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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.

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  • 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

1 Answer 1

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You already have the truth-table corresponding to the conjunction connective, so it should be transparent for you to recognize the answer to your question from the syntactic point of view of the calculus.

Asserting ¬(P.Q) [Premise 1] is logically equivalent to assigning a binary truth-value of zero to (P.Q). So, looking at the table, you see there are three possible assignments to the truth-values of P and Q that yield that result. Then, by asserting ¬(Q) [Premise 2] and using the same argument, you assign a truth-value of zero to Q; looking again at the truth table, this restricts the possibilities to two cases (namely, one in which P's truth-value is zero, and another in which it is one). So, the assertions do not imply a definite binary value for P's truth state.

From the semantical point of view, the interpretation of the conjunction connective in this case follows (informally) this way. You know that either P is false while Q is true, or Q is false while P is true, or P and Q are both false; and you also know that Q is false. This last statements eliminates the first possibility (Q being true while P was false), and you are left with the other two; so, what you seem to think to be justified (the second possibility, that Q being false then P must be true) excludes the third situation when both P and Q are false, and so is fallacious.

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  • 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?
    – cpx
    Oct 20, 2012 at 1:29
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    When you assert a premise, you assign a truth value of one to that formula. If you assert ¬(P.Q), then that wff is formally true, and that "knowledge" comes from the very act of assertion. What the logical system allows you to is to draw valid conclusions from hypothesis; it doesn't give you a clue to what premises are true or false in the first place. But once you have asserted a few premises, you can apply the rules of the calculus to get new (logically equivalent) formulas. So, from the assertion of ¬(P.Q), you can (syntactically) obtain (P.Q) by double negation, and assign a zero to it.
    – Mono
    Oct 20, 2012 at 3:34

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