Logic: a Brief CourseThis short book, geared towards undergraduate students of computer science and mathematics, is specifically designed for a first course in mathematical logic. A proof of Gödel's completeness theorem and its main consequences is given using Robinson's completeness theorem and Gödel's compactness theorem for propositional logic. The reader will familiarize himself with many basic ideas and artifacts of mathematical logic: a non-ambiguous syntax, logical equivalence and consequence relation, the Davis-Putnam procedure, Tarski semantics, Herbrand models, the axioms of identity, Skolem normal forms, nonstandard models and, interestingly enough, proofs and refutations viewed as graphic objects. The mathematical prerequisites are minimal: the book is accessible to anybody having some familiarity with proofs by induction. Many exercises on the relationship between natural language and formal proofs make the book also interesting to a wide range of students of philosophy and linguistics. |
Contents
| 3 | |
| 7 | |
3 The Resolution Method
| 12 |
4 Robinsons Completeness Theorem
| 19 |
5 Fast Classes for DPP
| 27 |
6 Godels Compactness Theorem
| 31 |
Syntax
| 35 |
Semantics
| 40 |
Part II Predicate Logic
| 55 |
11 The Quantifiers There Exists and For All
| 56 |
12 Syntax of Predicate Logic
| 63 |
13 The Meaning of Clauses
| 70 |
14 Godels Completeness Theorem for the Logic of Clauses
| 79 |
15 Equality Axioms
| 89 |
16 The Predicate Logic L
| 95 |
17 Final Remarks
| 117 |
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Common terms and phrases
Apply DPP atomic formula binary connective Brief Course Caius CNF formula Completeness Theorem conclusion congruence axiom conjunction Definition denoted disjunction DNF equivalent elements empty clause empty set equality axioms equivalent to F example Exercise existential quantifier exists find a model finite set finite subset finitely satisfiable Formalise formula F formula in DNF function symbol Gödel graph Herbrand universe Horn clauses idempotent induction hypothesis instantiation ISBN Krom clauses Lemma literal logical calculus logical consequence Martian Matematica mathematical means model with equality mortal Mundici n-tuple natural numbers negation number of steps obtain the empty occur open parentheses parsing certificate pivot predicate logic premises prenex normal form problem Proof propositional logic Prove Quarteroni refutational method satisfies F set of clauses Skolemisation Solution Springer-Verlag Italia 2012 step of DPP tautology unsatisfiable variables vertex vertices votes for Octavio won the polls