7.2.6 The Maxim of Quality

Grice's Maxims.

We now show how we can use inference to check whether an utterance - given some previous discourse - conforms to the maxims of quantity and quality (or, more precisely, we show how to detect a lot of cases where it doesn't). We will formulate inference tasks that help us decide this question and that we can give to (for instance) a tableaux prover.

Quality

First, we shall look at the maxim of quality. An utterance must at least be consistent with the preceding discourse in order to be true. Now this is definitely something we can decide using a theorem prover.

An Inference Task

Let's suppose we want to check the consistency of an utterance (more precisely the formula representing the meaning of the utterance) with respect to a preceding (consistent) discourse, which as a first approximation, we take to be the conjunction of the logical forms of the n sentences uttered so far . How can we do this?

We proceed indirectly: We check whether is unsatisfiable. If so, then we know that is not consistent with (because we have assumed that the preceding discourse is consistent, we know that is to blame for the inconsistency). Otherwise, we know that is consistent with .

How can we use our tableaux calculus to find out if is unsatisfiable? Up to now, we've only seen how to prove theorems. But how can we reduce inconsistency checks to this task? We just have to look at the negation of the formula that we want to prove unsatisfiable. If this negation is a theorem, we know that the unnegated formula is unsatisfiable. So we will take the negation of the conjunction for the complete discourse (i.e. ), and check if it is a theorem. This theorem-check is where our tableaux-prover comes in. We feed the negated formula to it and try to construct a closed tableaux. If we manage to build one, we can ``infer backwards'' a little.

Here is how, step by step:

1. is a theorem (this is what we've proven on our tableaux).

2. Hence the unnegated must be unsatisfiable.

3. This means that the discourse corresponding to is inconsistent.

4. But the previous discourse is consistent (by assumption).

5. Hence the inconsistency can be traced back to adding utterance .

6. Finally, this means that uttering after having uttered violates the Maxim of Quality.

Let us look at a (very) small discourse as an example: ``If Mutz is a Siamese cat, then Mary likes her. Mutz is a Siamese cat.''. Given this ``discourse'', we can use our tableaux calculus to detect that the sentence ``Mary doesn't like Mutz'' violates the maxim of quality. We have to construct a closed tableaux for the following input (since we do not have any treatment of pronouns, we formalize ``her'' as if it was ``Mutz''):

This is equivalent to:

As an exercise, convince yourself that this formula really yields a closed tableaux.