We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.

We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.

A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional (...) coverage of introductory material such as sets. Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses. Reduced mathematical rigour to fit the needs of undergraduate students. (shrink)

Dzhaparidze, G., A generalized notion of weak interpretability and the corresponding modal logic, Annals of Pure and Applied Logic 61 113-160. A tree Tr of theories T1,...,Tn is called tolerant, if there are consistent extensions T+1,...,T+n of T1,...,Tn, where each T+i interprets its successors in the tree Tr. We consider a propositional language with the following modal formation rule: if Tr is a tree of formulas, then Tr is a formula, and axiomatically define in this language the decidable logics TLR (...) and TLRω. It is proved that TLR yields exactly the schemata of PA-provable sentences, if Tr is understood as “Tr is tolerant”. In fact, TLR axiomatizes a considerable fragment of provability logic with quantifiers over ∑1-sentences, and many relations that have been studied in the literature can be expressed in terms of tolerance. We introduce and study two more relations between theories: cointerpretability and cotolerance which are, in a sense, dual to interpretability and tolerance. Cointerpretability is a characterization of ∑1-conservativity for essentially reflexive theories in terms of translations. (shrink)