§ 7. Implication

One "truth-functional schema is said to imply another if there is no way of so interpreting the letters as to make the first schema true and the second false... whether a truth-functional schema S1 implies another, S2, can be decided always by taking S1 as antecedent and S2 as consequent of a conditional, and testing the conditional for validity... implication is validity of the conditional." (p. 46).

Once a logician encounters a false outcome of a conditional, it is clear that the conditional is invalid, and that its antecedent does not imply its consequent.

Quine lays out the following four rules of schematic implication:

(i) Any schema implies itself.
(ii) If one schema implies a second and the second a third then the first implies the third.
(iii) An inconsistent schema implies every schema and is implied by inconsistent ones only.
(iv) A valid schema is implied by every schema and implies valid ones only.

Implication always transmit validity.

I really like when Quine says

We must be able to think up schemata which imply or are implied by a given schema and promise well as links in a proposed chain of argument. Such products can be checked mechanically by truth-value analysis, but thinking them up is an unmechanical activity.

-p. 47

It brings to mind the worn-out quotation of Einstein, "Imagination is more important than knowledge". Here, even in the midst of pursuing the discipline of meticulous, exhaustive, systematic, and well-defined logical analyses, we see the glory of knack and of imagination and of art.

He continues

Facility in it depends on grasping the sence of a simple schemata clearly enough to be able, given a schema, to conjure up quite an array of fairly simple variants which imply or are implied by it. Given 'p v q', it should occur to us immediately that 'p' and 'q' and 'pq' and '~p→q' imply it and that 'p v q v r' and '~p→q' are implied by it... Such flashes need not be highly accurate, for we can check each hunch afterward by truth-value analysis. What is important is that they be prolific, and accurate enough to spare excessive lost motion.

No doubt repertoire is an aid to virtuosity in contriving implications, but understanding is the principal thing. When simple schema are sufficiently transparent to us, we can see through them by the light of pure reason to other schemata which must come out true if these do, or which can not come out true unless these do.

This is important to note, but I would like to comment that I enjoy Quine's writing style immensely and love the rhetorical beauty with which he teaches logic. He goes on to say "Readiness with implications is aided also, no doubt, by ease of checking" and offers a shortcut to what he calls the "full sweep" (truth-value analysis the long way"), which he calls the "fell swoop".

The fell swoop may be executed when a schema is "visibly verifiable by one and only one interpretation" of its literals (p. 48). "Now when S is such a schema, the question whether S implies a schema S' can be settled simply by supplanting [the literals] in S' by the values which make for truth of S, and resolving. If we come out with... a valid schema, then S implies S'; otherwise not" (ibid.).

The fell swoop may be reversed as well. When S' "is a schema thus falsifiable by one and only one interpretation, the question whether a schema S implies S' can be settled simply by supplanting [the literals] in S by the values which make for falsity of S', and resolving. If we come out with an inconsistent schema, then S implies S'; otherwise not" (ibid.)>