scholarly journals Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic

1998 ◽  
Vol 63 (3) ◽  
pp. 869-890 ◽  
Author(s):  
Wil Dekkers ◽  
Martin Bunder ◽  
Henk Barendregt
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Lambda Calculus ◽  
Preceding Paper ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
First Order ◽  
Two Systems

AbstractIllative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions.

scholarly journals Systems of illative combinatory logic complete for first-order propositional and predicate calculus

1993 ◽  
Vol 58 (3) ◽  
pp. 769-788 ◽  
Author(s):  
Henk Barendregt ◽  
Martin Bunder ◽  
Wil Dekkers
Keyword(s):  
Lambda Calculus ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
Consistent System ◽  
First Order ◽  
Base Logic

AbstractIllative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).

Embedding first order predicate logic in fragments of intuitionistic logic

1976 ◽  
Vol 41 (4) ◽  
pp. 705-718 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Propositional Calculus ◽  
Expressive Power ◽  
Predicate Calculus ◽  
First Order ◽  
Intuitionistic Propositional Calculus ◽  
Individual Variables ◽  
The One ◽  
Order Predicate Calculus

Some syntactically simple fragments of intuitionistic logic possess considerable expressive power compared with their classical counterparts.In particular, we consider in this paper intuitionistic second order propositional logic (ISPL) a formalisation of which may be obtained by adding to the intuitionistic propositional calculus quantifiers binding propositional variables together with the usual quantifier rules and the axiom scheme (Ex), where is a formula not containing x.The main purpose of this paper is to show that the classical first order predicate calculus with identity can be (isomorphically) embedded in ISPL.It turns out an immediate consequence of this that the classical first order predicate calculus with identity can also be embedded in the fragment (PLA) of the intuitionistic first order predicate calculus whose only logical symbols are → and (.) (universal quantifier) and the only nonlogical symbol (apart from individual variables and parentheses) a single monadic predicate letter.Another consequence is that the classical first order predicate calculus can be embedded in the theory of Heyting algebras.The undecidability of the formal systems under consideration evidently follows immediately from the present results.We shall indicate how the methods employed may be extended to show also that the intuitionistic first order predicate calculus with identity can be embedded in both ISPL and PLA.For the purpose of the present paper it will be convenient to use the following formalisation (S) of ISPL based on [3], rather than the one given above.

scholarly journals On the Definitional Embeddability of the Combinatory Logic Theory into the First-Order Predicate Calculus

2015 ◽  
Vol 21 (2) ◽  
pp. 9-14
Author(s):  
В. И. Шалак
Keyword(s):  
Predicate Logic ◽  
Computability Theory ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
Applied Theory ◽  
First Order ◽  
Computable Functions ◽  
Order Predicate Calculus ◽  
Classical Predicate

In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.

scholarly journals Higher-Order Illative Combinatory Logic

2013 ◽  
Vol 78 (3) ◽  
pp. 837-872 ◽  
Author(s):  
Łukasz Czajka
Keyword(s):  
Strong Consistency ◽  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Model Construction ◽  
Partial Answer ◽  
Combinatory Logic ◽  
First Order ◽  
Propositional Quantifiers

AbstractWe show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

On the formalisation of indirect discourse

1958 ◽  
Vol 23 (4) ◽  
pp. 417-419 ◽  
Author(s):  
R. L. Goodstein
Keyword(s):  
Predicate Logic ◽  
Logical Truth ◽  
Predicate Calculus ◽  
Inference Rules ◽  
First Order ◽  
Indirect Discourse ◽  
Image Position ◽  
Order Predicate Calculus

Mr. L. J. Cohen's interesting example of a logical truth of indirect discourse appears to be capable of a simple formalisation and proof in a variant of first order predicate calculus. His example has the form:If A says that anything which B says is false, and B says that something which A says is true, then something which A says is false and something which B says is true.Let ‘A says x’ be formalised by ‘A(x)’ and let assertions of truth and falsehood be formalised as in the following table.We treat both variables x and predicates A (x) as sentences and add to the familiar axioms and inference rules of predicate logic a rule permitting the inference of A(p) from (x)A(x), where p is a closed sentence.We have to prove that from

An intuitionistically plausible interpretation of intuitionistic logic

1977 ◽  
Vol 42 (4) ◽  
pp. 564-578 ◽  
Author(s):  
H. C. M. de Swart
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Calculus ◽  
Mathematical Treatment ◽  
First Order ◽  
The Past ◽  
Definition Of ◽  
Image Position ◽  
Order Predicate Calculus ◽  
Made In

Let IPC be the intuitionistic first-order predicate calculus. From the definition of derivability in IPC the following is clear:(1) If A is derivable in IPC, denoted by “⊦IPCA”, then A is intuitively true, that means, true according to the intuitionistic interpretation of the logical symbols. To be able to settle the converse question: “if A is intuitively true, then ⊦IPCA”, one should make the notion of intuitionistic truth more easily amenable to mathematical treatment. So we have to look then for a definition of “A is valid”, denoted by “⊨A”, such that the following holds:(2) If A is intuitively true, then ⊨ A.Then one might hope to be able to prove(3) If ⊨ A, then ⊦IPCA.If one would succeed in finding a notion of “⊨ A”, such that all the conditions (1), (2) and (3) are satisfied, then the chain would be closed, i.e. all the arrows in the scheme below would hold.Several suggestions for ⊨ A have been made in the past: Topological and algebraic interpretations, see Rasiowa and Sikorski [1]; the intuitionistic models of Beth, see [2] and [3]; the interpretation of Grzegorczyk, see [4] and [5]; the models of Kripke, see [6] and [7]. In Thirty years of foundational studies, A. Mostowski [8] gives a review of the interpretations, proposed for intuitionistic logic, on pp. 90–98.

Extensional interpretations of modal logics

1966 ◽  
Vol 31 (1) ◽  
pp. 23-45 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Predicate Logic ◽  
Modal Logics ◽  
Predicate Calculus ◽  
First Order ◽  
Image Position ◽  
Order Predicate Calculus ◽  
First Order Predicate Logic

By ΡL we shall mean the first order predicate logic based on S4. More explicitly: Let Ρ0 stand for the first order predicate calculus. The formalisation of Ρ0 used in the present paper will be given later. ΡL is obtained from Ρ0 by adding the rules the propositional constant □ and

A logic stronger than intuitionism

1971 ◽  
Vol 36 (2) ◽  
pp. 249-261 ◽  
Author(s):  
Sabine Görnemann
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Predicate Logic ◽  
Predicate Calculus ◽  
Theoretical Methods ◽  
Axiom Scheme ◽  
Simple Notion ◽  
Image Position ◽  
Quasi Ordering

S. A. Kripke has given [6] a very simple notion of model for intuitionistic predicate logic. Kripke's models consist of a quasi-ordering (C, ≤) and a function ψ which assigns to every c ∈ C a model of classical logic such that, if c ≤ c′, ψ(c′) is greater or equal to ψ(c). Grzegorczyk [3] described a class of models which is still simpler: he takes, for every ψ(c), the same universe. Grzegorczyk's semantics is not adequate for intuitionistic logic, since the formulawhere х is not free in α. holds in his models but is not intuitionistically provable. It is a conjecture of D. Klemke that intuitionistic predicate calculus, strengthened by the axiom scheme (D), is correct and complete with respect to Grzegorczyk's semantics. This has been proved independently by D. Klemke [5] by a Henkinlike method and me; another proof has been given by D. Gabbay [1]. Our proof uses lattice-theoretical methods.

The completeness of a predicate-functor logic

1985 ◽  
Vol 50 (4) ◽  
pp. 903-926 ◽  
Author(s):  
John Bacon
Keyword(s):  
Predicate Logic ◽  
Axiom System ◽  
Combinatory Logic ◽  
Completeness Proof ◽  
First Order ◽  
Full Theory ◽  
History Of ◽  
Quantificational Logic ◽  
Term Logic ◽  
Individual Variables

Predicate-functor logic, as founded by W. V. Quine ([1960], [1971], [1976], [1981]), is first-order predicate logic without individual variables. Instead, adverbs or predicate functors make explicit the permutations and replications of argument-places familiarly indicated by shifting variables about. For the history of this approach, see Quine [1971, 309ff.]. With the evaporation of variables, individual constants naturally assimilate to singleton predicates or adverbs, leaving no logical subjects whatever of type 0. The orphaned “predicates” may then be taken simply as terms in the sense of traditional logic: class and relational terms on model-theoretic semantics, schematic terms on Quine's denotational or truth-of semantics. Predicate-functor logic thus stands forth as the pre-eminent first-order term logic, as distinct from propositional-quantificational logic. By the same token, it might with some justification qualify as “first-order combinatory logic”, with allowance for some categorization of the sort eschewed in general combinatory logic, the ultimate term logic.Over the years, Quine has put forward various choices of primitive predicate functors for first-order logic with or without the full theory of identity. Moreover, he has provided translations of quantificational into predicate-functor notation and vice versa ([1971, 312f.], [1981, 651]). Such a translation does not of itself establish semantic completeness, however, in the absence of a proof that it preserves deducibility.An axiomatization of predicate-functor logic was first published by Kuhn [1980], using primitives rather like Quine's. As Kuhn noted, “The axioms and rules have been chosen to facilitate the completeness proof” [1980, 153]. While this expedient simplifies the proof, however, it limits the depth of analysis afforded by the axioms and rules. Mindful of this problem, Kuhn ([1981] and [1983]) boils his axiom system down considerably, correcting certain minor slips in the original paper.

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