Covalidity and cosatisfiability

Sažetak

Two formue of an elementary formal language are covalid it whenever one of them is true under all valuations compatible (i. e. agreeable on the common basic symbols) with a given valuation of the other formula, this other formula is true under that given valuation. This definition corrects the definition in Belt und Machover p. 98. Our definition of cosatisfiability agrees with that in Bell and Machover pp. 97-98: Two formulae are cosatisfiable if whenever one of them is true under some valution, there exists a compatible valution under which the other formula is true too. It is shown that neither does covalidity imply cosatisfiability nor does cosatisfiability imply covalidity. These semantic relations are studied in comparison with the semantic properties of validity and satisfiability. Any two formule which are both logically valid or both are contradictions, are also covaltd and cosatisfiable; but there are covalid or cosalisfiable formulae which are neither valid nor contradictory. Special cases are considered in which covalidity and cosatisf lability reduce to:
(i) bipartitions of the set of all formulae into valid and non-valid formulae and into contradictory and non-conctradictory formulae, respectively;
(ii) logical equivalence relation;
(iii) vatidivalence and satisfivalence relations (our terms) or (in a somewhat narrower sense) V-form (validity form) and S-form (satisfiability form) respectively (in terms of Beli and Machover).
We end the paper by briefly noting that what wc have setforth constitutes a framework for the study of Herbrand and Skolem forms, and the problem of rearranging quantifiers in a given formula.