Question on Conditionalizing

[Laure]

I have a question for the logic experts on the forum. (I'm not an expert, but I had a symbolic logic course in college and was the top student, and I had the benefit of a high school math curriculum that emphasized proofs. As a programmer, I use boolean logic just about every day.)

In looking back at my high school math book, it describes Modus Ponens, Modus Tollens, and all that good stuff. It describes "Conditionalization": a conditional sentence follows from its consequent. So, if Q is true, P-->Q is true for any old P.

1) Is my interpretation correct, or does P have to be something that was previously used in order to derive Q?

2) My textbook gives an example that has Q as "The moon is a satellite", and two possible P's: "The moon is made of green cheese" and "The moon is not made of green cheese". It says either of the green-cheese statements can be used as P, and P-->Q is true in either case. This kind of makes sense, since we KNOW that the moon is a satellite, no matter what, so if the moon is made of green cheese, it's a satellite, and if it's not made of green cheese, it's a satellite. My question is, what about "If the moon is not a satellite, the moon is a satellite"? True statement? Is this considered a paradox, or is it just a perfectly valid, if strange, application of conditionalizing?

(This post is not trying to start an argument; I just want to learn something from the experts. I defer to Ba'al & Merlin (and maybe even Dragonfly!).)

[BaalChatzaf]

I have a question for the logic experts on the forum. (I'm not an expert, but I had a symbolic logic course in college and was the top student, and I had the benefit of a high school math curriculum that emphasized proofs. As a programmer, I use boolean logic just about every day.)

In looking back at my high school math book, it describes Modus Ponens, Modus Tollens, and all that good stuff. It describes "Conditionalization": a conditional sentence follows from its consequent. So, if Q is true, P-->Q is true for any old P.

1) Is my interpretation correct, or does P have to be something that was previously used in order to derive Q?

2) My textbook gives an example that has Q as "The moon is a satellite", and two possible P's: "The moon is made of green cheese" and "The moon is not made of green cheese". It says either of the green-cheese statements can be used as P, and P-->Q is true in either case. This kind of makes sense, since we KNOW that the moon is a satellite, no matter what, so if the moon is made of green cheese, it's a satellite, and if it's not made of green cheese, it's a satellite. My question is, what about "If the moon is not a satellite, the moon is a satellite"? True statement? Is this considered a paradox, or is it just a perfectly valid, if strange, application of conditionalizing?

(This post is not trying to start an argument; I just want to learn something from the experts. I defer to Ba'al & Merlin (and maybe even Dragonfly!).)

Q -> (P -> Q) is a tautology where -> is material implication.

Let ^ mean "and" and ~ mean "not"

P -> Q == ~(P ^ ~Q) (that is a definition). P materially implies Q if it is never the case that P is true an Q is false. From the truth of Q one can infer that P ^ ~Q is false, hence P -> Q.

Ba'al Chatzaf

[Laure]

Thanks for the quick reply! So, given that the moon is a satellite, "if the moon is not a satellite, then the moon is a satellite" is just a funny little tautology and not a paradox.

[BaalChatzaf]

Thanks for the quick reply! So, given that the moon is a satellite, "if the moon is not a satellite, then the moon is a satellite" is just a funny little tautology and not a paradox.

That translates to the moon is a satellite or the moon is a satellite which happens to be true (contingently), but it is NOT a tautology. A tautology is a statement that is true in any possible case. For example the moon is a satellite or the moon is not a satellite. Now, that is a tautology since it is true whether or not the assertion the moon is a satellite is true or false. If we lived on Venus, the statement the moon is a satellite (of Venus) would be false, but the disjunction just prior would still be true. See http://en.wikipedia.org/wiki/Tautology_(logic)

Statements of the form Q -> (P ->Q); P -> P; (P ^ ~P) -> Q are tautologies. You will notice a false statement implies any statement and a true statement is implied by any statement. These are some of the discontents of material implication. There are other systems of logic and other types of entailment or implication in which such oddities do not occur. The system of strict implication is one such alternative system of logic.

See

http://en.wikipedia.org/wiki/Strict_implication

Ba'al Chatzaf

[Laure]

Interesting. The "strict implication" article goes to the heart of Thom's "Iffy" thread. I think what Thom is getting at is that "strict implication" is the way we intuitively interpret "if" in natural language. Taking an example from the article, "If Bill Gates went to medical school, then Elvis is still alive". A true statement, but intuitively false, since there's nothing about Bill Gates going to medical school that would cause Elvis to still be alive.

Clicking around in Wikipedia, I found this article: Paradoxes of Material Implication. (It has a few "broken" formulas, unfortunately, but I think it's an interesting article anyway.) It says, "The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic." which I'm not sure is a good definition of paradox. Just because something is intuitively problematic (like my example in the first post) doesn't mean it's a real paradox.

Anyway, I thought this article was relevant to some of the other discussions going on here. It points out that from a contradictory set of premises, anything follows. It says that such arguments are "valid" but not "sound". "Valid" = there's nothing wrong with the logic itself. But not "sound" because it contains faulty premises. That's why the lady said "check your premises" after all. Your argument can be perfectly valid, but if you have a false premise, you can "prove" anything!