A generalization of the adequacy theorem for the quasi-senses.
Cinzia Bonotto · Notre Dame Journal of Formal Logic · 1990
In the present paper, based on Bressan's sense language SL p a , a version of the adequacy theorem for quasi-senses is proved that is applicable in every case, even when SL v a collapses into an extensional language.Thus this version affords a new result also for Bressan's modal language ML", which is substantially identical to SL\.Furthermore, some conditions of the adequacy theorem are weakened: the basic well-formed expressions (wfes) can contain primitive constants.Then we consider a theory T based on SLa, a definition system Z>, and strong (weak) extensions of Γin connection with a semantics for which the senses of the wfes are (are not) preserved by the principles of λ-conversion.The designation rules for quasi-senses are given in a complete form, also for strong theories.In fact, by means of the notion of a /"-correspondent of a wfe, every defined constant has a quasisense.Synonymy relations are extended to strong and weak extensions of T. Finally, the previous version of the adequacy theorem is further generalized by making the wfes contain primitive and defined constants, and making the valuations be noninjective on their free variables.By means of this result it is possible to construct quasi-senses for any choice of a synonymy notion.