Originally Posted by

**xisnotx**
First one is identity.

Nope. Rule of thumb: if you can swap the premise and conclusion and the argument is still valid, then the premise and conclusion are logically equivalent. However, the following argument is invalid, so there is no identity:

Socrates is a man or Socrates is immortal.

Therefore,

Socrates is a man.

The premise merely states that Socrates is a man, Socrates is immortal, or both, but it doesn't specify which is true, so the conclusion does not follow. The premise could be true even if Socrates is not a man.

It makes sense because as soon as you say "x is true" then you can't say anything to not make "x" not true. (Even by contradiction).

Almost, but not quite. Consider the case of a contradictory premise:

Socrates is mortal and Socrates is immortal

Therefore,

Socrates is mortal.

The conclusion follows from its premise according to the inferential rules of classical logic. However, the argument cannot be a valid (or sound), because there is no way for the premise to be true. It's often referred to as the principle of * ex falso quodlibet*, or 'from a contradiction, anything follows.'

The third argument is valid because there is only one combination of truth and falsity where a material implication is false, e.g. 'P → Q' is false if, and only if, 'P' is true and 'Q' is false. Therefore, the statement 'If Socrates is immortal, then Socrates is mortal' is false only if the antecedent ('Socrates is immortal') is true and the consequent ('Socrates is mortal') is false. However, given our premise ('Socrates is mortal'), the antecedent cannot be true and the consequent cannot be false. If the material implication is not false, then it must be true by the law of excluded middle. In other words, while the antecedent is the negation of the premise, the antecedent itself is not being asserted as true, so there is no contradiction.

I don't know for sure but I think x->-x should lead to x.

Right, and that's why the third argument is the identity--the premise entails the conclusion *and* the conclusion entails the premise.

xisnotxisx

or it isn't,

in which case

xisnotxisnotx.

Did you just multiply yourself?