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Domain: $\{1, 2, 3, 4\}$
Referents: $a: 3, b: 4$
Extensions: $M:\{1, 2\}, F:\{3, 4\}, J:\{1, 3\}, S:\{2, 4\}$

How can I determine the truth value if there is more than one quantifier applied to the same expression? For example, how to determine the truth value of these four expressions:

  1. $\forall x \forall y(Mx \implies My)$
  2. $\forall x \exists y(Jy \iff Sx) \lor Jb$
  3. $\exists x \exists y(Jx \lor Fy \lor My)$
  4. $\exists y \forall x(Sy \iff Mx) \land Ja$
Hanlon
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  • What do these wffs actually mean, intuitively? What have you tried?? – Peter Smith Dec 26 '17 at 20:06
  • First probably means true for all x and all y in all combinations, second for all x at least one y, third at least one x and at least one y and fourth at least for one y all x. These are only my intuitive guesses, I can't solve any of these wffs in practice. I need to understand mechanics behind all these wffs. – Hanlon Dec 26 '17 at 20:11
  • " I need to understand mechanics behind all these wffs." Which textbooks have you taken a look at? – Peter Smith Dec 26 '17 at 20:13

1 Answers1

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Let's do a simpler example, assume the domain is simply $\{ 1,2 \}$ and evaluate $\forall x \exists y \ (Mx \land My)$

You can work this out one quantifier at a time. First the universal:

$\exists y (M1 \land My) \land \exists y (M2 \land My)$

And now both existentials:

$((M1 \land M1) \lor (M1 \land M2)) \land ((M2 \land M1) \lor (M2\land M2))$

Bram28
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