Springer Science & Business Media, 6 dic 2012 - 160 pagine
Except for this preface, this study is completely self-contained. It is intended to serve both as an introduction to Quantification Theory and as an exposition of new results and techniques in "analytic" or "cut-free" methods. We use the term "analytic" to apply to any proof procedure which obeys the subformula principle (we think of such a procedure as "analysing" the formula into its successive components). Gentzen cut-free systems are perhaps the best known example of ana lytic proof procedures. Natural deduction systems, though not usually analytic, can be made so (as we demonstrated in ). In this study, we emphasize the tableau point of view, since we are struck by its simplicity and mathematical elegance. Chapter I is completely introductory. We begin with preliminary material on trees (necessary for the tableau method), and then treat the basic syntactic and semantic fundamentals of propositional logic. We use the term "Boolean valuation" to mean any assignment of truth values to all formulas which satisfies the usual truth-table conditions for the logical connectives. Given an assignment of truth-values to all propositional variables, the truth-values of all other formulas under this assignment is usually defined by an inductive procedure. We indicate in Chapter I how this inductive definition can be made explicit-to this end we find useful the notion of a formation tree (which we discuss earlier).
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FirstOrder Logic Preliminaries 43
FirstOrder Analytic Tableaux
A Unifying Principle
Axiom Systems for Quantification Theory
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A-consistent analytic consistency property analytic tableaux atomic valuation axiom scheme axiom system Boolean descendent Boolean valuation called Chapter clashing pair closes with weight completeness proof completeness theorem concludes the proof configuration conjugate consider construction define dyadic tree E-complete equivalent Exercise false finite set finite subset First-Order Logic formation tree free occurrence Fundamental Theorem Gentzen hence Hintikka set Hintikka's lemma induction inference rules infinite interpolation formula Interpolation Lemma interpretation König's lemma least one element magic set maximally consistent mean modified block tableau modus ponens negation obvious occur open branch order logic order valuation prenex normal form prenex sentences propositional logic propositional variable provable in Q quantification theory quantificational rules regular set result sequent signed formulas successor Suppose systematic tableau tableau method tautology term true truth value truth-functional truth-functionally implied truth-functionally satisfiable unsatisfiable unsigned valid verify Vx)Px weak subformula weight k2