J.M. Smith. On a Nonconstructive Type Theory and Program Derivation. To appear in the proceedings of Conference on Logic an d its Applications, Bulgaria 1986 (Plenum Press).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....operator : The motivation comes from Russell s work on denoting (see [27, 26] 3.2.3 Excluded Middle The last principle we shall consider is the principle of excluded middle. PiA : Set)A : A) The extension of Martin Lof s set theory with this principle has been considered by J. Smith in [29]. It is direct to check that this principle is equivalent to ( PiA : Set) A) A: 3.3 An application We can now state the application of the inconsistency of polymorphic higher order logic. Lemma: The set (9f : o o o) Pix; y : B) T (f(x; y) j [T (x) T (y) IMP ) and, for each set A; the ....

Smith J. "On a Nonconstructive Type Theory and Program Derivation." The Proceedings of Conference on Logic and its Applications, Bulgaria, Plenum Press, 1986.

....(v 0 , v 0 , w) # u 0 (v 0 ) # 0 # (#y#N)T (v 0 , v 0 , y) 2 Remark. Note that this proof shows that if we require our theory to be constructive, then we cannot expect (#) to hold for all predicates P (x) However, if we to ETT add the axiom A # A # (A) for each type A , as suggested in [9], then (#) becomes provable for all predicates since the law of the excluded middle implies that all predicates are stable. 6 Conclusions A straightforward introduction of subset types in type theory is problematic because the subset type is di#cult to integrate with propositions as types. ....

J.M. Smith. On a Nonconstructive Type Theory and Program Derivation. To appear in the proceedings of Conference on Logic an d its Applications, Bulgaria 1986 (Plenum Press).

....type theory as a computational system, since, for instance, the proof that every object of a type can be computed to normal form cannot be formalized in first order arithmetic. The nonderivability of :Eq(N; 0; 1) for the version of type theory given in Martin Lof [4] was already shown in Smith [6] as a corollary to a somewhat less straightforward construction made with a different purpose. The proofs in this paper will work for any of the different formulations of Martin Lof s type theory. 2 The construction of the interpretation We define a truth valued function on the types of ....

J.M. Smith. On a Nonconstructive Type Theory and Program Derivation. To appear in the proceedings of Conference on Logic and its Applications, Bulgaria 1986 (Plenum Press).

....(v 0 ; v 0 ; w) fu 0 g(v 0 ) 0 : 9y2N)T (v 0 ; v 0 ; y) 2 Remark. Note that this proof shows that if we require our theory to be constructive, then we cannot expect ( to hold for all predicates P (x) However, if we to ETT add the axiom A 2 A ( A) for each type A , as suggested in [9], then ( becomes provable for all predicates since the law of the excluded middle implies that all predicates are stable. 6 Conclusions A straightforward introduction of subset types in type theory is problematic because the subset type is difficult to integrate with propositions as types. ....

J.M. Smith. On a Nonconstructive Type Theory and Program Derivation. To appear in the proceedings of Conference on Logic and its Applications, Bulgaria 1986 (Plenum Press).

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J.M. Smith. On a Nonconstructive Type Theory and Program Derivation. To appear in the proceedings of Conference on Logic an d its Applications, Bulgaria 1986 (Plenum Press).

No context found.

Jan M. Smith. On a Nonconstructive Type Theory and Program Derivation. Proceedings of Conference on Logic and its Applications, Bulgaria, Plenum Press, 1986.

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