# Hierarchy of operators of propositional logic

I have a first order logic clause and I have to transform it to its normal clausular form. $$\forall x \exists y \left[A(x) \land \lnot B(x) \implies C(x,y) \land\exists zD(z)\right]$$

But I have several problems with the hierarchy when I have an implication in the middle of the formula.

So, I imagine these relationships as if they were of propositional logic: $a\lor b \implies c \lor d$, but now my question is:

$a\lor v \implies c \lor d$ is equal to $(a \lor b) \implies (c \lor d)$? or is equal to $a \lor (b \implies c) \lor d$?

Thanks.

• The first option is correct. The scope of $\to$ is larger which is to say both $\lor$ and $\land$ take precedence over $\to$. This is just a convention. – Git Gud Jun 22 '14 at 20:59
• But, if I have $a \lor b \implies c$ it isn't equal to $(a \lor b ) \implies c$? – Tomi Jun 22 '14 at 21:03
• Yes, that's what I'm saying. – Git Gud Jun 22 '14 at 21:04
• So, finally, mi Normal Clausular Form is $\forall x[(\lnot A(x) \lor B(x)) \lor (C(x,f(x)) \land D(c))]$? Thanks! – Tomi Jun 22 '14 at 21:07
• I don't know what Normal Clausular Form is, sorry. – Git Gud Jun 22 '14 at 21:10

Yes, it is right.

The formula, written with parentheses :

$∀x∃y[(A(x)∧¬B(x)) \rightarrow (C(x,y)∧∃zD(z))]$

is equivalent to :

$∀x∃y[\lnot (A(x) \land \lnot B(x)) \lor (C(x,y) \land ∃zD(z))]$

[by equivalence between $p \rightarrow q$ and $\lnot p \lor q$] which in turn is equivalent to :

$∀x∃y[(\lnot A(x) \lor B(x)) \lor (C(x,y) \land ∃zD(z))]$

[by De Morgan and double negation].