On a synonymy relation for extensional first order theories. III: A necessary and sufficient condition for synonymy (Q801898)

[For parts I and II see ibid. 69, 63-76 (1983; Zbl 0525.03019) and ibid. 70, 13-19 (1983; Zbl 0538.03025) respectively.] The work to which the paper belongs concerns synonymy in connection with an extensional (scientific) formal language, which expresses a typical axiomatic theory \({\mathcal T}.\) The definition of a synonymy relation - see Part I, {\S} 6 - is not of the usual inductive type. This may give rise to difficulties, e.g. in proving that some given expressions are not synonymous. Therefore in Part III we state two necessary and sufficient conditions for two wfes of \({\mathcal T}\) to be synonymous. To reach this aim, we introduce an auxiliary theory \(\dot {\mathcal T}\), which is substantially capable to speak of the (\({\mathcal I},V)\)-senses of the wfes of \({\mathcal T}\), i.e. the senses of these wfes relative to the interpretation \({\mathcal I}\) of \({\mathcal T}\) (which need not be a model of \({\mathcal T})\) and to an \({\mathcal I}\)-valuation V (which assigns \({\mathcal T}'s\) variables with value belonging to the domain of \({\mathcal I})\). Lastly, \(\sim p\) and \(\sim \sim \sim p\), with p atomic, are proved not to be synonymous. By the result \((8.2)_ 3\) in Part II, this shows that a rather simple condition proved in Part II to be sufficient for non-synonymy, is not necessary for this.