LE MOLTE VERSIONI DELLA CONSEGUENZA LOGICA

This thesis is on the concept of logical consequence (LC) and it is divided into two parts. In the first one, I show that LC: (1) was not important in eminent logicians (like Aristotle) (2) has been described in several different ways (preservation of truth from premises to conclusion, formality, necessity of thought, following a rule, to transform what is a ground for the premises into a ground for the conclusion, …) and by different methods (predication, natural language, formal language, per se entities, Theory of Types, Set-theory, derivation in a formal calculus, variation of the non-logical parts of the sentences, …). I explain how LC became one of the central notions of contemporary logic, why it was not important in many authors (in certain cases, until very recent years), which forms had the logics in which LC was not important, the many and important relations among LC and extra-logical (metaphysical, epistemological, pragmatic, …) notions. It shows that we cannot simply take for granted that there is an intuitive concept of LC or even a natural concept of LC, since it has always been formulated and become understandable and important only in connection with non-logical notions and different scientific aims. The authors or the schools studied in this first section are: Aristotle, Descartes, Kant, Bolzano, Frege, the algebra of logic, the axiomatic study of mathematical theories, Brouwer, Gentzen, Tarski, Etchemendy and Prawitz. In the second part of my thesis, I explore the seminal Tarski’s idea to consider LC as a closure operator and further developed by Łos, Suszko, Wójcicki, Czelakowski and the Barcelona Group. I define LC as a structural closure relation on the algebra of formulas, without taking into account its syntactical or semantic definition. I examine the philosophical ideas lying behind this conception and I explore different definitions (e.g., non-monotone consequence). Then I explore how we can define LC by a calculus (Hilbert-calculus and Natural Deduction Calculus) or by a semantic system (I consider predicative language too) and I explain the philosophical implications of these different points of view. In the last chapter I explore how we can define LC by matrices and the philosophical implication of this method. I study Lindenbaum matrices, Lindenbaum bundles, Lindenbaum-Tarski algebras and I investigate the relation among some properties of logical systems and Lindenbaum matrices.