Introduction to LogicThis well-organized book was designed to introduce students to a way of thinking that encourages precision and accuracy. As the text for a course in modern logic, it familiarizes readers with a complete theory of logical inference and its specific applications to mathematics and the empirical sciences. Part I deals with formal principles of inference and definition, including a detailed attempt to relate the formal theory of inference to the standard informal proofs common throughout mathematics. An in-depth exploration of elementary intuitive set theory constitutes Part II, with separate chapters on sets, relations, and functions. The final section deals with the set-theoretical foundations of the axiomatic method and contains, in both the discussion and exercises, numerous examples of axiomatically formulated theories. Topics range from the theory of groups and the algebra of the real numbers to elementary probability theory, classical particle mechanics, and the theory of measurement of sensation intensities. Ideally suited for undergraduate courses, this text requires no background in mathematics or philosophy. |
Contents
PRINCIPLES OF INFERENCE AND DEFINITION | 1 |
THE SENTENTIAL CONNECTIVES | 3 |
SENTENTIAL THEORY OF INFERENCE | 20 |
SYMBOLIZING EVERYDAY LANGUAGE | 43 |
GENERAL THEORY OF INFERENCE | 58 |
FURTHER RULES OF INFERENCE | 101 |
POSTSCRIPT ON USE AND MENTION | 121 |
TRANSITION FROM FORMAL TO INFORMAL PROOFS | 128 |
THEORY OF DEFINITION | 151 |
ELEMENTARY INTUITIVE SET THEORY | 175 |
SETS | 177 |
RELATIONS | 208 |
FUNCTIONS | 229 |
SETTHEORETICAL FOUNDATIONS OF THE AXIOMATIC METHOD | 246 |
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Common terms and phrases
algebra ambiguous names apply argument arithmetic assertion atomic sentences axiomatic axioms binary operation binary relation Chapter conclusion conditional proof consider corresponding defined definiens derived rule discussed domain of individuals elementary empty set example EXERCISES existential quantifier fallacious false formal derivation formula free variables function f given H₁ hypothesis identity element implication individual constants informal proof instance introduced intuitive isomorphic logically equivalent mathematics method negation non-empty notation obtain occurrence operation symbols ordered couples particle mechanics positive integers primitive notions primitive symbols principle probability space problem properties prove quasi-ordering R₁ real numbers relation symbol replace restriction rules of inference satisfied sentential connectives sentential interpretation set of premises set theory set-theoretical statement strict partial ordering subset substitute tautologically tautologically equivalent tautologically imply theorem of logic tion transitive translated true truth table universal quantifiers valid vector
Popular passages
Page xii - ... which is adequate to deal with all the standard examples of deductive reasoning in mathematics and the empirical sciences. The concept of axioms and the derivation of theorems from axioms is at the heart of all modern mathematics. The purpose of this...