bluuebunny/crossref_metadata_2025_split huggingface dataset:
The theory of quaternialty
Authors: Gottschalk W. H. 
Published: May 2006

It is well-known that every involution in a logical or mathematical system gives rise to a theory of duality; for example, negation in the sentential calculus and predicate calculus, complementation in the calculus of classes, complementation and conversion in the calculus of relations, etc. The purpose of this note is to call attention to the fact that every involution in a logical or mathematical system gives rise to a theory of quaternality and that the square of quaternality, of which the classical squares of opposition are special cases, provides a diagrammatic representation for much of the theory of quaternality. For the sake of definiteness we expose the theory of quaternality in the context of the lower predicate calculus.Let ϕ and ψ be formulas of the lower predicate calculus. The constants {T and F} {∧ and ∨} {→ and ⇷} {← and ⇸}{↔ and ⇹} {↑ and ↓} {∧ (all) and ∨ (some)} are dual. The constant ~ is self-dual. The negational of ϕ, denoted ϕN, is the formula obtained from ϕ by interchanging negated and unnegated variables (that is, sentence variables and predicate variables) and by interchanging dual constants. The contradual of ϕ, denoted ϕC, is the formula obtained from ϕ by interchanging negated and unnegated variables. The dual of ϕ, denoted ϕD, is the formula obtained from ϕ by interchanging dual constants. The four formulas ϕ, ϕN, ϕC, ϕD, are quaternals of each other.
New notes on simplicity
Authors: Goodman Nelson 
Published: May 2006

The treatment of simplicity in my book and in earlier articles needs one important amplification and certain minor amendments. The basic method for computing complexity values nevertheless remains the same.New studies show that so to adjust the actual technique of computation as to take full account of the factor of reflexivity would result in unwieldy rules. Fortunately, however, this is unnecessary; for the method already prescribed works for thoroughly irreflexive predicates, and every other predicate is replaceable by such predicates. Thus we need only require that before the complexity of a basis is evaluated every primitive be replaced by the simplest adequate set of irreflexive predicates.Consider, for example, two-place predicates that are not irreflexive. Any two-place (or indeed any n-place) predicate that is satisfied only by identity pairs (or identity n-ads) can obviously be replaced by a one-place predicate. Any two-place predicate that is satisfied by some non-identity pairs and that is either exactly reflexive (i.e. satisfied also by just the identity pairs of elements in such non-identity pairs) or totally reflexive (i.e. satisfied also by all identity pairs) can be replaced by a two-place irreflexive predicate. In the former case, for instance, we replace “P” by “Q”, so explained that Qx, y if and only if Px, y and x ≠ y; and then define:Finally, a non-irreflexive two-place predicate “P” that is of none of these three kinds can be replaced by a one-place and an irreflexive two-place predicate as follows:—let Qx if and only if Px,x, and let Rx,y if and only if Px,y andx ≠ y; then define:Predicates of more than two places offer much greater variety since they may exhibit any sort of reflexivity with respect to any selection of places.
Conditions affecting the application of symbolic logic
Authors: Berkeley Edmund C. 
Published: May 2006

In 1513 the Spanish adventurer Balboa discovered an ocean three times bigger than any other ocean in the world. When he returned to Europe at the end of the year and told the Spaniards, they probably did not fully believe him. Like Balboa, the symbolic logicians have recently discovered a new vast ocean of thought, and like the Europeans of his day, many people are hardly aware of the discovery or its full implications.The newly discovered ocean, symbolic logic, is, in its broadest sense, a new science which studies through use of efficient symbols the nature and properties of all nonnumerical relations, seeking precise meanings and necessary conclusions. As an applied science, it holds immense promise. For example, it may give us an unambiguous language for political, economic, and social fields, which will conveniently reflect the structure of these fields and make discussion and analysis easy.
Models of logical systems
Authors: Kemeny John G. 
Published: May 2006

The purpose of this paper is to show that some of the difficulties confronting logicians are caused by too narrow an interpretation of the function of mathematical logic. I hope to show that very fruitful results can be obtained by considering a much wider range of problems than usually found in the literature.I believe that the function of mathematical logic is to study all logical systems. The problems now considered usually concern some particular system or set of systems. Of course we would have to agree first on what a logical system is. I shall suggest a tentative definition. The rest of my paper will proceed on the assumption that we have agreed on this question. However, it is only necessary for this paper that some definition should be accepted, it is not necessary that it should be the one I suasest.
On confirmation
Authors: Hosiasson-Lindenbaum Janina 
Published: May 2006

Taking for the relation of confirmation the following obvious axioms, we obtain several more or less well-known theorems and are able to solve in a definite and strict manner several problems concerning confirmation.Let a, b, and c be variable names of sentences belonging to a certain class, the operations a·b, a + b, and ā the (syntactical) product, the sum, and the negation of them. Let us further assume the existence of a real non-negative function c(a, b) of a and b, when b is not self-contradictory. Let us read ‘c(a, b)’ ‘degree of confirmation of a with respect to b' and take the following axioms:Axiom I. If a is a consequence of b, c(a, b) = 1.Axiom II. If is a consequence of c,Axiom III. c(a·b, c) = c(a, c)·c(b, a·c).Axiom IV. If b is equivalent to c, c(a, b) = c(a, c).As may be easily seen, the interval of variation for c is (0, 1); this is quite conventional.
On the axiom of extensionality – Part I
Authors: Gandy R. O. 
Published: May 2006

In part I of this paper it is shown that if the simple theory of types (with an axiom of infinity) is consistent, then so is the system obtained by adjoining axioms of extensionality; in part II a similar metatheorem for Gödel-Bernays set theory will be proved. The first of these results is of particular interest because type theory without the axioms of extensionality is fundamentally rather a simple system, and it should, I believe, be possible to prove that it is consistent.Let us consider — in some unspecified formal system — a typical expression of the axiom of extensionality; for example:where A(h) is a formula, and A(f), A(g) are the results of substituting in it the predicate variagles f, g for the free variable h. Evidently, if the system considered contains the predicate calculus, and if h occurs in A(h) only in parts of the form h(t) where t is a term which lies within the range of the quantifier (x), then 1.1 will be provable. But this will not be so in general; indeed, by introducing into the system an intensional predicate of predicates we can make 1.1 false. For example, Myhill introduces a constant S, where ‘Sϕψχω’ means that (the expression) ϕ is the result of substituting ψ for χ in ω.
Toward a calculus of concepts
Authors: Quine W. V. 
Published: May 2006

By concepts will be meant propositions (or truth-values), attributes (or classes), and relations of all degrees. The degree of a concept will be said to be 0, 1, or n (> 1), and the concept will be said to be medadic, monadic, or n-adic, according as the concept is a proposition, an attribute, or an n-adic relation. The common procedure in systematizing logistic is to treat these successive degrees as ultimately separate categories. The partition is not rested upon properties of the thus classified elements within the formal system, but is imposed rather at the metamathematical level, through stipulations as to what combinations of signs are to be accorded or denied meaning. Each function of the formal system is restricted, thus metamathematically, to one degree for its values and to one for each of its arguments. The theory of types imports a further scheme of infinite partition, imposed by metamathematical stipulations as to the relative types of admissible arguments of the several functions and stipulations as to the types of the values of the functions relative to the types of the arguments.The elaborateness of the metamathematical grillwork which thus underlies formal logistic accounts in part for the tendency of those interested in logistic less for the matters treated than for the structures exemplified to limit their attention to the propositional calculus and the Boolean calculus of attributes (or classes), which, taken separately, are independent of the partitioning. A second reason for the algebraic appeal of these departments is their freedom from bound (apparent) variables: for use of bound variables fuses systematic considerations with notational or metamathematical ones in a way which resists ordinary formulation in terms of fixed functions and their arguments. Freedom from bound variables may be regarded, indeed, as the feature distinguishing algebra from analysis.
A note on universal sets
Authors: Lachlan A. H. 
Published: May 2006

In this note is proved the following:Theorem.Iƒ A × B is universal and one oƒ A, B is r.e. then one of A, B is universal.Letα, τbe 1-argument recursive functions such thatxgoes to (σ(χ), τ(χ)) is a (1–1) map of the natural numbers onto all ordered pairs of natural numbers. A set A of natural numbers is calleduniversalif every r.e. set is (many-one) reducible to A; A × B is calleduniversalif the set
A note on many·one reducibility
Authors: Liu Shih-Chao 
Published: May 2006

In this note we partly answer a question left open in [1, p. 52] by proving the following theorem.Theorem. Supposep0≧2, ωn+1·p0≦σ<ωn+1(p0+1). Then W(σ) is uniformly many-one reducible to W(τ) for τ≧(ωn+1·p0)+1.To prove this theorem we need only show that there is a recursive function f(e) such that if ∣e∣<σ, then ∣f(e)<(ωn+1·p0)+1, while if ∣e∣ ≮ σ, f(e)∉W. (For the notations see [2].)
Replacement in some modal systems
Authors: Thomas Ivo 
Published: May 2006

In the original formulations of Lewis's S1–S5 in [1], Feys's S1°, S2° in [2], and Sobociński's S3°, S4° in [3], a primitive rule is that of replacement for strict equivalents which it is enough to express formally as:RI: where γ′ is like γ except for having β at one place where γ has α.
Proof of the law of infinite conjunction using the perfect disjunctive normal form
Authors: Thomson James 
Published: May 2006

The law of infinite conjunction (Quine's name, see Methods of logic, rev. ed., Holt & Co., New York, p. 254) says that a set of truthfunctional formulas is satisfiable provided every finite subset is. We give a proof of the law which uses some properties of the perfect disjunctive normal form.Any satisfiable formula can be written as a disjunction of conjunctions of literals (a literal is a statement-letter or the negation of one) in which a statement-letter which occurs in any conjunction occurs exactly once in each; a literal by itself counts as a conjunction and a conjunction by itself as a disjunction.
The undecidability of intuitionistic theories of algebraically closed fields and real closed fields
Authors: Gabbay Dov M. 
Published: May 2006

Let T be a set of axioms for a classical theory TC (e.g. abelian groups, linear order, unary function, algebraically closed fields, etc.). Suppose we regard T as a set of axioms for an intuitionistic theory TH (more precisely, we regard T as axioms in Heyting's predicate calculus HPC).Question. Is TH decidable (or, more generally, if X is any intermediate logic, is TX decidable)? In [1] we gave sufficient conditions for the undecidability of TH. These conditions depend on the formulas of T (different axiomatization of the same TC may give rise to different TH) and on the classical model theoretic properties of TC (the method did not work for model complete theories, e.g. those of the title of the paper). For details see [1]. In [2] we gave some decidability results for some theories: The problem of the decidability of theories TH for a classically model complete TC remained open. An undecidability result in this direction, for dense linear order was obtained by Smorynski [4]. The cases of algebraically closed fields and real closed fields and divisible abelian groups are treated in this paper. Other various decidability results of the intuitionistic theories were obtained by several authors, see [1], [2], [4] for details.One more remark before we start. There are several possible formulations for an intuitionistic theory of, e.g. fields, that correspond to several possible axiomatizations of the classical theory. Other formulations may be given in terms of the apartness relation, such as the one for fields given by Heyting [5]. The formulations that we consider here are of interest as these systems occur in intuitionistic mathematics. We hope that the present methods could be extended to the (more interesting) case of Heyting's systems [5].
Elementary intuitionistic theories
Authors: Smorynski C. 
Published: May 2006

The present paper concerns itself primarily with the decision problem for formal elementary intuitionistic theories and the method is primarily model-theoretic. The chief tool is the Kripke model for which the reader may find sufficient background in Fitting's book Intuitionistic logic model theory and forcing (North-Holland, Amsterdam, 1969). Our notation is basically that of Fitting, the differences being to favor more standard notations in various places.The author owes a great debt to many people and would particularly like to thank S. Feferman, D. Gabbay, W. Howard, G. Kreisel, G. Mints, and R. Statman for their valuable assistance.The method of elimination of quantifiers, which has long since proven its use in classical logic, has also been applied to intuitionistic theories (i) to demonstrate decidability ([9], [15], [17]), (ii) to prove the coincidence of an intuitionistic theory with its classical extension ([9], [17]), and (iii), as we will see below, to establish relations between an intuitionistic theory and its classical extension. The most general of these results is to be obtained from the method of Lifshits' quantifier elimination for the intuitionistic theory of decidable equality.Since the details of Lifshits' proof have not been published, and since the proof yields a more general result than that stated in his abstract [15], we include the proof and several corollaries.
On axiomatizing fragments
Authors: Smorynski C. 
Published: May 2006

The main purpose of this note is to present new semi-model-theoretic proofs of some axiomatization results. Principally, we prove the result of [van Dalen and Statman] on the axiomatization of the equality fragment of the intuitionistic theory of apartness by ω-fold stability axioms. Further examples are also discussed.The intuitionistic theories of equality and apartness are given by the following nonlogical axioms:EQ .AP .In AP, one can define a notion of equality by means of the abbreviation:.The corresponding interpretation of E Q in AP is not faithful — since equality is a negation, we have stability of equality:.The equality fragment of AP is axiomatized by a suitable generalization of the stability of equality. To state it, one first defines a sequence of partial apartness relations:.A simple induction shows, for all n,whence.
The inadequacy of the neighbourhood semantics for modal logic
Authors: Gerson Martin 
Published: May 2006

We present two finitely axiomatized modal propositional logics, one betweenTandS4 and the other an extension ofS4, which are incomplete with respect to the neighbourhood or Scott-Montague semantics.Throughout this paper we are referring to logics which contain all the classical connectives and only one modal connective □ (unary), no propositional constants, all classical tautologies, and which are closed under the rules of modus ponens (MP), substitution, and the rule RE (fromA↔Binfer αA↔ □B). Such logics are calledclassicalby Segerberg [6]. Classical logics which contain the formula □p∧ □q→ □(p∧q) (denoted byK) and its “converse,” □{p∧q)→ □p∧ □q(denoted byR) are called regular;regularlogics which are closed under the rule of necessitation, RN (fromAinfer □A), are callednormal. The logics that we are particularly concerned with are all normal, although some of our results will be true for all regular or all classical logics. It is well known thatKandRand closure under RN imply closure under RE and also that normal logics are also those logics closed under RN and containing □{p→q) → {□p→ □q).
On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model
Authors: Mckenzie Ralph 
Published: May 2006

An algorithm has been described by S. Burris [3] which decides if a finite set of identities, whose function symbols are of rank at most 1, has a finite, nontrivial model. (By “nontrivial” it is meant that the universe of the model has at least two elements.) As a consequence of some results announced in the abstracts [2] and [8], it is clear that if the restriction on the ranks of function symbols is relaxed somewhat, then this finite model problem is no longer solvable by an algorithm, or at least not by a “recursive algorithm” as the term is used today.In this paper we prove a sharp form of this negative result; showing, by the way, that Burris' result is in a sense the best possible result in the positive direction. Our main result is that in a first order language whose only function or relation symbol is a 2-place function symbol (the language of groupoids), the set of identities that have no nontrivial model, is recursively inseparable from the set of identities such that the sentence has a finite model. As a corollary, we have that each of the following problems, restricted to sentences defined in the language of groupoids, is algorithmically unsolvable: (1) to decide if an identity has a finite nontrivial model; (2) to decide if an identity has a nontrivial model; (3) to decide if a universal sentence has a finite model; (4) to decide if a universal sentence has a model. We note that the undecidability of (2) was proved earlier by McNulty [13, Theorem 3.6(i)], improving results obtained by Murskiǐ [14] and by Perkins [17]. The other parts of the corollary seem to be new.
Extending Gödel's negative interpretation to ZF
Authors: Powell William C. 
Published: May 2006

In [5] Gödel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Gödel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Gödel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with

Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.
On the inconsistency of systems similar to
Authors: Bunder M. W. , Meyer R. K. 
Published: May 2006

This note shows that the inconsistency proof of the system of illative combinatory logic given in [1] can be simplified as well as extended to the absolute inconsistency of a more general system.One extension of the result in [1] lies in the fact that the following weakened form of the deduction theorem for implication will lead to the inconsistency:Also the inconsistency follows almost as easily foras it does for ⊢ H2X for arbitrary X, so we will consider the more general case.The only properties we require other than (DT), (1) and Rule Eq for equality are modus ponens,andLet G = [x] Hn−1x⊃: . … H2x⊃ :Hx⊃ . x ⊃ A, where A is arbitrary. Then if Y is the paradoxical combinator and X = YG, X = GX.Now X ⊂ X, i.e.,
Theories with models of prescribed cardinalities
Authors: Mekler Alan 
Published: May 2006

The Löwenheim–Skolem theorem states that if a theory has an infinite model it has models of all cardinalities greater than or equal to the cardinality of the language in which the theory is defined. A natural question is what happens if there is a model whose cardinality is less than that of the language.If κ is an infinite cardinal less than the first measurable cardinal and κ < κω, the Rabin–Keiler theorem [1, p. 139] gives an example of a theory which has a model of cardinality κ in which every element is the interpretation of a constant and all other models have cardinality μ ≥ κω. Keisler has also shown that if a theory has a model of cardinality κ it has models of all cardinalities μ ≥ κω. We will show that within the bounds of the above theorems anything can happen.The main result is as follows.
Categories of frames for modal logic
Authors: Thomason S. K. 
Published: May 2006

§1. A complete atomic modal algebra (CAMA) is a complete atomic Boolean algebra with an additional completely additive unary operator. A (Kripke) frame is just a binary relation on a nonempty set. If is a frame, then is a CAMA, where mX = {y ∣ (∃x)(y < x Є X)}; and if is a CAMA then is a frame, where is the set of atoms of and b1 < b2 ⇔ b1 ∩ mb2 ≠∅.Now , and the validity of a modal formula on is equivalent to the satisfaction of a modal algebra polynomial identity by and conversely, so the validity-preserving constructions on frames ought to be in some sense equivalent to the identity-preserving constructions on CAMA's. The former are important for modal logic, and many of the results of universal algebra apply to the latter, so it is worthwhile to fix precisely the sense of the equivalence.The most important identity-preserving constructions on CAMA's can be described in terms of homomorphisms and complete homomorphisms. Let and be the categories of CAMA's with homomorphisms and complete homomorphisms, respectively. We shall define categories and of frames with appropriate morphisms, and show them to be dual respectively to and . Then we shall consider certain identity-preserving constructions on CAMA's and attempt to describe the corresponding validity-preserving constructions on frames.The proofs of duality involve some rather detailed calculations, which have been omitted. All the category theory a reader needs to know is in the first twenty pages of [7].
λ-scales, κ-Souslin sets and a new definition of analytic sets
Authors: Busch Douglas R. 
Published: May 2006

SummaryThe notions of sets of reals being κ-Souslin (κ a cardinal) and admitting a λ-scale (λ an ordinal) are due respectively to D. A. Martin and Y.N. Moschovakis. A set is ω-Souslin if and only if it is Σ11 (analytic). We show that a set is ω-Souslin if and only if it admits an (ω + l)-scale. Jointly with Martin and Solovay we show that if κ is uncountable and has cofinality ω, then being κ-Souslin is equivalent to admitting a κ-scale. Our results together with those of Kechris give a new simultaneous characterization of Σ11 and Δ11 (Borel) sets (a set is Σ11 if it admits an (ω + 1)-scale and Δ11 if it admits an ω-scale) and determine completely the relation between the κ-Souslin sets and the sets admitting λ-scales.