First-Order LogicSpringer 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 [3]). 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). |
Sommario
3 | |
Analytic Tableaux | 15 |
Compactness | 30 |
FirstOrder Logic Preliminaries | 42 |
FirstOrder Analytic Tableaux | 52 |
The SkolemLöwenheim and Compactness Theorems for FirstOrder Logic | 63 |
The Fundamental Theorem of Quantification Theory | 70 |
Axiom Systems for Quantification Theory | 79 |
Gentzen Systems for Propositional Logic | 99 |
Block Tableaux and Gentzen Systems for FirstOrder Logic | 109 |
Prenex Tableaux | 117 |
Craigs Interpolation Lemma and Beths Definability Theorem | 127 |
Symmetric Completeness Theorems | 133 |
Systems of Linear Reasoning | 141 |
156 | |
Magic Sets | 86 |
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4-consistent a₁ a₂ analytic consistency property analytic tableaux atomically closed axiom scheme axiom system B₁ B₂ block tableau Boolean descendent Boolean valuation C₁ called Chapter clashing pair closed tableau closes with weight completeness theorem concludes the proof configuration conjugate consider construction define dyadic tree E-complete equivalent finite set finite subset First-Order Logic free occurrence Fundamental Theorem Gentzen hence Hintikka set Hintikka's lemma induction inference rules infinite interpolation formula Interpolation Lemma interpretation k₁ König's lemma magic set maximally consistent mean modus ponens negation obvious occur open branch order valuation P₁ prenex normal form propositional logic propositional variable provable prove Q₁ quantification theory S₁ S₂ satisfiable sequent signed formulas ẞ₁ successor Suppose systematic tableau tableau method tautology term true truth set truth value truth-functional truth-functionally implied truth-functionally unsatisfiable unsigned valid weak subformula weight k₂ X₁ Y₁