Eodnhoj7 wrote: ↑Mon Jan 11, 2021 5:21 am
If (P=P)=(-P=-P) then P=-P through the law of identity which both P and -P share.
If (P=P)≠(-P=-P) then the law of identity is not equal to itself given both P and -P exist through the law of identity.
I think Terrapin gave some great examples of the problem when he used addition operation in place of your choice of the '=' within the brackets. You are confusing the meaning of the parts to the wholes and vice versa. The whole expression, "(P
=P)=(-P
=-P) means that the '=' sign between the two parts as wholes are LOGICALLY EQUIVALENT by the bolded versions of '
=', not to whatever
P or
-P refers to. The truth of the bracketed '=' signs are what the unbolded '=' refers to. You are incorrectly presuming the '=' in both cases to mean the same. This is why most systems use alternate symbols when expressing this, such as....
(P = P) ≡ (-P = -P)
The ONLY concern for referencing the "Law of identity" with respect to logic, though, is to assert that for WHICHEVER system you are using or defining, SOMETHING has to remain '
consistent'. The law is the reference to WHATEVER 'consistency' is needed in ANY system. The other two universal laws are rephrasing this by different perspectives. For instance, the "Law of the Excluded Middle" refers to expressing what is '
not consistent' as one may define something in the given system by contrast:
P ≢ -P ...says that wherever
P is assumed,
-P cannot be assumed also.
That is, there has to be some rule(s) in any system of logic that discriminates between what it means for something to be 'inconsistent' so that the system can
decide what to eliminate. If there is no elimination rule, then the system is not defined as 'logical' because it cannot be used to 'consistently' decide anything. You don't want a calculator, for instance, that has a 'logic' that has different answers for the same inputs, otherwise, it lacks any usefulness. [Well, perhaps one could use such a possible inconsistent machine, if it were possible, to find 'random' numbers. But this too still requires at least some meaning of 'consistency', such as the use of such a particular machine to be used 'consistently' to find random numbers.]
The third law of any 'logic' is just another extension of the other two that asserts one must decide whether to accept (or reject) contradictions in some 'consistent' manner. The last law defines what is 'inconsistent' with respect to the first law's defining of what is 'consistent'; then the third asserts that the first two rules in essence have to evaluate ONE of the two as what one expects of the system. In essence, if one wants to deny contradictions, then it says we toss out things that match to the second law. If you wanted a machine to favor 'inconsistency', like to find a machine that seeks random things, you might want such a mechanism to keep 'contradictions' as a means for an output.
Either way, the Law of Identity is not about the symbols specifically but to the agreement in the MEANING of the symbols you choose. You have to have something that is defining of "the same", something that clarifies "the difference", and whether you accept only one or both. Selecting
both is the same as asserting that you accept contradictions for some practical purpose.
Note that selecting to
keep contradiction is more about an 'open' system, such as what you might think of a sense organ to do: it doesn't necessarily require DENYING input because it is not the COMPLETE closed system of logic that the sense-to-brain-to-motor system of the whole deals with. So you can have a PARTIAL 'logic' system that is OPEN to accepting information without actually eliminating input values.
