The logic just corresponding to involutive quantales, which was introduced by Wendy MacCaull, is reconsidered in order to obtain a cut-free sequent calculus formulation, and the completeness theorem for this logic is proved using a new admissible rule.

Abstractabstract:In the following, I argue that Hegel took concepts—not propositions, judgments, or spatiotemporal objects—as the primary truth-bearer in his logic and attempt to offer a defensible interpretation of what it means for an individual concept (or "thought-determination") to be assessed as true or untrue. Along the way, I consider the shortcomings of several alternative interpretations of truth in Hegelian logic, paying particular attention to the now-common contention that a commitment to something like Frege's context principle prevents Hegel from (...) assessing concepts independently of the role that they play in judgments. (shrink)

Chapter One: Constructions. Entities, constructions, and functions When one travels from Los Angeles to New York, going, say, by way of St. Louis, Chicago, ...

Is there any link between the doctrine of logical fatalism and prime numbers? What do logic and prime numbers have in common? The book adopts truth-functional approach to examine functional properties of finite-valued Łukasiewicz logics Łn+1. Prime numbers are defined in algebraic-logical terms and represented as rooted trees. The author designs an algorithm which for every prime number n constructs a rooted tree where nodes are natural numbers and n is a root. Finite-valued logics Kn+1 are specified that they (...) have tautologies if and only if n is a prime number. It is discovered that Kn+1 have the same functional properties as Łn+1 whenever n is a prime number. Thus, Kn+1 are 'logics' of prime numbers. Amazingly, combination of logics of prime numbers led to uncovering a law of generation of classes of prime numbers. Along with characterization of prime numbers author also gives characterization, in terms of Łukasiewicz logical matrices, of powers of primes, odd numbers, and even numbers. (shrink)

Reasoning about concurrent programs involves representing the information that concurrent processes manipulate disjoint portions of memory. In sophisticated applications, the division of memory between processes is not static. Through operations, processes can exchange the implied ownership of memory cells. In addition, processes can also share ownership of cells in a controlled fashion as long as they perform operations that do not interfere, e.g., they can concurrently read shared cells. Thus the traditional paradigm of distributed computing based on locations is replaced (...) by a paradigm of concurrent computing which is more tightly based on program structure. Concurrent Separation Logic with Permissions, developed by O’Hearn, Bornat et al., is able to represent sophisticated transfer of ownership and permissions between processes. We demonstrate how these ideas can be used to reason about fine-grained concurrent programs which do not employ explicit synchronization operations to control interference but cooperatively manipulate memory cells so that interference is avoided. Reasoning about such programs is challenging and appropriate logical tools are necessary to carry out the reasoning in a reliable fashion. We argue that Concurrent Separation Logic with Permissions provides such tools. We illustrate the logical techniques by presenting the proof of a concurrent garbage collector originally studied by Dijkstra et al., and extended by Lamport to handle multiple user processes. (shrink)

An incisive analysis of Deleuze's philosophy of eventsSean Bowden shows how the Deleuzian event should be understood in terms of the broader metaphysical thesis that substances are ontologically secondary with respect to events. He achieves this through a reconstruction of Deleuze's relation to the history of thought from the Stoics through to Simondon, taking account of Leibniz, Lautman, structuralism and psychoanalysis along the way.This exciting new reading of Deleuze focuses firmly on his approach to events. Bowden also examines and clarifies (...) a number of Deleuze's most difficult philosophical concepts, including sense, problematic Ideas and intensive individuation, and engages with material by Lautman and Simondon that has not yet been translated into English. (shrink)

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

The logics BN4 and E4 can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively. The logic BN4 was developed by Brady in 1982 and the logic E4 by Robles and Méndez in 2016. The aim of this paper is to investigate the implicative variants (of both systems) which contain Routley and Meyer’s logic B and endow them with a Belnap-Dunn type bivalent semantics.

Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a b a for all b, then a a* a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style (...) representation theorem for complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic. (shrink)

Frege believes that the real power to determine the sentence to show the logic of interest to "true." And with the word "truth" of redundancy on the same, he believes the lack of everyday language and the ability to determine the appropriate symbol. However, Frege, it seems that he invented the concept of text assigned to a special force to determine the sign: determine the bar. This paper attempts to demonstrate, although Frege until the final stage of his academic (...) career to determine the bar still insist to retain its logic into the language system, but the concepts and words in this design does not meet him on the relationship between logic and fuzzy real insight. A key step in the argument, we will by means of Husserl on the "faith of variation" to clarify the doctrine of "judge" and "determined" by Frege between the neglected difference. On this basis, we finally prompted a brief on the logic of Frege and true insight into the relationship between the interpretation of ambiguous direction. (shrink)

In Artemovʼs Justification Logic, one can make statements interpreted as “t is evidence for the truth of formula F.” We propose a variant of this logic in which one can say “I have degree r of confidence that t is evidence for the truth of formula F.” After defining both an axiomatic approach and a semantics for this Logic of Uncertain Justifications, we will prove the usual soundness and completeness theorems.

The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?

This paper intends to explain key differences between Aristotle’s understanding of the relationships between nous, epistêmê, and the art of syllogistic reasoning(both analytic and dialectical) and the corresponding modern conceptions of intuition, knowledge, and reason. By uncovering paradoxa that Aristotle’s understanding of syllogistic reasoning presents in relation to modern philosophical conceptions of logic and science, I highlight problems of a shift in modern philosophy—a shift that occurs most dramatically in the seventeenth century—toward a project of construction, a pervasive desire (...) for rational certainty, and a general insistence on the reducibility of the sciences. The major motivation of this analysis is my intention to show that modern attempts to reduce science/epistêmê to a single science/method of inquiry occlude dialectical and ethico-political dimensions of “reason” and, hence, also impoverish philosophy’s critical capacities. (shrink)

Frege’s conceptions of sense, truth, and falsehood, assertion, thought, the logical connectives, the laws of thought, and the laws of logic is described as Wittgenstein understood them. Wittgenstein’s criticisms of Frege’s conceptions as elaborated in the Tractatus and associated early writings are described and analysed. The criticisms are shown to be powerful and to undermine Frege’s conception of logic and the primary supports of his philosophy of logic.

We investigate a subsystem of minimal logic related to D. Vakarelov’s logic \, using the framework of subminimal logics by A. Colacito, D. de Jongh and A. L. Vargas. In the course of it, the relationship between the two semantics in the respective frameworks is clarified. In addition, we introduce a sequent calculus for the investigated subsystem, and some proof-theoretic properties are established. Lastly, we formulate a new infinite class of subsystems of minimal logics.