An individual or whole we understand to be whatever is represented in any given discourse by signs belonging to the lowest logical type of which that discourse makes use. What is conceived as an individual and what as a class is thus relative to the discourse within which the conception occurs. One task of applied logic is to determine which entities are to be construed as individuals and which as classes when the purpose is the development of a comprehensive systematic discourse.

The concept of an individual and that of a class may be regarded as different devices for distinguishing one segment of the total universe from all that remains. In both cases, the differentiated segment is potentially divisible, and may even be physically discontinuous. The difference in the concepts lies in this: that to conceive a segment as a whole or individual offers no suggestion as to what these subdivisions, if any, must be, whereas to conceive a segment as a class imposes a definite scheme of subdivision—into subclasses and members.

The relations of segments of the universe are treated in traditional logistic at two places, first in its theorems concerning the identity and diversity of individuals, and second in its calculus of membership and class-inclusion. But further relations of segments and of classes frequently demand consideration. For example, what is the relation of the class of windows to the class of buildings? No member of either class is a member of the other, nor are any of the segments isolated by the one concept identical with segments isolated by the other. Yet the classes themselves have a very definite relation in that each window is a part of some building. We cannot express this fact in the language of a logistic which lacks a part-whole relation between individuals unless, by making use of some special physical theory, we raise the logical type of each window and each building to the level of a class—say a class of atoms—such that any class of atoms that is a window will be included (class-inclusion) in some class that is a building. Such an unfortunate dependence of logical formulation upon the discovery and adoption of a special physical theory, or even upon the presumption that such a suitable theory could in every case be discovered in the course of time, indicates serious deficiencies in the ordinary logistic. Furthermore, a raising of type like that illustrated above is often precluded in a constructional system by other considerations governing the choice of primitive ideas.

§1. In [1] Gödel proves the non-existence of a finite matrix characteristic for the intuitionist propositional calculus IC by the use of the finite matrices , where n is a natural number and

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 disjunction

and the factors of the conjunction

Every 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:

.

In 1922 Skolem delivered an address before the Fifth Congress of Scandinavian Mathematicians in which he pointed out what he called a “relativity of set-theoretic notions”. This “relativity” has frequently been regarded as paradoxical; but today, although one hears the expression “the Löwenheim-Skolem Paradox”, it seems to be thought of as only an apparent paradox, something the cognoscenti enjoy but are not seriously troubled by. Thus van Heijenoort writes, “The existence of such a ‘relativity’ is sometimes referred to as the Löwenheim-Skolem Paradox. But, of course, it is not a paradox in the sense of an antinomy; it is a novel and unexpected feature of formal systems.” In this address I want to take up Skolem's arguments, not with the aim of refuting them but with the aim of extending them in somewhat the direction he seemed to be indicating. It is not my claim that the “Löwenheim-Skolem Paradox” is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.

The structure of my argument will be as follows: I shall point out that in many different areas there are three main positions on reference and truth: there is the extreme Platonist position, which posits nonnatural mental powers of directly “grasping” forms (it is characteristic of this position that “understanding” or “grasping” is itself an irreducible and unexplicated notion); there is the verificationist position which replaces the classical notion of truth with the notion of verification or proof, at least when it comes to describing how the language is understood; and there is the moderate realist position which seeks to preserve the centrality of the classical notions of truth and reference without postulating nonnatural mental powers.

In this paper we shall use a logic with truth values ranging over all the real numbers x such that 0 ≦ x ≦ 1.1 will be “complete truth” and 0 will be “complete falsity.” The primitive sentential connectives are ‘⊃’ and ‘∼’; other connectives are ‘∨’ and ‘·’. Assume that ‘p’ and ‘q’ are sentential variables, whose truth values are respectively x and y. Then

1.1. ‘p ⊃ q’ has the value min(1 − x + y, 1),

1.2. ‘∼p’ has the value 1 − x,

1.3. ‘p∨q’ has the value max(x, y), and

1.4. ‘p·q’ has the value min (x, y).

‘∨’ and ‘·’ can be defined as follows:

It is the purpose of this paper to prove a theorem which will be stated in the next section. The following symbolism and convention will be used throughout the paper:

S is a logical formula.

ν (S) is the value of S.

‘p’, ‘pi1, ’p2, …, ‘q’, are sentential variables.

ν(p) = x and ν(x1) = x1, etc.

ν(S) = σ and ν(S1) = σ1, etc.

If S contains the sentential variables ‘p1’, ‘p2’, …, then we write for S, S(p1, P2, …). Also ν{S(p1, p2, …)) = σ(x1, x2, …).

A logical formula is defined in the usual manner. 1. A sentential variable is a logical formula; 2. if S is a logical formula then ·S is a logical formula; and 3. if S and S′ are logical formulae then (S ⊃ S′) is a logical formula.

Nach allgemeiner Ueberzeugung können gewisse Sätze der Analysis nicht konstruktiv bewiesen werden. Man denke etwa an den folgenden Satz: Eine monotone und beschränkte Folge a(n) von rationalen Zahlen konvergiert, d.h. es gibt eine solche ganzzahlige Funktion k(m), daß

Unter einem konstruktiven Beweis dieses Satzes hätte man etwa die Angabe eines Verfahrens zu verstehen, das gestattet, die Funktion k(m) auf Grand der Folge a(n) rekursiv zu definieren. Wenn ein solcher Beweis vorläge, wäre insbesondere gezeigt, daß es zu einer rekursiv definierten Folge a(n) mit den verlangten Eigenschaften stets eine rekursiv definierbare Konvergenzfunktion k(m) gibt.

Es soil nun im folgenden von einigen grundlegenden Sätzen der reellen Analysis gezeigt werden, daß sie nicht konstruktiv bewiesen werden können—konstruktiv in dem Sinn, der eben angedeutet wurde. Wir stützen uns dabei auf die Begriffe der rekursiven und der berechenbaren Funktion; rekursive und berechenbare Funktionen sind zahlentheoretische Funktionen. Unter einer rekursiven Funktion möge vorläufig eine primitiv rekursive Funktion verstanden werden (vgl. etwa D. Hilbert und P. Bernays [3], S.286); berechenbar meint berechenbar im Sinne von A. Church [1], S. C. Kleene [4], A. M. Turing [6]. Der Begriff der “konstruktiv definierten Funktion” wird durch die Bestimmung als “rekursive Funktion” eher eng, durch die Bestimmung als “berechenbare Funktion” eher weit gefaßt.

Definition I. Eine Folge rationaler Zahlen Φ(n) heißt rekursiv,(berechenbar), wenn es solche rekursiven (berechenbaren) Funktionen ϕ{n), ψ(n) und χ(n) gibt, daß χ(n) ≧1 und

.

Definition II. Eine Folge Φ(n) von rationalen Zahlen heißt rekursiv (berechenbar) konvergent, wenn es eine solche rekursive (berechenbare) Funktion v(m) gibt, daß

Wir werden zeigen (Satz I): Es gibt eine rekursive, monotone und beschränkte Folge rationaler Zahlen, die nicht berechenbar konvergiert. Bei unserer Auffassungsweise ist darin enthalten, daß der zu Beginn erwähnte Satz nicht konstruktiv bewiesen werden kann.

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not.

The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in the limit if, after some finite point in the sequence, all the guesses are correct and the same (in case there is more than one correct answer). Functions, sets, and functionals which are decidable by such infinite algorithms will be called limiting recursive. These, together with classes of objects which can be identified in the limit, are the subjects of this report.

Without qualification, set will mean set of numbers; function will mean number-theoretic function of 1 variable, possibly partial; functionals will take numerical values and have any number of numerical and/or function variables, the latter ranging solely over total functions of 1 variable. Thus a function is a special case of a functional, x will invariably stand for a numerical variable; φ for a function variable; g for a guess (a number); n for the numerical variable which indexes the guesses, referred to as the time.

In § 1 we present conceptual material concerning the notion of a Gödel numbering of the partial recursive functions. § 2 presents a theorem about these concepts. § 3 gives several applications. The material in § 1 and § 2 grew out of attempts by the author to find routine solutions to some of the problems discussed in § 3. The author wishes to acknowledge his debt in § 2 to the fruitful methods of Myhill in [M] and to thank the referee for an abbreviated and improved version of the proof for Lemma 3 in § 2.

In the literature of mathematical logic, “Gödel numbering” usually means an effective correspondence between integers and the well-formed formulas of some logical calculus. In recursive function theory, certain such associations between the non-negative integers and instructions for computing partial recursive functions have been fundamental. In the present paper we shall be concerned only with numberings of the latter, more special, sort. By numbers and integers we shall mean non-negative integers. Our notation is, in general, that of [K]. If ϕ and ψ are two partial functions, ϕ = ψ shall mean that (∀x)[ϕ(x)≃(ψx)], i.e., that ϕ and ψ are defined for the same arguments and are equal on those arguments. We consider partial recursive functions of one variable; applications of the paper to the case of several variables, or to the case of all partial recursive functions in any number of variables, can be made in the usual way using the coordinate functions (a)i of [K, p. 230]. It will furthermore be observed that we consider only concepts that are invariant with respect to general recursive functions; more limited notions of Gödel numbering, taking into account, say, primitive recursive structure, are beyond the scope of the present paper.

We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass, et al.

It is the purpose of this paper to make explicit the connection between J.-Y. Girard's “linear logic” [4], and certain models for the logic of quantum mechanics, namely Mulvey's “quantales” [9]. This will be done not only in the case of commutative linear logic, but also in the case of a version of noncommutative linear logic suggested, but not fully formalized, by Girard in lectures given at McGill University in the fall of 1987 [5], and which for reasons which will become clear later we call “cyclic linear logic”.

For many of our results on quantales, we rely on the work of Niefield and Rosenthal [10].

The reader should note that by “the logic of quantum mechanics” we do not mean the lattice theoretic “quantum logics” of Birkhoff and von Neumann [1], but rather a logic involving an associative (in general noncommutative) operation “and then”. Logical validity is intended to embody empirical verification (whether a physical experiment, or running a program), and the validity of A & B (in Mulvey's notation) is to be regarded as “we have verified A, and then we have verified B”. (See M. D. Srinivas [11] for another exposition of this idea.)

This of course is precisely the view of the “multiplicative conjunction”, ⊗, in the phase semantics for Girard's linear logic [4], [5]. Indeed the quantale semantics for linear logic may be regarded as an element-free version of the phase semantics.

In 1965 Zadeh introduced the concept of fuzzy sets. The characteristic of fuzzy sets is that the range of truth value of the membership relation is the closed interval [0, 1] of real numbers. The logical operations ⊃, ∼ on [0, 1] which are used for Zadeh's fuzzy sets seem to be Łukasiewciz's logic, where p ⊃ q = min(1, 1 − p + q), ∼ p = 1 − p. L. S. Hay extended in [4] Łukasiewicz's logic to a predicate logic and proved its weak completeness theorem: if P is valid then P + Pn is provable for each positive integer n. She also showed that one can without losing consistency obtain completeness of the system by use of additional infinitary rule.

Now, from a logical standpoint, each logic has its corresponding set theory in which each logical operation is translated into a basic operation for set theory; namely, the relation ⊆ and = on sets are translation of the logical operations → and ↔. For Łukasiewicz's logic, P Λ (P ⊃ Q). ⊃ Q is not valid. Translating it to the set version, it follows that the axiom of extensionality does not hold. Thus this very basic principle of set theory is not valid in the corresponding set theory.

The purpose of this paper is to provide a partial logical analysis of a few concepts that may be classified as value concepts or as concepts that are closely related to value concepts. Among the concepts that will be considered are striving for, doing, believing, knowing, desiring, ability to do, obligation to do, and value for. Familiarity will be assumed with the concepts of logical necessity, logical possibility, and strict implication as formalized in standard systems of modal logic (such as S4), and with the concepts of obligation and permission as formalized in systems of deontic logic. It will also be assumed that quantifiers over propositions have been included in extensions of these systems.

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.

In 1949 Nelson [5] proposed a constructive logic in which falsity is conceived in a fashion analogous to that for intuitionistic truth. The predicate calculus N (for strong negation) was characterized by the usual axioms and rules for positive intuitionistic connectives (see Kleene [4, p. 82, la–7 and 9–12]), with the additional axiom schemata for strong negation: A ⊃ (¬A ⊃ B),

Nelson's paper showed that N may be interpreted with concepts for constructive truth (P-realizability) and falsity (N-realizability). The paper also gave mappings between the intuitionistic and strong negation systems of arithmetic.

These mappings are easily adapted to pure predicate calculus. One shows that intuitionistic predicate calculus I is a subsystem of N by reading the intuitionistic negation of A as A ⊃ B & ¬B in N. It is possible to map N into a subsystem of I using the definition of A′ in [5, p. 19] changing only the definition of (¬A)′ for atomic A to A ⊃ B & ¬B.

We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.— exist in space-time; but we mean more than this. We renounce them altogether.

We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write ‘x is a dog,’ or ‘x is between y and z’; for here ‘is a dog’ and ‘is between … and’ can be construed as syncate-gorematic: significant in context but naming nothing. But we cannot use variables that call for abstract objects as values. In ‘x is a dog,’ only concrete objects are appropriate values of the variable. In contrast, the variable in ‘x is a zoölogical species’ calls for abstract objects as values (unless, of course, we can somehow identify the various zoological species with certain concrete objects). Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.

Let be a finite set of (nonlogical) predicate symbols. By an -structure, we mean a relational structure appropriate for . Let be the set of all -structures with universe {1, …, n}. For each first-order -sentence σ (with equality), let μ n (σ) be the fraction of members of for which σ is true. We show that μ n (σ) always converges to 0 or 1 as n → ∞, and that the rate of convergence is geometrically fast. In fact, if T is a certain complete, consistent set of first-order -sentences introduced by H. Gaifman [6], then we show that, for each first-order -sentence σ, μ n (σ) → n 1 iff T ⊩ ω. A surprising corollary is that each finite subset of T has a finite model. Following H. Scholz [8], we define the spectrum of a sentence σ to be the set of cardinalities of finite models of σ. Another corollary is that for each first-order -sentence a, either σ or ˜σ has a cofinite spectrum (in fact, either σ or ˜σ is “nearly always“ true).

Let be a subset of which contains for each in exactly one structure isomorphic to . For each first-order -sentence σ, let ν n (σ) be the fraction of members of which a is true. By making use of an asymptotic estimate [3] of the cardinality of and by our previously mentioned results, we show that v n(σ) converges as n → ∞, and that lim n ν n (σ) = lim n μ n (σ).If contains at least one predicate symbol which is not unary, then the rate of convergence is geometrically fast.

Guarded fragments of first-order logic were recently introduced by Andréka, van Benthem and Németi; they consist of relational first-order formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful model-theoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of first-order logic.

Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of first-order logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is Exptime-complete. We further establish a tree model property for both the guarded fragment and the loosely guarded fragment, and give a proof of the finite model property of the guarded fragment.

It is also shown that some natural, modest extensions of the guarded fragments are undecidable.

In this paper we shall prove three theorems about recursively enumerable sets. The first two answer questions posed by Myhill [1].

The three proofs are independent and can be presented in any order. Certain notations will be common to all three. We shall denote by “Re” the set enumerated by the procedure of which e is the Gödel number. In describing the construction for each proof, we shall suppose that a clerk is carrying out the simultaneous enumeration of R0, R1, R2, …, in such a way that at each step only a finite number of sets have begun to be enumerated and only a finite number of the members of any set have been listed. (One plan the clerk can follow is to turn his attention at Step a to the enumeration of Re where e+1 is the number of prime factors of a. Then each Re receives his attention infinitely often.) We shall denote by “Rea” the set of numbers which, at or before Step a, the clerk has listed as members of Re. Obviously, all the Rea are finite sets, recursive uniformly in e and a. For any a we can determine effectively the highest e for which Rea is not empty, and for any a, e we can effectively find the highest member of Rea, just by scanning what the clerk has done by Step a. Additional notations will be introduced in the proofs to which they pertain.

Gentzen's sequent calculus LJ, and its variants such as G3 [21], are (as is well known) convenient as a basis for automating proof search for IPC (intuitionistic propositional calculus). But a problem arises: that of detecting loops, arising from the use (in reverse) of the rule ⊃⇒ for implication introduction on the left. We describe below an equivalent calculus, yet another variant on these systems, where the problem no longer arises: this gives a simple but effective decision procedure for IPC.

The underlying method can be traced back forty years to Vorob′ev [33], [34]. It has been rediscovered recently by several authors (the present author in August 1990, Hudelmaier [18], [19], Paulson [27], and Lincoln et al. [23]). Since the main idea is not plainly apparent in Vorob′ev's work, and there are mathematical applications [28], it is desirable to have a simple proof. We present such a proof, exploiting the Dershowtiz-Manna theorem [4] on multiset orderings.

Consider the task of constructing proofs in Gentzen's sequent calculus LJ of intuitionistic sequents Γ⇒ G, where Γ is a set of assumption formulae and G is a formula (in the language of zero-order logic, using the nullary constant f [absurdity], the unary constant ¬ [negation, with ¬A =defA ⊃ f] and the binary constants &, ∨, and ⊃ [conjunction, disjunction, and implication respectively]). By the Hauptsatz [15], there is an apparently simple algorithm which breaks up the sequent, growing the proof tree until one reaches axioms (of the form Γ⇒ A where A is in Γ), or can make no further progress and must backtrack or even abandon the search. (Gentzen's argument in fact was to use the subformula property derived from the Hauptsatz to limit the size of the search tree. Došen [5] improves on this argument.)