Meanings? I am not looking for meaning. Logic is a tool of self-expression.
When you are attempting to express simple ideas Classical logic is fine.
When you are attempting to express complex ideas about a complex reality - you need higher order logic.
There are things I am unable to express in classical logic Like A != A.
If I were to express A !=A I contradict the Identity axiom.
More precisely, every time I express "A != A" a Classical logician (mistakenly) asserts that I have contradicted myself when I haven't.
Speaking different languages is not conducive to communication.
Forcing me to speak your language is not conducive to self-expression.
So I need a logic in which "for all x: x = x" is NOT an axiom.
Or more precisely. I need a logic in which we have MULTIPLE identity axioms.
for all x: x = x
for all y: y ! = y
for all z: z == z
for all p: p rabbit p
The code I provided in Python (Lambda calculus - same thing) demonstrates that I can have a logic in which A != A is a valid, non-contradictory expression.
I have removed the "foundation" and the skyscraper remains standing. Plato was right.
Logic has nothing to do with nature. Logic/mathematics is invented. Man-made.
I most definitely do not adhere to LEM. No constructive mathematician does.
Non-contradiction stays .
No it doesn't. Lambda calculus is consistent AND complete. Read/understand Curry-Howard isomorphism.
It is consistent. It contains no contradictions. If it was inconsistent the computer would throw an error.
Humans can deal with (ignore? Overlook? fail to naturally detect?) inconsistency - computers can't.
What I am claiming is that Classical logic is inconsistent. Because the meaning of "=" is vague and inconsistent.Scott Mayers wrote: ↑Sun Feb 24, 2019 7:05 am What you appear to be claiming is most extreme: that NO CONSISTENT systems exist or are relevant to intellectual reasoning. If this were true, then everyone would already be correct to randomly agree or disagree with you WITHOUT a need to argue. Why would you even try to "reason" beginning with your implied default of 'reasoning' that 'reasoning == non-reasoning' (A == non-A)?
I didn't imply that reasoning == non-reasoning? You are thinking like an absolutist. (and maybe that's my error, I am also thinking like an absolutist but at the other end of the spectrum).
The point is that we don't need a "foundational" axiom. Both of these statements can be true within the same logic-system
A = A => True
B = B => False
I believe in consistency and non-contradiction. That is why I am rejecting classical logic.Scott Mayers wrote: ↑Sun Feb 24, 2019 7:05 am 0 = 0 and 1 as [inconsistent totality = inconsistent subsets of totality AND consistent subsets of totality]
You are acting as though a totality that has no law is something that has some 'law' that disallows it to also be contradictory.....thus contradicting your belief in contradiction. Then you further contradict the left over consistent reality in a way that cannot ever be resolvable.
When dealing with integers the meaning of "=" is "absolute distance from 0 on the number line"
When dealing with abstracts (Sky is blue) the "=" does NOT mean "absolute distance from 0".
The "=" requires a human, making contextual choices/interpretations.
That's why I am down to one axiom. Non-contradiction.
Identity is inconsistent.
Read/understand the Curry-Howard isomorphism.Scott Mayers wrote: ↑Sun Feb 24, 2019 7:05 am Then, if you want to show there are degrees or variations as a part of this world, do so without denying that there are discrete realities. Note, for instance that mathematical calculus is about a continuous logic but it proves it through a discrete form of reasoning. And it also shows that you can reverse this, begin with a continuous default to show how and where discrete realities (non-continuous) ones fit in [more difficult but possible].
Shannon Information bridges the discrete/continuous gap: https://en.wikipedia.org/wiki/Nyquist–S ... ng_theoremScott Mayers wrote: ↑Sun Feb 24, 2019 7:05 am This is the same kind of problem with physics regarding the discrete concept of 'quantification' (treating the very small as having discrete whole or integral measures) versus relativity which begins with a continuous (non-discrete) system of reasoning by nature. Both are still understood to be essential. You may find difficulty making sense of one or the other but both are still 'true' of reality. But we don't know which or what universal system includes them both or that some third more inclusive explanation is more conclusive.
OK. Classical logic is consistent sometimes and inconsistent sometimes. That means you have to understand WHEN it is inconsistent and make sure you aren't falling for the trap.
Are you doing that?
Incorrect. I think the reason we are drifting apart is that you probably (like most people who have studied philosophy but not computer science) assume set theory as the foundation of Mathematics.Scott Mayers wrote: ↑Sun Feb 24, 2019 7:05 am Denying it would require also denying even 'fuzzy' logic. [I read some books that try to 'sell' out classic logic as being false rather than to simply assert it incomplete. I'm guessing that the authors were not the actual 'fuzzy logicians' because they don't realize that fuzzy logic is an EXTENSION of classical binary-valued logic into multi-variable forms.] It's sad when even authors who may intend to advocate for the concept of multi-variable logics tend to act as the very 'classical' thinkers they accuse 'classical logic' to be about: that to be logical, you have to pick ONLY-classical binary-valued logic XOR non-classical multi-valued logic. Do you see how expressing this is itself is hypocritical?
I don't. I consider Type theory/Lambda calculus as foundation. Computation. From there all rules/axioms are synthetic. Models.
Turing-completeness + recursion gives me ALL logics. https://en.wikipedia.org/wiki/Chomsky_hierarchy