By Ladislav Rieger

ISBN-10: 1483231232

ISBN-13: 9781483231235

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If Z denotes the expression x = yb then W(Z) = μ=(χ, y). We say that a basic sentential expression Z is true (or false) according as W(Z) = 1 or W(Z) = 0. Let us now formulate the recursive semantic rules corresponding to (b) of Sect. 1: (B) Suppose that sentential expressions X and 7 have already been given meanings (in dependence on the meanings of individual variables in Ω). Then: Each of the expressions (X) v (7), (X)&(Y), (X) -► (7) assumes the meaning o/the corresponding compound statement or condition, this meaning being determined by the meaning of the components X, 7 and by the familiar meaning of the logical connectives v , &, -».

3; then u, n, ' stand for the join, meet and complement in this algebra): W[(X) v (Y)] = W(X) u W(Y) = max (W(X), W(Y)) , W[(X) & (7)] = W(X) n W(Y) = W(X) . W{Y) , w(i (*)] = (w(x))' = i - w(x), W[(X) - (7)] = (W(X)y u W(y) = max (1 - W(X), W(Y)) , W[3 x(X)] = sup, W(X) , w\y x{x)~\ = MX w(x). ) To summarize: The expression (X) v (Y) is true if and only if at least one of the expressions X, Y is true; the expression (X) & (y) is true if and only if both the expressions X9 Y are true; the expression (X) -> (Y) is false if and only if the first expression is true and the second false*); the expression ~l (X) is true if and only if the expression X is false; the expression 3x (X) is true if and only if by a suitable choice of the meaning (value) of the variable x the expression X is true; the expression Vx (X) is true if and only if for every choice of the meaning of the variable x the expression X is true.

For the distinction between the so-called applied and pure predicate calculi see p. 66, Remark 2. 3. , order theory. , An, which are symbolized sentences. For instance, in the case of the theory of dense order without the smallest element, we would have n = 8 (cf. Example 1 in Sect. 2) with A^ A2: A3: A4: A5: A6: A7: A8: — (irreflexivity), Vx (~l (x < x)) Vx (Vy (Vz ((x < y) & (y < z) -+ (x < z)))) - (transitivity), Vx (Vy ( ( l (x = y)) -> ((x < y) v (y < x)))) - (dichotomy), "Ί (3y (Vx ((y < x) v (y = x)))) — (nonexistence of a smallest object), Vx (Vy (3z ((x < y) -> ((x < z)&(z < y))))) - (density), Vx (x = x) — (reflexivity of equality), Vx (Vy ((x = y) -> (j = x))) — (symmetry of equality), Vx (V>; (Vz ((x = y)&(y = z)) -► (x = z))) - (transitivity of equality).

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Algebraic Methods of Mathematical Logic by Ladislav Rieger

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