A Structuralist Theory of Logic
Professor Koslow advances a new account of the basic concepts of logic. A central feature of the theory is that it does not require the elements of logic to be based on a formal language. Rather, it uses a general notion of implication as a way of organizing the formal results of various systems of logic in a simple, but insightful way. The study has four parts. In the first two parts the various sources of the general concept of an implication structure and its forms are illustrated and explained. Part 3 defines the various logical operations and systematically explores their properties. A generalized account of extensionality and dual implication is given, and the extensionality of each of the operators, as well as the relation of negation and its dual are given substantial treatment because of the novel results they yield. Part 4 considers modal operators and studies their interaction with logical operators. By obtaining the usual results without the usual assumptions this new approach allows one to give a very simple account of modal logic minus the excess baggage of possible world semantics.
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A V B accessibility relation antecedent assigns Barcan formula Belnap binary relation bisection implication characterization classical implication structure Cn(X component conjunction operator Consequently conservative extension converse deﬁned Deﬁnition direct answer disjunction operator elements equivalent example Exercise exist existential quantiﬁers extended implication structure extensional with respect fails ﬁnite ﬁrst condition ﬁrst-order ﬁxed follows formula function G-L modal Gentzen holds hypothetical operator implication relation incomplete with respect interrogatives intuitionistic K4 modal logical operators mapping modal character modal operator modal systems modal with respect modus ponens necessitation modal negation is classical negation operator NN(A nonclassical structures nonextensional notion preceding theorem predicate Proof reﬂexive relativized respect to negation result rules of inference satisﬁes satisfy the ﬁrst second condition sentences sentential calculus sequence strong bisection strongly closed subset Suppose syntactic Tarski Tarskian theories thesis topological spaces truth operator truth values universal quantiﬁer weakest member