Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
The 'classical' versus 'non-classical' definitions of
logic is about 'consistency'. The classical ones are all consistency based whereas all the other variations accept some form of denial of 'consistency' as a requirement by their own extra additions of the classical versions.
This is the roseA = roseB problem. What is the difference between a consistent and an inconsistent system?
What is your paragon for 'consistency'?
By the Curry-Howard isomorphism mine is Turing-completeness.
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
All computers by default are 'classical' and thus, as you noted, cannot 'contradict'.
Correlation is not causation. They cannot contradict BECAUSE when you implement the mathematics of a Boolean gate in a transistor it is IMPOSSIBLE for a bit to be low AND high voltage at the same time. If that were to happen it's either a hardware problem or your 'classical' machine just went Quantum...
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
So how or why you go against the very 'classical' forms begs why?
a
Because it is undecidable. Consistent-but-undecidable is not Turing-complete and if it's not Turing-complete it is not representative of MY mind.
Because I make choices when I think. I recall memories. I store memories for later retrieval.
I think factually and counter-factually. The process most definitely does NOT fit in the structures of classical logic.
I don't like prison cells for my mind.
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
The non-classical forms that allow contradiction are similar to the quantum computing. They may 'cheat' to create the illusion of contradicting or utilize newer design to approach this. Some believe it is possible via the weirdness of quantum entanglement. THAT is "non-classical"!
Here is a paraconsistent logic which violates identity while preserving LEM and LNC.
https://repl.it/repls/StrangeLiquidPolyhedron
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
All computers and programs run through them are 'consistent' in that if the design is not malfunctioning, only the people are at fault (Gi/Go concept you embrace).
OK great! We are on the same page. If you accept the Curry-Howard isomorphism then working computer programs are proofs.
And so if you are calling a non-malfunctioning program 'consistent' this is synonymous meaning with 'valid'.
Proofs compute. A Program without errors is DEDUCTIVELY valid and contains no grammatical/semantic errors.
That is ALL validity buys you. Guarantee that you haven't made a LINGUISTIC error. You have good grammar! **pats back**
It makes absolutely no guarantees whether your system produces anything of value whatsoever.
It definitely does not give you "correspondence".
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
So this indicates you are misinformed about what 'classical' logic is.
It is complete-but-undecidable and therefore NOT Turing-complete.
Its grammar and semantics do not allow for expressing things like storing/recollecting memories OR if-then-else type expressions and therefore it is not representative of the way I think.
What am I misinformed about?
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
I assumed by "LEM" you are meaning "Law of Excluded Middle"? but can't see how this fits with the motive question(s)?
These are a response to the second motivating question which just adds to the first a question about whether we can extend logic beyond validity by beginning with
apriori matters. I disagree with your answer but my opinion is distinct from the original motivating questions. This was a real question asked and a part of much of the controversies between scientific and logical/mathematical philosophy at the same time as the first question. "Permit" was about the fact that this is relatively 'political', not actually able to be deemed false. If laws of nature have no 'logic', you'd have to step back to question whether computers actually require being 'consistent'.
Lets take a step back here though. By Curry-Howard EVERY SINGLE WORKING PROGRAM is valid AND complete.
This is a valid/complete logical system.
All 15 million lines of the Linux kernel. Valid and Complete logical system.
Neither my 1-liner above nor the linux kernel are exactly representative of anything pertaining to reality (I think)?
So it seems to me that consistency (validity?) is necessary-but-insufficient.
A logical system's STRUCTIRE being internally consistent does not (in any way) suggest that the system's structure corresponds to the structure of reality in any useful or meaningful way.
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
The point is that these questions historically gave rise to limitation-theorems in logic. Pioneers of the 19th century begun to realize paradoxes in logic, science, and reality in general, which was summed up by 1900 (for logic more specifically) as "
Hilbert's Program". Basically, he listed a set of problems within mathematics (which was 'logic' limited to number and measure) that acted as a sort of manifesto to solve all of math in one unified umbrella. It was a hope to unify all laws of reality into a simple formulation based on a minimum set of common postulates, of which the fewest would ideally be zero or one. [Zero is preferable; but, just as HOW you responded above with absolution about logic to NEVER be able to prove anything about reality, this steps into politics because that kind of response is itself paradoxical and unable to be resolved without logic: what 'logic' can we allow/permit to use to 'disprove' that logic is itself not at the core of reality? There is none other than religious-type beliefs about limits based on emotion and our senses.]
Either way, paradoxes, which are
real contradictions (until/unless resolved) are what the 'limits' pertain to.
The most interesting of Hilbert's problems (as far as I can tell) is his Entscheidungsproblem.
Which is exactly the roseA = roseB problem
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
And this is precisely proof of your own that the questions of the past were worthy of challenging. If there is a 'universal' logic, then all things could be unified to one standard of proof. This would require a base-logic or 'first-order' one of which all other different types can resort to as a foundation.
Your use of 'proof' is ambiguous here. Curry-Howard (computaqbility) is an objective and universal standard (PROOF OF) validity.
If standardization you want - that is solved.
This is precisely WHY I insisted on tackling the is logic A better than logic B problem? Define "better"
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
A 'proof' only deals with structure, not the 'truth' of the premises but the theorems that are validated of them require the very first inputs to be gambled as true. If the domain of the first input premises being used are true though, formal logic (deductive) requires the conclusion to follow or, regardless of the 'reality' of the claims of the inputs, those conclusions about reality are also invalid. So logic IS necessary for proofs about reality except for the core 'observations' that cannot be questioned by anyone other than the subjective individuals senses. Anything else beyond that is 'politics'. Science, for instance, is a 'politic' in that it requires AGREEMENT of subjective observers who vote on which definitions, interpretations, and logic systems that are 'permitted' to be used in the study of reality.
Does not not bother you though that all Mathematical logic is deductive in an inductive reality?
It sure strikes me as using the wrong tool for the job...
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
Now you are 'transferring' definitions in your head to 'forbidden' when you link it to 'permitted' (the actual word I used). What is 'permitted' about my statement on motives was to the political option to allow some particular system of reasoning (logic) to be used within a field, like science.
Permit/forbid.
Allow/deny.
Potato - potato.
Science is a pragmatic institution, not a prescriptive one.
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
What is 'permitted' deals with whether upon arguing between people if we must accept that logic is 'allowed' to prove or disprove something about reality, ....not about whether nature 'permits' variable logic systems to be used in principle.
That's an easy answer. No.
Logic computes consequences. With a certain margin of error.
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
You are jumping over the motivational problem here. The motivation of the limit theorems were about whether a universal system of logic as a 'parent' logic....a metalogical system....could be found that all other complex ones could be rooted in.
The answer is yes. The Type 0 Chomsky grammar is the super-set of all regular grammars.
https://en.wikipedia.org/wiki/Chomsky_hierarchy
https://en.wikipedia.org/wiki/Regular_language
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
It doesn't concern itself with whether one can create different architectures or programs that are 'free' to define its terms distinctly.
Unfortunately this paragraph is contradictory with the one before.
Type 0 Chomsky grammars are Turing complete and therefore isomorphic to Lambda calculus.
And so you have infinite freedom to define any and all of your terms.
Of course, this is a very intuitive conclusion.
Logic is just language and so without some form of limits (STRUCTURE) anything goes.
Where do you get STRUCTURE from? Your mind - of course. You DESCRIBE the structure of your experiences.
More limits arrive when you marry the software with physical reality: hardware.
Then things like time, energy, space (memory) become things to be taken into account.
Scott Mayers wrote: ↑Tue Feb 26, 2019 8:17 pm
We are not on the same page. The "garbage in- garbage out" issue relates but logic calls this 'validity'. The logic is only as "sound" (meaning maps to reality) if it is BOTH valid and its content 'true' to reality.
I thought we agreed that validity/consistency was the same as 'working software'?
https://en.wikipedia.org/wiki/Decidability_(logic)
Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines the notion of logical validity.
If validity is strictly a semantic notion (leaning towards Tarski's work) and all meaning is subjective then logical semantics cannot originate anywhere else but with the person CONSTRUCTING the logic.
Of course - this is to be expected because logic is just a Metalanguage all language ever was (is?) is a tool for self-expression.
That we are trying to model reality IN language, well... Reality doesn't care what we want.
And so all you can ever define IN language is your metaphysical experiences OF reality. NOT reality itself.
Still! Computer are useful for other things than 'defining reality'....