Garrel Pottinger. Definite descriptions and excluded middle in the theory of constructions. Circulated electronically to TYPES mailing list, November 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Middle without Definite Descriptions in the Theory of Constructions Jonathan P. Seldin # Department of Mathematics Concordia University Montreal, Quebec, Canada seldin alcor.concordia.ca November 15, 1998 1 Introduction In his posting to the TYPES network [3], Pottinger shows that if excluded middle and definite descriptions are added to Coquand s calculus of constructions, then any two terms in a small type (i.e. a type in Prop) are equal (in the sense of Leibniz equality) This conclusion is called proof degeneracy . Although in general proof ....

....under the Curry Howard isomorphism, have the disjunction property. Since this property is known to be characteristic of constructive logic and incompatible with classical logic, this result of Coquand is really a confirmation of what we should expect of classical logic. The result of Pottinger [3], on the other hand, is unwelcome, since both excluded middle and definite descriptions are desirable in some circumstances. The result proved here shows that we are more likely to have to give up definite descriptions than excluded middle. An ASCII version of this document was circulated on the ....

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Garrel Pottinger. Definite descriptions and excluded middle in the theory of constructions. Circulated electronically to TYPES mailing list, November 1989.

....topos theoretic result into UTT ( 34] Assuming the axiom of choice, SET is a boolean topos. Thus every proposition of the topos is logically equivalent to the topos truth values or . As a consequence, we can derive p :p for every proposition p of UTT. The second implication follows from [45]. For the sake of completeness, we give a derivation of the first fact entirely in type theoretical terms. Let p be a proposition. Define the terms ff def = x : Prop:x ( x p) fi def = x : Prop: x (x p) fl def = Sigmay : Prop Prop: Pix : Prop: yx) ffx) Pix : Prop: yx) ....

G.Pottinger. Definite descriptions and excluded middle in the theory of constructions, TYPES mailing list, November 1989.

....proof, somewhat more direct and based in a different idea than Reynolds , has been given by S. Berardi, and checked in the proof checker LEGO of R. Pollack. The present result, which concerns the principle of definite descriptions, generalises and was motivated by a result of G. Pottinger [20]. 4.3 Consistency and Independence Results S. Berardi has shown by a model theoretic argument that the axiom of description, and hence the axiom of choice, is not provable in impredicative Type Theory (personal communication. A syntactic version of this model is described in [1] It is ....

Pottinger G. "Definite Descriptions and Excluded Middle in the Theory of Constructions." Communications in the TYPES electronic forum. October 1, 1989.

....The resulting type system is called S e . Note that effectiveness has very strong logical consequences: Lemma 10 [4] In ECC e , the axiom of choice 11 implies excluded middle and proof irrelevance. Proof: the first part is a straightforward adaptation of [14] The second part follows from [20], see also [6] Congruence types Quotient types do not capture, when it exists, the computational content of the equivalence relation. In [2, 4] the author introduces a variant of quotient types, congruence types, which preserve the computational content of the quotienting relation. If T is an ....

G.Pottinger. Definite descriptions and excluded middle in the theory of constructions, TYPES mailing list, November 1989.

....a greater expressibility then higher order predicate logic so we may also put in the context axioms which do have a meaning but can not be expressed in the logic, for example an axiom that makes a statement about all domains. An example of this is the axiom of definite descriptions as described in [Pottinger 1989], DD : 8ff:Prop:8P :ff Prop:8z: 9 x:ff:Px) P ( ffP z) where 9 x:ff:P x : 9x:ff:P x) 8x; y:ff:P x Py (x = ff y) and is a term of type 8ff:Prop:8P :ff Prop: 9 x:ff:Px) ff. One can take some fixed closed term for but also declare it as variable in the context. We assume the intended ....

....:ff Prop: 9 x:ff:Px) ff. One can take some fixed closed term for but also declare it as variable in the context. We assume the intended meaning of DD in HOPL to be clear. Together with classical logic, the axiom of definite descriptions has an unexpected side effect in CC. Proposition 4. 13 [Pottinger 1989] Classical logic and definite descriptions yield proof irrelevance in CC We have already encountered the semantical notion of proof irrelevance in the discussion of the model in 4.7. It can also be expressed in purely syntactical terms as the phenomenon that for all propositions , all ....

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G. Pottinger, Definite descriptions and excluded middle in the theory of constructions, TYPES network, November 1989.

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