Definition talk:Therefore

I honestly have yet to figure out how this is different from the conditional. Is there a simple explanation, or do I just need to take a course in mathematical logic? --Cynic (talk) 03:25, 27 June 2009 (UTC)

Not sure there really is one ultimately. 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability doesn't even mention the "therefore" concept, but their approach is unusual.

In a slightly different context, but with the same message (and I've translated the symbols to match what we have on this site), M. Ben-Ari's "Mathematical Logic for Computer Science" says: "Isn't $\dashv \vdash$ a boolean operator? The answer is no. It is simply a shorthand for the phrase "is logically equivalent to", unlike $\iff$ which is a boolean operator in the logic that we are describing."

Another thing: $\neg \left({p \implies q}\right)$ has a specifically defined behaviour which is the negation of $\implies$, which is true only when $p$ is true and $q$ is false, whereas $p \not \vdash q$ just means "$p$ does not imply $q$, that is, it does not necessarily follow that when $p$ is true, then $q$ must be false.

Easier to follow what I mean when I discuss the difference between $p \iff q$ and $p \dashv \vdash q$.

Again, $\neg \left({p \iff q}\right)$ is the exclusive or operator, whereas the negation of $p \dashv \vdash q$ does not necessarily mean that $p$ and $q$ have opposite truth values, it just mean it doesn't automatically have to be the case that they are the same. --Prime.mover 10:24, 27 June 2009 (UTC)