Logic: lecture notes

Basic notions What is logic? Arguments and their formulation Complex reasoning Truth and falsity Ambiguity, vagueness, context sensitivity Validity Some set-theoretical notions True premises Validity as necessary truth preservation Other logical properties and relations Some important facts about validity Validity is not everything Formality Validity and formal validity Validity and formal invalidity Formal language Formal system Object language and metalanguage Some further set-theoretical notions The symbols of propositional logic Sentence letters Sentential connectives Brackets Expressive completeness Truth-functionality and substitutivity Formalization in a propositional language The language L Formation rules Syntactic trees Some basic syntactic notions Interpretation Truth tables Logical consequence in L Definition of logical consequence Some important facts about logical consequence Logical consequence as a test for validity Other logical properties and relations Effective computability The system Sn Derivation Rules for negation Rules for the conditional Rules for conjunction Rules for disjunction Derivability in Sn Definition of derivability Some important facts about derivability Some tips Derived rules Other natural deduction systems The system Sa Axioms and inference rule Deduction theorem Some syntactic properties of Sa Deductive equivalence between S-p and Sa Systems and theories Consistency, soundness, completeness Consistency of Sa Correspondence between syntax and semantics Soundness of Sa Completeness of Sa Extension to S-n Quantification Quantified sentences A brief historical survey Existential import Multiple generality Definite descriptions The symbols of predicate logic non-logical expressions Logical constants and auxiliary symbols Other symbols Numerical expressions Multiple generality and scope ambiguity Existence The language Lp Syntax Basic semantic notions Satisfaction Truth Logical consequence Undecidability The system Sp Axioms and inference rule Derivability in Sp Validity and derivability Deduction theorem and other syntactic results Alphabetic variants Consistency, soundness, completeness Consistency of Sp Equivalent formulations of soundness and completeness Soundness of Sp Completeness of Sp Final remarks Undecidability and related results Undecidability of Sp Gödel numbering Effective enumerability of the theorems of Sp Effective enumerability of the valid formulas of Lp A further corollary Recursive axiomatization and recursive decidability First-order logic First-order language First-order logic with identity First-order theory The language of basic arithmetic Peano Arithmetic Theories and Models Cardinality Compactness theorem Löwenheim-Skolem theorem Isomorphism Categoricity Gödel's incompleteness theorems Overview The arithmetization of syntax The Gödel sentence First incompleteness theorem: semantic version First incompleteness theorem: syntactic version Second incompleteness theorem Rudiments of modal logic Modal operators A modal propositional language Systems of modal propositional logic A modal predicate language Systems of modal predicate logic Soundness and completeness