Logical parenting, balloons, and Abelard's insights on quantifiers

My 3.5 year old daughter has apparently been learning about opposites at nursery, because all weekend she was popping out such gems as "You know what are opposites? Big and little!" (Hot and cold, up and down, in and out, etc., etc., etc.). Sunday evening while we were getting supper read, she proceeded to play underfoot with a balloon she'd been given at a birthday party earlier in the day. This was increasingly irritating until she came out with:

"Do you know what are opposites? No balloon and some balloon!"

Logical parenting: Ur doin it right.

Of course, I was curious to know if she could extrapolate, so I asked her what the opposite of "All balloons" was. Her reply was "No balloon", which I couldn't complain about, because, after all, I hadn't specified whether I was looking for the contradictory opposite or the contrary opposite. Being the proud parent I was, I relayed the story on FB, and was amused at the selection of half-joking, half-serious suggestions I got for the opposite of "all balloons": Negative balloons? Impossible balloons? The square root of minus one balloons? i balloons? But it also made me think: The usual Aristotelian quantifier opposed to 'all' is 'some____not'. But "Some balloons not" doesn't make any sense. You can have "all balloons", you can have "no balloons", you can have "some balloons", but you can't have "some balloons not" [1]; if you want to use that quantifier, there needs to be more than just a quantified subject, there has to be a predicate, too. The same is not true of the non-Aristotelian form of the negation, 'not all': While you can't have "some balloons not", you can have #notallballoons.

Reflecting on this on the way home this evening, I was reminded of how Abelard made this very distinction, between non omnis and quidam non, arguing that these two are not equivalent with each other: non omnis does not have existential import, while quidam non always does. Many people think that making a distinction between 'not all' and 'some____not' is only necessary in a context where 'all' has existential import; but perhaps Abelard's insight that non omnis and quidam non are not equivalent reflects something deeper than just logical machinery to deal with a problematic assumption about universal quantifiers.

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[1] This is, essentially, just the well-known observation that there is no single natural language English term 'nall'.

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© 2015, Sara L. Uckelman