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".

  • Yes, they are. See PNF. – Mauro ALLEGRANZA Mar 23 '20 at 11:12
  • I'll check it out. Thank you! – Jorbnc Mar 23 '20 at 18:22
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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.

  • 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? – Jorbnc Mar 24 '20 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 Mar 24 '20 at 18:29
  • All right! Thank you! – Jorbnc Mar 25 '20 at 15:00

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