On the reduction of the decision problem. Third paper. Pepis prefix, a single binary predicate

1950 ◽  
Vol 15 (3) ◽  
pp. 161-173 ◽  
Author(s):  
László Kalmár ◽  
János Surányi
Keyword(s):  
Decision Problem ◽  
Predicate Calculus ◽  
Existential Quantifier ◽  
First Order ◽  
Binary Predicate ◽  
Image Position ◽  
Analogous Theorem ◽  
Order Predicate Calculus ◽  
Predicate Variable

It has been proved by Pepis that any formula of the first-order predicate calculus is equivalent (in respect of being satisfiable) to another with a prefix of the formcontaining a single existential quantifier. In this paper, we shall improve this theorem in the like manner as the Ackermann and the Gödel reduction theorems have been improved in the preceding papers of the same main title. More explicitly, we shall prove theTheorem 1. To any given first-order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.An analogous theorem, but producing a prefix of the formhas been proved in the meantime by Surányi; some modifications in the proof, suggested by Kalmár, led to the above form.

On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate

1939 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
László Kalmár
Keyword(s):  
Decision Problem ◽  
Reduction Process ◽  
The Other ◽  
First Order ◽  
Other Hand ◽  
Special Cases ◽  
Binary Predicate ◽  
Ternary Variables ◽  
Order Predicate Calculus ◽  
Predicate Variable

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

On the reduction of the decision problem

1947 ◽  
Vol 12 (3) ◽  
pp. 65-73 ◽  
Author(s):  
László Kalmár ◽  
János Surányi
Keyword(s):  
Special Form ◽  
Decision Problem ◽  
Predicate Calculus ◽  
Reduction Theorem ◽  
First Order ◽  
Title One ◽  
Order Predicate Calculus ◽  
Predicate Variable

In the first paper of the above main title, one of us has proved that any formula of the first order predicate calculus is equivalent (as to being satisfiable or not) to some binary first order formula having a prefix of the form (Ex1)(x2)(Ex3) … (xn) and containing a single predicate variable. This result is an improvement of a theorem of Ackermann stating that any first order formula is equivalent to another with a prefix of the above form but saying nothing about the number of predicate variables appearing therein. Hence the question arises if other theorems reducing the decision problem to the satisfiability question of the first order formulas with a prefix of a special form can be improved in like manner. In the present paper we shall answer this question concerning Gödel's reduction theorem stating that any first order formula is equivalent to another the prefix of which has the form

scholarly journals On a Propositional Calculus Whose Decision Problem is Recursively Unsolvable

1970 ◽  
Vol 38 ◽  
pp. 145-152
Author(s):  
Akira Nakamura
Keyword(s):  
Decision Problem ◽  
Propositional Calculus ◽  
Predicate Calculus ◽  
Reduction Theorem ◽  
List Type ◽  
First Order ◽  
Order Predicate Calculus

The purpose of this paper is to present a propositional calculus whose decision problem is recursively unsolvable. The paper is based on the following ideas: (1) Using Löwenheim-Skolem’s Theorem and Surányi’s Reduction Theorem, we will construct an infinitely many-valued propositional calculus corresponding to the first-order predicate calculus.(2) It is well known that the decision problem of the first-order predicate calculus is recursively unsolvable.(3) Thus it will be shown that the decision problem of the infinitely many-valued propositional calculus is recursively unsolvable.

A normal form theorem for first order formulas and its application to Gaifman's splitting theorem

1984 ◽  
Vol 49 (4) ◽  
pp. 1262-1267
Author(s):  
Nobuyoshi Motohashi
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Atomic Formula ◽  
Splitting Theorem ◽  
First Order ◽  
Usual Manner ◽  
Binary Predicate ◽  
Normal Form Theorem ◽  
Image Position ◽  
Order Predicate Calculus

Let L be a first order predicate calculus with equality which has a fixed binary predicate symbol <. In this paper, we shall deal with quantifiers Cx, ∀x ≦ y, ∃x ≦ y defined as follows: CxA(x) is ∀y∃x(y ≦ x ∧ A(x)), ∀x ≦ yA{x) is ∀x(x ≦ y ⊃ A(x)), and ∃x ≦ yA(x) is ∃x(x ≦ y ∧ 4(x)). The expressions x̄, ȳ, … will be used to denote sequences of variables. In particular, x̄ stands for 〈x1, …, xn〉 and ȳ stands for 〈y1,…, ym〉 for some n, m. Also ∃x̄, ∀x̄ ≦ ȳ, … will be used to denote ∃x1 ∃x2 … ∃xn, ∀x1 ≦ y1 ∀x2 ≦ y2 … ∀xn ≦ yn, …, respectively. Let X be a set of formulas in L such that X contains every atomic formula and is closed under substitution of free variables and applications of propositional connectives ¬(not), ∧(and), ∨(or). Then, ∑(X) is the set of formulas of the form ∃x̄B(x̄), where B ∈ X, and Φ(X) is the set of formulas of the form.Since X is closed under ∧, ∨, the two sets Σ(X) and Φ(X) are closed under ∧, ∨ in the following sense: for any formulas A and B in Σ(X) [Φ(X)], there are formulas in Σ(X)[ Φ(X)] which are obtained from A ∧ B and A ∨ B by bringing some quantifiers forward in the usual manner.

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.

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.

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

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.

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