On the proof theory of Coquand's calculus of constructions

Seldin, Jonathan P.

doi:10.1016/s0168-0072(96)00008-5

Cited by 12 publications

(18 citation statements)

References 8 publications

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“…This assumption and the assumption that (∀n : N)(¬(σ n = N 0)) are proved consistent in [7,Theorem 23]. Essentially the same method can be used to prove that the above classical assumption and ¬(T = Bool F) are consistent.…”

Section: Representing Logic With Equality Frege Stylementioning

confidence: 92%

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Interpreting HOL in the calculus of constructions

Seldin

2004

Journal of Applied Logic

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The purpose of this paper is to consider a representation of the HOL theorem-prover in the calculus of constructions with the property that consistency results from the calculus of constructions imply such results in HOL. This kind of representation is impossible using the propositions-as-types representation of logic and equality, but it is possible if a different representation is used.Since the basic formalism of higher-order logic (HOL) is weaker than that of the calculus of constructions, it seems natural to interpret the HOL theorem prover [1] in the latter. However, there are some problems in carrying out such an interpretation. The HOL theorem prover is based on classical logic, includes an -choice operator, and includes a principle of extensionality in the formThe calculus of constructions is based on constructive logic, and although it can be consistently extended to classical logic by postulating excluded middle, it has been known for some time that excluded middle together with an -choice operator leads to proof irrele-✩ This paper represents a presentation entitled "Logic and Type Theory in Theorem Provers" which was presented to the conference "Thirty-Five Years of AUTOMATH" held at

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Section: Representing Logic With Equality Frege Stylementioning

confidence: 92%

“…where case is defined along with the injections inl and inr for the disjoint sum (disjunction) in [7,Definition 21]. Note that it is shown in [7] that for A, B, C :…”

Section: Representing Logic With Equality Frege Stylementioning

confidence: 99%

Interpreting HOL in the calculus of constructions

Seldin

2004

Journal of Applied Logic

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“…Huet (1986, 1988), and the related metatheoretical analysis, e.g. Seldin (1997Seldin ( , 2000. Similarly, a sequent calculus can be formulated that reflects the operations of becoming informed as operations on antecedents of sequents.…”

Section: Going Modalmentioning

confidence: 99%

An epistemic logic for becoming informed

Primiero¹

2008

Synthese

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Various conceptual approaches to the notion of information can currently be traced in the literature in logic and formal epistemology. A main issue of disagreement is the attribution of truthfulness to informational data, the so called Veridicality Thesis (Floridi 2005). The notion of Epistemic Constructive Information (Primiero 2007) is one of those rejecting VT. The present paper develops a formal framework for ECI. It extends on the basic approach of Artemov's logic of proofs (Artemov 1994), representing an epistemic logic based on dependent justifications, where the definition of information relies on a strict distinction from factual truth. The definition obtained by comparison with a Normal Modal Logic translates a constructive logic for "becoming informed": its distinction from the logic of "being informed"-which internalizes truthfulness-is essential to a general evaluation of information with respect to truth. The formal disentanglement of these two logics, and the description of the modal version of the former as a weaker embedding into the latter, allows for a proper understanding of the Veridicality Thesis with respect to epistemic states defined in terms of information.

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“…Illative systems of combinatory logic such as these of Bunder [11], [3], [4], and [7], the later "Frege Structures" of Aczel [1], and the version of the Calculus of Constructions in Coquand [14] and Seldin [17] are slightly more general than the SAIs that we have developed here in that in…”

Section: Ptss and Icls In The Literaturementioning

confidence: 99%

Equivalences between Pure Type Systems and Systems of Illative Combinatory Logic

Bunder¹,

Dekkers²

2005

Notre Dame J. Formal Logic

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Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds of PTSs. Most importantly PTSs are about statements of the form M : A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold. Abstract Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds of PTSs. Most importantly PTSs are about statements of the form M : A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold. Disciplines Physical Sciences and Mathematics

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On the proof theory of Coquand's calculus of constructions

Cited by 12 publications

References 8 publications

Interpreting HOL in the calculus of constructions

Interpreting HOL in the calculus of constructions

An epistemic logic for becoming informed

Equivalences between Pure Type Systems and Systems of Illative Combinatory Logic

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