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. 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."

I'll get back to you on this, I have a domestic duty to perform ... --Matt Westwood 10:07, 27 June 2009 (UTC)

Back again. 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. --Matt Westwood 10:24, 27 June 2009 (UTC)