scholarly journals An elimination theorem of uniqueness conditions in the intuitionistic predicate calculus

1982 ◽  
Vol 85 ◽  
pp. 223-230 ◽  
Author(s):  
Nobuyoshi Motohashi
Keyword(s):  
Free Variable ◽  
The Other ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Classical Predicate

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.

Solution of a problem of Leon Henkin

1955 ◽  
Vol 20 (2) ◽  
pp. 115-118 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Number Theory ◽  
Recursive Function ◽  
Formal System ◽  
Free Variable ◽  
Formal Proof ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Order Predicate Calculus

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2].One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula (Ex)(x, a), with Gödel-number a, is provable or not. Here (x, y) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y.In this note we present a solution of the previous problem with respect to the system Zμ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function (k, l) used below is definable.The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Zμ containing no free variables, whose Gödel number is a, then ({}) stands for (Ex)(x, a) (read: the formula with Gödel number a is provable in Zμ); if is a formula of Zμ containing a free variable, y say, ({}) stands for (Ex)(x, g(y)}, where g(y) is a recursive function such that for an arbitrary numeral the value of g() is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing (), where is an arbitrary numeral, for (Ex){x, ).

Resolution systems and their applications I

1980 ◽  
Vol 3 (2) ◽  
pp. 235-268
Author(s):  
Ewa Orłowska
Keyword(s):  
Theorem Proving ◽  
Sufficient Conditions ◽  
Predicate Calculus ◽  
Resolution Method ◽  
First Order ◽  
Deductive Systems ◽  
Resolution System ◽  
Resolution Principle ◽  
Necessary And Sufficient ◽  
Classical Predicate

The central method employed today for theorem-proving is the resolution method introduced by J. A. Robinson in 1965 for the classical predicate calculus. Since then many improvements of the resolution method have been made. On the other hand, treatment of automated theorem-proving techniques for non-classical logics has been started, in connection with applications of these logics in computer science. In this paper a generalization of a notion of the resolution principle is introduced and discussed. A certain class of first order logics is considered and deductive systems of these logics with a resolution principle as an inference rule are investigated. The necessary and sufficient conditions for the so-called resolution completeness of such systems are given. A generalized Herbrand property for a logic is defined and its connections with the resolution-completeness are presented. A class of binary resolution systems is investigated and a kind of a normal form for derivations in such systems is given. On the ground of the methods developed the resolution system for the classical predicate calculus is described and the resolution systems for some non-classical logics are outlined. A method of program synthesis based on the resolution system for the classical predicate calculus is presented. A notion of a resolution-interpretability of a logic L in another logic L ′ is introduced. The method of resolution-interpretability consists in establishing a relation between formulas of the logic L and some sets of formulas of the logic L ′ with the intention of using the resolution system for L ′ to prove theorems of L. It is shown how the method of resolution-interpretability can be used to prove decidability of sets of unsatisfiable formulas of a given logic.

scholarly journals Elementary-base cirquent calculus II: Choice quantifiers

2020 ◽  
Author(s):  
Giorgi Japaridze
Keyword(s):  
Proof Theory ◽  
Predicate Calculus ◽  
First Order ◽  
Complete Axiomatization ◽  
Calculus I ◽  
Soundness And Completeness ◽  
Classical Predicate

Abstract Cirquent calculus is a novel proof theory permitting component-sharing between logical expressions. Using it, the predecessor article ‘Elementary-base cirquent calculus I: Parallel and choice connectives’ built the sound and complete axiomatization $\textbf{CL16}$ of a propositional fragment of computability logic. The atoms of the language of $\textbf{CL16}$ represent elementary, i.e. moveless, games and the logical vocabulary consists of negation, parallel connectives and choice connectives. The present paper constructs the first-order version $\textbf{CL17}$ of $\textbf{CL16}$, also enjoying soundness and completeness. The language of $\textbf{CL17}$ augments that of $\textbf{CL16}$ by including choice quantifiers. Unlike classical predicate calculus, $\textbf{CL17}$ turns out to be decidable.

Sufficient conditions for the undecidability of intuitionistic theories with applications

1972 ◽  
Vol 37 (2) ◽  
pp. 375-384 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Decision Problem ◽  
Sufficient Conditions ◽  
Kripke Model ◽  
The Other ◽  
Predicate Calculus ◽  
Kripke Models ◽  
Symmetric Relation ◽  
Undecidable Theory ◽  
Classical Models ◽  
Classical Predicate

Let Δ be a set of axioms of a theory Tc(Δ) of classical predicate calculus (CPC); Δ may also be considered as a set of axioms of a theory TH(Δ) of Heyting's predicate calculus (HPC). Our aim is to investigate the decision problem of TH(Δ) in HPC for various known theories Δ of CPC.Theorem I(a) of §1 states that if Δ is a finitely axiomatizable and undecidable theory of CPC then TH(Δ) is undecidable in HPC. Furthermore, the relations between theorems of HPC are more complicated and so two CPC-equivalent axiomatizations of the same theory may give rise to two different HPC theories, in fact, one decidable and the other not.Semantically, the Kripke models (for which HPC is complete) are partially ordered families of classical models. Thus a formula expresses a property of a family of classical models (i.e. of a Kripke model). A theory Θ expresses a set of such properties. It may happen that a class of Kripke models defined by a set of formulas Θ is also definable in CPC (in a possibly richer language) by a CPC-theory Θ′! This establishes a connection between the decision problem of Θ in HPC and that of Θ′ in CPC. In particular if Θ′ is undecidable, so is Θ. Theorems II and III of §1 give sufficient conditions on Θ to be such that the corresponding Θ′ is undecidable. Θ′ is shown undecidable by interpreting the CPC theory of a reflexive and symmetric relation in Θ′.

A note on a theorem of Vaught

1971 ◽  
Vol 36 (3) ◽  
pp. 439-440 ◽  
Author(s):  
Joseph G. Rosenstein
Keyword(s):  
Predicate Calculus ◽  
Complete Theory ◽  
First Order ◽  
Free Variables ◽  
Order Predicate Calculus ◽  
Countable Models

In [2] Vaught showed that if T is a complete theory formalized in the first-order predicate calculus, then it is not possible for T to have exactly (up to isomorphism) two countable models. In this note we extend his methods to obtain a theorem which implies the above.First some definitions. We define Fn(T) to be the set of well-formed formulas (wffs) in the language of T whose free variables are among x1 x2, …, xn. An n-type of T is a maximal consistent set of wffs of Fn(T); equivalently, a subset P of Fn(T) is an n-type of T if there is a model M of T and elements a1, a2, …, an of M such that M ⊧ ϕ(a1, a2, …, an) for every ϕ ∈ P. In the latter case we say that 〈a1, a2, …, an〉 ony realizes P in M. Every set of wffs of Fn(T) which is consistent with T can be extended to an n-type of T.

On an Ackermann-type set theory

1973 ◽  
Vol 38 (3) ◽  
pp. 410-412
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinal ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus ◽  
Universal Closure

Ackermann's set theory A* is usually formulated in the first order predicate calculus with identity, ∈ for membership and V, an individual constant, for the class of all sets. We use small Greek letters to represent formulae which do not contain V and large Greek letters to represent any formulae. The axioms of A* are the universal closures ofwhere all free variables are shown in A4 and z does not occur in the Θ of A2.A+ is a generalisation of A* which Reinhardt introduced in [3] as an attempt to provide an elaboration of Ackermann's idea of “sharply delimited” collections. The language of A+ is that of A*'s augmented by a new constant V′, and its axioms are A1–A3, A5, V ⊆ V′ and the universal closure ofwhere all free variables are shown.Using a schema of indescribability, Reinhardt states in [3] that if ZF + ‘there exists a measurable cardinal’ is consistent then so is A+, and using [4] this result can be improved to a weaker large cardinal axiom. It seemed plausible that A+ was stronger than ZF, but our main result, which is contained in Theorem 5, shows that if ZF is consistent then so is A+, giving an improvement on the above results.

Undecidability of modal and intermediate first-order logics with two individual variables

1993 ◽  
Vol 58 (3) ◽  
pp. 800-823 ◽  
Author(s):  
D. M. Gabbay ◽  
V. B. Shehtman
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Predicate Calculus ◽  
Single Individual ◽  
First Order ◽  
Short Summary ◽  
Binary Predicate ◽  
Individual Variables ◽  
Image Position ◽  
Classical Predicate

The interest in fragments of predicate logics is motivated by the well-known fact that full classical predicate calculus is undecidable (cf. Church [1936]). So it is desirable to find decidable fragments which are in some sense “maximal”, i.e., which become undecidable if they are “slightly” extended. Or, alternatively, we can look for “minimal” undecidable fragments and try to identify the vague boundary between decidability and undecidability. A great deal of work in this area concerning mainly classical logic has been done since the thirties. We will not give a complete review of decidability and undecidability results in classical logic, referring the reader to existing monographs (cf. Suranyi [1959], Lewis [1979], and Dreben, Goldfarb [1979]). A short summary can also be found in the well-known book Church [1956]. Let us recall only several facts. Herein we will consider only logics without functional symbols, constants, and equality.(C1) The fragment of the classical logic with only monadic predicate letters is decidable (cf. Behmann [1922]).(C2) The fragment of the classical logic with a single binary predicate letter is undecidable. (This is a consequence of Gödel [1933].)(C3) The fragment of the classical logic with a single individual variable is decidable; in fact it is equivalent to Lewis S5 (cf. Wajsberg [1933]).(C4) The fragment of the classical logic with two individual variables is decidable (Segerberg [1973] contains a proof using modal logic; Scott [1962] and Mortimer [1975] give traditional proofs.)(C5) The fragment of the classical logic with three individual variables and binary predicate letters is undecidable (cf. Surańyi [1943]). In fact this paper considers formulas of the following typeφ,ψ being quantifier-free and the set of binary predicate letters which can appear in φ or ψ being fixed and finite.

Natural models and Ackermann-type set theories

1975 ◽  
Vol 40 (2) ◽  
pp. 151-158 ◽  
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus

Our results concern the natural models of Ackermann-type set theories, but they can also be viewed as results about the definability of ordinals in certain sets.Ackermann's set theory A was introduced in [1] and it is now formulated in the first order predicate calculus with identity, using ∈ for membership and an individual constant V for the class of all sets. We use the letters ϕ, χ, θ, and χ to stand for formulae which do not contain V and capital Greek letters to stand for any formulae. Then, the axioms of A* are the universal closures ofwhere all the free variables are shown in A4 and z does not occur in the Θ of A2. A is the theory A* − A5.Most of our notation is standard (for instance, α, β, γ, δ, κ, λ, ξ are variables ranging over ordinals) and, in general, we follow the notation of [7]. When x ⊆ Rα, we use Df(Rα, x) for the set of those elements of Rα which are definable in 〈Rα, ∈〉, using a first order ∈-formula and parameters from x.We refer the reader to [7] for an outline of the results which are known about A, but we shall summarise those facts which are frequently used in this paper.

An undecidable two sorted predicate calculus

1969 ◽  
Vol 34 (1) ◽  
pp. 21-23 ◽  
Author(s):  
A. B. Slomson
Keyword(s):  
Decision Procedure ◽  
The Other ◽  
Predicate Calculus ◽  
First Order ◽  
Predicate Language

Let L be a first order predicate language with two sorts of variables and a single dyadic predicate letter whose first place is to be filled by variables of one sort and whose second place is to be filled by variables of the other sort. In answer to a question of M. H. Löb we show that there is no decision procedure for determining whether or not a sentence of L is universally valid.

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