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UNIVERSITÀ DEGLI STUDI DI SASSARI DIPARTIMENTO DI STORIA, SCIENZE DELL’UOMO, E DELLA FORMAZIONE Laboratorio di Filosofia Analitica SEMINARI DEL LABORATORIO DI FILOSOFIA ANALITICA Giovedì 26 maggio 2016 Prof. Raymond Turner (University of Essex, UK) THE COMPLETENESS OF FIRST ORDER LOGIC AND THE JUSTIFICATION OF DEDUCTION Two of the central results of classical first order logic relate its proof theory with its model theory. One result insists that whatever is provable is universally valid i.e. true in all models. This is the soundness result. The converse states that what is universally valid is provable: in other words, a sentence that is true in all models is derivable using the rules of the logic. This is the completeness theorem. In this lecture we provide an account of the logic, its model theory, its soundness, and its completeness and, as a corollary, prove the Lowenheim Skolem theorem. One the central questions concerns the philosophical significance, if any, of these results. In particular, is there any sense in which they justify the rules of the logic? Does the soundness theorem do so? On the face of it the proof of soundness uses the very rules under consideration for their justification. Thus it has an air of circularity about it: how can one justify rules by using the very reasoning codified by the rules? And yet the theorem sare in some way enlightening: they tell us something about the proposed semantic interpretation and the operational rules of the logic. But what? Dummett in a very influential article argues that the theorems do not justify the logical rules but they explain them. We shall spend some time exploring this perspective.