First-Order Logic

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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 [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).
 

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Indice

Preliminaries
3
Analytic Tableaux
15
Compactness
30
FirstOrder Logic Preliminaries 43
42
FirstOrder Analytic Tableaux
52
A Unifying Principle
65
Axiom Systems for Quantification Theory
79
Magic Sets
86
Gentzen Systems 101
99
Elimination Theorems
110
Prenex Tableaux
117
Craigs Interpolation Lemma and Beths Definability Theorem
127
Symmetric Completeness Theorems
133
Systems of Linear Reasoning
141
References
156
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