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In propositional logic we had studied the following rules:

(P • Q)   ->  P, Q   
~(P ∨ Q)  ->  ~P, ~Q   
~(P ⊃ Q)  -> P, ~Q  

and say, the inference rules:

~(P • Q), P -> ~Q
(P ∨ Q), ~P -> Q
(P ⊃ Q), ~Q -> ~P

Now in predicate logic you can use these rules but my question is how do I decide where does the negation go? This is unfortunately not clearly stated in the book.

For example:

If I have:

~((x)Pa ⊃ (x)Qa)

I can deduce two inferences from this such as firstly:

(x)Pa

But for the second inference where does the negation go?

Is it (x)~Qa or ~(x)Qa ?

4

¬(P → Q) is equivalent to : P ∧ ¬Q.

Thus, ¬((x)Px → (x)Qx) must be : (x)Px ∧ ¬(x)Qx.

  • Is it true in all the cases that the negation always goes before the quantifier? I am confused about where does the negation of this (x)~Qa comes from if the negation of (x)(Qx) is ~(x)Qx? – cpx Oct 9 '15 at 11:29
  • @cpx - It depends on what your are expressing : "not all men are Greeks" is ¬(x)Qx (that is true) and it is different from "all men are not-Greeks", i.e. (x)¬Qx (that is false). – Mauro ALLEGRANZA Oct 9 '15 at 11:35
  • I see. So, I guess that particular case is not actually a negation of something but rather an expression itself. In other words, all the negations should go in the front of quantifier if it is of something else, right? – cpx Oct 9 '15 at 11:41
  • @cpx - YES; the negation of an expression A is always ¬A. Then, there are "equivalences", like those between ¬(x)Qx and (∃x)¬Qx and between (x)¬Qx and ¬(∃x)Qx. – Mauro ALLEGRANZA Oct 9 '15 at 12:01

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