In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

Interest has steadily increased among logicians and philosophers in versions of quantification theory which meet the following criteria: (1) no existence assumptions are made with respect to individual constants, and (2) theorems are valid in every domain including the empty domain. Logics meeting the former of these criteria are called free logics by Lambert and have been investigated in a series of papers by him and by van Fraassen, and by Leblanc and Thomason.1

Although it is natural to impose (2) in the presence of (1), the criteria are independent.2 Hence we baptize logics which meet both criteria universally free.

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.

In a paper that was written some time ago but appeared only recently [1], I indicated that Quine's system of quantification in [2], as well as his inclusive system of [3] can be modified so as to make it possible to prove *119 of [2] by another method than the one discovered by Berry. The reason originally given in [1, p. 136] for preferring the modified systems was wrong. (See [1, p. 158] for correction of previous remark.) But, at the cost of a slight complication of one of the axiom schemata, a considerable simplification of the proof of *119 is achieved in the modified systems, by dispensing with the use of *115 and *117 as intermediaries.

In [11] the constructive ordinals were extended to constructive finite number classes by using systems of notations where mappings at limit ordinals are just partial recursive functions. It turned out that these systems are equivalent, both in terms of ordinals represented and the forms of the sets of notations, to extensions obtained by using mappings at limit ordinals which are partial recursive in (sets of notations for) previously defined number classes. In this article these results are extended to constructive transfinite number classes. We present a system (F, ||) which, in terms of our analogy with the classical ordinals, provides notations for the ordinals less than the first “constructively inaccessible” ordinal, and show that the above equivalence holds at least this far.

It is well known that Gödel's famous undecidability result may be viewed in the following strong form. Suppose we are given a specific presentation (i.e., a specific formulation in terms of axioms and rules of inference) of number theory. Then there exists an effective method which, when applied to a consistent axiomatizable extension of the theory yields an undecidable sentence of this extension. For distinct presentations the undecidable sentences obtained would be distinct. This is because the sentence constructed depends upon the notion of proof and hence ultimately upon the axioms and rules of inference—i.e., upon the specific presentation.

The well-known incompleteness results of Gödel assert that there is no recursively enumerable set of sentences of formalized first order arithmetic which entails all true statements of that theory. It is equally well known that by introducing sufficiently nonconstructive rules, such as the ω-rule of induction, completeness can be re-established.

Starting from the work of Turing [4] Feferman in [1] developed another method, viz. the study of transfinite recursive progressions of theories, for closing the gap between Gödel (recursively enumerable sets of axioms yield incompleteness) and Tarski (number-theoretic truth is not arithmetically definable).

§1. The set Cn(R, A + X) is the set of all expressions deducible, on the ground of formulas A, from the class X of formulas of the implicational propositional calculus, by means of rules belonging to the set R. In other words, Cn(R, A + X) is the set of expressions obtained from the set X by the help of a two-parameter function Cn[R, A]. The formula is called the classical deduction theorem.

The classical deduction theorem is true for the system 〈R, A〉 (where R is the set of primitive rules, A is the set of axioms of the propositional calculus) if it holds for the function Cn[R, A].

“Apart from merits or defects of PA1, however, its existence demonstrates the feasibility of a new approach to the logic of propositions involving the principle of subjunctive contrariety. We thus have good reason to investigate the effect this principle, and a concept of conditionality compatible with it, might exert if introduced into standard quantification theory, into set theory, into modal logic and into epistemology and the philosophy of science.”

We shall take “strong negation” to be a unary operator with some of the properties usually associated with negation, and such that the strong negation of a statement implies the “ordinary” negation ofthat statement, but not vice-versa. This paper will consider two varieties of strong negation: first, strong negation set in the context of the propositional calculus, specifically, the intuitionist PC, and secondly, strong negation in modal systems, that is, strong negation as impossibility. In the latter part of the paper, we shall present axiomatizations of several classical Lewis-modal systems having only material implication and impossibility primitive.

The present paper is a sequel to [1]. It is our purpose to formulate a general theory derived from the methods used to obtain three results for the field of real algebraic numbers in [1]. As there, we shall concern ourselves almost exclusively with fields of characteristic zero; thus we assume a convenient formulation of first order logic with extralogical constants E(x, y), S(x, y, z), F(x, y, z), F(x, y), N(x) and 0, whose intended interpretations are equality, sum, product, y is the successor of x, x ∈ (where is a substructure satisfying all first order truths of the natural numbers) and zero, respectively. In addition, we shall use Q(x, y) for x ≤ y in those cases where our field is ordered, e.g. the field of real algebraic numbers.

It is extremely difficult to make general statements about functional completeness. (For the main reference on the subject see Post [2].) In this paper we restrict ourselves to the case of unary functions in a finite valued logic, and prove a result concerning minimal functionally complete sets, along with a necessary and sufficient condition for completeness. A basic familiarity with group theory would be helpful.

Given a class U which is the result of closing a finite class of expressions with respect to a binary mode of combination, there exists, according to RFBL,1 a recursively enumerable subclass K* of U, and concepts of representation (5.3) and complete representation (6.2) such that:

(a) A relation among members of U is recursively enumerable if and only if it is represented in K*. (6.7.)

(b) A relation among members of U is recursive if and only if it is completely represented in K*. (6.10.)

In [1], William H. Jobe has shown that a certain three-valued proper logic is canonically complete, i.e., each first order J is expressible by a formula that is in M-normal form. The first order operator Jk(p)1 represents the truth table of a possible formula which has the truth value k in the ith row and the truth value 1 in all other rows. “The M-normal form of a given formula is another formula which has the same truth table as the given formula but which contains no operation symbols other than those of conjunction, disjunction, and undefined unary operations, with no nonunary operation, i.e., conjunction or alternation, included in the scope of an operation which is unary, and no alternation included in the scope of a conjunction.” Here it is shown that it is possible to construct a canonically complete proper logic for each choice of M which contains, in addition to alternation and conjunction, exactly two unary operations. Moreover, it is further shown that at least two unary operations are necessary in order for such a logic to be canonically complete.