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I've a couple of questions.

From MIT notes:

$\frac{NOT(P)\;IMPLIES\;NOT(Q)}{P\;IMPLIES\;Q}$

is not sound: if P is assigned T and Q is assigned F, then the antecedent is true and the consequent is not.

Well, if I get it right, this is not sound because not all the premises are true (i.e., $P$ is assigned to true and $Q$ is assigned to false) and the argument is not valid since $Q$ should imply $P$ in the consequent.

1- How can the antecedent be evaluated to true in this case?

or How can $NOT(true)$ imply $NOT(false)$?

Again from the notes:

Note that a propositional inference rule is sound precisely when the conjunction (AND) of all its antecedents implies its consequent.

2- Does this mean the same as, "the rule is sound if all the antecedents are true" and consequently so as the consequent" or I just misunderstand?

Thanks in advance!

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In checking validity of an argument, it's not a good practice to just say "the right conclusion would be this, rather than the given conclusion." What you need to do in showing an argument is not valid, is to come up with truth assignments for the variables which make all premises true but make the conclusion come out false.

In your example the conclusion is P implies Q, which (if we are trying to show the argument is invalid) we make this conclusion false, and the only way in this case to do that is to make $P$ true and $Q$ false.

We now know the truth values of the letters, and we plug into the premise. The premise here says "if not P then not Q". Well since we have assigned P = true, we have that (not P) is false. We also know that "false implies anything". SO the premise comes out true.

Altogether this shows the given aqrgument is invalid.

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