By Roy T. Cook

This necessary reference introduces undergraduate and post-graduate scholars to the most difficulties and positions of philosophical good judgment. parts contain the most important figures, positions, terminology, and debates inside of philosophical common sense in addition to concerns that pertain to similar, overlapping disciplines, equivalent to set idea and the philosophy of arithmetic. Entries are largely cross-referenced for identity in the context of wider debates.

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We know m is a member of A, so this simplifies to: m is a member of B if and only if m is not a member of f(m). Since f(m) = B, this is then equivalent to m is a member of B if and only if m is not a member of B. This is a contradiction, so by reductio ad absurdum we know that there is no m such that f(m) = B. In other words, f “misses” B, and is not onto. Since f was completely arbitrary, this shows that there can be no onto function from A to the powerset of A. Thus, the cardinal number of the powerset of A is greater than the cardinal number of A.

If f is a morphism from X to Y, then we write f: X → Y. (2) For any X, Y, and Z in C, and any f: X → Y in Hom(X, Y) and g: Y → Z in Hom(Y, Z), the composition of f and g exists; that is, there is a binary mapping º from Hom(X, Y) × Hom (Y, Z) to Hom(X, Z). The composition of f and g is g º f, and is in Hom(X, Z). (3) For any object X in C, there is a morphism idX in Hom(X, X), the identity on X, such that idX º f = f = f º idX. (4) Composition of morphisms is associative: For any f: X → 1004 02 pages 001-322:Layout 1 44 16/2/09 15:11 Page 44 c at e g o ry t h e o ry Y, g: Y → Z, and h: Z → W, we have: h º (g º h) = (f º g) º h.

From a semantic perspective, classical logic accepts both bivalence and the law of non-contradiction – that is, classical logic assumes that each statement receives exactly one of the true and the false as its semantic value. Classical logic also requires truth functionality – that is, that each compound statement receives its 1004 02 pages 001-322:Layout 1 16/2/09 15:11 Page 48 classical mereology 48 semantic value as a function of the semantic values of its constituent statements. Typically, when the term “classical logic” is used without modification it is classical propositional or classical first-order logic that is being referred to.