Algebraic Methods of Mathematical Logic by Ladislav Rieger

By Ladislav Rieger

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4* 51 3. Construct/on of the relation of consequence freely in the sentential expression 7. ) We have thus presented the recursive rules for the extension of the syntactic relation of formal consequence, including a (rough and general) semantic com­ mentary. g. in our sym­ bolized order theory) obtains if and only if there exists a finite sequence {Xi9 7^"= l of pairs of sentential expressions (the so-called formal derivation), such that: 1) Xn = X, Yn= 7. , n, the relation Xf => 7f is valid either as basic, according to 0: —6:, or as derived according to rules I: —III:*) from (one or two) preceding pairs of the sequence.

On the other hand, it should be mentioned here that there exists a number of equivalent procedures for constructing the same syntactic relation of consequence. Here we have chosen one which, as shall be seen, is parti­ cularly suitable for an algebraic treatment. The basic points in this construction are: (a) A determination of the forms of the simplest evident instances of the syntactic relation of consequence. The recursive step (ß), together with (γ), then determines how to construct further instances of this relation from given instances.

Fundamental descriptively-syntactic rules basic symbols. We shall clarify this construction by including the corresponding semantic explanations. But to be able to speak even about the basic symbols of a given theory, we need to have signs for these symbols. It is of course possible to introduce (exhibit) the symbol itself instead of having a special sign (name) for the individual symbol dealt with*) — and as a rule we shall follow this practice. g. , one of the objects or a special order relation or a logical disjunction of some theorems).

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