bluuebunny/crossref_metadata_2025_split huggingface dataset:
Alternative postulate sets for Lewis's S5
Authors: Lemmon E. J. 
Published: May 2006

Professor Prior (Formal logic (1955), pp. 305–307) gives alternative postulate sets for Lewis's S5, one substantially Lewis's own set (p. 305, 6.1) and another, considerably simpler, due to Prior himself (pp. 306–307, 6.5). This note aims at showing that the systems based on these two postulate sets are in fact equivalent. We label the system based on Lewis's postulates ‘S5’, as usual, and that on Prior's postulates ‘P’.The form of the postulate set for P considered here is as follows. We add to the full propositional calculus one definition:Df. M: M = NLN;and two special rules:L1: ⊦Cαβ → ⊦CLαβ;L2: ⊦Cαβ → ⊦CαLβ, if α is fully modalized (a propositional formula is fully modalized, if and only if all occurrences of propositional variables in it lie within the scope of a modal operator).Prior's text (pp. 201–205) indicates sufficiently how it could be shown that any thesis of S5 is a thesis of P and that the rules and definitions of S5 are obtainable as derived rules of P. We turn, therefore, to the problem of showing that any thesis of P is a thesis of S5 and that the rules and definition of P are obtainable as derived rules of S5.
Quantification and the empty domain
Authors: Quine W. V. 
Published: May 2006

Quantification theory, or the first-order predicate calculus, is ordinarily so formulated as to provide as theorems all and only those formulas which come out true under all interpretations in allnon-emptydomains. There are two strong reasons for thus leaving aside the empty domain.(i) WhereDis any non-empty domain, any quantificational formula which comes out true under all interpretations in all domains larger thanDwill come out true also under all interpretations inD. (Cf. [4], p. 92.) Thus, though any small domain has a certain triviality, all but one of them, the empty domain, can be included without cost. To include the empty one, on the other hand, would mean surrendering some formulas which are valid everywhere else and thus generally useful.(ii) An easy supplementary test enables us anyway, when we please, to decide whether a formula holds for the empty domain. We have only to mark the universal quantifications as true and the existential ones as false, and apply truth-table considerations.Incidentally, the existence of that supplementary test shows that there is no difficulty in framing aninclusivequantification theory (i.e., inclusive of the empty domain) if we so desire. A proof in this theory can be made to consist simply of a proof in the exclusive (or usual) theory followed by a check by the method of (ii). We may, however, be curious to see a more direct or autonomous formulation: one which does not consist, like the above, of the exclusive theory plus a rule of expurgation. And, in fact, such formulations have of late been forthcoming: Mostowski [5], Hailperin [3], and, as part of a broader context, Church [2]. I shall not presuppose acquaintance with these papers, except in my final paragraph (and then only with [3]).
On the independence of Henkin's axioms for fragments of the propositional calculus
Authors: L'abbé Maurice 
Published: May 2006

A general system of axioms has been given by Henkin for a fragment of the propositional calculus having as primitive symbols, in addition to the usual parentheses, variables, and implication sign ⊃, an arbitrarily given truth function symbol ϕ. This system of axioms, which we shall denote by S(⊃, ϕ), contains the following three axiom schemataplus the 2m further axiom schemata involving the symbol ϕwhere ϕ is an m-placed function symbol. We refer to Henkin's paper, p. 43, for the detailed description of the axiom schemata (4).The remark was made in the above mentioned paper that each of the 2m axiom schemata of (4) is trivially independent of the rest of the axioms of S(⊃, ϕ), and it was conjectured that the axiom schemata (1), (2) and (3) are also independent. In this note, we prove the general independence of the axiom schemata (1) and (2). As for (3), we show on the one hand its independence in the systems S(⊃) and S(⊃, f), and, on the other hand, its dependence in the system S(⊃, ∼). The net result is, therefore, that in any of these systems of axioms S(⊃, ϕ) all the axiom schemata are independent, except possibly the axiom schema (3).
On formulas with valid cases
Authors: Quine W. V. 
Published: May 2006

Commonly, when we succeed in showing a formula of quantification theory to be consistent, we do so by producing a true interpretation. Sometimes we achieve the same effect without even exceeding the resources of quantification theory itself: we show a formula consistent by producing a valid formula from it by substitution. Example: ‘(∃x)Fx ⊃ (x)(∃y) (Gxy ▪ Fy)’ is seen consistent by noting its valid substitution case ‘(∃x)Fx ⊃ (x)(∃y)(Fx ∨ Fy ▪ Fy)’. How generally available is this latter method? I shall show that it is available if and only if the formula whose consistency is shown is satisfiable in a one-member universe.The “only if” part is immediate. For, if ψ is a substitution case of ϕ, then ϕ is satisfiable wherever ψ is; and ψ, if valid, is satisfiable in a one-member universe.Conversely, suppose a true interpretation of ϕ in a one-member universe. Each predicate letter of ϕ is thereby interpreted outright as true or false, in effect, since there is no varying the values of ‘x’, ‘y’, etc. Now form ψ from ϕ by substitution as follows: change each atomic formula ϕi(e.g. ‘Fx’, ‘Fy’, ‘Gxy’) to ⌜ϕi⊃ϕi⌝ if its predicate letter is one that was interpreted as true, or to ⌜ϕi ▪ − ⊃ϕi⌝ if its predicate letter is one that was interpreted as false. Clearly ψ under all interpretations even in large universes, will receive the truth value that ϕ received under the supposed interpretation in the one-member universe. So ψ is valid.
A formal proof of Gödel's theorem
Authors: Chwistek L. 
Published: May 2006

I. The present paper contains a formal proof of the following theorem of the elementary system ⊨ [ 1 0 ] c:This theorem means that if there were a theorem of ⊢0 ( 3 2 ) 0 stating that there is no contradiction in ⊢2 ( 5 4 ) 2, we would have a contradiction in ⊢2 ( 5 4 ) 2. (This is in accordance with the second theorem of Gödel.)Our proof is based on the calculus of the system ⊨ [ 1 0 ] c and of the metasystem ⊢0 ( 3 2 ) 0.Note that we could takeinstead of Ax E .1 ( L ) (.1 ( L ) .0 ( L ) L) Z. We would then have to do with the simple systems and metasystems of Hetper, which do not contain propositional variables or the logico-semantical axiom. Our method applies equally to the systems and metasystems of New Foundation and to the simple systems and metasystems of Hetper.Our proof can be considerably simplified by using a recent result of Hetper concerning ancestral functions. Hetper introduces the following abbreviations:If A ( Z1c), B ( x1cy1cZ1c) are propositions, the expression will be called an ancestral function (we prove without difficulty that this expression is a proposition). The expressions will be called respectively the principal term and the term of derivation of this function.
Implication and deducibility
Authors: Emch Arnold F. 
Published: May 2006

Lewis, in the presentation of his calculus of strict implication, contends that this calculus accords with the usual meaning of “implies” such that “p strictly implies q” is synonymous with “q is deducible from p” (pp. 126–127, 235–262). It is the purpose of this paper to present, in 1, certain considerations in the light of which this statement does not hold, and in 2, a new implicative relation upon which a calculus of propositions can be based such that it will accord with the relation of deducibility and from which it is possible to derive the calculi of strict and material implications.1. Strict implication regarded as synonymous with deducibility. In the presentation of his calculus of strict implication Lewis observes that despite inclusion of both intensional and extensional propositions in this calculus, the meaning of its elements, p, q, r, etc., “remain fixed, whether it is their extensional or their intensional relations which are in question” (p. 120). But if the meaning or content of the elements of this calculus is irrelevant to the logical truth which it contains, then, as Weiss has already remarked, the propositions of this calculus “are really extensionally treated, involving multiple values,” e.g., possibility, on the analogy of truth and falsity. The importance of this fact is that, in consequence, the relation of strict implication is subject to certain paradoxes, such as (19.74) “a proposition which is impossible strictly implies any proposition” or (19.75) “a proposition which is necessarily true is strictly implied by any proposition” (p. 174) or (19.84) “any two necessarily true propositions are strictly equivalent” (p. 176).
Some consequences of the axiom of power-set
Authors: Abian Alexander , Lamacchia Samuel 
Published: May 2006

In this paper we prove:Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.The proof of Theorem 2 will follow from the lemmas which we establish below.We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.Clearly, in every theory of sets we have:(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),(2) (x = y) ↔ (P(x) = P(y)),(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),(4) ⋃P(x) = x.In view of (2), (3) and the definition of equality, we have:Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.We have also, in view of (4):Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.
The decidability of certain intermediate propositional logics
Authors: Mckay C. G. 
Published: May 2006

Acquaintance with [11] and [8] will be assumed. Let be the alphabet →, &, ν, ┐, (,), A1, A2, … where →(I), &(C), ν (D) and ┐(N) are called the connectives. The letters appearing in parentheses are the names of the connectives. A1, A2 … are the atoms.Definition 1.1. A string S is an ICDN word iff there is a finite sequence of strings S1 … Sn = S such that for i, 1 ≤ i ≤ n, either S1 is an atom or S1 is one of the forms (S1→Sk), (S1 & Sk), (S1, ν Sk) or ┐S1,j, k ≤ i.
Short definitions of the ordinals
Authors: Brown Kenneth R. , Wang Hao 
Published: May 2006

In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms.The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions.
On a set theory of bernays
Authors: Tharp Leslie H. 
Published: May 2006

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema
where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.
Infinitary logic and admissible sets
Authors: Barwise Jon 
Published: May 2006

In recent years much effort has gone into the study of languages which
strengthen the classical first-order predicate calculus in various ways.
This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the
classical language, but(II) still simple enough to yield interesting general results. Languages
investigated include second-order logic, weak second-order logic, ω-logic,
languages with generalized quantifiers, and infinitary logic.
Axioms for strong reduction in combinatory logic
Authors: Hindley Roger 
Published: May 2006

In combinatory logic there is a system of objects which intuitively represent functions, and a binary relation between these objects, which represents the process of evaluating the result of applying a function to an argument. (This is explained fully in [1].) From this relation called weak reduction, “≥,” an equivalence relation is defined by saying that X is weakly equivalent to Y if and only if there exist n (with 0 ≤ n) and X0,…,Xη such that
It turns out that equivalent objects represent the same function, but two objects representing the same function need not be equivalent.
Generic expansions of structures
Authors: Knight Julia F. 
Published: May 2006

In this paper, Cohen's forcing technique is applied to some problems in model theory. Forcing has been used as a model-theoretic technique by several people, in particular, by A. Robinson in a series of papers [1], [10], [11]. Here forcing will be used to expand a family of structures in such a way that weak second-order embeddings are preserved. The forcing situation resembles that in Solovay's proof that for any theorem φ of GB (Godel-Bernays set theory with a strong form of the axiom of choice), if φ does not mention classes, then it is already a theorem of ZFC. (See [3, p. 105] and [2, p. 77].)The first application of forcing here is to the problem (posed by Keisler) of when is it possible to add a Skolem function to a pair of structures, one of which is an elementary substructure of the other, in such a way that the elementary embedding is preserved.It is not always possible to find such a Skolem function. Payne [9] found an example involving countable structures with uncountably many relations. The author [4], [6] found an example involving uncountable structures with only two relations. The problem remains open in case the structures are required both to be countable and to have countable type. Forcing is used to obtain a positive result under some special conditions.
An application of Σ40 determinacy to the degrees of unsolvability
Authors: Jockusch Carl G. 
Published: May 2006

If a and b are degrees of unsolvability, a is called (as in [1]) a minimal cover of b if b < a and no degree c satisfies b < c < a. A degree a is called a minimal cover if it is a minimal cover of some degree b. We show in ZF set theory that there is a degree d such that every degree a ≥ d is a minimal cover. In [1] it was remarked that this result follows via a lemma of D. A. Martin [4] from the determinacy of a certain Σ50 Gale-Stewart game (which in turn follows [5] from the existence of a measurable cardinal). The argument here is parallel to that of [1], but the essential new ingredient is the result of Paris [6] that Σ50 determinacy can be proved in ZF. Also it is necessary to replace the Σ50 game of [1] by a related Σ40 game and to use the method, rather than just the statement, of Martin's lemma.Theorem. There is a degreedsuch that every degreea ≥ dis a minimal cover.Proof. In a (Gale–Stewart) game on 2ω, players I and II construct a set of integers A as follows: At the nth stage, player I determines whether n ϵ A if n is even and player II if n is odd. Thus player I controls A0 and player II controls A1 where Ai = {n: 2n + i ϵ A}. Let Aij denote (Ai)j and let M (A, B) mean that the degree of A is a minimal cover of that of B. Consider now the particular game in which player I wins iff M(A, A00) holds.
On deciding the provability of certain fixed point statements
Authors: Boolos George 
Published: May 2006

Terminology. PA is Peano Arithmetic, classical first-order arithmetic with induction. ⌈A⌉ is the formal numeral in PA for the Gödel number of A. – A is the negation of A, (A&B) is the conjunction of A and B, and Bew(x) is the usual provability predicate for PA. neg(x), conj(x, y), bicond(x, y), and bew(x) are terms of PA such that for all sentences A and B of PA ⊢PA, neg(˹A˺) = ˹−A˺ ⊢PA Conj(˹A˺, ˹B˺)= ˹(A&B)˺ ⊢PA bicond(˹A˺, ˹B˺)= ˹(A ↔ B)˺, and ⊢PA bew(˹A˺) = ˹Bew(˹A˺)˺. T is the sentence ‘0 = 0’ and Con is the usual sentence expressing the consistency of PA. If A (x) is any formula of PA, then a fixed point of A(x) is a sentence S such that ⊢PAS ↔ A(˹S˺). (It is well known that every formula of PA with one free variable has a fixed point.) The P-terms are defined inductively by: the variable x is a P-term; if t(x) and u(x) are P-terms, so are neg(t(x)), conj(t(x), u(x)), and bew(t(x)). A basic P-formula is a formula Bew(t(x)), where t(x) is a P-term; and a P-formula is a truth-functional combination of basic P-formulas. An F-sentence is a member of the smallest class that contains Con and contains −A, (A&B), and −Bew(˹−A˺) whenever it contains A and B. In [B] we gave a decision procedure for the class of true F-sentences.−Bew(x), Bew(x), and Bew(neg(x)) are examples of P-formulas, and fixed points of these particular P-formulas have been studied by Gödel, Henkin [H] and Löb [L], and Jeroslow [J], respectively. In this note we show how to decide whether or not a fixed point of any given P-formula is provable in PA.
Partially ordered interpretations
Authors: Motohashi Nobuyoshi 
Published: May 2006

In this paper, we shall define the “partially ordered interpretation” of a first order theory in another first order theory and state some recent results. Although an exact definition will be given in §4 below, we now give a brief outline. First of all, let us recall the “interpretations” defined by A. Tarski et al. in [17] and the “parametrical interpretations” defined by P. Hájek in [6], [7] and U. Felgner in [3]. Since “interpretations” can be considered as a special case of “parametrical interpretations”, we consider only the latter type of “interpretations”. A parametrical interpretation I of a first order language L in a consistent theory T′ (formulated in another first order language L′) consists of the following formulas:(i) a unary formula C(p) (i.e. a formula with one designated free variable p), which is used to denote the range of parameters,(ii) a binary formula U(p, x), which is intended to denote the pth universe for each parameter p,(iii) an (n + 1)-ary formula Fp(p, x1 …, xn) for each n-ary predicate symbol P in L,such that the formulas (∃p)C(p) and (∀p)(C(p)→(∃x)U(p, x)) are provable in T". Then, given a formula A in L and a parameter p, we define the interpretation Ip (A ) of A by I at p to be the formula which is obtained from A by replacing every atomic subformula P(*, …, *) in A by Fp(p, *,…,*), and relativizing every occurrence of quantifiers in A by U(p, * ). A sentence A in L is said to be I-provable in T′ if the sentence (∀p) (C(p)→ Ip(A)) is provable in T′. Then, it is obvious that every provable sentence in L is I-provable in T′. This is a basic result of “parametrical interpretations” and is used to prove the “consistency” of a theory T in L by showing that every axiom of T is I-provable in T′ when I is said to be a parametrical interpretation of T in T′. As is shown above, the word “interpretation” is used in the following three senses: interpretations of languages, interpretations of formulas and interpretations of theories. So, in this introduction we let the word “interpretation” denote “interpretation of languages”, for short.
On certain types and models for arithmetic
Authors: Blass Andreas 
Published: May 2006

AbstractThere is an analogy between concepts such as end-extension types and minimal types in the model theory of Peano arithmetic and concepts such as P-points and selective ultrafilters in the theory of ultrafilters on N. Using the notion of conservative extensions of models, we prove some theorems clarifying the relation between these pairs of analogous concepts. We also use the analogy to obtain some model-theoretic results with techniques originally used in ultrafilter theory. These results assert that every countable nonstandard model of arithmetic has a bounded minimal extension and that some types in arithmetic are not 2-isolated.
Formalisations of further ℵ0-valued Łukasiewicz propositional calculi
Authors: Rose Alan 
Published: May 2006

It has been shown that, for all rational numbers r such that 0≤ r ≤ 1, the ℵ0-valued Łukasiewicz propositional calculus whose designated truth-values are those truth-values x such that r ≤ x ≤ 1 may be formalised completely by means of finitely many axiom schemes and primitive rules of procedure. We shall consider now the case where r is rational, 0≥r≤1 and the designated truth-values are those truth-values x such that r≤x≤1.We note that, in the subcase of the previous case where r = 1, a complete formalisation is given by the following four axiom schemes together with the rule of modus ponens (with respect to C),the functor A being defined in the usual way. The functors B, K, L will also be considered to be defined in the usual way. Let us consider now the functor Dαβ such that if P, Dαβ take the truth-values x, dαβ(x) respectively, α, β are relatively prime integers and r = α/β thenIt follows at once from a theorem of McNaughton that the functor Dαβ is definable in terms of C and N in an effective way. If r = 0 we make the definitionWe note first that if x ≤ α/β then dαβ(x)≤(β + 1)α/β − α = α/β. HenceLet us now define the functions dnαβ(x) (n = 0,1,…) bySinceit follows easily thatand thatThus, if x is designated, x − α/β > 0 and, if n > − log(x − α/β)/log(β + 1), then (β + 1)n(x−α/β) > 1.
General random sequences and learnable sequences
Authors: Schnorr C. P. , Fuchs P. 
Published: May 2006

AbstractWe formalise the notion of those infinite binary sequences z that admit a single program P which expresses the entire algorithmical structure of z. Such a program P minimizes the information which must be used in a relative computation for z. We propose two concepts with different strength for this notion, the learnable and the super-learnable sequences. We establish three different equivalent characterizations of learnable (super-learnable, resp.) sequences. In particular, we prove that a sequence z is learnable (super-learnable, resp.) if and only if there is a computable probability measure p such that p is Schnorr (Martin-Löf, resp.) p-random. There is a recursively enumerable sequence which is not learnable. The learnable sequences are invariant with respect to all total and effective transformations of infinite binary sequences.