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Find which of the following are well-formed formulas of predicate logic

a. $( \forall y)(Ay ⋁ (\exists y)By)$

b. $ ((\forall y)Ay ⋁ (\exists y)By)$

c. $ (\forall y)(\forall x)(Ay ⋁ Bx) $

d. $ Ay ⋁ Bx $

An : According to me options $ \ (b), \ (c) , \ (d) \ $ are well-defined predicate Logic except the first option $( \forall y)(Ay ⋁ (\exists y)By) \ $ .

Though I am not sure. Any help is there?

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    $\begingroup$ What about (d)? I'm not sure myself, but aren't quantifiers necessary? $\endgroup$
    – Shuri2060
    Commented Jul 24, 2017 at 10:10
  • $\begingroup$ yes you are at the right point $\endgroup$
    – MAS
    Commented Jul 24, 2017 at 10:12
  • $\begingroup$ @Shuri2060 No, you can have free variables in well-formed formulas. Not in sentences, but in wff's it is ok. $\endgroup$
    – Bram28
    Commented Jul 24, 2017 at 12:25
  • $\begingroup$ I'm not so familiar with the topic myself, so I'd made an assumption about that, mb $\endgroup$
    – Shuri2060
    Commented Jul 24, 2017 at 14:12
  • $\begingroup$ @Shuri2060 No problem! We're all here to learn! :) $\endgroup$
    – Bram28
    Commented Jul 24, 2017 at 14:33

1 Answer 1

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They are all well-formed formulas! It is in fact ok to have a variable quantified within the scope of a quantifier of that same variable (as in a). For a), you simply have that the $y$ in $Ay$ is quantified by the $\forall y$, and the $y$ in $By$ is quantified by the $By$. So, it is equivalent to $(\forall y)(Ay \lor (\exists x) Bx)$.

ALso, you are allowed to have free variables in well-formed formulas, as in d). What this means is that d) is not a sentence, but it is a well-formed formula.

Finally, someone may complain that b) has an extraneous pair of sentences, but others might say that d) should have an extra pair of sentences. This actually depends on how the well-formed formulas are exactly syntactically defined in your book (not every book does it the same ... in fact many books don't put parentheses around $\forall x$ like your formulas all do). But in both cases that would be some superficial nitpicking, so I would consider both of them well-formed.

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