The expression "material implication" was coined by Russell to refer to a special class of formulas in his logic, i.e., those of the form:

p q [material implication]

i.e., disjunctions with the first disjunct negated. At first sight, it is not clear how a disjunction could be understood as an implication. To say that p implies q is at a minimum to express a conditional, namely, that if p is true, q is also true, so that the truth-value of p depends on that of q. But a disjunction does not have the form of a conditional at all, nor does it express a relation of dependence.

The reason for calling a disjunction "implication" is indirect, having to do with the role the disjunction plays in inference, not the form of the disjunction in itself. Russell noticed that that the truth of the conditional p q requires that when, in addition to the disjunction, p is true, q is also true.

If p is true, p is false, and accordingly the only alternative left by the proposition p q is that q is true. In other words, if p and p q are both true, then q is true. In this sense, the proposition p q will be quoted as stating that p implies q. The idea contained in this propositional function is so important that it requires a symbolism which with direct simplicity represents the proposition as connecting p and q without the intervention of p. But "implies" as used here expresses nothing other than the connection between p and q also expressed by the disjunction "not-p or q." (PM, p. 7)

p q is defined as p q.

So the symbolism of "material implication" is a way of expressing the disjunction without the occurrence of a negation sign, allowing the move from the truth of p to the truth of q with the aid of p q, which contains no negation sign. Russell went on to caution his readers not to confuse implication, as defined here, with inference.

The process of inference is as follows: a proposition "p" is asserted, and a proposition "p implies q" is asserted, and then as a sequel the proposition "q" is asserted. The trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error. (PM, pp. 8-9)

It is worth noting that the kind of inference Russell has in mind is what is commonly called "disjunctive syllogism." Some logicians have argued that disjunctive syllogism, using the truth-functional form fo disjunction, is an invalid argument form.

Lewis accepted disjunctive syllogism but objected to the labeling of disjunctions with a negated left disjunct as implications. Material implication "is obviously a relation between the truth-values of propositions, not between any supposed content or logical import of propositions."

"p materially implies q" means "it is false tht p is true and q is false." . . . The one thing this relation has in common with other meanings of "implies" --a most important thing of course--is that if p is true and q is false, then p does not materially imply q" (SSL, p. 326).

These and other so-called "paradoxes of material implication" (several of which are found at the beginning of Bonavac's Chapter 9) were what first aroused Lewis's suspicions about Russell's system. With the first formula, given the assertion of the truth of q, one may infer the truth of p q; that is, given the truth of p, one may infer that p materially implies q.

"The moon is made of geen cheese" implies "2 + 2 = 4,"--because q (p q). Let q be "2 + 2 = 4" and p be "The moon is made of green cheese." Then, since "2 + 2 = 4" is true, its consequence above is demonstrated. (SSL, p. 326)

It can be argued that what is in dispute here is merely the proper terminology for describing formulas with the connective '' as the main operator. Perhaps it should be called a "material conditional," as is often the case in logic texts. (Note that Bonevac's uses the paradoxes of material implication to claim that it does not even correctly represent conditionals.) For Lewis, the issue is whether p q is "a relation which can validly represent the logical nexus of proof and demonstration" (SSL, 328). It was his view that a logic is incomplete unless it expresses in its symbolism the fundamental logical relation.

For this reason, Lewis introduced a new symbol to extend Russell's logic, with the intention of making the symbol stand for the kind of "strict implication" which is the basis for proof in logic. The new symbol is in the shape of a fish-hook:

p q [strict implication]

and is interpreted as meaning that it is not possible that p be true and q false. As with strict implication, Lewis allowed the semantical notion of possibility to be reflected in the symbolic language:

p q is equivalent to (p & q)

Thus, possibility is invoked to clarify the notion of implication. Indeed, students first introduced to the notion of a deductively valid argument are given a similar formulation: that it is not possible that all the premises be true and the conclusion false.

Necessity arises through another equivalence: what cannot possibly not be the case is necessarily true. Thus we have

p is equivalent to p

and with substitution and suppression of double negation we have,

(p q) is equivalent to (p & q).

Therefore,

(p q) is equivalent to p q.

So it turns out that Lewis's strict implication is the necessary truth of Russellian material implication.

Lewis recognized that there are some remnants of the "paradoxes of material implication" in his system. For example, we have a "paradox of strict implication" which is a modal version of the original paradox we looked at.

p (q p)

a necessary proposition is strictly implied by any proposition. Given that p is necessarily true, it cannot be false, and so it cannot be the case that q is true and p is false. (This assumes the validity of the inference from (p & q) to p, which in fact does not hold for Lewis's system, S1, which he developed some time later.) It has been objected that the fact that the a necessary proposition is said to be implied by another, irrelevant, proposition precludes the relation between the two from being one of implication. How can the connection be one of meaning or content, when the antecedent and consequent have nothing in common?