Some Contradiction notes

Contradiction, in mathematics-logic form, reads -(p&-p). There are a few different spins we can put on this, including throwing the negative and flipping the signs inside of the rule.

If you want to sound this out loud, it goes, not (any given proposition p as true and that same proposition as false). There's some modal stuffs that I do not know, in which case you would put a box, or a diamond, in front of these things. The general idea of contradiction is absolutely never.

One really cool thing that we do in mathematical logic is called "assuming the opposite" or Reductio Ad Absurdum, which is Latin, if you missed it.

Anyway, you do the RAA by assuming the opposite thing of what you want to prove. If it leads to a contradiction from the direct impediment or derived impediment of your axioms, lemmas, and previous proofs, then you just proved the opposite of your opposite is true. Namely, you just proved what you wanted to prove.

Of course, like nerdy economists, we might say that nobody ever is passionate about anything, just like the economy is never guided by passion, no one wants to prove anything so much as they feel the scientific tug of the universe at their fingertips. Truth be told: it's either something is or it is not, I suppose. Here's some heady philosophy for you: the nature of prescription, description, creation, and synthesis in rational science, such as math and logic. (Note: One of the big problems with my favorite philosopher Spinoza was that he had created a system which was wholly inapplicable to real life. What good is it, really, to say God is substance? or any of the other unverifiable stuff?).

So at any given time, some philosophy, science, and folk wisdom-popular culture, might use descriptions of so-called real life, might pose prescriptions, might pose synthesis of both of those, and creation based on all or none (but the none part is also arguable: Creatio ex nihilo?).

Here's a quote from the local Philosopher of Mathematics, "We're hardly sure about anything!"

There's some problems with the Law of Contradiction, as powerful as it is, that we'll go over later.