In teaching symbolic logic to beginners, it is a not infrequent experience to encounter difficulties in the transition from the informal “if …, then …” construction of common discourse to its analysis in terms of material implication and its truth-table. There is no trouble with the idea that “if p, then q” must be so understood that “if p/T1, then we must have (p⊃q)T when q/T, and (P⊃q)/F when q/F. There is thus no problem in leading the explanation to the point.

Let Pp, Pd, and N be the intuitionistic formal systems of prepositional calculus, predicate calculus, and elementary number theory, respectively.1 Consider the following six propositions.8

(1) ├A V B only if ├A or ├B.

(2) ├∋xA(x) only if ├Ã(t) for some formula Ã(x) congruent to A(x) and some term t free for x in Ã(x).

The “implicational paradoxes” are treated by most contemporary logicians somewhat as follows:

“The two-valued prepositional calculus sanctions as valid many of the obvious and satisfactory inferences which we recognize intuitively as valid, such as

(A→.B→C)→.A→B→.A→C,2

and

A→B→.B→C→.A→C;

it consequently suggests itself as a candidate for a formal analysis of implication.

For the m-valued (2<w<∞) logics of Łukasiewicz as described in [2] using the values 1,2, …, m and based on the functions f(p, q) = max(1,q—p+1) and g(p) = m—p+1, Evans and Schwartz have essentially2 proved in [1] that the addition of a constant function i, 1<i<m, yields functional completeness if and only if (m—1, i—1)= 1.3 In this note it will be shown that if n constant functions i1, …, in, 1<ik<m and 1≦k≦n, are added, functional completeness obtains if and only if (m—1, i1—1, …, in—1) = 1.

We assume familiarity with the canonical languages of Post (cf. [1], [2]).

A set S of strings is said to be representable in a canonical system (F) if there is a string π such that for every string X, X ∈ S if and only if πX is provable in (F).

Suppose that K is a finite alphabet containing at least 2 symbols, and that W is a set of strings in (the symbols of) K. If W is representable in some canonical system, is it necessarily representable in a canonical system which uses only the symbols of K ? We answer this question affirmatively. Thus, e.g., it is possible to construct a 2-sign universal system over its own alphabet.2

Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.

In the familiar first-order predicate logic generality is expressed by means of variables and quantifiers, and each interpretation of the system is based upon the choice of some class over which the variables may range, the only restriction placed on this ‘domain of individuals’ being that it should not be empty.