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To make the antecedent true at least one thing that is or is not in the extension of $F$ must also be or not be in the extension of $G$.

But then this seems to automatically make at least one case of the second bi-conditional, true. And to find an interpretation where the first is true and the second false, I need there to be no cases where what is or is not $F$ is not the same as what is or is not $G$.

One idea I had was to set $F$'s extension to {1} and $G$'s extension to $\{∅\}$.

In general I am not clear on what it takes to make the antecedent false. For the bi-conditioanl to be false, the truth of '$\exists x(Fx)$' and '$\exists x(Gx)$' need to mismatch. And the truth of either requires at least one member of the domain to be in the extension of the predicate letter. So if I satisfy that requirement, then to get the mismatch that falsifies the bi-conditional I need to have nothing from the domain in the extension of the other predicate letter.

Thanks for any help.

Physor
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beachtrees
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    "For the bi-conditional to be false, the truth of '∃x(Fx)' and '∃x(Gx)' need to mismatch." Correct. Thus the only way is that e.g. there is something that is F (like F(1)) and there is nothing that is G (i.e. the interpretation of G is $\emptyset$). – Mauro ALLEGRANZA Oct 09 '20 at 10:53
  • To satisfy the other formula, it is enough to find an element such that both are true or both are false. $2$ will do, because F(2) is False as well as G(2). – Mauro ALLEGRANZA Oct 09 '20 at 10:56

1 Answers1

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One idea I had was to set $F$'s extension to {1} and $G$'s extension to $\{∅\}$.

That's correct.
With $x \mapsto 2$, there is an object of which neither $F$ nor $G$ is true, so the biconditional is true, so $\exists x (F(x) \leftrightarrow G(x))$ is true.
However, with $x \mapsto 1$ there is an object of which $F$ is true, so $\exists x F(x)$ is true, but since the extension of $G$ is empty, $\exists x G(x)$ is false, so $\exists x F(x) \leftrightarrow \exists x G(x)$ is false, as desired.

In general I am not clear on what it takes to make the antecedent false

If by antecedent you mean the left-hand side of the biimplication (typically the antecedent/succedent terminology is only used with one-directional implication), $\exists x F(x)$, then the requirement would be that the extension of $F$ is empty. But you don't necessarily need the left-hand side false, but, as you say, a mismatch between the two sides, which is achieved by having $\exists x F(x)$ true but $\exists x G(x)$ false (or vice versa).

Natalie Clarius
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