H. P. Grice on P. F. Strawson on the conversational implicata of the Square of Opposition

In the twentieth century there were many creative uses of logical tools and techniques in reassessing past doctrines. One might naturally wonder if there is some ingenious interpretation of the square that attributes existential import to the O form and makes sense of it all without forbidding empty or universal terms, thus reconciling traditional doctrine with modern views. Peter Geach, 1970, 62–64, shows that this can be done using an unnatural interpretation. Peter Strawson, 1952, 176–78, had a more ambitious goal. Strawson’s idea was to justify the square by adopting a nonclassical view of truth of statements, and by redefining the logical relation of validity. First, he suggested, we need to suppose that a proposition whose subject term is empty is neither true nor false, but lacks truth value altogether. Then we say that Q entails R just in case there are no instances of Q and R such that the instance of Q is true and the instance of R is false. For example, the A form ‘Every S is P’ entails the I form ‘Some S is P’ because there is no instance of the A form that is true when the corresponding instance of the I form is false. The troublesome cases involving empty terms turn out to be instances in which one or both forms lack truth value, and these are irrelevant so far as entailment is concerned. With this revised account of entailment, all of the “traditional” logical relations result, if they are worded as follows:

Contradictories:	The A and O forms entail each other’s negations, as do the E and I forms. The negation of the A form entails the (unnegated) O form, and vice versa; likewise for the E and I forms.
Contraries:	The A and E forms entail each other’s negations
Subcontraries:	The negation of the I form entails the (unnegated) O form, and vice versa.
Subalternation:	The A form entails the I form, and the E form entails the O form.
Converses:	The E and I forms each entail their own converses.
Contraposition:	The A and O forms each entail their own contrapositives.
Obverses:	Each form entails its own obverse.

These doctrines are not, however, the doctrines of [SQUARE]. The doctrines of [SQUARE] are worded entirely in terms of the possibilities of truth values, not in terms of entailment. So “entailment” is irrelevant to [SQUARE]. It turns out that Strawson’s revision of truth conditions does preserve the principles of SQUARE (these can easily be checked by cases),[27] but not the additional conversion principles of [SQUARE], and also not the traditional principles of contraposition or obversion. For example, Strawson’s reinterpreted version of conversion holds for the I form because any I form proposition entails its own converse: if ‘Some A is B’ and ‘Some B is A’ both have truth value, then neither has an empty subject term, and so if neither lack truth value and if either is true the other will be true as well. But the original doctrine of conversion says that an I form and its converse always have the same truth value, and that is false on Strawson’s account; if there are As but no Bs, then ‘Some A is B’ is false and ‘Some B is A’ has no truth value at all. Similar results follow for contraposition and obversion.

The “traditional logic” that Strawson discusses is much closer to that of nineteenth century logic texts than it is to the version that held sway for two millennia before that.[28] But even though he literally salvages a version of nineteenth century logic, the view he saves is unable to serve the purposes for which logical principles are formulated, as was pointed out by Timothy Smiley in a short note in Mind in 1967.[29] People have always taken the square to embody principles by which one can reason, and by which one can construct extended chains of reasoning. But if you string together Strawson’s entailments you can infer falsehoods from truths, something that nobody in any tradition would consider legitimate. For example, begin with this truth (the subject term is non-empty):

No man is a chimera.

By conversion, we get:

No chimera is a man.

By obversion:

Every chimera is a non-man.

By subalternation:

Some chimera is a non-man.

By conversion:

Some non-man is a chimera.

Since there are non-men, the conclusion is not truth-valueless, and since there are no chimeras it is false. Thus we have passed from a true claim to a false one. (The example does not even involve the problematic O form.) All steps are validated by Strawson’s doctrine. So Strawson reaches his goal of preserving certain patterns commonly identified as constituting traditional logic, but at the cost of sacrificing the application of logic to extended reasoning.