They note a point, deriving from work by Timothy Smiley, about explaining negation in basic logical forms like modus tollens in terms of rejection. Instead of standard modus tollens,
if p, q
not q
therefore not p
you could use questions and answers:
a. Is it the case that if p, q? Yes!
b. Is it the case that q? No!
c. Therefore is it the case that p? No!
Or to take another case,
a. Is it the case that if not p, not q? Yes!
b. Is it the case that p? No!
c. Therefore is it the case that q? No!
Both of these are clearly valid, and work because the Yes's and No's indicate assertions and rejections or denials.
They then say ((6) and (7) are the above arguments):
But now we have a puzzle. The speech act indicators (5) that Smiley postulates are nonembeddable—if is it the case that p? No!, then... borders on incomprehensible—but as shown by (6) and (7) they can nonetheless feature in inferences. To simply reduce not p to Is it p? No! seems unsatisfactory due to the divergent embedding behaviour of these phrases (Rumfitt, 2000). But if (7b) is not reducible to the antecedent of (7a), then—following the structure of the Frege–Geach argument—how can (7) be valid?
The argument, in other words, is that "not p" cannot be reduced to "Is it p? No!" because these two have different logical behaviors: I can say, "If not p" but I cannot say, "If is it p? No!"
I find this argument extremely unconvincing. People do not in fact go around saying "If not p" unless they have been trained to do so by logicians; it's not a common way of speaking English, but an arbitrary convention invented by logicians for convenience. (While English does sometimes allow similar constructions, e.g., "Not all that glitters is gold" or "Not all who wander are lost", these are rare, and the construction would merely confuse people for most propositions, e.g., "Not some people are going to the movies today" or "Not if this is true, that is true". (Although, of course, we could say in colloquial English, "Some people are going to the movies today -- Not!") And if we are going to allow logicians to get away with bizarre barbarisms like that in the one case, it's unclear why they can't get away with it in the question case.
Moreover, English does actually have a way to handle this kind of situation. Sometimes in English, you are arguing in questions. It would be odd syntactically to put your question in the antecedent of a conditional, but it can be understood to be there. We do this all the time. Suppose I say:
Is it the case that Daniel bought milk? If no, then I want to stop by the store.
(We would often say 'not', but 'no' would not be very unusual.) What is the logical antecedent of the conditional statement? It's not merely 'no'; the consequent does not follow from negation in general. It's 'no' as an answer to the question. It's just that English takes the question to be understood. We could very well represent it as,
If [Is it the case that Daniel bought milk?] No, then I want to stop by the store.
You wouldn't say that explicitly in colloquial English as a matter of custom; but it has to be understood or it doesn't mean anything. The same can be said for when the consequent is just 'yes' or 'no', as well. We wouldn't usually do it for both the antecedent and the consequent simultaneously, but that seems obviously because, since the questions are merely understood, it quickly becomes confusing which answer goes with which question. But nothing about it is impossible, given the right set up.
So, in short, there seems no real puzzle here. If we are embedding as logicians embed, it's not more of a solecism than 'not p' is. If we are embedding as English speakers embed, once one recognizes that 'yes' and 'no' often have understood questions it becomes clear that English can in fact embed simple versions of this. If "Is it the case that p? No!" differs from "Not p", it's not due to any purely logical embedding behavior; they just seem to differ practically.
In yes-or-no questions, in fact, 'yes' and 'no' don't seem to be any different than assigning truth values to propositions. The real parallel is to
It is true that if p, q
It is false that q
Therefore it is false that p.
