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tarski introduction logic deductive sciences 1994

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Textbook in PDF format On the Use of Variables Constants and variables Expressions containing variables—sentential and designatory functions Construction of sentences in which variables occur—universal and existential sentences Universal and existential quantifiers; free and bound variables The importance of variables in mathematics Exercises On the Sentential Calculus Logical constants; the old logic and the new logic Sentential calculus; the negation of a sentence, the conjunction and the disjunction of sentences Implications or conditional sentences; implications in the material meaning The use of implications in mathematics Equivalence of sentences The formulation of definitions and its rules Laws of sentential calculus The symbolism of sentential calculus; compound sentential functions and truth tables An application of laws of sentential calculus in inference Rules of inference, complete proofs Exercises On the Theory of Identity Logical concepts outside sentential calculus; the concept of identity Fundamental laws of the theory of identity Identity of objects and identity of their designations; the use of quotation marks Equality in arithmetic and in geometry, and its relationship to logical identity Numerical quantifiers Exercises On the Theory of Classes Classes and their elements Classes and sentential functions with one free variable The universal class and the null class The fundamental relations among classes Operations on classes Equinumerous classes, the cardinal number of a class, finite and infinite classes; arithmetic as a part of logic Exercises On the Theory of Relations Relations, their domains and counter-domains; relations and sentential functions with two free variables The algebra of relations Several kinds of relations Relations which are reflexive, symmetric, and transitive Ordering relations; examples of other relations Many-one relations or functions One-one relations or bijective functions, and one-to-one correspondences Many-place relations; functions of several variables and operations The importance of logic for other sciences Exercises On the Deductive Method Fundamental constituents of a deductive theory—primitive and defined terms, axioms and theorems Models and interpretations of a deductive theory The law of deduction; formal character of deductive sciences Selection of axioms and primitive terms; their independence Formalization of definitions and proofs, formalized deductive theories Consistency and completeness of a deductive theory; the decision problem The widened conception of the methodology of deductive sciences Exercises Construction of a Mathematical Theory: Laws of Order for Numbers The primitive terms of the theory to be constructed; the axioms concerning the fundamental relations among numbers The laws of irreflexivity for the fundamental relations; indirect proofs Further theorems on the fundamental relations Other relations among numbers Exercises Construction of a Mathematical Theory: Laws of Addition and Subtraction The axioms concerning addition; general properties of operations, the concept of a group and the concept of an Abelian group Commutative and associative laws for larger numbers of summands The laws of monotonicity for addition and their converses Closed systems of sentences A few consequences of the laws of monotonicity The definition of subtraction; inverse operations Definitions whose definiendum contains the identity sign Theorems on subtraction Exercises Methodological Considerations on the Constructed Theory Elimination of superfluous axioms from the original axiom system Independence of the axioms of the reduced system Elimination of a superfluous primitive term and the subsequent simplification of the axiom system; the concept of an ordered Abelian group Further simplification of the axiom system; possible transformations of the system of primitive terms The problem of consistency of the constructed theory The problem of completeness of the constructed theory Exercises Extension of the Constructed Theory: Foundations of Arithmetic of Real Numbers The first axiom system for the arithmetic of real numbers Closer characterization of the first axiom system; its methodological advantages and didactic disadvantages The second axiom system for the arithmetic of real numbers Closer characterization of the second axiom system; the concept of a field and that of an ordered field Equipollence of the two axiom systems; methodological disadvantages and didactic advantages of the second system Exercises Index

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tarski introduction logic deductive sciences 1994