Showing papers in "Journal of Symbolic Logic in 1970"
TL;DR: Methodological preliminaries of generative grammars as theories of linguistic competence; theory of performance; organization of a generative grammar; justification of grammar; descriptive and explanatory theories; evaluation procedures; linguistic theory and language learning.
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TL;DR: In this article, Henkin made the observation that certain second-order existential formulas may be thought of as the Skolem normal forms of formulas of a language which is first-order in every respect except its incorporation of a form of partially-ordered quantification.
Abstract: In [3] Henkin made the observation that certain second-order existential formulas may be thought of as the Skolem normal forms of formulas of a language which is first-order in every respect except its incorporation of a form of partially-ordered quantification. One formulation of this sort of language is the closure of a first-order language under the formation rule that Qφ is a formula whenever φ is a formula and Q, which is to be thought of as a quantifier-prefix, is a system of partial order whose universe is a set of quantifiers. Although he introduced this idea in a discussion of infinitary logic, Henkin went on to discuss its application to finitary languages, and he concluded his discussion with a theorem of Ehrenfeucht that the incorporation of an extremely simple partially-ordered quantifier-prefix (the quantifiers ∀x, ∀y, ∃v, and ∃w, with the ordering {〈∀x, ∃v〉, 〈∀y, ∃w〉}) into any first-order language with identity gives a language capable of expressing the infinitary quantifier ∃zκ0x.
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TL;DR: The present paper shall provide a strong completeness theorem for RM by generalizing the semantics of Meyer, and shall obtain the promised results for extensions of RM.
Abstract: Schiller Joe Scroggs in [9] established remarkable facts concerning “normal” extensions of the modal sentential calculus S5, the most notable of these facts being that all such proper extensions have finite characteristic matrices. The major import of the present paper is that like facts hold for the relevant sentential calculus R-Mingle (RM). Robert K. Meyer in [6] has obtained an important completeness result for RM, which will play a central role in our results. However, in §2 we shall obtain a new proof of Meyer's result as a by-product of the algebraic logic that we develop in §1. Also in §2 we shall obtain the promised results for extensions of RM. In §3 we shall provide a strong completeness theorem for RM by generalizing the semantics of Meyer.
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TL;DR: A denumerable structure is said to be recursive iff its universe is a recursive subset of the natural numbers and its relations and operations are recursive.
Abstract: A denumerable structure is said to be recursive iff its universe is a recursive subset of the natural numbers and its relations and operations are recursive. For example, the standard model of number theory is recursive. A structure is said to be recursively presentable iff it is isomorphic to a recursive structure. For example, a Boolean algebra generated by ℵ0 free generators is easily seen to be recursively presentable. (For basic facts concerning Boolean algebras, the reader is referred to R. Sikorski [9] and A. Tarski and A. Mostowski [10].)
83 citations
TL;DR: It is shown that the Kreisel-Putnam logic has the finite model property, and since it is finitely axiomatizable, it is decidable by [4]; the decidability of Scott's system was proved by J. G. Anderson in his thesis in 1966.
Abstract: The intuitionistic propositional logic I has the following disjunction property
This property does not characterize intuitionistic logic. For example Kreisel and Putnam [5] showed that the extension of I with the axiom has the disjunction property. Another known system with this propery is due to Scott [5], and is obtained by adding to I the following axiom: In the present paper we shall prove, using methods originally introduced by Segerberg [10], that the Kreisel-Putnam logic is decidable. In fact we shall show that it has the finite model property, and since it is finitely axiomatizable, it is decidable by [4]. The decidability of Scott's system was proved by J. G. Anderson in his thesis in 1966.
TL;DR: The notion of a “partition relation”, as it has been studied in the context of set theory for the past several years, was inspired by the following theorem of F. P. Ramsey: there exists an infinite subset x of the nonnegative integers all of whose n-element subsets are contained in only one of A or B.
Abstract: The notion of a “partition relation”, as it has been studied in the context of set theory for the past several years, was inspired by the following theorem of F. P. Ramsey [14]: Theorem 0.1. Let n be a positive integer and let {A, B} be a partition of those subsets of the nonnegative integers containing exactly n elements. Then there exists an infinite subset x of the nonnegative integers all of whose n-element subsets are contained in only one of A or B. (Any such set x is said to be “homogeneous” for the partition.)
TL;DR: Two theorems are proved that there exists an axiomatizable, essentially undecidable theory in standard formalization such that all axiom atizable extensions of are finite extensions.
Abstract: In this paper we prove two theorems. They answer questions raised by Myhill in 1956. (We recall the well-known fact that Myhill's invention of the maximal set in 1956 [2] stemmed from his attempt to prove I below.) I. There exists an axiomatizable, essentially undecidable theory in standard formalization such that all axiomatizable extensions of are finite extensions. II. There exists an axiomatizable but undecidable theory in standard formalization such that (a) has a consistent, complete, decidable extension , (b) If is an axiomatizable extension of then either (i) is a finite extension of , or (ii) is a finite extension of .
TL;DR: In this paper, a topological enlargement of a separated Hausdorff topological group *G is shown to possess the same formal properties as *G in the sense explained in [8], and every set of subsets of *G with the finite intersection property satisfies ∩*Aν ≠ o, where the *Aν are the extensions of the Aν in *G.
Abstract: Let G be a separated (Hausdorff) topological group and let *G be an enlargement of G (see [8]). Thus, *G (i) possesses the same formal properties as G in the sense explained in [8], and (ii) every set of subsets {Aν} of G with the finite intersection property—i.e. such that every nonempty finite subset of {Aν} has a nonempty intersection—satisfies ∩*Aν ≠ o, where the *Aν are the extensions of the Aν in *G, respectively.
TL;DR: This paper shows that the decision problem is unsolvable for formulas that are like those considered by Maslov except that they have prefixes of the form ∀ x ∃ y 1 … ∀ y k ∀ z and describes some special properties of formulas in which all disjunctions are binary that implies that any proof of the result, that a class of formulas is a reduction class for satisfiability, is necessarily indirect.
Abstract: In [8] S. J. Maslov gives a positive solution to the decision problem for satisfiability of formulas of the formin any first-order predicate calculus without identity where h, k, m, n are positive integers, αi, βi are signed atomic formulas (atomic formulas or negations of atomic formulas), and ∧, ∨ are conjunction and disjunction symbols, respectively (cf. [6] for a related solvable class). In this paper we show that the decision problem is unsolvable for formulas that are like those considered by Maslov except that they have prefixes of the form ∀x∃y1 … ∃yk∀z. This settles the decision problems for all prefix classes of formulas for formulas that are in prenex conjunctive normal form in which all disjunctions are binary (have just two terms). In our concluding section we report results on decision problems for related classes of formulas including classes of formulas in languages with identity and we describe some special properties of formulas in which all disjunctions are binary including a property that implies that any proof of our result, that a class of formulas is a reduction class for satisfiability, is necessarily indirect. Our proof is based on an unsolvable combinatorial tag problem.
TL;DR: By language the authors understand a lower predicate calculus with identity and (perhaps) relation and function symbols, and admit s = t as well formed, no matter what the sorts of s and t.
Abstract: By language we understand a lower predicate calculus with identity and (perhaps) relation and function symbols. It is convenient to allow for more than one sort of variable. Now each individual constant (if there are any) is of a specified sort, the formal expressions R(t1, … tn), f(t1,…, tn) are well formed only if the terms t1, …, tn are of specified sorts determined by the relation symbol R and the function symbol f, and the term f(t1, …, tn) (if well formed) is of a sort determined by f. We admit s = t as well formed, no matter what the sorts of s and t.
TL;DR: The strong homogeneity conjecture asserts that, for any Turing degree, a , there is a jump preserving isomorphism from the upper semilattice of degrees to the upper seminar of degrees above a .
Abstract: The strong homogeneity conjecture asserts that, for any Turing degree, a, there is a jump preserving isomorphism from the upper semilattice of degrees to the upper semilattice of degrees above a. Rogers [3, p. 261] states that this problem is open and notes that its truth would simplify many proofs about degrees. It is, in fact, false. More precisely, let 0 be the smallest degree and let 0(n) be the nth iterated jump of 0, as defined in [3, pp. 254–256].
TL;DR: It is proved that if 9 is an ultrafilter and X0 ?
Abstract: We shall prove that if 9 is an ultrafilter and X0 ? A = ni/9, Ato = A. This affirms a conjecture of Keisler.
TL;DR: This paper gives a related embedding of (first order) classical logic directly into ( first order) S4, with or without the Barcan formula.
Abstract: There are well-known embeddings of intuitionistic logic into S4 and of classical logic into S5. In this paper we give a related embedding of (first order) classical logic directly into (first order) S4, with or without the Barcan formula. If one reads the necessity operator of S4 as ‘provable’, the translation may be roughly stated as: truth may be replaced by provable consistency. A proper statement will be found below. The proof is based ultimately on the notion of complete sequences used in Cohen's technique of forcing [1], and is given in terms of Kripke's model theory [3], [4].
TL;DR: If L is a finite distributive lattice then L is isomorphic to an initial segment of hyperdegrees, and as a consequence the elementary theory of the ordering ofhyperdegrees is recursively undecidable.
Abstract: An initial segment of hyperdegrees is a set S of hyperdegrees such that whenever h ∈ S and k ≦ h then k ∈ S The main results of this paper affirm the existence of initial segments having certain order types In particular, if L is a finite distributive lattice then L is isomorphic to an initial segment of hyperdegrees [Theorem 1]; as a consequence the elementary theory of the ordering of hyperdegrees is recursively undecidable [Corollary 1]
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