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Certainty of outcomes Vs possibility of outcomes?

I'm sill a little new to this whole wiki editing thing so i thought i'd better make it a discussion rather than put my foot in it by changing the page willy-nilly. Where you've written "if P is false, then P → Q is true." would it not be more accurate to say "if P is false, then P → Q may still be true."? after all, if (P), or (P and Q) are both false as per the last two rows in the table, we don't necessarily know that P → Q is true, just that it might be true and that we have insufficient data to rule it out. --Murphy 20:04, 7 November 2009 (CST)

I think I know where you're going with this...
Material implication explores the possibility of Q and ¬Q in the presence of P and ¬P.
Logical implication explores the causative effect of P and ¬P on Q and ¬Q.
The information and example is attempting to explain both material and logical implication in the context of material implication alone. Thus, the article as a whole could seem to be saying ¬P ⇒ (P ⇒ Q), which is false.
Material Implication Logical Implication
P → Q P ⇒ Q
P Q Valid P demands Q Valid P causes Q
P ¬Q Invalid P prevents ¬Q Invalid P cannot cause ¬Q
¬P Q Valid ¬P allows Q Invalid ¬P is not the cause of Q
¬P ¬Q Valid ¬P allows ¬Q Invalid ¬P is not the cause of ¬Q
¬P allows either Q or ¬Q, but does not cause either.
In either case, we've shown logically that ¬P cannot be used to establish either Q or ¬Q. Only P can.
You could explain the difference between the two, and perhaps even include the chart/info I just wrote. Another level of confusion for the already confused creationists :) --Jaban 15:52, 8 November 2009 (CST)
I think i kind of get it, but even by that definition of material implication (as in not a causal link), isn't the truth of the statement (¬P allowes ¬Q) a completely separate issue to the truth of the original statement that (P → Q).
As far as i can see, (¬P allows ¬Q) would be similar to (¬fruitbowl allows ¬apple) which isn't necessarily false, but it doesn't really tell us anything about the truth of the statement (fruit bowl → apple). I just don't see how based on (particularly in row four) insufficient data, you can't make a definitive claim about the truth of statement (P → Q) one way or the other.
Likewise in the logical implication side of your table, the last two rows don't necessarily invalidate the statement that (P ⇒ Q). In row three, perhaps in this case Q was in this case caused by X, but that doesn't necessarily mean Q can't also caused by P (when p occurs). And in row four we have the same problem. Insufficient data to say one way or the other. You could could make the same statement as in the material implication. (¬P allows ¬Q), but that doesn't necessarily tell us whether (P ⇒ Q) is actually true or false.
I guess what I'm getting at is that if we've shown logically that (¬P cannot be used to establish either Q or ¬Q. Only P can), then how can we make any True, false, valid, or invalid claims about the statements (P → Q) or (P ⇒ Q) based on the last two rows of the table. Shouldn't they read "insufficient data" or "unknown" or something (@_@?) Sorry, i don't mean to be an ass or anything, I'm just not quite seeing it.--Murphy 17:12, 8 November 2009 (CST)
If I can summarize what you're thinking:
If '¬P ⇒ Q' is false, that doesn't speak about 'P ⇒ Q'. So if ¬P don't we automatically lose the ability to speak about 'P ⇒ Q'?
My answer is that we ARE asking what you think we should be. You could rewrite the table like this:
Logical Implication
P Q Question Answer
true true Does +P ⇒ +Q? yes
true false Does +P ⇒ ¬Q? no
false true Does ¬P ⇒ +Q? no
false false Does ¬P ⇒ ¬Q? no
But that's not necessary. P and Q are containers for values, not values themselves. "Does P ⇒ Q?" means "Does the value of P in this case imply the value of Q in this case?" It does not dictate the values as positive.
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