Fourman, M., The logic of topoi, in Handbook of Mathematical Logic (ed. Barwise), pp. 1053-1090, North-Holland, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an ill typed formula has some truth value. If this seems unsatisfactory, observe that a traditional theory of Peano arithmetic specifies no value for division by zero, yet the term a 0 denotes some number in each model, for each a. Of course, there are alternative semantics. Fourman and Scott [11, 33] can reason about whether a b exists, but their existence predicate involves some complexity. Their logic has a topos semantics, which is a categorical generalization of set theory. Martin Lof s Type Theory [22] has a constructive, operational semantics. An ambitious type theory can even be based ....

....can always be satisfied: #x : #.P(x) #x : #. #y : #.P(y) 7.4 Alternative formulations The above quantifier and description rules are adopted for the present formulation of simple type theory. Here are two other ways both based on topos theory of admitting empty types. Fourman [11] and Dana Scott [33] formalize the notion of existence: a term can have a valid type and yet be undefined. A type is empty if it has no defined elements. The # elimination rule can only be applied to a defined term. The description #x : #.P(x) exists only if P (x) is satisfied by a unique value, ....

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Michael P. Fourman. The logic of topoi. In J. Barwise, editor, Handbook of Mathematical Logic, pages 1053--1090. North-Holland, 1977.

....v) ffi g) u = f ffi F X:F;A (v; u) ffi ud) oe v = coit(g; f) u = it(g; f) This logic is essentially an extension of that of Plotkin [18] with new axioms for recursion. Three minor differences are worth noting. First, Plotkin works with a logic of existence [7] or a logic of partial elements [20, 9]) where free variables are thought of as ranging over a domain of possibly non existing elements. Second, he axiomatises approximation (v, rather than equality. Finally, as here we are interested only in the minimal framework in which our results hold, we have adopted a weaker axiomatisation ....

M.P. Fourman. The logic of topoi. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic. North Holland, 1977.

....notation than a first introduction to the concepts just listed. For that a much longer text is needed, and we refer the reader to the following list of core references. Fourman Scott 77] Mac Lane Moerdijk 92] Johnstone 77] Troelstra van Dalen 88] Wyler 91] Fourman 74] and also [Fourman 77] Rosolini 80] Ambler 92] and [Nawaz 85] The intention behind the notion of an m set is to model sets in a constructive universe with truth values in . Thus operations like equality (between members of sets) and set membership that usually yield values in 2 should now yield values in . The ....

....prefers is naturally a matter of taste, although at the first glance, the diagram way of saying things seem rather complicated. However, diagrams have proven their value in terms of facilitating proofs on many occasions. For a good example of lots of nice proofs using categorical diagrams, see [Fourman 77] We define products and function spaces for apos as expected. Definition 4.20 Given apos A and B the sheaf part of the product A x B is the product (in h(R) of the sheaf parts of the components, and the relation part A x B is defined as A x B( a,b) a , b ) A(a,a )A B(b,b ) Definition ....

Fourman, M., The logic of topoi, in Handbook of Mathematical Logic (ed. Barwise), pp. 1053-1090, North-Holland, 1977.

....notation than a first introduction to the concepts just listed. For that a much longer text is needed, and we refer the reader to the following list of core references. Fourman Scott 77] Mac Lane Moerdijk 92] Johnstone 77] Troelstra van Dalen 88] Wyler 91] Fourman 74] and also [Fourman 77] Rosolini 80] Ambler 92] and [Nawaz 85] The intention behind the notion of an Omega Gamma 28 is to model sets in a constructive universe with truth values in Omega Gamma Thus operations like equality (between members of sets) and set membership that usually yield values in 2 should now ....

....prefers is naturally a matter of taste, although at the first glance, the diagram way of saying things seem rather complicated. However, diagrams have proven their value in terms of facilitating proofs on many occasions. For a good example of lots of nice proofs using categorical diagrams, see [Fourman 77] We define products and function spaces for apos as expected. Definition 4.20 Given apos A and B the sheaf part of the product A Theta B is the product (in Sh( Omega Gamma5 of the sheaf parts of the components, and the relation part A Theta B is defined as A Theta B( a; b) a 0 ; b 0 ....

Fourman, M., The logic of topoi, in Handbook of Mathematical Logic (ed. Barwise), pp. 1053-1090, North-Holland, 1977.

....to provide a completeness theorem. Since categorical logic is essentially intuitionistic, the equivalence between typed lambda theories (de ned using the traditional axiom system) and arbitrary cartesian closed categories could be considered an intuitionistic completeness theorem (see, e.g. [Fou77, Lam80, LS86]) However, we prefer the completeness theorem using only Kripke models for several reasons. For one, Kripke models are relatively easy to picture, and they seem to support a set like intuition about the lambda terms better than arbitrary cartesian closed categories. In addition, predicate logic ....

....a de nition from rst principles, our model de nition and many of our results may be developed using a paradigm that is well known to researchers in categorical logic. We are grateful to Edmund Robinson and Pino Rosolini for some helpful discussion of this point of view, and refer the reader to [Fou77, Sco80, LS86] for related discussion. In short, the usual de nition of semantics of typed lambda calculus, as in [Bar84, Fri75, Hen50] may be formalized in the language of set theory: a model is a collection of sets satisfying several properties easily described by logical formulas. While we 4 usually ....

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M.P. Fourman. The logic of topoi. In Handbook of Mathematical Logic, pages 1053-1090, North-Holland, 1977.

....only in functional languages. On the other hand, let x(e in e 0 ) cannot be treated as syntactic sugar for [e=x]e 0 (involving only the more primitive substitution) without collapsing computations to values. The existence predicate e # is inspired by the logic of partial terms elements (see [Fou77, Sco79, Mog88]) however, there are important di erences, e.g. strict x: pl p(e) # 2 x: pl e # 1 p: 1 2 is admissible for partial computations, but not in general. For certain notions of computation there may be other predicates on computations worth considering, or the existence predicate ....

M.P. Fourman. The logic of topoi. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic. North Holland, 1977.

....further strengthening of the type formation and derivation rules in the internal logic. Much effort was expended in the Seventies in the systematic exploration of the interaction between logical properties and categorical structure, culminating in the elaboration of the internal logic of toposes [28,49,10,53,5]. The most significant development in this regard must be Lawvere and Tierney s observation that the representability of the notion of subobject (via the so called subobject classifier) is what makes set theory possible. Significant for later authors in the development of 1 I am grateful to ....

....a function will use the proof of Sx in an essential way. For the purposes of extending the work presented here, it seems a natural definition, with a highly constructive flavour. This seems to be a stronger constructive notion of partial map that those defined in logics with an existence predicate [28,99], which are rather more flexible in how one obtains proofs that a given term denotes. Pragmatically, and theoretically, one would hope to do rather better in reconciling partiality and constructive type theory. 1 Strong normalisation for the calculus extended with inductive types, and fi ffi ....

M.P.Fourman, The logic of topoi, in: The Handbook of Mathematical Logic, ed. J.Barwise, North-Holland, 1977.

....v) ffi g) u = f ffi F X:F;A (v; u) ffi ud) oe v = coit(g; f) u = it(g; f) This logic is essentially an extension of that of Plotkin [18] with new axioms for recursion. Three minor differences are worth noting. First, Plotkin works with a logic of existence [7] or a logic of partial elements [20, 9]) where free variables are thought of as ranging over a domain of possibly non existing elements. Second, he axiomatises approximation (v, rather than equality. Finally, as here we are interested only in the minimal framework in which our results hold, we have adopted a weaker ....

M.P. Fourman. The logic of topoi. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic. North Holland, 1977.

....that something exists. Ordinary intuitionistic higher order logic can be interpreted in it. We describe models for it, and we prove soundness of its inference rules. There is a straightforward constructive form of higher order logic whose alphabet consists of ) the existence predicate E of [2] and [6] and predicate variables. The introduction and elimination rules of conventional intuitionistic logic can be interpreted in it if one adds a condition to the 8E rule, that the term substituted for the bound variable must exist. This seems a very natural assumption. Several similar forms ....

....(see [3] 4] and at the BMC at Canterbury in 1996, but the draft I presented then was not sound. This paper describes the same language but with modified inference rules, and it includes an account of models and a soundness theorem. 1 A brief outline of syntax: The language of ) and E As in [2], there is a language called LE involving a (meta)set called Sort, with a (meta)function called the power type map from S n0 Sort n to Sort written as (A 1 ; An ) Gamma [A 1 ; An ] The sort [ is thought of as the sort of truth values. When n 1, A 1 ; An ] is the ....

M. P. Fourman. The logic of topoi. In J. Barwise, editor, Handbook of Mathematical Logic, pages 1053--1090. North-Holland / Elsevier, 1977.

....between cartesian closed categories and the typed calculus) McLarty [1992] and Mac Lane and Moerdijk [1992] all describe the internal language of a topos. Other papers worth looking at in this area are [Boileau and Joyal, 1981] Bunge, 1984] Osius, 1975b] Osius, 1975a] Poign e, 1986a] [Fourman, 1977] and [Vickers, 1993] Bell [1988] develops topos theory completely in terms of its language. Lawvere [1975] discusses some of the early history of the subject. In a somewhat subtle sense, sketches are equivalent in expressive power to first order logic [Guitart and Lair, 1982b] Makkai and Par e, ....

Michael Fourman. The logic of topoi. In Handbook of Mathematical Logic, J. Barwise et al., editors. North-Holland, 1977. (9)

....1 Previous research track record A. Personnel Michael Fourman is a Professor of Computer Science at the University of Edinburgh, where he is currently head of the Department of Computer Science and of the Informatics Planning Unit. He has contributed to the development of categorical logic [Fou77], the application of category theory to semantics [PF92, FT95] the promulgation and commercial exploitation of Standard ML, and the application of computer assisted formal reasoning to system design. As a founding Director of Abstract Hardware Limited, he was responsible for AHL s adoption and ....

M.P. Fourman. The logic of topoi. In J. Barwise, editor, Handbook of Mathematical Logic. North-Holland, 1977.

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Fourman, M., The logic of topoi, in Handbook of Mathematical Logic (ed. Barwise), pp. 1053-1090, North-Holland, 1977.

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Fourman M.P.: The Logic of Topoi, in: Barwise J. (ed.), Handbook of Mathematical Logic, r. 4, North-Holland, (1977), p. 1053-1090. 49

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Fourman, M.P., `The Logic of Topoi', in Barwise, J. (ed.), Handbook of Mathematical Logic, North-Holland, 1977.

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