Intuitionistic Predicate Calculus Research Articles

A Generalized Realizability and Intuitionistic Logic

Let V be a set of number-theoretical functions. We define a notion of V -realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V -realizability if and only if some natural conditions for V hold.

Type inhabitation of atomic polymorphism is undecidable

Abstract Atomic polymorphism $\mathbf{F_{at}}$ is a restriction of Girard and Reynold’s system $\mathbf{F} $(or $\lambda 2$) which was first introduced in Ferreira [ 2] in the context of a philosophical commentary on predicativism. $\lambda 2$ is a well-known and powerful formal tool for studying polymorphic functional programming languages and formal methods in program specification and development, but its computational power far exceeds the recursive level of interest in applications. Hence, the interest of studying subsystems of $\lambda 2$ with weaker computational power. $\mathbf{F_{at}}$ is defined by restricting instantiation to atomic variables only. It turns out that the type system is still sufficiently powerful to possess embeddings of full intuitionistic propositional calculus [ 3, 4], and since the calculus has fewer connectives and strong normalizability is simple to prove [ 3], this result allows us to circumvent many of the extra computational complexities present when dealing with the proof theory of IPC. It is natural to inquire whether type inhabitation, i.e. provability in the corresponding fragment of second-order intuitionistic propositional logic, is decidable or not and in general to see whether the negative results involving the undecidability of type inhabitation, typability and type-checking for $\mathbf{F}$ still hold in this fragment. A further theme would be to study the result of adding type constructors, recursors or even dependent types to $\mathbf{F_{at}}$. In this paper, we show that type inhabitation for $\mathbf{F_{at}}$ is undecidable by codifying within it an undecidable fragment of first-order intuitionistic predicate calculus, adapting and modifying the technique of Urzyczyn’s [ 1, 7] purely syntactic proof of the undecidability of type inhabitation for $\mathbf{F}$.

The principle of reductio ad absurdum as an ontological problem

The ontological status of evidence is reduced to reducing to absurdity. The problem is that in such evidence, an impossible (or still possible?) Situation is allowed as the initial one. What is the ontological status of such situations, do they have an ontological justification? The answer to this question involves considering the structure of evidence by reducing to absurdity, choosing the appropriate logic and searching for an adequate ontology. The article shows that in the classical calculus of predicates of the first order, proofs by reduction to absurdity are included as a special case of proof by contradiction. The intuitionistic predicate calculus uses evidence to reduce to absurdity, but evidence from the contrary is not accepted in it. Further, the key concept of “absurdity” is discussed. It is shown that the treatment of absurdity as nonsense leads to a dead end, because the problem of meaning does not have an adequate solution in modern science. We can not guarantee the presence of a semantic meaning for correctly constructed expressions of the sign system, but we can ensure the presence of a denotational value for such expressions in artificial sign systems. As applied to our problem, this indicates the need to search for the denotational significance of contradictions. Since the models of logical calculi are constructed by means of set theory, the domain of the search is narrowed to the choice of an appropriate set theory. The modified set theory ZF is considered in which the axiom of the existence of an empty set is replaced by its negation. In this case, it is possible to give the denotational significance to the contradictions, but then the contradictions of the type A and not- A get an ontological justification, since both A and non- A are fulfilled.

Approximations on intuitionistic fuzzy predicate calculus through rough computing

In 1986, Attansov introduced the intuitionistic fuzzy set which is the generalized approach of fuzzy set model introduced by Zadeh. Considering the importance of these two theories, various hybridizations with the available mathematical tools have been derived and they are in practice. Related to the concept of rough sets, the hybridized work of Nakamura, Dubois and Prade are noteworthy. In 2005, G. Ganesan et al., have introduced the concept of thresholds in Rough Fuzzy Hybridizations. Later, in 2008, G. Ganesan et al, derived a naive approach on approximating the connectives in fuzzy predicate calculus through rough sets. This concept was developed by introducing a threshold on the membership of the arguments in the fuzzy predicates. This work yielded the Two way approximations for the fuzzy connectives. In this work, it is noticed that the complement and the negation of the predicates are found in a unique way, whereas in case of intuitionistic fuzziness, the complement and the negation are to be found in two different ways namely membership based and non membership based. Using this concept, in this paper, we introduce Two way approach on Intuitionistic Fuzzy Logic. Also, in this paper, we derive Two way approximations on each intuitionistic fuzzy connectives using rough sets.

Reverse Mathematics and Uniformity in Proofs without Excluded Middle

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis.

Commuting Conversions vs. the Standard Conversions of the “Good” Connectives

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions.

Läuchli’s Completeness Theorem from a Topos-Theoretic Perspective

We prove a variant of Lauchli’s completeness theorem for intuitionistic predicate calculus. The formulation of the result relies on the observation (due to Lawvere) that Lauchli’s theorem is related to the logic of the canonical indexing of the atomic topos of \(\mathbb{Z}{\text{ - sets}}\). We show that the process that transforms Kripke-counter-models into Lauchli-counter-models is (essentially) the inverse image of a geometric morphism. Completeness follows because this geometric morphism is an open surjection.

On Some Kripke Complete and Kripke Incomplete Intermediate Predicate Logics

The Kripke-completeness and incompleteness of some intermediate predicate logics is established. In particular, we obtain a Kripke-incomplete logic (H* +A+D+K) where H* is the intuitionistic predicate calculus, A is a disjunction-free propositional formula, D = ∀x(P(x) V Q) ⊃ ∀xP(x) V Q, K = ¬¬∀x(P(x) V ¬P(x)) (the negative answer to a question of T. Shimura).

Two remarks on the Lifschitz realizability topos

The purpose of this note is to clarify two points about the topos Lif, introduced in [13] as a generalization of Lifschitz' realizability ([9, 12]). Lif is a subtopos of Hyland's Effective topos ([1]). The points I want to make are:Remark 1. Lif is the largest subtopos of satisfying the axiom (O):where denotes partial recursive application, and “∈ Tot” means that e and f range over codes for total recursive functions. One may read (O) as the statement “The union of two -sets is again a -set”. That is, let be a subtopos of . Then (O) is true in for the standard interpretation (the variables range over the natural numbers object of , etc.) if and only if the inclusion ↣ factors through the inclusion Lif ↣ .Remark 2. Like , Lif contains at least two weakly complete internal full subcategories, thus providing us with more models of polymorphism and other impredicative type theories.The principle (O) has some standing in the history of constructive mathematics:- H. Friedman has proved that (O) is equivalent to a formulation of intuitionistic completeness of the intuitionistic predicate calculus for Tarskian semantics; see [8]. This is not to imply that this result is of immediate relevance to Lif: Friedman works in a system of analysis, a theory of lawless sequences with an axiom of “open data” for arithmetical formulas, which at least for the domain of all functions from N to N fails in Lif. However, there might exist a “nonstandard model” of arithmetic and a corresponding system of analysis, in which we may be able to carry out his proof.- Moreover, Remark 2 entails that Lif should provide us with models of synthetic domain theory (for an exposition, see [3]), and one with the nice property that the dual of one of the axioms (axiom 7 in [3]) comes for free, by (O).

Intuitionistic ϵ‐ and τ‐calculi

AbstractThere are several open problems in the study of the calculi which result from adding either of Hilbert's ϵ‐ or τ‐operators to the first order intuitionistic predicate calculus. This paper provides answers to several of them. In particular, the first complete and sound semantics for these calculi are presented, in both a “quasi‐extensional” version which uses choice functions in a straightforward way to interpret the ϵ‐ or τ‐terms, and in a form which does not require extensionality assumptions. Unlike the classical case, the addition of either operator to intuitionistic logic is non‐conservative. Several interesting consequences of the addition of each operator are proved. Finally, the independence of several other schemes in either calculus are also proved, making use of the semantics supplied earlier in the paper.

A semantical proof of De Jongh's theorem

In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ 2 0 -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA.

Constructible falsity and inexact predicates

In 1949 Nelson [5] proposed a constructive logic in which falsity is conceived in a fashion analogous to that for intuitionistic truth. The predicate calculus N (for strong negation) was characterized by the usual axioms and rules for positive intuitionistic connectives (see Kleene [4, p. 82, la–7 and 9–12]), with the additional axiom schemata for strong negation: A ⊃ (¬A ⊃ B),Nelson's paper showed that N may be interpreted with concepts for constructive truth (P-realizability) and falsity (N-realizability). The paper also gave mappings between the intuitionistic and strong negation systems of arithmetic.These mappings are easily adapted to pure predicate calculus. One shows that intuitionistic predicate calculus I is a subsystem of N by reading the intuitionistic negation of A as A ⊃ B & ¬B in N. It is possible to map N into a subsystem of I using the definition of A′ in [5, p. 19] changing only the definition of (¬A)′ for atomic A to A ⊃ B & ¬B.

A free IPC is a natural logic: Strong completeness for some intuitionistic free logics

IPC, the intuitionistic predicate calculus, has the property (i) Vc(Γ⊢Ac/x) ⇒ Γ⊢∃xA.Furthermore, for certain important Γ, IPC has the converse property (ii) Γ⊢∃xA ⇒ Vc(Γ⊢Ac/x). (i) may be given up in various ways, corresponding to different philosophic intuitions and yielding different systems of intuitionistic free logic. The present paper proves the strong completeness of several of these with respect to Kripke style semantics. It also shows that giving up (i) need not force us to abandon the analogue of (ii).

Three ways of recognizing inessential formulas in sequents

The first method is based on the familiar method of lowering thinnings downwards and is a further development of the lemmas on “weeding” of [1]. The second method is based on the use of sufficiently wide classes of sequents, for which derivability in the intuitionistic predicate calculus coincides with derivability in the classical predicate calculus and the familiar property of disjunction is true. By this method one can get, e.g., a syntactic proof of the following assertion. If the positive formula A is derivable in the theory of groups under additional assumptions of the form then A is also derivable in the theory of groups without these assumptions. As the third method there is proposed a syntactically formulated test for the conservativeness of extensions of intuitionistic axiomatic theories. With the help of this test one can get, for example, a syntactic proof of the hereditary undecidability of the intuitionistic theory of equality, with additional axioms which are the formula , all formulas of the form and all negations of formulas derivable in the classical predicate calculus.

Primitive recursive estimate of strong normalization for predicate calculus

The strong normalization theorem asserts that any sequence of reductions (standard steps of cut elimination) stops at an object in normal form, i.e., at an object to which further reductions are inapplicable. The most intuitive proofs of strong normalization for various systems, including first- and second-order arithmetic, use, essentially, the concept of hereditary normalizability of an object. This concept, for objects of unbounded complexity, cannot be expressed in the language of arithmetic, so that the proofs mentioned leave its domain. Howard's proof for arithmetic, using nonunique assignment of ordinals, apparently, can be modified so as to get a primitive recursive proof of strict normalizability for the $$(\forall , \supset )$$ -fragment of the intuitionistic predicate calculus, but the author of the present paper has not succeeded in overcoming the combinatorial difficulties. Our goal is to give an intuitive proof for the predicate calculus, from which one can extract a primitive recursive estimate of the number of reductions, and a proof in primitive recursive arithmetic of the fact that this estimate is proper. The proof of normalizability, that is, the construction of a special reduction sequence stopping at a normal term, is well known. In this sequence one first converts subterms of the highest levell, and first the innermost of them. (Corresponding to this, in the proof one can apply reduction with respect to level.) Here the number of convertible terms of maximal level, suitable for reduction, is lowered and the newly arising convertible terms have lower level.

An elimination theorem of uniqueness conditions in the intuitionistic predicate calculus

This paper is a sequel to Motohashi [4]. In [4], a series of theorems named “elimination theorems of uniqueness conditions” was shown to hold in the classical predicate calculus LK. But, these results have the following two defects : one is that they do not hold in the intuitionistic predicate calculus LJ, and the other is that they give no nice axiomatizations of some sets of sentences concerned. In order to explain these facts more explicitly, let us introduce some necessary notations and definitions. Let L be a first order classical predicate calculus LK or a first order intuitionistic predicate calculus LJ. n-ary formulas in L are formulas F(ā) in L with a sequence ā of distinct free variables of length n such that every free variable in F occurs in ā.

Equivalence between semantics for intuitionism. I

It is probably because intuitionism is founded on the concept of (abstract) proof that it has been possible to develop various kinds of models. The following is but a partial list: Gabbay [5], Beth [2], Kripke [8], Kleene [7], Läuchli [9], McKinsey and Tarski [10], Rasiowa and Sikorski [14], Scott [15], de Swart [16], and Veldman [17].The original purpose for having the models appears to have been for obtaining independence or consistency results for certain formalizations of intuitionism [see Beth [2], Prawitz [13]]; of course, if the models could be also justified as being plausible interpretations of intuitionistic thinking, so much the better. In fact, having some kind of plausible interpretation makes it much easier to work with the models. Occasionally the models were used to suggest possible extensions of the formal systems; for example, the Kripke models with constant domains have motivated interest in the formal logic CD which extends the Intuitionistic Predicate Calculus (IPC) by having the axiom schema

On the interpolation theorem for the logic of constant domains

In Gabbay [1] it is stated as an open problem whether or not Craig's Theorem holds for the logic of constant domains CD, i.e. for the extension of the intuitionistic predicate calculus, IPC, obtained by adding the schema; . Then in the later article, [2], Gabbay gives a proof of it. The proof given in [2] is via Robinson's (weak) consistency theorem and depends on relatively complicated (Kripke-) model-theoretical constructions developed in [1] (see p. 392 of [1] for a brief sketch of the method). The aim of this note is to show that the interpolation theorem for CD can also be obtained, by simple proof-theoretic methods, from §80 of Kleene's Introduction to Metamathematics [3].GI is the classical formal system whose postulates are given on p. 442 of [3]. Let GD be the system obtained from GI by the following modifications: (1) the sequents of GD are to have at most two formulas in their succedents and (2) the intuitionistic restriction that Θ be empty is required for the succedent rules (→ ¬) and (→ ⊃). It is a simple matter to show that: , x not free in . It then follows that, using Theorem 46 of [3], if then .

The ${\bf Q}$-consistency of ${\cal F}_{22}$.

In his [CSC],1 Curry proved the consistency of a system, which he there defines and calls 922, and which is closely related to the system 9$λ of [CLg.Il] §15C.2 This is essentially a type-free intuitionistic predicate calculus without conjunction, alternation, or negation but with quantification over propositions and propositional functions. However, Curry's consistency proof is rather weak, since it only proves that every theorem of the system belongs to a class of obs (terms) which are defined to be canonical (called canobs) and since the canobs are those obs which are to be interpreted as propositions this proof leaves open the possibility that every ob which is to be interpreted as a proposition is a theorem of the

Another intuitionistic completeness proof

In March 1973, W. Veldman [1] discovered that, by a slight modification of a Kripke-model, it was possible to give an intuitionistic proof of the completeness-theorem for the intuitionistic predicate calculus (IPC) with respect to modified Kripke models. The modification was the following: Let f represent absurdity, then we allow the possibility that and we agree that, for all sentences ϕ, , if . Just one modified Kripke model is constructed such that validity in implies derivability in IPC. While usually one thinks of as some subset of ⋃nNatn and of as the discrete natural ordering in ⋃nNatn, in Veldman's model , is a spread and , where Γα and Γβ are sets of sentences associated with α, resp. β, is a nondiscrete ordering.In the completeness-proofs, both for Beth and for Kripke models that we present here, we consider only models over ⋃nNatn, with the natural discrete ordering and we need validity in all models, not just in one, to get derivability in IPC. Also we have to modify the definition of a model in a somewhat different way than Veldman did. We agree that if ∨s[M⊨sf], then M⊨sϕ for each s ∈ ⋃nNatn and for each sentence ϕ.One can view a single model of the type constructed in [1] as the result of throwing together all the models of (the type constructed in) this paper into one big model, which has the somewhat strange properties mentioned above.