We consider the very weak paracomplete and paraconsistent logics that are obtained by a straightforward weakening of Classical Logic, as well as some of their maximal extensions that are a fragment of Classical Logic. We prove that these logics may be faithfully embedded in Classical Logic, and that the interpolation theorem obtains for them.

We define bisimulations for temporal logic with Since and Until. This new notion is compared to existing notions of bisimulations, and then used to develop the basic model theory of temporal logic with Since and Until. Our results concern both invariance and definability. We conclude with a brief discussion of the wider applicability of our ideas.

This paper establishes a connection between structure sensitive categorial inference and classical modal logic. The embedding theorems for non-associative Lambek Calculus and the whole class of its weak Sahlqvist extensions demonstrate that various resource sensitive regimes can be modelled within the framework of unimodal temporal logic. On the semantic side, this requires decomposition of the ternary accessibility relation to provide its correlation with standard binary Kripke frames and models.

We describe a categorial system (PPTS) based on partial proof trees(PPTs) as the building blocks of the system. The PPTs are obtained byunfolding the arguments of the type that would be associated with a lexicalitem in a simple categorial grammar. The PPTs are the basic types in thesystem and a derivation proceeds by combining PPTs together. We describe theconstruction of the finite set of basic PPTs and the operations forcombining them. PPTS can be viewed as a categorial system incorporating someof (...) the key insights of lexicalized tree adjoining grammar, namely the notionof an extended domain of locality and the consequent factoring of recursionfrom the domain of dependencies. PPTS therefore inherits the linguistic andcomputational properties of that system, and so can be viewed as a ’middleground‘ between a categorial grammar and a phrase structure grammar. We alsodiscuss the relationship between PPTS, natural deduction, and linear logicproof-nets, and argue that natural deduction rather than a proof-net systemis more appropriate for the construction of the PPTs. We also discuss howthe use of PPTs allows us to ’localize‘ the management of resources, therebyfreeing us from this management as the PPTs are combined. (shrink)