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.

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.

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.

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.

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.

This paper presents several results whose common element is their connection with Hubert's tenth problem.1 The one that seems most striking (perhaps because it can be stated with a minimum of technical apparatus) is this: there does not exist an algorithm for determining whether or not a polynomial (in n variables) represents every integer.2

In addition, the present paper (i) gives a simple characterization of the diophantine sets (of positive, non-negative, and arbitrary integers), and (ii) gives a rigorous proof of Myhill's theorem [6] that there is no decision method for statements of the form

According to Gödel's completeness theorem, every consistent theory1 has a model whose domain is a set of natural numbers. The objects of the model corresponding to the predicate symbols of the theory are then predicates of natural numbers. Kleene [5] p. 398 showed that, if the theory is axiomatizable,2 then the model can be chosen so that these predicates are in both two-quantifier forms, i.e., they can be expressed in both the forms (x)(Ey)R and (Ex)(y)S with R and S recursive. An alternative proof has been given by Hasen jaeger [3].

Let the prepositional calculus (PC) be cast in the following form: (a) The primitive signs of PC are to be a denumerably infinite hst of propositional letters, the two connectives ‘∼’ and ‘&’, and the two parentheses ‘(‘and’)’; (b) The formulas of PC, referred to hereafter by ‘A’, ‘B’, ‘C’, and ‘D,’ are to be all finite sequences of primitive signs of PC; (c) The well-formed formulas (wffs) of PC are to be all propositional letters, all formulas of the form ∼A, where A is a wff of PC, and all formulas of the form (A&B), where A and B are wffs of PC; (d) (A⊃B) is to be short for ∼(A&∼B), (AνB) short for ∼(∼A&∼B), and (A ≡ B) short for (((A ⊃B)&(B⊃A)).