By Ladislav Rieger
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Additional resources for Algebraic Methods of Mathematical Logic
Let us note that the definition given above of the principal syntactic notions, "sentential expression" and "sentence", could be taken over to any (however complicated) symbolized idealized theory. Only convention (a) would change; it would be replaced by the introduction of those signs for the basic (primitive) relations or properties (instead of < and = ) which are dealt with in the theory being treated. For example, such a complicated and rich theory as is the axiomatic set theory in the sense of von Neumann-Bernays-Gödel, contains the equality = and only one other additional primitive binary relation, viz.
4. Logical syntax and logical semantics If the investigation of the symbolized mathematical language is restricted to logico-combinatorial properties of the language and the meaning, actual or potential*), of symbols and theorems is disregarded, then we are concerned with the so-called (logical) syntax of this language. e. g. problems of the validity or truth of a theorem), then the investigations belong to the (logical) semantics of the language in question. By declaring mathematical logic to be the study of the relation of con sequence, and by not taking the (symbolic) language of mathematics itself to be the primary bearer of this relation, we in fact rank logical semantics above logical syntax (similarly as one ranks theoretical physics above mathematical physics).
Under the convention preceding item 6: in Sect. \ νξ (X) => X. On the other hand, 3ξ (X) => X may be derived from X => X according to III:, and X => V£ (X) in the same manner according to III*:. e. X => 3ξ (X), 3ξ(Χ)=>Χ, χ=>νξ (x), v£ (x) => x. R e m a r k a n d c o n v e n t i o n . The relation of formal consequential equi valence between expressions X and Y, stating that both X => Y and Y => X are valid, is evidently (cf. 0: and I:) reflexive, symmetric and transitive; thus it is an equivalence.
Algebraic Methods of Mathematical Logic by Ladislav Rieger