When propositions become complex, one must keep track of the grammatical structures that medieval logicians use. The notation “Linguish” is devised. Syntactical rules generate logical forms based on grammatical roles; the roles are supplied by verbs and participles; denoting phrases are allowed into a structure only by filling a specific role. An algorithm that converts logical forms into the surface sentences of Latin that logicians made use of. The logical forms encode only grammatical structure, since there is no logical form in addition to grammatical form in the medieval tradition. Rules of inference are formulated for Linguish logical forms. Using the medieval notions of signification and supposition a modern semantics is developed that agrees with medieval logicians’ judgments about truth conditions of propositions. A notion of validity is defined, and a completeness theorem is proved for the stated rules of inference.
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