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A Semantics for Static Type Inference
 Information and Computation
, 1993
"... Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model cl ..."
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Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model classes. We show completeness for Curry's system itself, relative to an extended notion of model that validates reduction but not conversion.
Semantically Constrained Condensed Detachment is Incomplete
, 1995
"... In reporting on the theorem prover SCOTT (Slaney, SCOTT: A Semantically Guided Theorem Prover, Proc. IJCAI, 1993) we suggested semantic constraint as as an appropriate mechanism for guiding proof searches in propositional systems where the rule of inference is condensed detachmenta generalisation ..."
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In reporting on the theorem prover SCOTT (Slaney, SCOTT: A Semantically Guided Theorem Prover, Proc. IJCAI, 1993) we suggested semantic constraint as as an appropriate mechanism for guiding proof searches in propositional systems where the rule of inference is condensed detachmenta generalisation of Modus Ponens. Such constrained condensed detachment is closely analogous to semantic resolution. This paper exhibits an example which shows that semantically constrained condensed detachment is incomplete. That is, there are formulae deducible by means of condensed detachment which are not deducible when the semantic constraint is imposed. This answers an open question from our 1993 paper. Semantically Constrained Condensed Detachment is Incomplete John Slaney and Timothy J. Surendonk August 8, 1995 Abstract In reporting on the theorem prover SCOTT (Slaney, SCOTT: A Semantically Guided Theorem Prover, Proc. IJCAI, 1993) we suggested semantic constraint as as an appropriate mechanism f...
www.elsevier.com/locate/apal Ternary relations and relevant semantics
"... Modus ponens provides the central theme. There are laws, of the form A → C. A logic (or other theory) L collects such laws. Any datum A (or theory T incorporating such data) provides input to the laws of L. The central ternary relation R relates theories L; T and U, where U consists of all of the ou ..."
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Modus ponens provides the central theme. There are laws, of the form A → C. A logic (or other theory) L collects such laws. Any datum A (or theory T incorporating such data) provides input to the laws of L. The central ternary relation R relates theories L; T and U, where U consists of all of the outputs C got by applying modus ponens to major premises from L and minor premises from T. Underlying this relation is a modus ponens product (or fusion) operation ◦ on theories (or other collections of formulas) L and T, whence RLTU i L◦T ⊆ U. These ideas have been expressed, especially with Routley, as (Kripke style) worlds semantics for relevant and other substructural logics. Worlds are best demythologized as theories, subject to truthfunctional and other constraints. The chief constraint is that theories are taken as closed under logical entailment, which clearly begs the question if we are using the semantics to determine which theory L is Logic itself. Instead we draw the modal logicians ’ conclusion—there are many substructural logics, each with its appropriate ternary relational postulates. Each logic L gives rise to a Calculus of Ltheories, on which particular candidate logical