In most published systems of modal logic only one or two modal operators occur as undefined constants alongside the primitives of non-modal logic. But in a single informal argument many non-interdefinable modal operators may occur. E.g. in philosophy we may have ‘it is logically true that’, ‘it is analytic that’, and ‘it is physically necessary that’; and in jurisprudence or private international law we may have statements about what it is obligatory or not obligatory to do under several different systems of rules. Moreover even if the contexts of such operators are all referentially opaque, in Quine's sense [1] of being exempt from the substitutivity of truth-functional biconditionals, some are, as it were, opaquer than others, in being exempt from the substitutivity of certain non-truth-functional biconditionals as well.

In my paper [10] I introduced the syntactical concepts “positive part” and “negative part” of logical formulas in first-order predicate calculus. These concepts make it possible to establish logical systems on inference rules similar to Gentzen's inference rules but without using the concept “sequent” and without needing Gentzen's structural inference rules. Proof-theoretical investigations of several formal systems based on positive and negative parts are published in [11]. In this paper I consider a similar formal system of simple type theory.

A syntactical concept of “strict derivability” results from the formal system in [10] by generalization of the axioms and inference rules from first to higher-order predicate calculus and by addition of inference rules for set abstraction by means of a λ-symbol which allows us to form set expressions of arbitrary types from well-formed formulas.

The extent to which we can grasp the content of a (non-logical) theory, expressing it economically by means of an axiom system or basis, varies greatly. In this paper we shall investigate what degree of economy, or at least regularity, can be achieved for all recursively axiomatizable first-order theories. A useful approach, also of interest in its own right, turns out to be the study of bases for subtheories, where a subtheory of a given theory consists of those theorems from which certain predicate symbols are absent. These predicate symbols might be thought of as the formal counterparts of the “purely theoretical” terms employed by a science, the theory corresponding to the science itself and the subtheory to its “observational consequences”. Roughly speaking, the types of operations involving such predicate symbols will be reduced to a minimum, so that their syntactical role in deductions will emerge more clearly.

Axiomatizations of the sentential calculus which use Rmp (modus ponens), have been shown equivalent to axiomatizations similar in all respects except that Rmp is replaced by the less restricted rule Re (rule of excision)1:

Re. If S and (…(S⊃S') …), then (… S' …).

We shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of (i ≦ n—2), and each is an extension of (i ≦ n—2). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.

Consider functions whose variables, finite in number, range over a fixed finite set N and whose values are elements of N. The elements of N are denoted simply by the natural numbers 1,2, …, n. There are nnm distinct m-place functions. If N is chosen to be the set of n truth-values then the functions considered are obviously truth-functions in n-valued logic.

McKinsey and Tarski [3] described Gödel's proof that the number of Brouwerian-algebraic functions is infinite. They gave an example of a sequence of infinitely many distinct Brouwerian-algebraic functions of one argument, which means that there are infinitely many non-equivalent formulas of one variable in the intuitionistic propositional calculus LJ of Gentzen [1]. However they did not completely characterize such formulas. In § 1 of this note, we define a sequence of basic formulas P∞(X), P0(X), P1(X), … and prove the following theorems.

In a previous paper [1] it was proved, among other results, that a closed disjunction of intuitionistic elementary number theory N can be proved if and only if at least one of its disjunctands is provable and that a closed formula of the type (Ex)B(x) is provable in N if and only if B(n) is provable for some numeral n. The method of proof was to show that, as far as closed formulas are concerned, N is equivalent to a calculus N1 for which the result is immediate. The main step in the proof consisted in showing that the set of provable formulas of N1 is closed under modus ponens. This was done by obtaining a subset of the set which is closed under modus ponens and contains all members of the original set, with which it is therefore identical.

Margaris has shown1 that for every triple 〈s,t,m〉 of integers such that 1 ≦ s < t < m it is possible to construct a formalisation of an m-valued propositional calculus satisfying the following conditions: I. Every statement which takes only truth-values belonging to the set {1, …,s} is provable. II. Every provable statement takes only truth-values belonging to the set {1, …, t}. III. There exist statements Pk, Qk which take only truth-values belonging to the set {1, …, k) and neither of which takes only truth-values belonging to the set {1, …, k—1} such that Pk is provable and Qk is unprovable (k = s+1, …, t). The systems of Margaris are all functionally incomplete and he appears to suggest2 that it is impossible to construct functionally complete systems having the required properties.

Those who object to the identification of strict implication and entail-ment speak in such a way that for A to entail B, A must be relevant to B. This view of entailment agrees not only with that of the naive student of logic (who tends to think the phrase paradoxes of “implication” more accurate than “paradoxes” of implication), but also with that expressed in the initial chapters of many textbooks in logic under the heading “Fallacies of Relevance.”2

In regard to formal theories proposed as explications of the notion of entailment, it becomes important to have at our disposal some formal account of the demand for relevance as between antecedent and consequent of an entailment.

Basic notation in this paper is as in [3]. From [5] and [9] the following additional notation is derived, ϕi is the partial recursive function with index i, Wi its range. ∅ is the empty set. ‘≡’ denotes isomorphism between sets, ‘≡m’ many-one equivalence, ‘≡T’ Turing equivalence, ‘≦1’ and ‘≦m’ signify one-one and many-one reducibility respectively. ‘Recursive’ is used throughout for ‘general recursive’.

The notion of constructibility introduced by Gödel in [2] has been generalized by Hajnal [3], [4] in order to prove the conditional independence of the generalized continuum hypothesis and related axioms. A very similar construction was used independently by Shoenfield [11], [12]2 and the author [7] to prove the conditional independence of V = L and related axioms. Here we shall prove results further in the latter direction than those in Shoenfield [12].

This paper presents a class of plausible definitions for validity of formulas in the infinitely-many-valued extension of the Łukasiewicz predicate calculus, and shows that all of them are equivalent. This extended system is discussed in some form in [3] and [4]; the questions discussed here are raised rather briefly in the latter.

We first describe the formal framework for the validity definition. The symbols to be used are the following: the connectives + and ‐, which are the strong disjunction B of [2] and negation respectively; the predicate variables Pi for i ∈ I, where I may be taken as the integers; the existential quantifiers E(J), where J⊆I, and I may be thought of as the index set on the individual variables, which however do not appear explicitly in this formulation.

Let P0, …,Pq be predicates, of which at least one has two or more places. By a formula in P0, …, Pq (or simply a formula, when the list P0, …, Pq is fixed, as in this section) is meant any formula whose only symbols, other than sentential connectives, quantifiers, and (individual) variables, are among P0, …, Pq. A realization (or possible model) of such a formula is a system where A is a non-empty set and each Pk is a relation among the elements of A, having the same number of places as Pk.

Full classes, which play such a crucial role in various definitions of the ordinals, seem not to have been studied on their own right. We shall discuss here some properties of full classes and provide new criteria for distinguishing the ordinals among the full classes.