Scott, D.S., Identity and existence in intuitionistic logic, in proccedings, Durham 1977, Applications of Sheaves, Lecture Notes in Mathematics, vol. 753, eds.: Fourman, Mulvey, Scott. Springer Verlag 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for explicit elementary types with join. Its logic is Beeson s [5] classical logic of partial terms for individuals and classical logic for types. The logic of partial terms takes into account the possibility of undefined terms, i.e. terms which represent nonterminating computations. Scott [34] has given a logic similar to the logic of partial terms, but he treats existence like an ordinary predicate. Troelstra and van Dalen [42] give a discussion about the di#erent approaches to partial terms. Among the main features of the logic of partial terms are its strictness axioms stating that ....

Dana S. Scott. Identity and existence in intuitionistic logic. In M. Fourman, C. Mulvey, and D. Scott, editors, Applications of Sheaves, volume 753 of Lecture Notes in Mathematics, pages 660--696. Springer, 1979.

....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 ....

....(# bool ) # bool ) The subtype containing just the injections is the sum type # #. Because a subtype may depend upon bound variables, we must consider introducing dependent types: general products and sums. These cause no semantic di#culties, but seem unnecessary (see also Dana Scott [33]) The term : #.R(z,x) y has no legal type. Its type could be z:# . x : #.R(z,x) ## if we added dependent types. However : #.R(z,x) P(y) 7 has type # bool because the body of the abstraction is a formula. Traditionally a term has a unique type, but each element of a subtype ....

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Dana Scott. Identity and existence in intuitionistic logic. In M. P. Fourman, editor, Applications of Sheaves, pages 660--696. Springer, 1979. Lecture Notes in Mathematics 753.

.... 6] on a sound treatment of skolemization in higher order logic without choice (see also [10, Section 7] One might also consider merging Nominal Logic s novel treatment of atoms and freshness with some conventional treatment of the logic of partial expressions (such as [2, Section VI.1] or [37]) We leave such considerations to the future and turn instead to a brief survey of existing work more directly related to the concerns of this paper. 9. RELATED WORK One can classify work on fully formal treatments of names and binding according to the mathematical construct used to model the ....

D. S. Scott. Identity and existence in intuitionistic logic. In M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, Proceedings, Durham 1977.

....such things as terms with locally fresh atoms, concretions of atom abstractions at atoms and maybe more besides, it seems that one should merge Nominal Logic s novel treatment of atoms and freshness with some conventional treatment of the logic of partial expressions (such as [1, Sect. VI.1] or [26]) 8 Related Work One can classify work on fully formal treatments of names and binding according to the mathematical construct used to model the notion of an abstraction over names: Abstractions as (name, term) pairs. Here one tries to work directly with parse trees quotiented by alpha ....

D. S. Scott. Identity and existence in intuitionistic logic. In M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, Proceedings, Durham 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 ....

D.S. Scott. Identity and existence in intuitionistic logic. In M.P. Fourman, C.J. Mulvey, and D.S. Scott, editors, Applications of Sheaves, volume 753 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

.... in an otherwise classical predicate logic (perhaps with identity) as chronicled in the work of some of the eld s pioneers [18, 20] The relevance of free logic to some branches of mathematics and related areas of mathematical computer science was explicitly argued beginning in the 1960 s [29, 30, 24, 23]. As noted elsewhere [15] free logic has been implicit from the beginning in a central area of computer science, program speci cation and veri cation [17] The term free logic has not been current in the mathematics and computer science literature, however, because some authors object to the ....

Dana S. Scott. Identity and existence in intuitionistic logic. In [9], pages 660-696, 1979.

....reflexive ) relation, and considering all possible such. One obtains a topos, and the validity of a formula in the internal logic of this topos is connected to the validity in the underlying model of many sorted predicate logic of a translation of into the logic of identity and existence ([84]) Hyland, Johnstone and Pitts discovered a useful generalization of the first step in this construction, calling it tripos for topos representing indexed preordered set 24 . The Theory of triposes is the subject matter of Andy Pitts thesis [73] but a major application of the idea is ....

D.S. Scott. Identity and existence in intuitionistic logic. In M. Fourman, C.J. Mulvey, and D.S. Scott, editors, Applications of Sheaves, pages 660--696, Berlin, 1979. Springer (LNM 753).

....be interpreted. Hyland was strongly motivated in his work by a then recent approach to the Boolean Valued models used in Set Theory, originally introduced in an obscure paper by Higgs [9] but subsequently exposed much more widely in substantial articles by Fourman and Scott [8] and Scott [20]. Suppose one has a complete Boolean algebra B. Then a B valued predicate on a set X can simply be taken to be a function P : X B. If we have a binary predicate on X and Y , this can be taken to be a function Q: X Y B. The propositional operators can be interpreted using the boolean ....

D.S. Scott. Identity and existence in intuitionistic logic. In M.P. Fourman, C.J. Mulvey, and D.S. Scott, editors, Applications of Sheaves, number 753 in Lecture Notes in Mathematics, pages 660-696. Springer-Verlag, 1979.

....j j The j j operation is called erasure. It erases all proof terms from the expressions. The operation is an auxiliary operation involved in the proof of the main theorem. It lifts a T proof to a D analogue. 1. 3 Related work There are many logics of partial terms, like Scott s E logic [17, 19] and Beeson s LPT [2] See [10] for an overview of the eld. However our approach is not a logic of partial terms because we don t allow unde ned terms. We think that it should be illegal to write 1 0 and not just meaningless. Our paper integrates the type theoretical way to model partial ....

D.S. Scott. Identity and existence in intuitionistic logic. In M.P. Fourman, C.J. Mulvey, and D.S. Scott, editors, Applications of Sheaves, volume 753 of Lecture Notes in Mathematics, pages 660-696, Berlin, 1979. Springer-Verlag.

....= a) c) V a2A 9y(y = a) 8x i (x i ) 8r 9x i (x i = c) c) 9x i (x i = c) 8x i (x i ) 8l Here, as usual, we require that c is free for x i in , and in 8r and 9l c and x i do not occur free in and . 1 Premises of the form 9y(y = a) are reminiscent of E logic (see [4]) 8 In addition to 9r; l and 8r; l, S fd contains two more axioms, x i ) 9x i (x i ) and 8x i (x i ) x i ) or two additional rules: x i ) 9x i (x i ) 9r 0 ; x i ) 8x i (x i ) 8r 0 Theorem 4.1 1. S fd is sound and complete for L fd . 2. For a class of ....

D. Scott. Identity and Existence in Intuitionistic Logic. In M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, Springer Lecture Notes in Mathematics 753, 660-696 1979.

....E(t) Logics of partial terms and precursors of it have been used for the foundation of explicit and Appeared in: Journal of Functional Programming, 8(2) 97 129, 1998. 1 constructive mathematics [4, 1] Troelstra and van Dalen [28] compare the logic of partial terms with the logic of existence [25]. The Russian constructivist school of N. A. Shanin used similar logics [22] The second view of partial functions leads to D. Scott s Logic for Computable Functions (LCF) which includes in its language constants # [9] Di#erent kinds of LCF s have been mechanized for formal proofs of properties ....

D. S. Scott. Identity and existence in intuitionistic logic. In M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, pages 660--696. Springer-Verlag, Lecture Notes in Mathematics 753, 1979.

....[Farmer 1990] Domain theory goes even further, adding a more defined than relation between functions [Gunter and Scott 1990] In our experience [Paulson 1985] the benefits of these approaches do not justify their complexity. Abstract Incorporated s LAMBDA system moved from a definedness logic [Scott 1979] to conventional higher order logic for similar reasons. 2.7 Examples The data structures and related operations found in programming and specification languages are easily represented in set theory. We show how to represent three of these structures: finite lists, records, and objects. 2.7.1 ....

Scott, D. 1979. Identity and existence in intuitionistic logic. In M. P. Fourman (Ed.), Applications of Sheaves, pp. 660--696. Springer. Lecture Notes in Mathematics 753.

....occurrences of x in A. A partial congruence need not satisfy any of the monotonicity laws, but monotonicity (together with symmetry and transitivity) implies the replacement axioms. Partial congruences are useful for axiomatizing partial functions. They occur in Scott s logic of partial equality [Scott 1979] and also in type theory where t T t in a type T can be proven only if t is provably a member of T . For example, the natural numbers with a partial predecessor function can be specified as follows: nat(x) x x nat(0) nat(x) nat(s(x) 0 s(x) nat(x) p(s(x) x nat(p(x) ....

Scott, D. S. 1979. Identity and existence in intuitionistic logic. In Applications of Sheaves, Proc. Durham, Volume 753 of Lecture Notes in Mathematics, pp. 660--695.

....logic can be strengthened (perhaps not very elegantly) to cope with the other properties. The logic is a first order theory of higher order programs. The theory is 1 In a logic of partial terms [2] variables (free or bound) only range over defined terms, whereas in a logic of partial existence [18], free variables range over all terms but quantified variables range only over defined terms 6 specified for a type free language with a call by value mode of evaluation, but it can easily be adapted to a typed or a call by name framework. The theory is built on a logic of partial terms [2] ....

D. S. Scott. Identity and existence in intuitionistic logic. In M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, volume 753 of Lecture Notes in Mathematics, pages 660--696, Berlin, 1979. Springer-Verlag.

....for explicit elementary types with join. Its logic is Beeson s [4] classical logic of partial terms for individuals and classical logic for types. The logic of partial terms 5 takes into account the possibility of undefined terms, i.e. terms which represent non terminating computations. Scott [33] has given a logic similar to the logic of partial terms, but he treats existence like an ordinary predicate. Troelstra and van Dalen [40] give a discussion about the di#erent approaches to partial terms. Among the main features of the logic of partial terms are its strictness axioms stating that ....

Dana S. Scott. Identity and existence in intuitionistic logic. In M. Fourman, C. Mulvey, and D. Scott, editors, Applications of Sheaves, volume 753 of Lecture Notes in Mathematics, pages 660--696. Springer, 1979.

....the closest work to our own is that on the specification language COLD ( FJ92] KdL89] and [dL94] However, the free logic underlying COLD is infinitary, and research on it has focused on strict predicates and functions. The design of COLD has been influenced by the work of Scott ( Sco67] [Sco79]) 4 In this case, one might think of t i as being undefined. There is another sense of undefined in computer science that means not fully specified . Hoare ( Hoa69] p. 580) has 2 to an existent if t i does not. A predicate (function) is called strict if it is strict in all of its ....

....is given the generality interpretation. Corcoran argues that slightly augmented firstorder logic is of little mathematical interest even though, when it is taken as the underlying logic, Peano arithmetic is categorical. Slightly augmented first order logic is similar to the system described in [Sco79], except that the latter is a constructive theory of types, has infinitely many free variables in each type, and imposes the generality interpretation on free variables in every type. 19 Suppes takes definitions to be open sentences in which free variables are given the generality ....

[Article contains additional citation context not shown here]

Dana S. Scott. Identity and existence in intuitionistic logic. In

....reflexive ) relation, and consider all possible such. One obtains a topos, and the validity of a formula in the internal logic of this topos is connected to the validity in the underlying model of many sorted predicate logic of a translation of into the logic of identity and existence ([78]) Hyland, Johnstone and Pitts discovered a useful generalization of the first step in this construction, calling it tripos for topos representing indexed preordered set 21 . The Theory of triposes is the subject matter of Andy Pitts thesis [68] but a major application of the idea is ....

D.S. Scott. Identity and existence in intuitionistic logic. In M. Fourman, C.J. Mulvey, and D.S. Scott, editors, Applications of Sheaves, pages 660--696, Berlin, 1979. Springer (LNM 753).

....sheaves, topoi, and the logic in topoi. However, the exposition should serve more as a reminder and a statement about 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 ....

....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 yield values in Omega . The standard theory of Omega Gamma 117 ( Fourman Scott 77] is based on Omega having stronger properties than those of a quantale. Definition 4.1 A complete Heyting algebra (cHa) is a complete lattice in which the operation a has a right adjoint for every element a . We call this adjoint a , and the operation , implication. 2 This means that a ....

[Article contains additional citation context not shown here]

Scott, D.S., Identity and existence in intuitionistic logic, in proccedings, Durham 1977, Applications of Sheaves, Lecture Notes in Mathematics, vol. 753, eds.: Fourman, Mulvey, Scott. Springer Verlag 1977.

....M) Sub) is a partial hyperdoctrine. Partial hyperdoctrines 13 4. Beeson s Partial Function Logic Traditionally, it is assumed that terms (over some signature) always denote. However, one frequently uses functions, such as x 7 x Gamma1 on the real numbers, that are only partially defined. Scott (1979) recognized that there is a need for a logic which can deal with partiality (c.f. Van Dalen and Troelstra 1988) In order to do so, he introduced an existence predicate E, where Et has the intuitive interpretation t exists . In his logic, free variables are treated purely schematic so that any ....

Scott, D.S. (1979), Identity and existence in intuitionistic logic. In M.P. Fourman, C.J. Mulvey, and D.S. Scott, editors, Applications of Sheaves.

No context found.

Scott, D.S., Identity and existence in intuitionistic logic, in proccedings, Durham 1977, Applications of Sheaves, Lecture Notes in Mathematics, vol. 753, eds.: Fourman, Mulvey, Scott. Springer Verlag 1977.

....sheaves, topoi, and the logic in topoi. However, the exposition should serve more as a reminder and a statement about 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 ....

....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 standard theory of m sets ( Fourman Scott 77] is based on having stronger properties than those of a quantale. Definition 4.1 A complete Heyting algebra (cHa) is a complete lattice in which the oper ation a A has a right adjoint for every element a . We call this adjoint a , and the operation , implication. This means that ....

[Article contains additional citation context not shown here]

Scott, D.S., Identity and existence in intuitionistic logic, in proccedings, Durham 1977, Applications of Sheaves, Lecture Notes in Mathematics, vol. 753, eds.: Fourman, Mulvey, Scott. Springer Verlag 1977.

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Scott, D.S.: Identity and existence in intuitionistic logic. In: Application of Sheaves. Volume 753 of Lecture Notes in Mathematics. Springer Verlag (1979) 660--696

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Scott, Dana. (1979). Identity and existence in intuitionistic logic. Pages 660--696 of: Fourman, Michael, Mulvey, Chris, & Scott, Dana (eds), Applications of sheaves. SpringerVerlag LNM 753.

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D.S. Scott. Identity and Existence in Intuitionistic Logic. In M.P. Fourman,

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D. S. Scott. Identity and existence in intuitionistic logic. In Applications of Sheaves, volume 753 of Lecture Notes in Mathematics, pages 660--696. Springer, 1979.

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