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Natural language quantifiers such as some and a are formalised for compositional semantics as λP.λQ.∃x.P(x)∧Q(x). Compared to the formalisation of quantifiers such as every or all I do not quite understand why ∧ is used rather than ⇒.

Obviously λP.λQ.∀x.P(x)∧Q(x) would mean all x satisfy P and Q, which is not the correct meaning, but how in the case of λP.λQ.∃x.P(x)∧Q(x) would it make a difference if ∧ was replaced by ⇒, obtaining λP.λQ.∃x.P(x)⇒Q(x)?

The one difference I see is that λP.λQ.∃x.P(x)∧Q(x) actually asserts that at least one x exists for which P and Q hold. λP.λQ.∃x.P(x)⇒Q(x) on the other hand would mean that if there is an x for which P holds it must also hold for Q. It does not assert that there actually is any x at all though for which P holds.

Now, when I compare this to the intuitive meaning of every or all I would say that if some asserts the existence of an x for which P and Q hold, then surely so does every?

So my question can be split up like so:

Assuming I identified the difference between defining some with an implication or defining it with a conjunction correctly:

  • What is the reasoning behind defining all to NOT assert existence of an x to satisfy P and Q

or conversely

  • What is the reasoning behind defining some to DO assert existence of an x to satisfy P and Q

If I did not not identify the difference between defining some with an implication or defining it with a conjunction correctly:

  • What is the actual difference and what is the reason to use a conjunction rather than an implication?
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  • Might be worth nothing that the NSM says that SOME and ALL are semantic primes, and so any definition of them will be inherently worse than just using the word directly. – curiousdannii Nov 14 '17 at 1:18
  • It makes no sense to connect "some" with implication. "Some man is mortal" does not mean that something's manhood implies its mortality. – Greg Lee Nov 14 '17 at 2:03
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The reason is as follows:

  • For ALL we need "for all objects, if they are P, then they are also Q". If we would use logical AND, it would mean "for all objects, they are P and they are Q" which is obviously not what we want.

  • For SOME we need "for some object, it is P and it is Q". If we would use implication, it would mean "for some object, if it is P, then it is Q" and it is possible to construct a counterintuitive example where this is the wrong translation:

Assume we have "some apple is green" and we have a world where there are only red apples and green bananas. Then for each apple the implication apple(X) -> green(X) is false because the apple is not green, but for all bananas the implication apple(X) -> green(X) is true, because the banana is green (see also https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference) ).

This means the green banana makes "some apple is green" true, which is clearly not the right translation.

With logical and, we need to find an object that is green and an apple, and this is the right translation.

I hope this example is helpful to understand the reason for using conjunction in one case and implication in the other case.

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"... λP.λQ.∃x.P(x)⇒Q(x) on the other hand would mean that if there is an x for which P holds it must also hold for Q."

No. You're being careless about the scope of the existential quantifier. In your informal rendering into English, you take the scope to be P(x), so that the antecedent of if-then is ∃xP(x). Then in the expression you're discussing, Q(x) is not in the scope of any x quantifier, and the expression is not closed.

That is why I commented above that your formula corresponds only to English nonsense. The closest pseudo-English to ∃x.P(x)⇒Q(x) would be: "There is something for which if it is a P, then it is a Q." so that both P(x) and Q(x) are in the scope of "there is some x".

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