# Denying a conjuct in propositional logic

According to Conjunctive syllogism, We accept one part of the statement as true and reject the other part as false.

Truth table for AND

``````_____________
| P Q | (P.Q)|
| 1 0 |   0  |
| 0 1 |   0  |
| 1 1 |   1  |
| 0 0 |   0  |
|_____|______|
``````

For example:

Suppose, we have,

``````Premise 1: ~(P.Q)
Premise 2: P
Therefore, ~Q
``````

I think we have it so far so good up to this point, but why can't we have the other way?

For example:

``````Premise 1: ~(P.Q)
Premise 2: ~Q
No conclusion.
``````

Why can't we draw our conclusion as `P` here since `~(P.Q)` means at least one is false? I'm not clear with what it means by at least one false because we can only attach `~` with either P or Q so not both are true. I'd like the reason to be explained with an example with truth table just to be clear.

• The logical formula you seem to be looking for is "XOr" - the exclusive or. Informally, that's P or Q, but not both. If that is true, and you know the value of one of P or Q, then we know the other has the opposite value. Truth tables are difficult in comments, as there's no line breaks, but I'll try: P, Q, P Xor Q - 0,0,0|0,1,1|1,0,1|1,1,0
– Ryno
Oct 22, 2012 at 8:41

• How do we know about `~(P.Q) = 0`? Is this because we take `P=1` and `Q=1` and then apply the negation? If so, how do we know the truth value of the letter?