With all seven of ‘∼’, ‘⊃’, ‘&’, ‘ν’, ‘≡’, ‘∀’, and ‘∃’ understood to serve as primitive signs, let an expression of the sort

A1, A2, … , An → B1, B2, … , Bm,

where A1, A2, …, Aη (η≥ 0), B1, B2, …, and Bm (m > 0 if n = 0, otherwise m ≥ 0) are wffs of the first-order quantificational calculus (QC), count as an L-sequent of QC; let an L-sequent of QC of the sort

A1, A2, … , An → B

count as an N-sequent of QC;2 and let an L-sequent S of QC count as a Gentzen L-sequent of QC if no individual variable of QC occurs both bound and free in S.

In [3], [4], and [5] Joyce Friedman formulated and investigated certain rules which constitute a semi-decision procedure for wffs of first order predicate calculus in closed prenex normal form with prefixes of the form ∀x1 … ∀xκ∃y1 … ∃ym∀z1 … ∀zn. Given such a wff QM, where Q is the prefix and M is the matrix in conjunctive normal form, Friedman's rules can be used, in effect, to construct a matrix M* which is obtained from M by deleting certain conjuncts of M.

The eventual purpose of this paper is to provide certain specific representations of ordinals and develop their basic properties as needed for the proofs of the results announced in [3]. Since the choice of these representations is really intelligible only if the general principles involved are made explicit, we devote the first half of this paper (§§1 and 2) to such questions. The entire treatment is self-contained.

Remark 1. Finite sums and products of recursive equivalence types have been defined (see [1, p. 73 for sums and p. 130 for products]) and many of their algebraic properties revealed. It seems natural to ask if infinite sums and products can be defined in a natural manner and an algebra of such sums and products developed. The initial work in this direction is to be found in [2], [3] and [4]. In this paper we extend the above studies, concentrating for the most part on products.

As those of us who instruct him are well aware, customary accounts of validity and implication for quantificational logic often bewilder the novice. For his benefit, I present here an account of validity, due (in effect) to the late E. W. Beth, and two accounts of implication, one my own, the other Jaakko Hintikka's, which add up to what the better textbooks say, but, making no mention whatever of domains, say it far more simply.

This paper is an exposition of the author's [2] investigations of a class of infinitary first order logics. These logics are sometimes characterized by saying that the syntactical role played by the positive integers in first order predicate logic is taken over by a nonstandard model of arithmetic. The result is then called a nonstandard logic.1 In particular the operations of conjunction, disjunction and quantification may be iterated infinitely (nonstandard finitely) often.

When the nonstandard model of arithmetic is an ultra power, the natural class of structures in which to interpret the nonstandard wffs is a class of ultra products. Within this framework, many model-theoretic properties of ultra products become manifest in the logical structure of the nonstandard language.

It is known that Leśniewski constructed an interpretation of mereology in the real number system using binary expansions.2 Unfortunately, this construction is no longer extant. The following paper, except for a slight variation, is an attempt to reconstruct this interpretation.

Leśniewski probably considered sequences of 0's and I's (except for the sequence of all 0's) and defined a first sequence as an element of the second if every place in which the first has a I, so also does the second.

Acquaintance with [11] and [8] will be assumed. Let be the alphabet →, &, ν, ┐, (,), A1, A2, … where →(I), &(C), ν (D) and ┐(N) are called the connectives. The letters appearing in parentheses are the names of the connectives. A1, A2 … are the atoms.

Definition 1.1. A string S is an ICDN word iff there is a finite sequence of strings S1 … Sn = S such that for i, 1 ≤ i ≤ n, either S1 is an atom or S1 is one of the forms (S1→Sk), (S1 & Sk), (S1, ν Sk) or ┐S1,j, k ≤ i.

Kanger [4] gives a set of twelve axioms for the classical prepositional calculus which, together with modus ponens and substitution, have the following nice properties:

(0.1) Each axiom contains =⊃, and no axiom contains more than two different connectives.

(0.2) Deletions of certain of the axioms yield the intuitionistic, minimal, and classical refutability1 subsystems of propositional calculus.

In his paper [3] Henkin proved for a first order language with identity symbol but without operation symbols the following version of the Craig-Lyndon interpolation theorem:

Theorem 1. If Γ╞Δ then there is a formula θ such that Γ ├Δand

(i) any relation symbol with a positive (negative) occurrence in θ has a positive (negative) occurrence in some formula of Γ.

The semantical account that Meyer and Lambert give in [7] of their quantificational calculus FQ can be considerably simplified, and—supposing as the authors do at the close of their paper that ‘ = ’ counts as a primitive sign—so can their axiom system for FQ.

(1) As the authors remark, axiom schema 102 can be simplified to read : A ⊃ (∀Χ)A, where Χ does not occur free in A. Following Tarski's [9], it can be further simplified: A ⊃ (∀Χ)A, where Χ does not occur in A.