By Ladislav Rieger
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Thus, it depends on the definition (interpretation) ofthe primitive predicate constants (for instance < and = ) ofthe theory. The construction must be in accordance with this meaning; in particular, a true sentence is to have only true consequences. • However, the recursive definition of the relation => itself should not be based on this property, but rather on purely combinatorial properties of sentential expressions, regarded merely as finite sequences of signs, devoid of any meaning. (a) Let X, Y, Z be arbitrary sentential expressions (of our theory).
The sentential expression 3x (X) acquires the sentential meaning of the condition or assertion that, for a suitable choice of the meaning (in Ω) of the free individual indeterminate x in X, the sentential meaning of the expression X is true. 2. Fundamental descriptively-semantic rules The sentential expression Vx (X) acquires the sentential meaning of the condition or assertion that, for every choice of the meaning (in Ω) of the free individual indeterminate x, the sentential meaning of the expression X is true.
Thus we have established the semantic rules corresponding to point (a) of Sect. 1. Let us formulate these explicitly, together with the rules concerning the truth of sentences (having in mind our example — symbolized order theory — for the sake of definiteness). (A) The meaning of a basic sentential expression is given, first, by the domain of objects Ω, over which the individual indeterminates (variables) may vary; and second, by the meaning of the basic predicate constants (as relations in that domain); we assume that these meanings are given and do not change.