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Could you tell me if these two sentences are equivalent? If they aren't, what would be the correct sentence that is equivalent to (1)? Please explain. Thank you!

(1) (∀x)[Ax→(∃y)(By & Txy)]

(2) (∀x)(∃y)[(Ax & By)→Txy]

The expression that is supposed to be symbolized is "Every A takes at least one B".

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  • Yes, they are. See PNF. Commented Mar 23, 2020 at 11:12
  • I'll check it out. Thank you!
    – jn_br
    Commented Mar 23, 2020 at 18:22
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    – J D
    Commented Mar 24, 2020 at 18:04

1 Answer 1

1

The two are not equivalent. You want to change your second sentence to be

(∀x)(∃y)[Ax → (By & Txy)]

One way to see that your 1 and 2 are not equivalent is that under an interpretation in which there are some As but no Bs, then sentence 1 is false, but sentence 2 is true, because the antecedent of the material conditional in 2 is always false.

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  • Uhmm, I see. But, what if now the expression to be symbolized is "Every A takes every B"...in that case, would (∀x)[Ax→(∀y)(By → Txy)] be equivalent to (∀x)(∀y)[(Ax & By)→Txy]? Would be correct to have an interpretation in which there are some As but no Bs?
    – jn_br
    Commented Mar 24, 2020 at 16:24
  • That would work OK. Bear in mind that in predicate logic 'all' does not imply 'some', and 'all' statements are trivially true when there is nothing for them to apply to. So it is true that all unicorns love me, and also that I love all dragons. It is also true that all meerkats love all unicorns, which is an analog of your last sentence.
    – Bumble
    Commented Mar 24, 2020 at 18:29
  • All right! Thank you!
    – jn_br
    Commented Mar 25, 2020 at 15:00

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