Robert A. G. Seely. Hyperdoctrines, natural deduction and the Beck condition. Zeitschr. f. math. Logik und Grundlagen d. Math., 29:505-- 542, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....done. Case (mono ) implies (mono) The rule (mono ) yields Gamma j 2(Val(M) x:x = M) Now apply (2e) to this along with the hypothesis of (mono) 2 75 76 Categorical Semantics of the FIX Logic 5. 1 FIX Hyperdoctrines For background on hyperdoctrines and indexed categories see [JP78] [See83] and [Pit89] Definition 5.1.1 A FIX hyperdoctrine is specified by a FIX category C (referred to as the base category) together with a C indexed poset, C: C Poset ; where if f : A B is a morphism in the base category C we denote the corresponding pullback function by f : C(B) C(A) with ....

R.A.G. Seely. Hyperdoctrines, natural deduction and the beck condition. Zeitschr. f. math. Logik und Grundlagen d. Math, 29:505--542, 1983.

....from the latter. Section 3 will discuss how to model other features by additional structure over an indexed category. In Categorical Logic there is a similar use of indexed categories to capture that types and terms of rst order logic are given independently from formulas and proofs (see [See83]) while to enforce the principle of formulas as types one must be able to map bers down to the base (see [See84] The general de nition of indexed category is fairly complicated, since it involves the notion of canonical isomorphism. However, for representing languages it is more appropriate to ....

....3 Intermezzo At this point we review the category theoretic structures used for interpreting some typed calculi and discuss the additional structures needed to model various features of programming languages. Hyperdoctrines model the proof theory of intuitionistic rst order logic (see [See83]) They are indexed categories C: B op Cat, where morphisms in the base correspond to terms and morphisms in the bers correspond to derivations. Moreover, the base B has nite products, the bers C[X] are bicartesian closed, the functors C[f ] preserve such structure and for every rst ....

R.A.G. Seely. Hyperdoctrines, natural deduction and the Beck condition. Zeitschr. f. math. Logik und Grundlagen d. Math., 29, 1983.

....is in fact a hyperdoctrine. The various axioms that we found necessary in our analysis of constraint programming languages [20] all follow from the adjunction between existential quantification and substitution. There is nothing original in this section; we follow the ideas in Seely s discussion [23] of the connection between natural deduction and hyperdoctrines. Hyperdoctrines have, until recently, not received a great deal of attention, the main arena for categorical logic being elementary toposes. There has, however, been a surge of interest starting with the recent categorical description ....

....For each f : A Gamma B in B and OE 2 P(A) and 2 P(B) we have 9 f (f OE) 9 f OE. In the case where we have a constraint system syntactically presented as a concrete theory in first order conjunctive logic with existential quantification the hyperdoctrine conditions are easy to check [23]. In the next several paragraphs we describe the passage from a syntactically presented first order theory to a (hyperdoctrinal) constraint system. At the least, the base category is generated by a set of basic data values V : the objects of B are all finite products of V , including the empty ....

[Article contains additional citation context not shown here]

R. A. G. Seely. Hyperdoctrines, natural deduction and the beck conditions. Zeitschr. f. math. Logik und Grundlagen d. Math., 29:505--542, 1983.

....is in fact a hyperdoctrine. The various axioms that we found necessary in our analysis of constraint programming languages [20] all follow from the adjunction between existential quantification and substitution. There is nothing original in this section; we follow the ideas in Seely s discussion [23] of the connection between natural deduction and hyperdoctrines. Hyperdoctrines have, until recently, not received a great deal of attention; the main arena for categorical logic being elementary toposes. There has, however, been a surge of interest starting with the recent categorical description ....

....pseudofunctors but in the posetal situation that we consider it does not make any difference. In the case where we have a constraint system syntactically presented as a concrete theory in first order conjunctive logic with existential quantification the hyperdoctrine conditions are easy to check [23]. In the next several paragraphs we describe the passage from a syntacticall presented first order theory to a (hyperdoctrinal) constraint system. At the least, the base category is generated by a set of basic data values V : the objects of B are all finite products of V , including the empty ....

[Article contains additional citation context not shown here]

R. A. G. Seely. Hyperdoctrines, natural deduction and the beck conditions. Zeitschr. f. math. Logik und Grundlagen d. Math., 29:505--542, 1983.

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Robert A. G. Seely. Hyperdoctrines, natural deduction and the Beck condition. Zeitschr. f. math. Logik und Grundlagen d. Math., 29:505-- 542, 1983.

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R.A.G. Seely, Hyperdoctrines, natural deduction and the Beck condition, Zeitschr. f. math. Logik und Grundlagen d. Math., 29(1983) 505--542

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R.A.G. Seely, Hyperdoctrines, natural deduction and the Beck condition, Zeitschr. f. math. Logik und Grundlagen d. Math., 29(1983) 505--542

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R. A. G. Seely, Hyperdoctrines, Natural Deduction and the Beck Condition, Zeitschr. f. math. Logik und Grundlagen d. Math. 29 (1983) 505--542.

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