What is the rationale behind denying the law of excluded middle as universally true? I can understand people having issues with AC or equivalent axioms, but the law of excluded middle seems to be too fundamental and undeniable.
What is the rationale behind denying the law of excluded middle as universally true?
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A stronger notion of disjunction for which excluded middle fails.
Double negated excluded middle is still valid.
Some who want to discount it imagine that it somehow would allow for the study of more esoteric topics like what's called "consciousness" or "life after death." However, the reality is that none of them have done any research along those lines and if they have, they have failed to provide any tangible and reproducible results from such a paradigm.
Is not complex
You just realize that a concious thing can go from A to C figuring B out
Where are the technological results of studying "consciousness?" New Age and Hindu gurus, and susceptible schizos like those in the consciousness generals have been posting and talking about all the esoteric nonsense for several decades, some even claim you need to abandon the law of the excluded middle to do so, and that somehow you'd find a boom in science. Meanwhile in reality, no one seems to have every actually pursued this and if they have, we haven't seen any new technology or anything along those lines. There's nothing stopping any of them from pursuing this. It's all sophistry.
>Where are the technological results of studying "consciousness?"
he doesn't know
Holy shit that pic looks so regressive.
Information theory, propaganda, hypnosis, psychedelics, mass communication, the attention based economy, etc
What the fuck are you talking about?
Its usually because there are lots of existence proofs in mathematics that work by showing that non existence is absurd and then inferring existence of a particular thing. The problem comes with that some of these existence proofs could never be proven by explicitly building the thing you want to build. There are some statements that you can say NOT NOT A, but may not be able to explicitly construct A.
As is happens tho. If yoy allow AC and the axiom of specification in ZFC then you can actually derive the law of the excluded middle. So you might not want to toss out choice so easily.
>If yoy allow AC and the axiom of specification in ZFC then you can actually derive the law of the excluded middle
How are you going to do that without the law of the excluded middle? The use of those axioms in any proof requires the law of the excluded middle.
Its called diaconescus theorem. Those axioms do not require the excluded middle. Here is the proof
>Those axioms do not require the excluded middle
Kek. He uses proof by contradiction. The technique of proof by contradiction requires that one assume the LEM. Otherwise there is no basis to assert he found a contradiction from his assumptions since you can have a statement that is both true and false or neither true and false without LEM. So he proves the LEM by using LEM. This is called "circular reasoning." It's not a proof. It's simply just sophistry. If you believe this is a proof of anything, you're deeply mistaken and you're not going to make it.
He does not use proof by contradiction, he uses proof by negation. The law of the excluded middle means you cant go NOT NOT A implies A. It doesnt mean you cant go A leads to absurdity, therefore NOT A. There is actually a large difference between the two. You are a pseud
Sorry typo. The law of excluded middle means that you CAN go NOT NOT A implies A
NOT adopting LEM means that you CANT go NOT NOT A implies A. Then the rest of the post continues.
>The law of the excluded middle means you cant go NOT NOT A implies A
Let us stick to the foundations instead of getting lost in things you don't understand since [math] A implies B [/math] is defined in terms of [math] A lor neg B [/math]. LEM means that either a statement or its negation is true. That is a statement is either true or false. Moreover, the proof relies on sentential logic which is foundational dependent on the LEM. The entirety of the axioms of sentential logic, that is the logic set theory is based upon, is entirely dependent on LEM. That is, there is no way to conclude an element is in or is not in a set (which is invoke in the proof repeatedly) without LEM. Moreover, even if you somehow assume contradiction can be used to within a proof and negation exists, LEM is equivalent to the concept of contradiction. That is [math] A lor neg A = neg (A land neg A ) [/math] by DeMorgan's law. You therefore can't assume [math] Aland neg A = F [/math] without also assuming [math] A lor neg A = T [/math].
>That is A∨¬A=¬(A∧¬A) by DeMorgan's law.
The relevant half of this classical identity requires LEM, retard.
>The problem comes with that some of these existence proofs could never be proven by explicitly building the thing you want to build. There are some statements that you can say NOT NOT A, but may not be able to explicitly construct A.
Why is this problematic, exactly? Does LEM lead to inconsistencies or something?
>Does LEM lead to inconsistencies or something?
Nope it's just constructivist pseuds seething as always
>Why is this problematic, exactly? Does LEM lead to inconsistencies or something?
Due to incompleteness, there are statements for which you can prove A∨¬A by LEM, but you cannot prove neither A nor ¬A.
So formal systems that use LEM and promise more than they can give.
Again, how is that problematic? Just because it hurts your feelings?
It makes the formal system less useful. I'm for more useful formal systems.
the person you need to put this question to is the nut who calls Gödel a Nazi thug
>Why is this problematic, exactly?
For vaguely the same reasons asserting the existence of unfalsifiable entities is frowned upon in science.
Science doesn't like unfalsifiable entities because it is based on empirical observations. Logic doesn't care about that shit. If you can show that something must necessarily exist, then it does.
>it is based on empirical observations.
Why? What's wrong with non-empirical claims? Why are they problematic? Just because it hurts your feelings?
There's nothing wrong with it, but if it's not observable, then it's not scientific. I'm all for developing non-empirical scientific models, but they must still be verifiable by observation to be any useful.
>There's nothing wrong with it, but if it's not observable, then it's not scientific.
There's nothing wrong with it, but if it's not provable, then it's not mathematical. I'm all for proving existence using LEM, but it must still be constructible to be any useful.
The fact that I can do useful math with LEM proves you wrong. Now, let's see your useful unobservable scientific theories.
> I can do useful math with LEM
And I can do useful science without accepting falsifiability.
Good for you.
I guess abstract reasoning isn't your strong point.
Either it is or it isn't.
>but they must still be verifiable by observation to be any useful
Can the statement 'but they must still be verifiable by observation to be any useful' itself verifiable by observation? If not, then by your own standards of epistemic justification the method of observation by the senses it self is not able to justify itself. This is part of the self defeating nature of being a strict empiricist.
Let me tell you what isn't necessary. It's the retarded philosophical mumbo jumbo smoke green you're spouting. It has never, not even once, led to scientific advancements.
>A solution exists, and here's how to find it.
>A solution exists... but you can never find it! But it definitely exists, in the Platonic realm, within the infinite mind of God...
It's that simple.
Sounds like two different ways of reaching the same conclusion to me.
A real tangible solution is the same as an imaginary solution that provably does NOT actually exist, since you can never construct it? Sounds like retarded philosophical mumbo jumbo to me.
Without LEM you could only say a solution both exists and does not exist simultaneously, so you can both find it and not find it at the same time and it both will and will not be useful for everything and not everything you apply it to.
>Without LEM you could only say a solution both exists and does not exist simultaneously,
LEM is not the law of non contradiction.
Basically its a philosophical inclination that if you can show A is true, its because you can explicitly construct A. No LEM does not make a theory explicitly incomplete, its about the idea that if you can say " A is true" its because you can actually construct A, not just say "well it cant not be true". If you accept ZFC then you accept LEM, its a theorem, ZFC implies lem. But for people who don't accept choice and want to explicitly construct shit you want to reject LEM because LEM allows you to infer existence without construction.
Strict adherence to the law of excluded middle would require its practitioners to rigorously define, categorize, and disseminate the meaning and usage of every term they utter when engaging in dialogue. The fact of the matter is that language is widely successful because it is inherently ambiguous or flexible in its usage, and these are highly desired traits for normal use. It allows people to generally be loose and relaxed without the world falling apart around them.
Definition deterioration is absolutely real though, and sometimes it occurs with malicious intent. I think many other people have been making the credible claim that current-day liberals employ this egregiously for manipulating public opinion and swaying policy for short-term political gain. The brewing consequences are absolutely clear with growing disdain and intolerance of people towards each other across the aisle.
tl;dr Law of excluded middle requires effort, and people are lazy and can get away without adherence to it (for a time).
>Strict adherence to the law of excluded middle would require its practitioners to rigorously define, categorize, and disseminate the meaning and usage of every term they utter when engaging in dialogue.
>The fact of the matter is that language is widely successful because it is inherently ambiguous or flexible in its usage, and these are highly desired traits for normal use. It allows people to generally be loose and relaxed without the world falling apart around them.
Yet in technical fields, it often doesn't matter.
>but the law of excluded middle seems to be too fundamental and undeniable.
That's precisely the reason why we need to deny it.
>There are logics that don't have the law of excluded middle. So your idea that mathematics is founded upon LEM is false.
Where is the math that actually uses a third or fourth truth value? We both know that it's not the math in set theory or in any of the math anyone knows. Mathematics, and particularly set theory is entirely dependent on LEM even at a primative level. The idea that there is a negation of elementhood alone depends on the LEM.
>LEM is explicitly only equivalent to NOT NOT A = A. That's it. LEM is not equivalent to contradiction, it's equivalent to double negation elimination. De Morgan's laws rely on law of excluded middle dipshit, so your can't invoke them to prove anything.
I accept your concession that sentential logic depends on LEM. This the logic explicit used in this alleged "proof." Thank you for finally conceding. By the way retard, the concept of negation used in the "proof" you posted the link do explicitly depends on the LEM.
>You should stop commenting on things that you clearly don't understand
You really do lack self-awareness
>Where is the math that actually uses a third or fourth truth value? We both know that it's not the math in set theory or in any of the math anyone knows. Mathematics, and particularly set theory is entirely dependent on LEM even at a primative level
No its not. Constructive mathematics is a well known field of mathematics, it uses intuitionist logic as its foundations which has the 3rd truth value of NOT NOT A. Its a thing. You can google it. It usually plays more into type theory tho, so you need to learn about that.
>The idea that there is a negation of elementhood alone depends on the LEM.
This sentence came out of a crackpipe
>I accept your concession that sentential logic depends on LEM. This the logic explicit used in this alleged "proof." Thank you for finally conceding. By the way retard, the concept of negation used in the "proof" you posted the link do explicitly depends on the LEM.
No the concept of negation I posted does not depend on double negation elimination. It shows that a leads to a contradiction, so therefore NOT A. It doesnt say that NOT A leads to a contradiction, therefore A. There is a difference between the two. No part of that proof relies on LEM. If you really cant look at the proof and see for yourself you can google diaconescus theorem or "choice implies LEM" and you dont get people saying that the theorem is bullshit and wrong, but agreeing with it and trying to understand the finer points of it. If you cant see the difference between A leads to contradiction therefore NOT A and NOT A leads to a contradiction therefore A then we are done here as you don't understand LEM.
statements such as "there is nothing in X place". Which is the same as "there isn't anything in X place". It's the negation of the statement "there is something in X place"
these are never contradictory with saying that there is something called Y in that exact same X place.
It's another schizo/midwit cope
>be a 'smart boy' but lazy
>envy real mathematicians/physicists/???
>pretend [thing] is a bad and the whole field is wrong in order to cope
There are lots of examples on this board, notably muh real numbers, muh relativity, flat earth
arguing against the construction of the real numbers has a proper motivation. it hasn't been proven whether space is continuous of discrete. if space is continuous, real numbers would be applicable as a measure. if space is discrete, they would not. you can't just assume that space is continuous when it hasn't ever ever been proven.
why space matters is that the core of maths is supposed to reflect the real world, not fanciful ones or specific constructs useful in obscure situations. you can have real numbers as some abstract concept but don't ever go around saying they're superior to other constructions in real life applications because there's no proof they even apply to real life space and time.
>you can't just assume that space is continuous when it hasn't ever ever been proven.
because it would be a religious belief
Every worldview must have either some underlying assumed belief or an "I don't know" in it somewhere, otherwise there would be an infinite regress
>why space matters is that the core of maths is supposed to reflect the real world,
WRONG. Math transcends "real" world.
so do unicorns and fairies.
Unless I can study them using a priori methods, I'm not interested.
Unicorns and fairies ARE a priori. Look:
>Axiom 1. Unicorns and fairies exist.
By all means, keep going. Expand your axiomatic system and deduce some interesting theorems.
That's called writing fiction. You postulate axioms of a fictional world, and you explore their logical consequences. Maybe you'll find yourself more at home in the humanities department?
It's not fictional if you can write the axioms in a consistent way.
That's called literary realism.
>You postulate axioms of a fictional world, and you explore their logical consequences.
literally mathematics, even at the elementary school level
>Sarah has 2 apples and Billy has 2 apples
>But Mrs Lesson there aren't any kids named Sarah or Billy in this class. You are merely postulating axioms of a fictional world.
Yes. You also learn morals from Aesop's fables, but they often feature talking animals that don't exist.
Likewise, some mathematical-fictional stories are more useful than others: those that give real answers when applied in analogy to the real world, i.e. those that are constructive by avoiding LEM. Sarah gives 2 apples to Billy, now he has 4.
So in the end with this analogy you agree with
Hey fuck you, don't tell me what analogy I agree with. Even if space were discrete -- and the only reason pseuds see this as possible is because of popsci quantum mechanics which relies on the construction of real numbers anyway -- real numbers would still be valuable as a computationally-light approximation to small-scale discrete behavior.
>the core of maths is supposed to reflect the real world
ah, another one filtered by etymology, my condolences to your children for enduring you
nowhere was anything mentioned about the word real in real numbers and nowhere was it implied that it had anything to do with the word real in the phrase real world. you're reading into things that were never meant.
they are certainly superior at making mathematical zeteticists(formally known as finitist) like you seethe, and id say that is quite an enjoyable irl application
>mathematicians be like:
>"liar's paradox therefore all math is incomplete, excluded middle is false, you will never have a universal unified maths and you will know nothing and be happy"
why not just add the axiom of excluded liar's paradox statements?
we literally did, that's called Zermelo-Fraenkel
they could just expand propositional logic to become natural language logic.
it would be the intersection between linguistics and logic.