No deal is better than a bad deal

My friend Michelle reminded me that Theresa May said this back in January, and has kept on saying it since then. Most people have been non-pedantic enough to let it go by without comment (after all, there's enough politics happening for us to talk about) but she, I and Chris Maslanka in the Guardian all noticed that it was ambiguous.

Negation, as I've mentioned in previous posts, is usually ambiguous because it takes scope over different bits of the sentence it's in. Here's Chris Maslanka's explanation of the two meanings:

Incidentally, there's a long bit in Alice Through The Looking Glass (or the other one) where a messenger pouts that 'nobody walks faster than I do' and the king says 'he can't, or he'd be here already'. Lewis Carroll was keen on logic and semantics jokes.

After some back and forth with my colleague (& friend) Christina, we think we've translated the two meanings into what looks like gibberish to non-linguists, but is actually a formal representation of the meaning. The notation explains how the bits of the sentence interact to give two different meanings from the same set of words in the same order. I've translated them underneath into increasingly more idiomatic English.

The meaning Theresa means, presumably, is this:

∀x∀y.[NO.DEAL(x) & BAD.DEAL(y) -> x > y]

(Roughly,

"For all x and for all y, if x is no deal and y is a bad deal, then x is better than y"
or "If x is no deal and y is a bad deal, then x is better than y"
or "Having no deal at all is better than having a bad deal".)

Whereas the meaning that's much more salient to me, and which made the sentence seem quite bizarre, is this one:

 ~∃x.[DEAL(x) & ∃y.[BAD.DEAL(y) & x > y] ]

(Roughly,

"There is no x such that x is a deal and there is some y such that y is a bad deal and x is better than y"
or "There is no deal which is better than a bad deal"
or "A bad deal is the best deal".)

Incidentally, I'm pretty sure that this particular type of negation scope ambiguity is regional - it seems less obvious to US speakers (see this post and the comments). But that's a purely anecdotal observation so do let me know if you have anecdata to add to that.