## Negative exhaustification?

Here's an idea that might explain a number of puzzling linguistic phenomena, including neg-raising, the homogeneity presupposition triggered by plural definites, the difficulty of understanding nested negations, and the data often assumed to support conditional excluded middle.

### Introducing nexh

An utterance of

(1a) We will not have PIZZA tonight

conveys two things. Unsurprisingly, it conveys that we will not have pizza tonight. But it also conveys, due to the focus on 'PIZZA', that we will have something else. By comparison,

(1b) We will not have pizza TONIGHT

conveys that we will have pizza at some other time.

My conjecture, in short, is that whatever explains this effect also explains the other linguistic phenomena.

I don't know how linguists model the effect in (1a) and (1b). It looks like the kind of thing that is best tackled in a dynamic framework. But let's try a static explanation first.

The explanation I'm going to assume is that (1a) and (1b) contain a tacit operator 'nexh'.

'nexh' is a cousin of the popular exhaustification operator 'exh'. Recall that $$\text{exh}(\phi)$$ is equivalent to $$\phi \land (\neg{}\phi'\land\neg\phi''\land\ldots)$$, where $$\phi', \phi'', \ldots$$ are certain alternatives to (the expression) $$\phi$$. Letting $$\text{alt}(\phi)$$ abbreviate $$\phi'\lor \neg\phi''\lor \ldots$$, $$\text{exh}(\phi)$$ is equivalent to $$\phi \land \neg\text{alt}(\phi)$$.

The 'nexh' operator (which I've made up) conjoins its argument $$\phi$$ disjunctively, (rather than conjunctively) with the negation of the alternatives: $$\text{nexh}(\phi)$$ is equivalent to $$\phi \lor \neg\text{alt}(\phi)$$.

It follows that $$\neg\text{nexh}(\phi)$$ is equivalent to $$\neg(\phi \lor \neg\text{alt}(\phi))$$, which is equivalent to $$\neg\phi \land \text{alt}(\phi)$$.

So $$\neg\text{nexh}(\phi)$$ says that $$\phi$$ is false while some alternative to φ is true.

Let's assume that 'not' expresses $$\neg\text{nexh}$$. (More minimally, let's assume that English 'not' behaves as if it had this meaning.)

We can then explain why (1a) and (1b) convey not only that 'we will have pizza tonight' is false, but also that some alternative to that sentence is true.

The role of intonational focus, on this view, is to narrow down the available alternatives: the alternatives to 'we will have PIZZA tonight' are sentences in which 'PIZZA' is replaced by an expression for some other kind of food.

This is independently plausible. While I've made up the 'nexh' operator, I haven't made up $$\text{alt}(\phi)$$. We know that the evaluation of scalar implicatures, as well as the interpretation of words like 'also' and 'only', requires rules for determining relevant alternatives, and we know that these rules are sensitive to intonation. (Compare 'I only eat PIZZA tonight' and 'I only eat pizza TONIGHT'.)

Let's turn to the other puzzling phenomena.

### "Homogeneity presupposition"

'The pigs are in the barn' entails that all the pigs are in the barn. One would therefore expect that if some but not all the pigs are in the barn, then

(2a) It is not the case that the pigs are in the barn

is true. But (2a) seems false. It conveys that none of the pigs are in the barn. How come?

Answer: English negation expresses $$\neg\text{nexh}$$.

Let $$\phi$$ be 'the pigs are in the barn'. (2a) appears to express $$\neg\phi$$, but suppose it expresses $$\neg\text{nexh}(\phi)$$, which is equivalent to $$\neg\phi \land \text{alt}(\phi)$$.

On that assumption, (2a) says not only that 'the pigs are in the barn' is false, but also that some alternative to 'the pigs are in the barn' is true.

Now assume the relevant alternatives hold fixed 'the pigs', but replace the location expression 'in the barn' with expressions for alternative locations. (2a) then entails that all the pigs are in some location other than the barn. Which entails that none of them are in the barn.

Why do the alternatives to 'the pigs are in the barn' hold fixed 'the pigs'? Perhaps that's a brute fact about how alternatives are computed. Or perhaps it's because 'the pigs' is unfocussed in ordinary utterances of (2a). Or perhaps it's because 'the pigs' isn't even in the scope of the 'nexh' operator in (2a), because plural definites set up a (plural) discourse referent xx for the pigs, and (2a) says of the xx that they are not in the barn.

We need to allow for relatively unspecific alternatives to 'in the barn', such as 'outside the barn'. Otherwise (2a) would imply that all the pigs are together at some specific location. But (2a) doesn't imply this. Or rather, it implies this only if there is a strong focus on 'barn' – as we would predict.

We can also explain why the "homogeneity presupposition" effect in (2a) depends on the absence of an explicit quantifier. Compare:

(2b) It is not the case that all of the pigs are in the barn.

This does not convey that none of the pigs are in the barn. So there's an important difference between negating 'the pigs are in the barn' and negating 'all of the pigs are in the barn', even though these sentences are truth-conditionally equivalent!

The explanation is that the sentences have different alternatives.

If 'not' expresses $$\neg\text{nexh}$$, then (2b) implies that some alternative to 'all of the pigs are in the barn' is true. And the alternatives to 'all of the pigs are in the barn' include statements like 'two of the pigs are in the barn', at least if there's no special focus on 'the barn'. If there is such a focus, we predict that the alternatives hold fixed 'all', and that we see the same effect as in (2a). As indeed we do.

### Neg-raising

We're at a restaurant. There's no particular reason for or against ordering the pizza. So 'I should order the pizza' and 'you should order the pizza' are both false. This suggests that

(3a) None of us should order the pizza

is true. Yet it seems false. (3a) conveys that we should both refrain from ordering the pizza. How come?

[Edit 28/07/2022: I have changed (3a) in response to helpful comments by Patrick Todd. The sentence I originally looked at plausibly wasn't a case of neg-raising at all.]

Answer: 'None of us' involves a tacit 'nexh'.

Observe that 'none of us' triggers negative exhaustification just as much as 'not'. For example, 'none of us will have PIZZA tonight' conveys that every one of us will have something other than pizza.

We can explain this if we assume that the LF of 'none of us is F' is something like

$[\text{No }x: Ux]\, \text{nexh}(Fx).$

This is equivalent to

$[\text{Every }x: Ux]\neg\text{nexh}(Fx),$

which in turn is equivalent to

$[\text{Every }x: Ux](\neg Fx \land \text{alt}(Fx)).$

Accordingly, the relevant LF of (3a) is something like

$[\text{No }x: Ux]\; \text{nexh}(\Box Px),$

which is equivalent to

$[\text{Every }x: Sx](\neg\Box Px \land \text{alt}(\Box Px)).$

Assume the alternatives to $$\Box Px$$ hold fixed the box ('should') and replace the prejacent 'order the pizza' with expressions for alternative actions. We then predict that (3a) implies that we should all refrain from ordering the pizza.

(As before, we have to assume that the alternatives to 'order the pizza' can be highly unspecific, at least if there's no special focus.)

The "neg-raising" effect in (3a) does not arise for 'must' or 'have to'. For example,

(3b) None of us has to order the pizza

does not convey that we have to refrain from ordering the pizza. Why not?

Short answer: because 'have to' is not held fixed when computing the alternatives.

Why is 'should' held fixed, but not 'have to'?

To some extent, this might be a brute fact about how the alternatives are computed. Remember that 'hope' doesn't trigger neg-raising, but its German translation 'hoffen' does. The rules for $$\text{alt}(\cdot)$$ in German seem to say that 'hoffen' should normally be held fixed, (except in certain quantified constructions, as I pointed out here). The rules of English seem to say that 'hope' should normally not be held fixed. There's no deeper explanation.

But there might also be a more systematic reason why 'have to' is replaceable (in the computation of alternatives) and 'should' is not. 'Have to' is stronger than 'should'. Perhaps strong expressions tend to be more replaceable than weak ones. (This would be a useful convention, if you think about it.)

The present explanation for (3a) easily extends to other neg-raising expressions. For example,

(4a) Alice doesn't think that Bob is at home

conveys that Alice thinks that Bob is not at home. Why?

Short answer: because the alternatives to 'Alice thinks that Bob is at home' hold fixed 'Alice thinks' (unless there's a special focus on 'Alice' or on 'thinks', in which case the neg-raising effect really does go away).

One might wonder why we see neg-raising only with box-type expressions.

Look at what happens if $$\neg\text{nexh}$$ is applied to a diamond expression. $$\neg\text{nexh}(\Diamond\phi)$$ is equivalent to $$\neg\Diamond\Phi \land \text{alt}(\Diamond\phi)$$. Even if the diamond is held fixed in $$alt(\Diamond\phi)$$, the effect of 'nexh' in $$\neg\text{nexh}(\Diamond\phi)$$ merely is to say that while $$\phi$$ itself is not possible (in whatever sense of possibility is expressed by the diamond), some alternative to $$\phi$$ is possible. This hardly adds anything.

### Conditional excluded middle

There is currently no tiger in my office. What do you think of this hypothesis?

(5a) If there were a tiger in my office then it would have exactly 107 stripes.

I'd say the hypothesis is false. It's false not because the tiger would have 106 stripes, or 108, or any other exact number. Intuitively, there is no number such that a tiger in my office would have that number of stripes.

This suggests that

(5b) It is not the case that if there were a tiger in my office then it would have exactly 107 stripes

is true. But (5b) seems false. It seems equivalent to

(5c) If there were a tiger in my office then it would not have exactly 107 stripes.

How come?

Answer: English negation expresses $$\neg\text{nexh}$$.

Suppose (5b) expresses $$\neg\text{nexh}(5a)$$. It then doesn't merely say that (5a) is false, but also that some alternative to (5a) is true.

If the alternatives to (5a) hold fixed everything up to and including 'would', we can explain why (5b) is equivalent to (5c), without assuming that there is a particular number of stripes the tiger in my office would have.

In general, if 'not' expresses $$\neg\text{nexh}$$ and the alternatives to a conditional 'A > C' only replace elements in C, then (under plausible assumptions about '>') 'not(A > C)' is equivalent to 'A > not C' whenever A is consistent.

And yet the principle of "Conditional Excluded Middle" (CEM),

$(A > C) \lor (A > \text{not } C)$

may well be invalid. This is possible because 'not(A > C)' and 'A > C' can both be false.

Other data that appear to support CEM can be explained away along similar lines.

Robbie Williams pointed out that (6a) appears to entail (6b):

(6a) No student would have passed if they had goofed off.
(6b) Every student would have failed if they had goofed off.

Generalising, one might intuit that $$[\text{no }x: Fx](Ax > Cx)$$ entails $$[\text{every }x: Fx](Ax > \neg Cx)$$. Since $$[\text{no }x: Fx](Ax > Cx)$$ is equivalent to $$[\text{every }x: Fx]\neg(Ax > Cx)$$, this suggests that $$Ax > \neg Cx$$ is equivalent to $$\neg(Ax > Cx)$$.

The argument assumes that there is no 'nexh' in (6a).

Assume, instead, that the LF of (6a) is something like

$[\text{No }x: Sx]\; \text{nexh}(Gx > Fx).$

This is equivalent to

$[\text{Every }x: Sx]\; \neg\text{nexh}(Gx > Fx),$

which in turn is equivalent to

$[\text{Every }x: Sx](\neg(Gx > Fx) \land \text{alt}(Gx > Fx)).$

Assuming, as before, that the alternatives to a conditional 'if A would C' hold fixed everything up to and including 'would', it follows that (6a) entails (6b), whether or not CEM is valid.

(Why do the alternatives to 'if A would C' always hold fixed the antecedent A? Good question. The present explanation seems to run into trouble in cases where heavy focus is on some elements in the antecedent.)

### Double negation

If 'not' simply inverses truth-conditions it shouldn't be terribly difficult to evaluate sentences with nested 'not's. In reality, however, this often is terribly difficult.

Can you make sense of what is conveyed by the negation of (1a), for example?

(8) It is not the case that we will not have PIZZA tonight.

How come it so hard to process nested negations?

Answer: English negation expresses $$\neg\text{nexh}$$.

The truth-conditions of $$\neg\neg\phi$$ are trivial to compute. The situation is very different for $$\neg\text{nexh}(\neg\text{nexh}(\phi))$$.

If you expand the two occurrences of 'nexh', you'll find that $$\neg\text{nexh}(\neg\text{nexh}(\phi))$$ is equivalent to

$(\phi \lor \neg\text{alt}(\phi)) \land \text{alt}(\neg\phi \land \text{alt}(\phi)).$

Needless to say, even if you got that far, you will hardly find it obvious what this means.

In fact, what it means depends on what the relevant alternatives are, and thus on intonation and other contextual clues.

But we can work out an interesting special case. In easy cases, $$\text{alt}$$ is equivalent to $$\neg$$. (Roughly, this means that we're not holding anything fixed.) In that case, $$\neg\text{nexh}(\phi)$$ is equivalent to $$\neg\phi$$, and 'not not φ', understood as $$\neg\text{nexh}(\neg\text{nexh}(\phi))$$, is equivalent to $$\phi$$.

But even in that case, the equivalence is far from obvious.

No wonder ordinary speakers struggle with negations under negations.

### Disposing of nexh

I'm not convinced that there is an 'exh' operator in the logical form of English sentences.

And I don't think there is a 'nexh' operator either.

As I said earlier, I think it would be better to explain the data in a dynamic approach.

Roughly, the idea is that the update rule associated with 'not' does two things. First, a successful utterance of 'not φ' removes all φ possibilities from the conversational scoreboard. Second, it adds $$\text{alt}(\phi)$$.

If we assume that the two updates both affect the common ground, so that 'not φ' effectively adds $$\neg\phi \land \text{alt}(\phi)$$, we get the same predictions as on the above proposal that 'not' expresses $$\neg\text{nexh}$$, at least for unembedded occurrences.

On closer inspection, however, it looks like the two updates concern different parts of the conversational scoreboard.

Suppose someone says

(1a) We will not have PIZZA tonight.

You know that we will have no food at all tonight. Would you say that the utterance is false? I'd say that it is true.

Similarly, suppose someone says

(4a) Alice doesn't think that Bob is at home.

You know that Alice is a 1-year old who has never heard of Bob and certainly doesn't think that Bob is not at home. Intuitively, the utterance is true, not false.

This suggests that even though 'not $$\phi$$' tends to convey both $$\neg\phi$$ and $$\text{alt}(\phi)$$, the latter isn't part of "what is said". Its status is more like that of a conventional implicature.

('The CEO is a woman, but she is smart' conveys that one would expect a female CEO to be dumb, but this part of its meaning doesn't enter into the sentence's truth-conditions.)

I'm not sure how best to model this kind of meaning.

Anyway, I'm sure I am re-inventing the wheel.

Has anyone tried to explain neg-raising, homogeneity presupposition, and similar puzzles by the assumption that negation (somehow) expresses $$\neg\text{nexh}$$?

(Alternatively, what is the standard explanation of (1a) and (1b)?)

Any pointers would be much appreciated.