Not quite, I meant absent reductio ad absurdum (what is required to conclude P absent ~P, in the case of ~P → P), which is what I remember Identity being defined as. I'm going to guess that has either changed now or you are familiar with different semantics. It's clear now, at least.
A
reductio ad absurdum is not required to prove the argument. The proof is extremely simple:
Code:
P ⊨ ~P → P
(1) P Premise
(2) ~P Assumption for discharge
(3) ~P → P From (1) and (2), discharge assumption
Yes, the premise and the assumption are contradictory, but there needn't be a
reductio ad absurdum. It's a straightforward assume and discharge.
The law of identity,
P ≡ P
That is a symbol often used to assert identity, and it's customarily extended to cases of logical equivalence:
P ≡ ~(~P)
P ≡ P & P
P ≡ P v P
P ≡ ~(~P & ~P)
...
And so on
ad infinitum. These are not, of course, identical formulae, but they are
logically identical. That is, they have precisely the same truth conditions and consequence class. If the premise is true, then so must the conclusion be true and
vice versa--the '
vice versa' is what sets identity apart from deducibility. Deducibility is transitive, while identity is symmetrical. So you can add to the above list:
P ≡ ~P → P
Take ~P → P.
In the case of P, then P is true.
In the case of ~P, then P is true.
For all possibilities, P is true (so your definition of Identity fits), however in the second case both ~P and P are true, contradicting ~(P & ~P) and thus making it absurd.
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 by '~P → P'. In other words, '~P → P' does not contradict 'P v ~P', and so all this irrelevant except as a means to construct an unnecessarily complicated proof.
With the original premise, P, and the absurdity of the case of ~P, reductio ad absurdum discharges ~P.
Without the original premise, P, reductio ad absurdum is equally valid in discharging ~P → P as it is ~P, since both statements were used to reach the absurd P & ~P.
The same is true when going from the premise to the conclusion, P and ~P both equally valid for discharge.
If you want to construct a
reductio ad absurdum, then yes, both formulae are used to deduce 'P & ~P'. However, while all assumptions one uses must be discharged for a 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.
Well various areas of statistics and probability are commonly counterintuitive, and that is absent any mental attempt to give English meaning to a non-English term (which I did as well when learning about implication as well). If semantic confusion falls under "counterintuitive" then so be it, but I found when learning about the Monty Hall problem, for example, that my mind attempted to contradict the actual reasoning, and not the semantics, which was a much more jarring experience. That changed the way I thought, whereas formal logic (sentential, at least) did not, it just provided clarity (and as an anecdote I thought I witnessed this same reaction in several others).
Well, my primary goal with this thread was to mildly entertain people who are interested but have little familiarity with formal logic. If I just wanted to discuss logic, there are more interesting subjects and better forums. I was also curious to see if someone who was initially puzzled by these arguments could figure them out.
