Definition of logical validity

[Logik]

There's nothing to justify. The law of excluded middle, like all logical truths, is intuitive. You accept it or you don't, it's your problem.

I guess your intuition and my intuition are significantly different then

https://en.wikipedia.org/wiki/Intuitionistic_logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Intuitionistic logic is one example of a logic in a family of non-classical logics called paracomplete logics: logics that refuse to tautologically affirm the law of the excluded middle.

So we still have this problem to resolve and how: whose intuition is wrong?

Most people accept it, including all logicians since Aristotle, over a period of 2,400 years.

Good for them.

I put aside claims made in the context of modern mathematical logic and based on the notion of material implication as defined by a truth table, as it is inconsistent with our intuition of the logical implication.

You trust your intuition so much you won't even bet $100 to settle a matter. Are you afraid your intuition is wrong or... ?

If you think there is a problem with the law of excluded middle, it's up to you to articulate why it would be inconsistent.

It's a tautology.

If you don't, then there's no point going further.
So, you haven't justified your claim that Aristotle was an idiot.
EB

I don't need to justify it. From where I am looking it's obvious and intuitive. It's a logical truth.

It follows from the Curry-Howard isomorphism. It follows from lambda calculus/type theory and other high-order, para-consistent logics.

https://en.wikipedia.org/wiki/Paraconsistent_logic

[Speakpigeon]

There's nothing to justify. The law of excluded middle, like all logical truths, is intuitive. You accept it or you don't, it's your problem.

I guess your intuition and my intuition are significantly different then

Sure, mine is in line not only with Aristotle but also most people with no training in logic and who can therefore provide non-biased empirical evidence on the logical intuitions most people have.

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular,systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Intuitionistic logic is one example of a logic in a family of non-classical logics called paracomplete logics: logics that refuse to tautologically affirm the law of the excluded middle.

No information relevant to whether Aristotle was an idiot there.

So we still have this problem to resolve and how: whose intuition is wrong?

You think you know. Me, I consider the available empirical evidence.

Most people accept it, including all logicians since Aristotle, over a period of 2,400 years.

Good for them.

So, all of a sudden, you feel like disregarding empirical evidence. Cherry-picking special pleading. No good.

I put aside claims made in the context of modern mathematical logic and based on the notion of material implication as defined by a truth table, as it is inconsistent with our intuition of the logical implication.

You trust your intuition so much you won't even bet $100 to settle a matter. Are you afraid your intuition is wrong or... ?

Your attention span is surprisingly short.
You still haven't articulated your point that Aristotle would be an idiot.
The empirical evidence availablefor now is that's your skill at debating in a rational way are very close to zero.

If you think there is a problem with the law of excluded middle, it's up to you to articulate why it would be inconsistent.

It's a tautology.

What is? Please justify.

If you don't, then there's no point going further.
So, you haven't justified your claim that Aristotle was an idiot.
EB

I don't need to justify it. From where I am looking it's obvious and intuitive. It's a logical truth.

Good, exactly what I thought.

It follows from the Curry-Howard isomorphism. It follows from lambda calculus/type theory and other high-order, para-consistent logics.
https://en.wikipedia.org/wiki/Paraconsistent_logic

Yes? Your link says that Inconsistency-tolerant logics have already been discussed by Aristotle. Whoa.
Still, the rest shows your pet theory is still work in progress. When you're done, give me call on my iPhone, but chances are, I won't be around any more. That certainly explains why you can't articulate your point. You don't have one.
And as to prove Aristotle an idiot, finished your homework first.
EB

[Logik]

Sure, mine is in line not only with Aristotle but also most people with no training in logic and who can therefore provide non-biased empirical evidence on the logical intuitions most people have.

In other words - a bandwagon fallacy.

No information relevant to whether Aristotle was an idiot there.

None whatsoever? Not even the rejection of LEM?

You still fail to recognise that the rejection of LEM is what led to the Curry-Howard isomprphism.

You think you know. Me, I consider the available empirical evidence.

But if your method for evaluating the evidence is unreliable, then you have a problem. No?

All the available evidence of 20th century developments in logic demonstrate that Aristotle was wrong.
Turing, Godel, Kleene, Church, Tarski, Kolmogorov, Brouwer, Curry, Howard, Lembek etc. etc. etc. etc.

So, all of a sudden, you feel like disregarding empirical evidence. Cherry-picking special pleading. No good.

Strawman and a cheap parlour trick.

I am the empiricist in this discussion. You are appealing to the authority of Aristotle.
All while you are ignoring all the evidence which contradict Aristotle.

Let me also remind you that you are the one who kept appealing to deduction, whereas I only rely on induction.

I am curious as to how you "regard" any evidence without using some form of inference.
Especially since you don't seem to grasp probability theory, never mind Bayesian inference.

I put aside claims made in the context of modern mathematical logic and based on the notion of material implication as defined by a truth table, as it is inconsistent with our intuition of the logical implication.

Oh. OK. What does your intuition tell you about the logical implications of quantum mechanics? Should we trust your intuition in this regard or empiricism?

You still haven't articulated your point that Aristotle would be an idiot.

Because the law of excluded middle is a self-affirming tautology.
Because constructive mathematics recognises that logic is simply a modeling tool and that philosophical methods of the kind you are appealing to have failed to provide any "truth" for thousands of years.

For an empiricist you sure disregard the evidentiary weight of thousands of years of failure.

The empirical evidence availablefor now is that's your skill at debating in a rational way are very close to zero.

You have a biased sample. I have a long streak of successfully debating mathematicians, physicists and logicians.

I just can't debate you because you reject anything that is not classical logic.

You are dogmatic.

What is? Please justify.

Please read any introductory textbook on constructive mathematics or para-consistent logic.

Or if you can't be bothered to read actual books try this:

https://en.wikipedia.org/wiki/Church%E2 ... ing_thesis
https://en.wikipedia.org/wiki/Church%E2 ... _principle

And if you are too lazy to click on it I will quote the relevant part for you: The principle states that a universal computing device can simulate every physical process.

Do you think logic/reasoning is a physical process?

Yes? Your link says that Inconsistency-tolerant logics have already been discussed by Aristotle. Whoa.
Still, the rest shows your pet theory is still work in progress.

Except Aristotle hadn't heard of Alan Turing. The theory is not "in progress"

There is nothing theoretical about computer science, quantum physics and computation.
Computers exist. Quantum computers exist. They work.

Mathematical proofs compute. The Curry-Howard isomorphism is empirical. Recursion theory is empirical.

[Speakpigeon]

In other words - a bandwagon fallacy.
None whatsoever? Not even the rejection of LEM?
You still fail to recognise that the rejection of LEM is what led to the Curry-Howard isomprphism.
But if your method for evaluating the evidence is unreliable, then you have a problem. No?
All the available evidence of 20th century developments in logic demonstrate that Aristotle was wrong.
Turing, Godel, Kleene, Church, Tarski, Kolmogorov, Brouwer, Curry, Howard, Lembek etc. etc. etc. etc.
Strawman and a cheap parlour trick.
I am the empiricist in this discussion. You are appealing to the authority of Aristotle.
All while you are ignoring all the evidence which contradict Aristotle.
Let me also remind you that you are the one who kept appealing to deduction, whereas I only rely on induction.
I am curious as to how you "regard" any evidence without using some form of inference.
Especially since you don't seem to grasp probability theory, never mind Bayesian inference.Oh. OK. What does your intuition tell you about the logical implications of quantum mechanics? Should we trust your intuition in this regard or empiricism?
Because the law of excluded middle is a self-affirming tautology.
Because constructive mathematics recognises that logic is simply a modeling tool and that philosophical methods of the kind you are appealing to have failed to provide any "truth" for thousands of years.
For an empiricist you sure disregard the evidentiary weight of thousands of years of failure.
You have a biased sample. I have a long streak of successfully debating mathematicians, physicists and logicians.
I just can't debate you because you reject anything that is not classical logic.
You are dogmatic.
Please read any introductory textbook on constructive mathematics or para-consistent logic.
Or if you can't be bothered to read actual books try this: https://en.wikipedia.org/wiki/Church%E2 ... ing_thesis
Except Aristotle hadn't heard of Alan Turing. The theory is not "in progress"
There is nothing theoretical about computer science, quantum physics and computation.
Computers exist. Quantum computers exist. They work.
Mathematical proofs compute. The Curry-Howard isomorphism is empirical. Recursion theory is empirical.

I'm bored to death. Your posts get longer and longer and still nothing that would look like a point.
To say that the law of excluded middle is a self-affirming tautology is trivia. I already told you, I accept we take it on trust, from our intuitive sense of logic. You're free to do otherwise. My experience with real, untrained and therefore unbiased people, is that most of accept intuitively all of Aristotle's syllogisms, the law of excluded middle and all logical truths provided they are simple enough for them to decipher and you enunciate them in ordinary language rather than in any abstruse formalism.
As to the Curry-Howard isomorphism, big deal. Shannon realised in the 50's Boole's algebra could be used to describe the way computers work. Hallelujah. I'm talking about logic, and you still have to articulate what would be wrong with Aristotle's logic, let alone why Aristotle was an idiot. It is obvious that mathematical logic isn't bound by any logical intuition we have, except, ironically, to prove its own theorem. It's like saying our intuition about flat space is wrong just because some idiot mathematician invented curved geometry. If our intuition about flat space is wrong, it's because space is not flat.
And mathematicians cannot have anything to do with that.
Now, it's relatively easy to observe that space may be curved. It's immediately much more tricky to observe that our intuitive logic is wrong. We agree on one point, which is that it is an empirical matter. We disagree as to where exactly are the logical facts to be observed.
As to Turing, Godel, Kleene, Church, Tarski, Kolmogorov, Brouwer, Curry, Howard, Lembek etc. , they were all trained in mathematics. Mathematicians are not physicists. They don't care about the truth of their theories.
You still haven't coherently articulated how the implication would be problematic. Is the Modus tollens at all problematic? You seem to believe so, but your articulation of your point in this respect has been moronic. The perfect ignoramus. The perfect idiot. The guy who wants to argue the modus tollens is problematic and can't even write the modus tollens itself or the truth table of it.
What about Quantum Physics? Explain yourself, so I can have a laugh. Is it about a particular cat, by any chance? Or the indeterminacy of quantum states? I can't wait.
Computers exist, yes, what about it? Cars, air planes, tooth brushes, bread and butter. So what? You're unable to explain yourself because you seem to believe that the mere existence of computers proves Aristotle's logic wrong! Whoa. very intuitionistic, that!
And I'm not interested reading books since I can't argue with the author. Most books are terminally fluffy in their articulation of ideas. Books about facts are good and effective. Books about ideas can only be about opening up new perspectives. I don't need that. I'm fully open, thanks.
Anyway, thanks, you're just a waste of time.
EB

[Logik]

I'm bored to death. Your posts get longer and longer and still nothing that would look like a point.

It seems that if the point was an elephant in a room you'd still miss it.

To say that the law of excluded middle is a self-affirming tautology is trivia. I already told you, I accept we take it on trust, from our intuitive sense of logic. You're free to do otherwise.

And I am telling you that your acceptance of LEM on trust/faith is the problem.

By throwing LEM away constructive mathematics and computational logic become possible!
My throwing LNC away para-consistent logics become possible!

Useful logics! Logics that allow us to build mathematical models which PREDICT CORRECTLY!

My experience with real, untrained and therefore unbiased people, is that most of accept intuitively all of Aristotle's syllogisms, the law of excluded middle and all logical truths provided they are simple enough for them to decipher and you enunciate them in ordinary language rather than in any abstruse formalism.

You don't consider Mathematics to be an ordinary language? Then how are we ever going to agree on anything...

As to the Curry-Howard isomorphism, big deal. Shannon realised in the 50's Boole's algebra could be used to describe the way computers work. Hallelujah. I'm talking about logic, and you still have to articulate what would be wrong with Aristotle's logic

Uhhhh. No. You are far off from understanding Shannon's work. You need to learn some probability theory for that.

Information theory is about communication. ALL communication. Human-to-human included.

, let alone why Aristotle was an idiot. It is obvious that mathematical logic isn't bound by any logical intuition we have, except, ironically, to prove its own theorem. It's like saying our intuition about flat space is wrong just because some idiot mathematician invented curved geometry. If our intuition about flat space is wrong, it's because space is not flat. And mathematicians cannot have anything to do with that.

You are yet to tell us what your intuition says about the quantum realm and why it can only be understood/modeled mathematically.

Now, it's relatively easy to observe that space may be curved. It's immediately much more tricky to observe that our intuitive logic is wrong.

That is only correct a priori.
A posteriori it's trivial to observe that your intuition is wrong.

Your predictions are wrong when your intuition is wrong.

Show me a confidence interval on anything you conclude with classical logic.

[Speakpigeon]

To say that the law of excluded middle is a self-affirming tautology is trivia. I already told you, I accept we take it on trust, from our intuitive sense of logic. You're free to do otherwise.

And I am telling you that your acceptance of LEM on trust/faith is the problem.

Articulate what the problem is exactly.

By throwing LEM away constructive mathematics and computational logic become possible!
My throwing LNC away para-consistent logics become possible!

Good. I don't want to stop you.

Useful logics! Logics that allow us to build mathematical models which PREDICT CORRECTLY!

Good! Do it!
But that's irrelevant. That A is good doesn't mean that B is bad.

You don't consider Mathematics to be an ordinary language? Then how are we ever going to agree on anything...

I don't need to agree with you. You are making idiotic claims by the shitload. Want to convince me? Sure, just try.

You are yet to tell us what your intuition says about the quantum realm and why it can only be understood/modeled mathematically.

First explain how Quantum Physics would be a problem for deductive logic as you claimed without articulating any cogent explanation, let alone producing any proof.
And explain how mathematics could possibly exist without mathematicians relying to prove their theorems on at least somebody's logical intuitions.

A posteriori it's trivial to observe that your intuition is wrong.

It can happen for sure. Intuition is a real process, nothing magical. Stuff happens.
However, I don't think you understand at all what kind of intuition I am talking about. So, "trivial", definitely not. I myself still have no actual example of that which I would have been able to confirm to my satisfaction. Most people talk big and understand very little, mathematicians included.
And you have yet to provide any actual instance of the problem you nonetheless claim affects deductive logic.

Show me a confidence interval on anything you conclude with classical logic.

Show me any inference by scientists that does not rely on deductive logic.
Prove to me you don't use yourself, one way or the other, deductive logic.
EB

[Logik]

To say that the law of excluded middle is a self-affirming tautology is trivia. I already told you, I accept we take it on trust, from our intuitive sense of logic. You're free to do otherwise.

And I am telling you that your acceptance of LEM on trust/faith is the problem.

Articulate what the problem is exactly.

By throwing LEM away constructive mathematics and computational logic become possible!
My throwing LNC away para-consistent logics become possible!

Good. I don't want to stop you.

Useful logics! Logics that allow us to build mathematical models which PREDICT CORRECTLY!

Good! Do it!
But that's irrelevant. That A is good doesn't mean that B is bad.

You don't consider Mathematics to be an ordinary language? Then how are we ever going to agree on anything...

I don't need to agree with you. You are making idiotic claims by the shitload. Want to convince me? Sure, just try.

You are yet to tell us what your intuition says about the quantum realm and why it can only be understood/modeled mathematically.

First explain how Quantum Physics would be a problem for deductive logic as you claimed without articulating any cogent explanation, let alone producing any proof.
And explain how mathematics could possibly exist without mathematicians relying to prove their theorems on at least somebody's logical intuitions.

A posteriori it's trivial to observe that your intuition is wrong.

It can happen for sure. Intuition is a real process, nothing magical. Stuff happens.
However, I don't think you understand at all what kind of intuition I am talking about. So, "trivial", definitely not. I myself still have no actual example of that which I would have been able to confirm to my satisfaction. Most people talk big and understand very little, mathematicians included.
And you have yet to provide any actual instance of the problem you nonetheless claim affects deductive logic.

Show me a confidence interval on anything you conclude with classical logic.

Show me any inference by scientists that does not rely on deductive logic.
Prove to me you don't use yourself, one way or the other, deductive logic.
EB

I can’t even prove that the sky is blue to a dogmatist.

In natural selection we trust.

[Speakpigeon]

I can’t even prove that the sky is blue to a dogmatist. In natural selection we trust.

It is both a logical truth and an empirical fact that dogmatists resist evolution.
EB

[Garry G]

A permanent truth does not entail a temporary truth and vice versa. With that thought in mind:

Definition of validity: an argument is valid when truth does not lead to its denial.

Denial may mean either false or alternative forms of truth e.g. temporary truth does not lead to permanent truth.

[Speakpigeon]

Definition of validity: an argument is valid when truth does not lead to its denial.

So, the argument below must be valid, according to your definition:

A or B;
Therefore, A and B.

Or do you mean something else?

A permanent truth does not entail a temporary truth and vice versa. With that thought in mind:
Denial may mean either false or alternative forms of truth e.g. temporary truth does not lead to permanent truth.

I would need a lot of convincing before I take this to be at all about logic.
EB

[Garry G]

The standard truth tables for the antecedent "A or B" are true on two permutation of truth possibilities where the consequent "A & B" are false. Therefore the inference is invalid because truth leads to false on those permutations, or as I phrased it truth leads to its denial. The definition I gave only means something different in the context of alternative logics like the one I am wrestling with in another thread and which you have posted a response.