Converting formal proofs to conform to sound deduction

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Logik
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Re: Converting formal proofs to conform to sound deduction

Post by Logik » Tue Apr 30, 2019 11:36 pm

wtf wrote:
Tue Apr 30, 2019 11:27 pm
Not picking a fight, just making an observation. Perhaps we have different "background frames" for the concept of isomorphism.
What I mean by isomorphism as in functional equivalency. Different form - same function.

I mean it in the sense where a transfer/translation function exists between two seemingly different forms.

I mean it in the sense where "are" means ⇔

In the compsci sense I mean it as: https://en.wikipedia.org/wiki/Reversible_computing
in the physics sense I mean it as conservation of information: https://en.wikipedia.org/wiki/No-hiding_theorem

Phenomenologically they are all the same idea.

wtf
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Re: Converting formal proofs to conform to sound deduction

Post by wtf » Wed May 01, 2019 12:54 am

Logik wrote:
Tue Apr 30, 2019 11:36 pm
I mean it in the sense where "are" means ⇔
Oh well that explains your "all X are Y" remark. Classically "All men are mortal" means ==> and not <==>, we agree on that I hope. So if you said, "All X are Y and all Y are X" you'd at least have identity between sets. That would have been more clear to anyone who's ever heard "All men are mortal" and understood that to mean ==> but not necessarily <==.

I still wouldn't call it isomorphism. Isomorphism is between two essentially different things that are regarded as the same only with respect to certain abstract properties or structure. But I won't press the point, you can call "All men are mortal" an isomorphism if you like. But not if you want to be plainly understood. Only if you want to misspeak, get challenged, and then go "Gotcha! When I say "All men are mortal" I don't mean what everyone else in the world means by it."

Logik
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Re: Converting formal proofs to conform to sound deduction

Post by Logik » Wed May 01, 2019 2:22 am

wtf wrote:
Wed May 01, 2019 12:54 am
Oh well that explains your "all X are Y" remark. Classically "All men are mortal" means ==> and not <==>
https://en.oxforddictionaries.com/definition/mortal
mortal adjective of a living human being often in contrast to a divine being subject to death.

The dictionary definition constrains the context to beings. It ascribes the property "mortality" to human beings and immortality to divine beings. There is no undistributed middle. It means <==>.
wtf wrote:
Wed May 01, 2019 12:54 am
I still wouldn't call it isomorphism. Isomorphism is between two essentially different things that are regarded as the same only with respect to certain abstract properties or structure.
Precisely why I am calling it an isomorphism!

Code: Select all

B = { being1, being2, ... beingN }
{ b in B | Human(b) } ⇔ { b in B | Mortal(b) }
{ b in B | Divine(b) } ⇔ { b in B | Immortal(b) }
Python equivalent here: https://repl.it/repls/ThisBustlingDeal

Perhaps you have chosen to treat mortality/immortality as a broader category? One that applies beyond the set B above? Cool. Then "All men are mortal" means ==> indeed. But now you are necessarily talking about classification.

And as with any classification exercise you are going to have to tell me what your classification rule is.

Cats ==> mortal
Flowers ==> mortal
Planets ==> mortal
Universe ==> mortal

You could say "mortality is a property of living things" but that's the exact same problem.

Cats ==> living
Planets ==> living

Category errors, eh? Categories ARE errors.

All models are wrong - some are useful
Last edited by Logik on Wed May 01, 2019 3:27 am, edited 1 time in total.

PeteOlcott
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Re: Converting formal proofs to conform to sound deduction

Post by PeteOlcott » Wed May 01, 2019 3:27 am

Logik wrote:
Tue Apr 30, 2019 8:47 pm
I call bullshit that you can fit all knowledge from the Internet into one consistent system, but lets go with that for 2 seconds.
I'll even pretend that you have (somehow) figured out how to unify incompatible theories like General Relativity and Quantum Mechanics.

wtf already showed you a mathematical framework in which 1 = 0 is a perfectly valid expression.

So I ask again. WHO chose the axioms and why? Because their "common sense" clearly disagrees with my "common sense".
(1) Neither General Relativity and Quantum Mechanics are tautologies that are not in the category of Truth.

(2) You still don't understand the difference between valid and sound?

This is valid, yet unsound:
Unsound deductive inference (false premise)
(a) All dogs are office buildings
(b) All office buildings have windows
(c) Therefore all dogs have windows

(3) The ordered set of integers defines the tautology that 1 > 0.
Last edited by PeteOlcott on Wed May 01, 2019 3:34 am, edited 1 time in total.

Logik
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Re: Converting formal proofs to conform to sound deduction

Post by Logik » Wed May 01, 2019 3:29 am

PeteOlcott wrote:
Wed May 01, 2019 3:27 am
(2) You still don't understand the difference between value and sound?
Let me assure you the Aristotelian error is all yours. I am getting tired of explaining it to you, least you waste 21 years of my life.

Here is one last attempt.

https://en.wikipedia.org/wiki/Validity_(logic)
In logic, an argument is valid if and only if it takes a form that makes it IMPOSSIBLE for the premises to be true and the conclusion nevertheless to be false.

The notion of deductive validity is IN CONFLICT with the notion of empirical falsification.

Premise 1: If it rains the ground will be wet.
Premise 2: It is raining
Conclusion: The ground is wet

Empirical counter-example: https://www.youtube.com/watch?v=XjXvkIzUTtk

Empirical evidence DEMONSTRATES that it is, in fact, possible for the premises to be true, but the conclusion to be false. Therefore the argument is invalid.

You will find a longer list here: https://en.wikipedia.org/wiki/Paradoxes ... mplication

I will stick to empiricism and intuitionism over classical logic any day. Thank you very much.
Last edited by Logik on Wed May 01, 2019 3:53 am, edited 14 times in total.

PeteOlcott
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Re: Converting formal proofs to conform to sound deduction

Post by PeteOlcott » Wed May 01, 2019 3:33 am

Logik wrote:
Tue Apr 30, 2019 8:55 pm
PeteOlcott wrote:
Tue Apr 30, 2019 8:51 pm
This is also perfectly valid, it is, however, unsound:
Unsound deductive inference (false premise)
(a) All dogs are office buildings
(b) All office buildings have windows
(c) Therefore all dogs have windows
It's an Idiotic example because you have absolutely no way of introducing modal logic into your ontology.
(F ⊢ x) means ◻x

Logik
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Re: Converting formal proofs to conform to sound deduction

Post by Logik » Wed May 01, 2019 1:57 pm

PeteOlcott wrote:
Wed May 01, 2019 3:33 am
(F ⊢ x) means ◻x
I am going to remove myself from these discussions because you seem to be dogmatically doubling down on deduction as some absolute ideal for reasoning.

A am observing that a lot of our disagreements boils down to the reductionism vs holism distinction. Q.E.D you reduce the logical argument into two distinct properties: validity and soundness and you evaluate those in isolation.

I go the exact opposite way to you. I look at everything holistically: the logical definition of 'validity', the structure of the argument, the conclusion and the empirical evidence. They are all inter-connected and it is a reductionist mistake to treat them in isolation.

But there is really no point in me making an argument when Quine has already made it.

https://en.wikipedia.org/wiki/Is_Logic_Empirical%3F
What is the epistemological status of the laws of logic? What sort of arguments are appropriate for criticising purported principles of logic? In his seminal paper "Two Dogmas of Empiricism," the logician and philosopher W. V. Quine argued that all beliefs are in principle subject to revision in the face of empirical data, including the so-called analytic propositions. Thus the laws of logic, being paradigmatic cases of analytic propositions, are not immune to revision.
https://en.wikipedia.org/wiki/Two_Dogmas_of_Empiricism
"Two Dogmas of Empiricism" is a paper by analytic philosopher Willard Van Orman Quine published in 1951. According to University of Sydney professor of philosophy Peter Godfrey-Smith, this "paper [is] sometimes regarded as the most important in all of twentieth-century philosophy".[1] The paper is an attack on two central aspects of the logical positivists' philosophy. One is the analytic–synthetic distinction between analytic truths and synthetic truths, explained by Quine as truths grounded only in meanings and independent of facts, and truths grounded in facts. The other is reductionism, the theory that each meaningful statement gets its meaning from some logical construction of terms that refers exclusively to immediate experience.

PeteOlcott
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Re: Converting formal proofs to conform to sound deduction

Post by PeteOlcott » Wed May 01, 2019 3:25 pm

Logik wrote:
Wed May 01, 2019 1:57 pm

But there is really no point in me making an argument when Quine has already made it.

https://en.wikipedia.org/wiki/Is_Logic_Empirical%3F
https://en.wikipedia.org/wiki/Two_Dogma ... ircularity

Analyticity
(2) No bachelor is married.
A statement with this form can be turned into a statement with form
(1) by exchanging synonyms with synonyms, in this case "bachelor" with "unmarried man".
It is the second class of statements that lack characterization according to Quine.
In a language with the modal adverb "necessarily" the problem is solved, as salva veritate holds in the following case:
(4) Necessarily all and only bachelors are unmarried men

Meaning Postulates (1952) by Rudolf Carnap formalized natural language semantics:
http://liarparadox.org/Meaning_Postulat ... p_1952.pdf
∀x Bachelor(x) → ¬Married(x)
In a language with the modal adverb "necessarily" the problem is solved, as salva veritate holds in the following case:

(4) Necessarily all and only bachelors are unmarried men
We simply have a meaning postulate (axiom) that says whatever a bachelor is it is not married.

wtf
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Re: Converting formal proofs to conform to sound deduction

Post by wtf » Fri May 03, 2019 4:23 am

Logik wrote:
Wed May 01, 2019 2:22 am
mortal adjective of a living human being often in contrast to a divine being subject to death.
Thanks for clarifying your thinking. You are making the point that under your understanding of logic, we should analyze the word "mortal" and so forth.

I'd submit that if you show the phrase "All men are mortal" to a large body of philosophers, virtually all will identify that phrase with Aristotle's elucidation of the syllogism (though Ari didn't actually use that particular example, which came later). And under that interpretation, it's perfectly clear that the proper subclass relationship is implied.

Secondly, the universal quantifier ∀ works the same way. All rationals are real but not all reals are rational. All isosceles triangles are triangles but not all triangles are isosceles. So again, your usage is nonstandard. Not wrong in the context of your own (idiosyncratic IMO) interpretation; but definitely not what people think of when they hear the English phrase "All tuna are fish." Or use the universal quantifier in a logical or mathematical argument.
Logik wrote:
Wed May 01, 2019 2:22 am

Code: Select all

B = { being1, being2, ... beingN }
{ b in B | Human(b) } ⇔ { b in B | Mortal(b) }
{ b in B | Divine(b) } ⇔ { b in B | Immortal(b) }
Cute little language, what is it?

If you say that's an isomorphism, what two things are isomorphic and exactly what is the isomorphism map?


Logik wrote:
Wed May 01, 2019 2:22 am
Perhaps you have chosen to treat mortality/immortality as a broader category? One that applies beyond the set B above? Cool. Then "All men are mortal" means ==> indeed. But now you are necessarily talking about classification.
I get that you are on an anti-Aristotle kick, which is fine. But for CLARITY OF COMMUNICATION you should say, "I am using my own personal definition of "are" to mean logical equivalence or equality, and NOT the usual proper subclass relation." Then people wouldn't have to waste time trying to understand your own personal way of speaking.

By the way you used this exact same code fragment as an example of a universal mapping property in category theory. It's perfectly clear to me that you are trying to bluff your way through. The code fragment does not express an isomorphism nor is it an example of a universal mapping property.
Logik wrote:
Wed May 01, 2019 2:22 am
And as with any classification exercise you are going to have to tell me what your classification rule is.

Cats ==> mortal
Flowers ==> mortal
Planets ==> mortal
Universe ==> mortal
Etc. I get it. You don't like Aristotle's categories. But that doesn't excuse your deliberately obscure way of communicating. If you are making up your own definitions for things you should say so.

The New England Patriots are a pro football team but not every pro football team is the NE Patriots. All oranges are fruits but not all fruits are oranges. All squares are quadrilaterals but not all quadrilaterals are squares. You are off on your own trip here. You're wrong on the history and sociological context of logic and philosophy; you're wrong on the universal quantifier; and you're wrong regarding standard natural language usage of the word "are."

PeteOlcott
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Re: Converting formal proofs to conform to sound deduction

Post by PeteOlcott » Sat May 04, 2019 5:25 am

Logik wrote:
Wed May 01, 2019 1:57 pm
PeteOlcott wrote:
Wed May 01, 2019 3:33 am
(F ⊢ x) means ◻x
I am going to remove myself from these discussions because you seem to be dogmatically doubling down on deduction as some absolute ideal for reasoning.

A am observing that a lot of our disagreements boils down to the reductionism vs holism distinction. Q.E.D you reduce the logical argument into two distinct properties: validity and soundness and you evaluate those in isolation.

I go the exact opposite way to you. I look at everything holistically: the logical definition of 'validity', the structure of the argument, the conclusion and the empirical evidence. They are all inter-connected and it is a reductionist mistake to treat them in isolation.

But there is really no point in me making an argument when Quine has already made it.

https://en.wikipedia.org/wiki/Is_Logic_Empirical%3F
What is the epistemological status of the laws of logic? What sort of arguments are appropriate for criticising purported principles of logic? In his seminal paper "Two Dogmas of Empiricism," the logician and philosopher W. V. Quine argued that all beliefs are in principle subject to revision in the face of empirical data, including the so-called analytic propositions. Thus the laws of logic, being paradigmatic cases of analytic propositions, are not immune to revision.
https://en.wikipedia.org/wiki/Two_Dogmas_of_Empiricism
"Two Dogmas of Empiricism" is a paper by analytic philosopher Willard Van Orman Quine published in 1951. According to University of Sydney professor of philosophy Peter Godfrey-Smith, this "paper [is] sometimes regarded as the most important in all of twentieth-century philosophy".[1] The paper is an attack on two central aspects of the logical positivists' philosophy. One is the analytic–synthetic distinction between analytic truths and synthetic truths, explained by Quine as truths grounded only in meanings and independent of facts, and truths grounded in facts. The other is reductionism, the theory that each meaningful statement gets its meaning from some logical construction of terms that refers exclusively to immediate experience.
I learn this stuff by pondering it not by reading about it.

Most of Quine's argument against analyticity in the first four sections is focused on showing that different explanations of analyticity are circular. The main purpose is to show that no satisfactory explanation of analyticity has been given.

Not circular, not at all:
Analytical truth is simply truth that can be totally verified as completely
true entirely based on the meaning of the words in the sentence.
Copyright 2012 Pete Olcott
Last edited by PeteOlcott on Sat May 04, 2019 2:23 pm, edited 1 time in total.

wtf
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Re: Converting formal proofs to conform to sound deduction

Post by wtf » Sat May 04, 2019 7:38 am

PeteOlcott wrote:
Sat May 04, 2019 5:25 am
Copyright ???? Pete Olcott
It's not a legal copyright without a year. When I was a tech writer (a very long time ago) I learned to write "Copyright (C) 1920 Thomas Edison Company" in exactly that format.

Just some free legal advice. And of course I'm not a lawyer and the law might have changed since then. But for sure it's not a valid copyright without a year.

PeteOlcott
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Re: Converting formal proofs to conform to sound deduction

Post by PeteOlcott » Sat May 04, 2019 2:26 pm

wtf wrote:
Sat May 04, 2019 7:38 am
PeteOlcott wrote:
Sat May 04, 2019 5:25 am
Copyright 2012 Pete Olcott
It's not a legal copyright without a year. When I was a tech writer (a very long time ago) I learned to write "Copyright (C) 1920 Thomas Edison Company" in exactly that format.

Just some free legal advice. And of course I'm not a lawyer and the law might have changed since then. But for sure it's not a valid copyright without a year.
It a legal copyright as soon as it is written down even if only on a piece of paper that you keep at home.
A copyright notice (not required) would not be fully conformant without a year. 2012 was the oldest
published copy that I could find.

PeteOlcott
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Re: Converting formal proofs to conform to sound deduction

Post by PeteOlcott » Sat May 04, 2019 3:01 pm

Logik wrote:
Wed May 01, 2019 1:57 pm
PeteOlcott wrote:
Wed May 01, 2019 3:33 am
(F ⊢ x) means ◻x
I am going to remove myself from these discussions because you seem to be dogmatically doubling down on deduction as some absolute ideal for reasoning.
https://en.wikipedia.org/wiki/Radical_skepticism

Radical skepticism or radical scepticism is the philosophical position that knowledge is most likely impossible.[1] Radical skeptics hold that doubt exists as to the veracity of every belief and that certainty is therefore never justified. To determine the extent to which it is possible to respond to radical skeptical challenges is the task of epistemology or "the theory of knowledge".

https://en.wikipedia.org/wiki/Two_Dogmas_of_Empiricism
I agree that nothing empirical can be known with 100% perfectly justified complete certainty.
There is nothing empirical about ¬(1 = 0).

There is no chance what-so-ever that living_thing->animal->dog is really an office_building,
these are not beliefs they are definitions.

Axioms are nothing more than expressions of language that have been defined to have
the semantic value of Boolean True.

I don't know how Quine can think that analytical truth is circular when analytical truth is this:
Analytical truth is simply truth that can be totally verified as completely
true entirely based on the meaning of the words in the sentence.
Copyright 2012 Pete Olcott

The fundamental nature of Truth has been my philosophical quest since I was a young boy
peeved by the inability of people to properly distinguish between opinions and established facts.

Since that time I have raised the bar and eliminated empirical "facts" from the set
of 100% justified complete certainty. Only fact that can be verified as completely
true entirely based on the meaning of their words count as actual truth.

I was so pleased the very first time that I found this:
◇P ↔ ¬◻¬P Possibly(P) <--> Not(Necessarily(Not(P))
◻P ↔ ¬◇¬P Necessarily(P) <--> Not(Possibly(Not(P)))
Now I finally had a way to formally express the distinction between
possible and impossible.

Possible(p) is any expression of language p that
HAS NOT BEEN proved Necessarily(Not(P)) based on the meaning of the words of p.

Impossible(p) is any expression of language p that
HAS BEEN proved Necessarily(Not(P)) based on the meaning of the words of p.

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Speakpigeon
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Re: Converting formal proofs to conform to sound deduction

Post by Speakpigeon » Sat May 04, 2019 6:33 pm

Logik wrote:
Wed May 01, 2019 2:22 am
Perhaps you have chosen to treat mortality/immortality as a broader category? One that applies beyond the set B above? Cool. Then "All men are mortal" means ==> indeed. But now you are necessarily talking about classification.

And as with any classification exercise you are going to have to tell me what your classification rule is.

Cats ==> mortal
Flowers ==> mortal
Planets ==> mortal
Universe ==> mortal
Why would we need a classification rule?

All we need is to be able to decide for ourselves, on the moment, that a particular proposition "all x, Fx" is true.

We may believe one minute that all cats are fish and infer validly that our cat Aristotle is a fish. Then, we realise our mistake, and the next minute we believe all cats are birds, and so we infer, again validly, that our cat Aristotle is a bird. Both inferences were valid and both remain valid. Both are valid, even though both conclusions are false (but I'm not sure they are false).

As Quine says, we have a systems of beliefs and we can revise our beliefs.

Where's the problem already?
EB

PeteOlcott
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Re: Converting formal proofs to conform to sound deduction

Post by PeteOlcott » Sat May 04, 2019 9:14 pm

Speakpigeon wrote:
Sat May 04, 2019 6:33 pm
Logik wrote:
Wed May 01, 2019 2:22 am
Perhaps you have chosen to treat mortality/immortality as a broader category? One that applies beyond the set B above? Cool. Then "All men are mortal" means ==> indeed. But now you are necessarily talking about classification.

And as with any classification exercise you are going to have to tell me what your classification rule is.

Cats ==> mortal
Flowers ==> mortal
Planets ==> mortal
Universe ==> mortal
Why would we need a classification rule?

All we need is to be able to decide for ourselves, on the moment, that a particular proposition "all x, Fx" is true.

We may believe one minute that all cats are fish and infer validly that our cat Aristotle is a fish. Then, we realise our mistake, and the next minute we believe all cats are birds, and so we infer, again validly, that our cat Aristotle is a bird. Both inferences were valid and both remain valid. Both are valid, even though both conclusions are false (but I'm not sure they are false).

As Quine says, we have a systems of beliefs and we can revise our beliefs.

Where's the problem already?
EB
When beliefs contradict tautologies beliefs can be known to be lies with 100% perfect certainty.
Quine seems to be pretty thick-headed when he can't see that analytical truth is perfectly
delineated by expressions of language known to be true entirely based on the meaning of their words.

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