show that these arguments fail, albeit in illuminating ways. An appeal to constructive logic does not commit one to this strong epistemological thesis, but appeals to semantics of intuitionistic logic none the less do give us certain conclusions about the connections between warranted assertability and truth.     

Modal Logic,Truth, and the Master Modality

In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a so-called master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show that if attention is restricted to bidirectional frames, then the expressive power of the master modality is exactly what is needed to translate the bounded fragment of first-order logic into hybrid logic in a truth preserving way. We believe that this throws new light on Arthur Prior's fourth grade tense logic.     

Higher-Order Logic and Disquotational Truth

Journal of Philosophical Logic - Truth predicates are widely believed to be capable of serving a certain logical or quasi-logical function. There is little consensus, however, on the exact nature...     

An Interpolation Theorem for First Order Logic with Infinitary Predicates

An interpolation Theorem is proved for first order logic withinfinitary predicates. Our proof is algebraic via cylindricalgebras.¹     

Distributive Terms,Truth, and the Port Royal Logic

The paper shows that in the Art of Thinking (The Port Royal Logic) Arnauld and Nicole introduce a new way to state the truth-conditions for categorical propositions. The definition uses two new ideas: the notion of distributive or, as they call it, universal term, which they abstract from distributive supposition in medieval logic, and their own version of what is now called a conservative quantifier in general quantification theory. Contrary to the interpretation of Jean-Claude Parienté and others, the truth-conditions do not require the introduction of a new concept of ‘indefinite’ term restriction because the notion of conservative quantifier is formulated in terms of the standard notion of term intersection. The discussion shows the following. Distributive supposition could not be used in an analysis of truth because it is explained in terms of entailment, and entailment in terms of truth. By abstracting from semantic identities that underlie distribution, the new concept of distributive term is definitionally prior to truth and can, therefore, be used in a non-circular way to state truth-conditions. Using only standard restriction, the Logic’s truth-conditions for the categorical propositions are stated solely in terms of (1) universal (distributive) term, (2) conservative quantifier, and (3) affirmative and negative proposition. It is explained why the Cartesian notion of extension as a set of ideas is in this context equivalent to medieval and modern notions of extension.     

Epistemic Friction: Reflections on Knowledge, Truth, and Logic

Knowledge requires both freedom and friction. Freedom to set up our epistemic goals, choose the subject matter of our investigations, espouse cognitive norms, design research programs, etc., and friction (constraint) coming from two directions: the object or target of our investigation, i.e., the world in a broad sense, and our mind as the sum total of constraints involving the knower. My goal is to investigate the problem of epistemic friction, the relation between epistemic friction and freedom, the viability of foundationalism as a solution to the problem of friction, an alternative solution in the form of a neo-Quinean model, and the possibility of solving the problem of friction as it applies to logic and the philosophy of logic within that model.     

The Logic of Pragmatic Truth

The mathematical concept of pragmatic truth, first introduced in Mikenberg, da Costa and Chuaqui (1986), has received in the last few years several applications in logic and the philosophy of science. In this paper, we study the logic of pragmatic truth, and show that there are important connections between this logic, modal logic and, in particular, Jaskowski's discussive logic. In order to do so, two systems are put forward so that the notions of pragmatic validity and pragmatic truth can be accommodated. One of the main results of this paper is that the logic of pragmatic truth is paraconsistent. The philosophical import of this result, which justifies the application of pragmatic truth to inconsistent settings, is also discussed.     

Stoic Sequent Logic and Proof Theory

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.     

On Moderate Pluralism About Truth and Logic

According to moderate truth pluralism, truth is both One and Many. There is a single truth property that applies across all truth-apt domains of discourse, but instances of this property are grounded in different ways. Propositions concerning medium-sized dry goods might be true in virtue of corresponding with reality while propositions pertaining to the law might be true in virtue of cohering with the body of law. Moderate truth pluralists must answer two questions concerning logic: (Q1) Which logic governs inferences concerning each truth-apt domain considered separately? (Q2) Which logic governs inferences that involve several truth-apt domains? This paper has three objectives. The first objective is to present and explain the moderate pluralist’s answers to (Q1) and (Q2). The second objective is to argue that there is a tension between these answers. The answer to (Q1) involves a commitment to a form of logical pluralism. However, reflection on the moderate truth pluralist’s answer to (Q2) shows that they are committed to taking logic to be topic neutrality. This, in turn, forces a commitment to logical monism. It would seem that the moderate truth pluralist cannot have it both ways. The third objective is constructive in nature. I offer an account of what moderate truth pluralists should say about logic and how they might resolve the tension in their view. I suggest that, just like moderate truth pluralists distinguish truth proper and “quasi-truth,” they should endorse a distinction between logic proper and “quasi-logic.” Quasi-truth is truth-like in the sense that instances of quasi-truth ground instances of truth. Quasi-logic is logic-like in the sense that it concerns arguments that are necessarily truth-preserving but are not generally so in a topic neutral way. I suggest that moderate truth pluralists should be monists about truth proper and logic proper but pluralists about quasi-truth and quasi-logic. This allows them to say that logic proper is topic neutral while still accommodating the idea that, for different domains, different arguments may be necessarily truth-preserving.     

Proof Analysis in Modal Logic

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel–Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.     

Frege on Truth,Assertoric Force and the Essence of Logic

In a posthumous text written in 1915, Frege makes some puzzling remarks about the essence of logic, arguing that the essence of logic is indicated, properly speaking, not by the word ‘true’, but by the assertoric force. William Taschek has recently shown that these remarks, which have received only little attention, are very important for understanding Frege's conception of logic. On Taschek's reconstruction, Frege characterizes logic in terms of assertoric force in order to stress the normative role that the logical laws play vis-à-vis judgement, assertion and inference. My aim in this paper is to develop and defend an alternative reconstruction according to which Frege stresses that logic is not only concerned with ‘how thoughts follow from other thoughts’, but also with the ‘step from thought to truth-value’. Frege considers logic as a branch of the theory of justification. To justify a conclusion by means of a logical inference, the ‘step from thought to truth-value’ must be taken, that is, the premises must be asserted as true. It is for this reason that, in the final analysis, the assertoric force indicates the essence of logic, for Frege.     

Truth versus testability in Quantum Logic

We forward an epistemological perspective regarding non-classical logics which restores the universality of logic in accordance with the thesis of global pluralism. In this perspective every non-classical truth-theory is actually a theory of some metalinguistic concept which does not coincide with the concept of truth (described by Tarski's truth theory). We intend to apply this point of view to Quantum Logic (QL) in order to prove that its structure properties derive from properties of the metalinguistic concept of testability in Quantum Physics. To this end we construct a classical language L _cand endow it with a classical effective interpretation which is partially inspired by the Ludwig approach to the foundations of Quantum Mechanics. Then we select two subsets of formulas in L _cwhich can be considered testable because of their interpretation and we show that these subsets have the structure properties of Quantum Logics because of Quantum Mechanical axioms, as desired. Finally we comment on some relevant consequences of our approach (in particular, the fact that no non-classical logic is strictly needed in Quantum Physics).     

Logic: The Question of Truth

Philological background information is presented on the origin and composition of the text generally known as Kant's Logic. The text, which was not in the strict sense of the word written by Kant himself, but rather assembled by another writer whom Kant had authorized to do so on his behalf, is a mixture of materials, not all of which originate directly from Kant, and cannot claim full authenticity.