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Non-standard models for formal logics
Authors: Rosser J. Barkley , Wang Hao 
Published: May 2006

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex (for instance, if it is adequate for number theory), then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either:(a) The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model.(b) That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦.(c) That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦.
On the logic of quantification
Authors: Quine W. V. 
Published: May 2006

The notation that I shall use here for the logic of quantification is of the familiar kind wherein the letters ‘p’, ‘q’, etc. stand in place of unspecified statements and the letters ‘ƒ’, ‘g’, etc. stand in place of unspecified predicates. The present section will deal with the significance of this notation; the purpose and scope of the paper as a whole can better be indicated afterward, in §2.By “predicates” I mean, not properties (or classes) and relations, but merely certain notational expressions. As a first approach they may be thought of as expressions like ‘walks,’ ‘is red,’ ‘touches,’ ‘gives to.’ Where ‘ƒ’, ‘g’, ‘h’, and ‘k’ represent these four predicates, we may read ‘ƒx’ ‘gy’, ‘hxy’, and ‘κxyz’ as ‘x walks,’ ‘y is red,’ ‘x touches y,’ ‘x gives y to z.’
A set of axioms for logic
Authors: Hailperin Theodore 
Published: May 2006

One of the preeminent problems confronting logicians is that of constructing a system of logic which will be adequate for mathematics. By a system's being adequate for mathematics, we mean that all mathematical theorems in general use can be deduced within the system. Several distinct logical systems, all having this end in view, have been proposed. Among these perhaps the best known are the systems referred to as “Principia Mathematica” and “set theory.” In both of these systems (we refer to the revised and simplified versions) there is a nucleus of propositions which can be derived by using only the axioms and rules of the restricted predicate calculus. However, if anything like adequacy for mathematics is to be expected, additional primitives and axioms must be added to the restricted predicate calculus. It is in their treatment of the additional primitive ε, denoting class or set membership, that the above-mentioned systems differ.In addition to these two, a third and a stronger system has been proposed by W. V. Quine in his paper New foundations for mathematical logic. It is with this system of Quine's that our work is concerned and of which we now give a brief description.
Remarks on identity and description in first-order axiom systems
Authors: Hailperin Theodore 
Published: May 2006

Hilbert and Ackermann ([1], p. 107) define a first order axiom system as one in which the axioms contain one or more predicate constants, but no predicate variables. Here “axiom” refers to the specific subject-matter axioms and not to the rules of the restricted predicate calculus (quantification theory), which rules are presupposed for each first-order system. It is pointed out by them that an exception could be made for the predicate of identity; for the axiom scheme for this predicate, namelywhich has in (b) the variable predicate F, could nevertheless be replaced, in any given first-order system, by a finite set of axioms without predicate variables. Thus, for example, if Φ[x, y) is the one constant predicate of such a system then PId(b) could be replaced byThus one postulates, in addition to the reflexivity, symmetry, and transitivity of identity, the substitutivity of identical entities in each of the possible “atomic” contexts of a variable (occurrences in the primitive predicates). In this method of introducing identity it has to be taken as an additional primitive predicate and further axioms are consequently needed. In such a system having PId(a), (b1)−(b3) as axioms, the scheme PId(b) can be derived as a meta-theorem of the system, F(x) then being any formula of the system.
The gergonne relations
Authors: Faris J. A. 
Published: May 2006

In this paper I am going to set forth a formal system based on five inter-class relations. These relations exist respectively between a class of a's and a class of b's.(i) if and only if every a is a b and every b is an a,(ii) if and only if every a is a b and not every b is an a,(iii) if and only if it is not the case that either every a is a b or every b is an a or no a is a b,(iv) if and only if every b is an a and not every a is a b,(v) if and only if no a is a b.These relations between classes, which correspond, as will be seen, to the five relations between two circles a and b shown in the well-known Eulerian diagrams,are of course connected in an intimate way with the four forms of proposition, A, E, I, O, of the traditional syllogistic logic. The French mathematician, J. D. Gergonne, seems to have been the first to recognize these relations explicitly and to understand their importance in syllogistic theory. It is therefore appropriate that they should be called by his name.Gergonne first of all showed with reference to these relations what are the sufficient and necessary conditions of the truth of propositions of each of the four traditional forms: for example, an A proposition, ‘All a is b’ is true if and only if either the first or the second relation exists between the class of a's and the class of b's.
On confirmation and rational betting
Authors: Lehman R. Sherman 
Published: May 2006

The purpose of this paper is to analyze rational betting. In particular, we concentrate on one necessary feature of rational betting, the avoidance of certainty of losing to a clever opponent. If a bettor is quite foolish in his choice of the rates at which he will bet, ah opponent can win money from him no matter what happens.This phenomenon is well known to professional bettors—especially bookmakers, who must as a matter of practical necessity avoid its occurrence. Such a losing book is called by them a “dutch book.” Our investigations are thus concerned with necessary and sufficient conditions that a book not be “dutch.”De Finetti [3] has started with the same idea and used it as a foundation for the theory of probability. It is our aim to consider the same subject more precisely and attempt to answer some questions about desirable features of a confirmation function. We wish to connect the ideas of De Finetti with those of Carnap [1] and Hossiasson-Lindenbaum [6]. The results expressed by Theorem 1 are essentially contained in De Finetti's work. Theorems 3 and 4 seem to be new.The confirmation functions which we consider will be functions with two sentences as arguments taking real numbers as values. Intuitively, C(h, e) will represent the rate at which a bettor would be willing to bet on the hypothesis h if he knew the information expressed by the sentence e, the evidence.
Fair bets and inductive probabilities
Authors: Kemeny John G. 
Published: May 2006

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.
On weakened quantification
Authors: Heyting A. 
Published: May 2006

In this note I give an account of the relations between different notions of weakened quantification, which can be defined in intuitionistic logic. The reason for publishing it is the fact that the last gap in our knowledge of these relations has been filled in an article by Kleene [5].For the intuitionistic calculus of propositions I refer to my article [2]; besides the rules of substitution and of inference I need only a few formulas from this calculus, which I shall repeat below. The simplest way of formalizing the intuitionistic predicate calculus is obtained by adjoining to the calculus of propositions the symbols, rules, and axioms as given by Hilbert and Ackermann [4], Chapter 3, §§4-5, pp. 53-57. (Reference is made to this book as HA.) I repeat only the axioms (e) and (f) and the rule γ.
On sentences which are true of direct unions of algebras
Authors: Horn Alfred 
Published: May 2006

It is well known that certain sentences corresponding to similar algebras are invariant under direct union; that is, are true of the direct union when true of each factor algebra. An axiomatizable class of similar algebras, such as the class of groups, is closed under direct union when each of its axioms is invariant. In this paper we shall determine a wide class of invariant sentences. We shall also be concerned with determining sentences which are true of a direct union provided they are true of some factor algebra. In the case where all the factor algebras are the same, a further result is obtained. In §2 it will be shown that these criteria are the only ones of their kind. Lemma 7 below may be of some independent interest.We adopt the terminology and notation of McKinsey with the exception that the sign · will be used for conjunction. Expressions of the form ∼∊, where ∊ is an equation, will be called inequalities. In accordance with the analogy between conjunction and disjunction with product and sum respectively, we shall call α1, …, αn the terms of the disjunctionand the factors of the conjunctionEvery closed sentence is equivalent to a sentence in prenez normal form,where x1, …, xm distinct individual variables, Q1, …, Qm are quantifiers, and the matrix S is an open sentence in which each of the variables x1, …, xm actually occurs. The sentence S may be written in either disjunctive normal form:where αi,j is either an equation or an inequality, or in conjunctive normal form:.
Interpretations of sets of conditions
Authors: Quine W. V. 
Published: May 2006

The celebrated theorem of Löwenheim and Skolem tells us that every consistent set S of quantificational schemata (i.e., every set of well-formed formulas of the lower predicate calculus admitting of a true interpretation in some non-empty universe) admits of a true numerical interpretation (i.e., an interpretation of predicate letters such that all schemata of S come out true when the variables of quantification are construed as ranging over just the positive integers).Later literature goes farther, and shows how, given S, actually to produce a numerical interpretation which will fit S in case S is consistent. The general case is covered by Kleene (see Bibliography). The special case where S contains just one schema (or any finite number, since we can form their conjunction) had been dealt with by Hilbert and Bernays. Certain extensions, along lines not to be embarked on here, have been made by Kleene, Kreisel, Hasenjäger, and Wang.My present purpose is expository: to make the construction of the numerical interpretation, and the proof of its adequacy, more easily intelligible than they hitherto have been. The reasoning is mainly Kleene's, though closer in some ways to earlier reasoning of Gödel.
On propositions neither necessary nor impossible
Authors: Prior A. N. 
Published: May 2006

There occurs in Aristotle a thesis which is symbolised in I. M. Bocheński's La Logique de Théophraste, Chapter V, by the formula EMpMNp. We might read this as “It is possible that P (Mp) if and only if (E) it is possible that not P (MNp).” But as Bocheński explains, Aristotle is here using the word “possible” to mean, not merely “not impossible”, but “neither necessary nor impossible”. And if we take “ It is impossible that P” to mean “It is necessary that not P”, Aristotle's thesis seems evidently true. For “It is neither necessary nor impossible that P” (Mp) amounts to “It is not necessary that P, and it is not necessary that not P” (KNSpNSNp, where S is the symbol for “It is necessary that”); and “It is neither necessary nor impossible that not P” (MNp) amounts to “It is not necessary that not P, and it is not necessary that not not P” (KNSNpNSNNp); and this reduces easily to the other by the law of double negation (ENNpp) and the commutativeness of conjunction (EKpqKqp).Despite the apparently unexceptionable character of this thesis EMpMNp, when properly understood, Bocheński informs us, in paragraph 42 of this chapter, that Łukasiewicz has shown that it “implies a contradiction”. He goes on to explain that what he means by this is that it follows from this thesis, together with the assumption that some propositions are “possible” in the Aristotelian sense, that any proposition whatever is “possible” (in that sense). And this, when one thinks what it means, is a most extraordinary contention.
On ω-consistency and related properties
Authors: Orey Steven 
Published: May 2006

1. This paper grew out of an attempt to answer a question raised in [4]. Let a logic L containing “numerals” z1, z2, … and a certain statement N(x) (intended to express the proposition that x is a natural number) be called ω-inconsistent if there is a statement such that ⊦ F(zk) for k = 1, 2, …, and ⊦ ∼(x)·N(x) ⊃ F(x); then it is evident that L cannot have a model in which N(x) is satisfied by the images of the numerals and nothing else if L is ω-inconsistent.Question: If L is ω-consistent, i.e. not ω-inconsistent, must there be such a model? Calling a model of the kind just described a special model, we ask for necessary and sufficient conditions on L to insure the existence of a special model. We give several sets of such conditions, applicable to a certain very inclusive class of logics, in Theorem 1 and Theorems 3 and 4. Theorem 2 shows that a logic may be ω-consistent but still not have a special model.This paper was close to completion when [3] appeared. For systems with only denumerably many symbols our results include Henkin's, for, by adjoining a new predicate N(x) to each of the systems considered in [3] which have only a denumerable number of constant symbols and then adding as an axiom (x)N(x), these systems become special cases of the systems we consider. It is easily seen that Henkin's Theorem 7 essentially proves the equivalence of conditions (2) and (3) in our Theorem 1, and Theorem 3 of [3] corresponds to our Theorem 2. Incidentally, our argument of Theorem 2 could also serve to prove Henkin's Theorem 6.
A formalization of the theory of ordinal numbers
Authors: Takeuti Gaisi 
Published: May 2006

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.
Systems of predicative analysis
Authors: Feferman Solomon 
Published: May 2006

This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.
A note on a theorem of Vaught
Authors: Rosenstein Joseph G. 
Published: May 2006

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.
An algebraic study of tense logics with linear time
Authors: Bull R. A. 
Published: May 2006

In [2] Prior puts forward a tense logic, GH1, which is intended to axiomatise tense logic with time linear and rational; he also contemplates the tense logic with time linear and real. The purpose of this paper is to give completeness proofs for three axiom systems, GH1, GHlr, GHli, with respect to tense logic with time linear and rational, real, and integral, respectively.1 In a fourth section I show that GH1 and GHlr have the finite model property, but that GHli lacks it.GH1 has the operators of the classical propositional calculus, together with operators P, H, F, G for ‘It has been the case that’, ‘It has always been the case that’, ‘It will be the case that’, ‘It will always be the case that’, respectively.
An infinitistic rule of proof
Authors: Enderton H. B. 
Published: May 2006

In this paper we consider a fonnal system of second-order Peano arithmetic with a rule of inference stronger than the ω-rule [3]. We also consider the relation to a class of models for analysis (i.e. second-order arithmetic) which lies between the class of ω-models and the class of β-models [5].The notation used is largely that of [3] and [5]. We assume that the reader has some familiarity with at least the ideas of the former. The formal system (A) of Peano arithmetic employed in [3] includes the comprehension axioms and the second-order induction axiom.
A logic stronger than intuitionism
Authors: Görnemann Sabine 
Published: May 2006

S. A. Kripke has given [6] a very simple notion of model for intuitionistic predicate logic. Kripke's models consist of a quasi-ordering (C, ≤) and a function ψ which assigns to every c ∈ C a model of classical logic such that, if c ≤ c′, ψ(c′) is greater or equal to ψ(c). Grzegorczyk [3] described a class of models which is still simpler: he takes, for every ψ(c), the same universe. Grzegorczyk's semantics is not adequate for intuitionistic logic, since the formulawhere х is not free in α. holds in his models but is not intuitionistically provable. It is a conjecture of D. Klemke that intuitionistic predicate calculus, strengthened by the axiom scheme (D), is correct and complete with respect to Grzegorczyk's semantics. This has been proved independently by D. Klemke [5] by a Henkinlike method and me; another proof has been given by D. Gabbay [1]. Our proof uses lattice-theoretical methods.
Transcendence of cardinals
Authors: Takeuti Gaisi 
Published: May 2006

In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].
The decidability of the Kreisel-Putnam system
Authors: Gabbay Dov M. 
Published: May 2006

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 axiomhas 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.
Applications of Strict Π11 predicates to infinitary logic
Authors: Barwise Jon 
Published: May 2006

Consider the predicate of natural numbers defined by:
where R is recursive. If, as usual, the variable ƒ ranges over ωω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π11 sets. If, however, ƒ ranges over 2ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable.3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-König Infinity Lemma.