Introduction to Logic
This 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.
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PRINCIPLES OF INFERENCE AND DEFINITION
THE SENTENTIAL CONNECTIVES
SENTENTIAL THEORY OF INFERENCE
SYMBOLIZING EVERYDAY LANGUAGE
GENERAL THEORY OF INFERENCE
FURTHER RULES OF INFERENCE
POSTSCRIPT ON USE AND MENTION
TRANSITION FROM FORMAL TO INFORMAL PROOFS
THEORY OF DEFINITION
ELEMENTARY INTUITIVE SET THEORY
SETTHEORETICAL FOUNDATIONS OF THE AXIOMATIC METHOD
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algebra ambiguous names apply argument arithmetic assertion atomic sentences axiomatic axioms binary operation binary relation Chapter conclusion conditional deﬁnitions conditional proof consider corresponding deﬁned deﬁniens deﬁnition derived rule domain of individuals elementary empty set example exercises existential quantiﬁer false ﬁeld ﬁnal ﬁnd ﬁrst ﬁrst-order ﬁve ﬁxed ﬂagged formula free variables given hypothesis implication indiﬁerence individual constants informal proof introduced intuitive isomorphic logically equivalent mathematics method negation notation obtain occur operation symbols ordered couples partial ordering particle mechanics positive integers predicate predicate logic primitive notions primitive symbols principle probability space problem proper deﬁnition properties prove quasi-ordering real numbers reﬂexive relation symbol replace restriction rules of inference satisﬁed sentential connectives sentential interpretation set of premises set theory set-theoretical simple speciﬁcation statement strict partial ordering subset substitution tautologically tautologically imply tion transitive true truth table universal quantiﬁers valid vector
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...