Weird Logic

[erm]

This is irrelevant to the validity of the argument and the truth of the conclusion. Since '~P → P' is--in an important sense--just another way of writing 'P', of course, if '~P' is true, then 'P' is false and we have contradicted the premise, but '~P' is neither a premise nor entailed '~P → P'. In other words, '~P → P' does not contradict 'P v ~P', and so it's irrelevant except as a means to construct an unnecessarily complicated proof.

It's not an "unnecessarily complicated proof", it is a non-abbreviated proof, which does not hide the RAA needed for the complete logic when law of noncontradiction and excluded middle are taken into account.

If you want to construct a reductio ad absurdum, then yes, both formulae are used to deduce 'P & ~P'. However, while assumptions must be discharged for the deduction to be valid, premises need not be. In fact, the rule is customarily that premises are never discharged. Since 'P' is the only premise, then, only '~P' can be discharged by the reductio ad absurdum.

Yes, but if you arbitrarily favor a premise like that, it still requires a RAA after the aforementioned laws are applied. RAA would discharge the assumption instead, and without the assumption the conclusion is not valid.

I was just trying to get across why such a definition of Identity would be used when I said the premise would be equally worthy of discharge, not making a formal logical statement. As I said, if Identity doesn't include those two laws, then it's an Identity (the premises and conclusion entail each other either way).

 

[garbage]

A bit of both. To entertain, to educate, to ponder, and amuse.

Got it.

Right, and we're back to you explaining why your answer was ambiguous, and it's a fine explanation. But then why does it feel like you're trying to disagree with me about something?

My intent isn't to try to disagree; I just do. Perhaps we have different definitions for (or perceptions of) ambiguous, and I'm content to leave the disagreement at that.

 

[xisnotx]

All three of these arguments are valid (i.e. if the premise is true, then the conclusion must be true on pain of contradiction).

Socrates is a man.
Therefore,
Socrates is a man or Socrates is immortal.

Socrates is a man.
Therefore,
If Socrates is immortal, then Socrates is a man.

Socrates is mortal.
Therefore,
If Socrates is immortal, then Socrates is mortal.​

One of these arguments is actually an identity (i.e. A, therefore, A). Which one?

So if all men are mortal, and Socrates is a man, then each of the conclusions above must be true, but they all seem intuitively wrong.

First one is identity.

It's been too long since I've sat down and done this.

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). For example:
x
not x
?

I don't know for sure but I think x->-x should lead to x.
The examples in the op are trying to assert that.
so perhaps

xisnotxisx
or it isn't,
in which case
xisnotxisnotx.

meh, i'll just give up.

 

[reason]

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?

 

[Little_Sticks]

Does it ever bother you that "logic" has to first assert a truth in order to "prove" anything? Logic is supposed to be this thing that can verify truth, it is "assumed", but it doesn't seem able to verify its own initial assumptions. Those assumptions must be considered...obvious. Science over religion just seems to be the same shit with a different wrapper. Well, it looks like shit can come in any form then.