John Page

So they're not (the original) A at all, right. They're representations of A. This has interesting implications for the law of identity...... seems my little problem could turn into your big problem!

mac_philo

This is all nonlogical vocabulary. I have to agree with the previous poster that you are probably the only person who believes this. After a week I still don't even understand the objection.

There is no original 'a', nor is there a representation of a.

Consider 3=3. That is a true statement. Consider -(3=3). That is a contradiction; if you derive this in a proof you have disproved your assumptions by reductio ad absurdum.

There is no original 3. There are not two 3's. There is an identity statement that can either be a tautology or a contradiction.

John Page

mac_philo:

What is nonlogical vocabulary? Does that just mean it doesn't make sense to you? Are you saying a billion Christians can't be wrong? Let me try to explain more understandably what seems illogical to me.

The quantification of identity requires transformation from a represented existence to a representational value or stereotype/class. There can be multiple levels of such an abstraction process. These representational values are the raw materials for symbolic logic. However, such logics seem to ignore the fact that their underlying mechanisms (how they work in the first place) can only compare non-identical values. When you evaluate axioms, being part of a language, the symbols in the axioms become represented objects themselves. So, it doesn't make any difference whether you are comparing external realities, internal concepts or axioms, mathematical or truth values; the same laws apply. Moving from here, my trouble is with the law of non-contradiction because its representation in propositional logic ~(A&~A) cannot be input to the quantization process above - you simply never get to comparing two identical objects. Ergo my assertion that the axioms of logic as represented in propositional logic are inconsistent.

Additional comment 1: I don't understand how you can say there is no original of 'a' nor a representation of 'a' - are you a nihilist?

Additional comment 2: I'm not trying to prove that the mathematical proof of 3=3 is wrong. I'm trying to demonstrate that the axioms of logic are not universal or absolute truths - they're prodcued by fallible humans and you cannot prove them to be perfect (even if they were!). Besides, history is on my side, they've been refined before.

Small point - isn't an identity statement that is a contradiction demonstration that the law of identity is flawed? Identity can only be a tautology by definition.

Cheers!

[ March 01, 2002: Message edited by: John Page ]</p>

mac_philo

Formulas have truth values; they aren't representations, they don't necessarily assert that anything exists. There is no representation of 'a'; a is a variable that may or may not be functioning in a referential sense. If it refers to the letter a, then 'a' represents a. So what?

3=3 is not just a mathematical formula; '3' is a symbol. This is a logical formula. That you think -(3=3) says anything against the law of contradiction is inexplicable. The formula is false! It doesn't assert the existence of any objects, it doesn't represent any objects; it states that 3 is not 3, which is by definition a contradiction. This doesn't say anything interesting whatsoever, except that the formula is false. So it is for every identity function presented in this thread.

A representation is something that represents. In other words, there is a correspondence between the form of the referent and the form of the symbol. A logical variable is something that may or may not behave in a purely referential manner, but it only ever represents by accident, and such an accident is a nonrelevant property.

I don't understand what your reference to Christians or nihilism has to do with anything. You are making a point about the axioms of logic in a method completely removed from formal logic. If there is something faulty about the axioms, just string out a proof and be done with it.

John Page

I don't understand, I'm not asserting that a formula "claims" anything exists. I am asserting that logic must operate through some physical substrate of the mind. You ask "So what?" My response is that it seems the process through which logic is implemented (not logic itself) must violate the axioms of logic as stated in propositional logic.

Again, I must be failing to explain myself in some way. I never said -(3=3) violates the law of non-contradiction. The mathematical symbol 3 clearly represents a quantity of some unspecified type of object. The number three is an axiomatic concept that exists in the mind. The mind can compare sets of three to show they are numerically equivalent. I don't have a problem with that.

I agree that the symbol used to represent a variable is arbitrary. But what of the value behind it? If we were just comparing the symbols the result would be arbitrary, therefore we must be comparing some actual values. This goes back to my point about understanding in physical practice how logic really operates. I'm saying we will not be able to understand how this actually occurs (at the physical layer) by maintaining laws of logic that posit "identical" copies can't exist! There must be some additional 'indirection', I want to understand it.

Apart from the opinion stuff where you implied I must be wrong because pj did, you previously said "There is no original 'a', nor is there a representation of a." I concluded you were saying 'a' didn't exist, hence the nihilist remark.

I'm tring to understand how things work. Yes, I am trying to do this with a method different than formal logic but not inconsistent with it. Maybe I'm a simple engineer but in this case I can't see how to get from a logical to a physical model of propositional logic without violating the axioms of same. I have partial explanations, but having told you that my issue is with the physical implementation layer, I ask you please to read again my problem statement which I've tweaked to try and be clearer:

The quantification of identity requires transformation from a represented existence to a representational value or stereotype/class value. There can be multiple levels of such an abstraction process. These representational values are the raw inputs to logic processes, the values being labeled with a symbol. However, such logics seem to ignore the fact that their underlying mechanisms (how they work in the first place) can only compare non-identical values. When you evaluate axioms, being part of a language, the symbols in the axioms become represented objects themselves. So, it doesn't make any difference whether you are comparing external realities, internal concepts or axioms, mathematical or truth values; the same laws must apply for internal consistency. Moving from here, my trouble is with the law of non-contradiction because its representation in propositional logic ~(A&~A) cannot be input to a quantification process consistent with the strictures of prop. logic's axioms - I cannot present two identical objects to test against the law of non-contradiction without violating the law of identity. Ergo my assertion that the axioms of logic as represented in propositional logic are inconsistent. [Note: I can find little objection to logics requiring the manipulation of quantities only, represented by number symbols, because quantities are pre-defined as homogenous and thus defined as exceptions to the law of identity]

I think I have a work-around, but I'm not sure. The explanation requires axiomatic concepts such as truth, or indeed any meta-representation, to be treated differently through substitution of the actual property represented. I would galdly show you the draft but would feel more comfortable that his would be fruitful once I have communicated accurately what the issue is (or withdrawn it).

Regards

[ March 01, 2002: Message edited by: John Page ]</p>