Justification of the mathematical definition of logical validity?

[Speakpigeon]

Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
Here is the definition:

Validity
A deductive argument is said to be 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.
Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/

Thanks for your answers.
EB

 

[Angra Mainyu]

I wasn't going to reply to this thread, but since you keep changing the definition in the other one, let us do this.


First, I asked you in the other thread whether you believed all mathematical statements were either true or false. You replied:
https://talkfreethought.org/showthr...ct-mathematics&p=683824&viewfull=1#post683824

I think all statements are either true or false.

I will grant this for the sake of the argument.

Second, by 'CML-valid' or 'valid according to classical mathematical logic' or similar expressions, I mean that a deduction (or argument, inference, or whatever one calls it) is valid according to the definition provided by you in the OP

Namely, a deduction is CML-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.

In particular, if A is a CML-valid mathematical argument with true premises P1,...,Pn and conclusion C, then C is true. Why? Because the premises are true, and the argument takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Therefore, C is not false. Given the previous hypothesis (i.e., every statement is either true or false), C is true.

This gives us an important feature of CML-validity: it is truth-preserving. If one starts with truths, by CML-valid arguments one gets only more truths.

Third, let us consider another proposed system of validity that is truth-preserving, say V. Since V is truth-preserving, it never is the case that an argument is V-valid but it takes a form such that it is possible for the premises to be true but the conclusion false. Hence, V-validity implies CML-validity. This gives us another important point: CML-validity is the strongest form of truth-preserving derivation of conclusions from premises.

Fourth, while mathematicians sometimes make logical errors in applying CML, that is usually corrected before a paper is published, because the authors and other mathematicians check it repeatedly. Moreover, if some errors make it into a paper, readers - who are usually also mathematicians - will almost certainly sooner or later (very probably sooner, if the paper has readers) spot it. So, while the system is not perfert, it is generally very reliable in getting CML-validity right.

Fifth, in order to prove things in mathematics, we need to start with something, right? Well, our starting points are true. If we were wrong about that, mathematics would be pretty much hopeless regardless of what method of deduction is used. We could discuss whether those starting points are true because they're self-evident, or because we just set up a hypothetical abstract scenario and stipulate that such-and-such things hold, so they do hold in the scenario we are considering, or it depends on the case (e.g., the natural numbers for the former; a Banach space for the latter), or something else, but at least we have true starting points (or first statements, or axioms, or whatever one calls them.)

So, by CML, we find new mathematical truths, and we can find any truth that could be found from the same starting points by another truth-preserving method, but also more truths than any weaker method.

So, that's a good reason to adopt CML-validity as the way of deriving statements from others: we find true statements from true statements, and it's the strongest method for doing that.


Additionally, I would add that CML is intuitively right, for most mathematicians. This is not so because they were told so. Where I live, most mathematicians never take a course in mathematical logic (I think it would be a good thing if they did, but anyway), but those who do (or who decide to study it on their own), when they first encounter it, usually find it very intuitive, and in particular, they find the definition very intuitive.


Now, do non-mathematicians find it intuitive?
It depends on the person. But for that matter, there is a difference between trained and untrained intuitions, and the former are often better. For example, there are plenty of cases where the intuitions of people with no previous training, in nearly all cases, go wrong (purely for example, the Monty Hall Problem, where the folk probabilistic intuitions nearly always fail). On the other hand, after learning mathematics (more specifically in the example, probability theory), humans are considerably less likely to go wrong.

Regardless, if CML-validity is not the same as the folk conception of validity, then for the previously given reasons, either CML-validity is stronger, or the folk conception fails to be truth-preserving. Either way, CML-validity is a superior tool for finding mathematical truths. On the other hand, if CML-validity is just the folk conception of validity, then no problem, either.

 

[steve_bank]

Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
Here is the definition:

Validity
A deductive argument is said to be 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.
Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/

Thanks for your answers.
EB

Math is not philosophy. Mathematicians develop rule based systems that are consistent, which means no ambiguities. By applying the rules in different ways and combinations you can never get two different answers. Given a problem in algebra no mater the order you apply addition, multi[plication, division, and subtraction you will always get the same valid result. It is consistent. Algebra is not a logical argument, it is a system based on rules and definitions.

Arithmetic
Geometry
Linear Algebra
And so on...

The Incompleteness Theorem, which has a proof, says in any consistent rule based system there are truths improvable in the system. In plane geometry it is impossible to prove that the shortest distance between two points is a straight line. That requires calculus.

As to deductive reasoning, it is generally not just either or deductive or inductive. Other than straightforward problems it is aleays a combination of the two.

Inductive vs deductive is also called top down vs bottom up and backwards vs forward. It is all the same Aristotuilan logic, the difference being the starting point.

The specific to the general and general to the specific is another desorption.

When actually solving problems you pick a staring point and ether try inductive and deductive, If yiou start with inductive but di not get a solution, you then try deductive based aon the inductive results.

When attempting a new math proof for which it is unknown if a proof exists, one goes back and forth, backwards forwards iterations looking for a solution..

In practice for other than obvious solutions the direct application pf logic as in a syllogism does not work.

Math like scientific theory on the end is demons red by usage over time. There is no guarantee that a proof in genral can have a lurking problem.

 

[A Toy Windmill]

Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
Here is the definition:

Validity
A deductive argument is said to be 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.
Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/

Thanks for your answers.
EB

The definition you quoted is a definition which works for the sort of Tarskian, truth-functional classical notion of validity. It's one way, arguably shallow and uninteresting, to define validity for a particular logic.

Logics which deviate from this, such as relevance logic, have their own definitions of validity.

 

[Speakpigeon]

Do you know of any proper justification by any specialist of mathematical logic, e.g. mathematicians, philosophers and computer scientists, that the definition of logical validity used in mathematical logic since the beginning of the 20th century would be the correct one?
Here is the definition:

Validity
A deductive argument is said to be 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.
Internet Encyclopedia of Philosophy - https://www.iep.utm.edu/val-snd/

Thanks for your answers.
EB

The definition you quoted is a definition which works for the sort of Tarskian, truth-functional classical notion of validity. It's one way, arguably shallow and uninteresting, to define validity for a particular logic.

Logics which deviate from this, such as relevance logic, have their own definitions of validity.

Yep, but I'm not sure you even understand the question.

Anyway, you should perhaps know that this question came after an argument with Angra Mainyu about validity.

I had started a thread on the validity of an argument (repeated below). Nearly all posters have voted either not valid or nonsensical, which amounts to not valid. So, I take this as a demonstration that the empirical evidence is that the argument is seen by most people as not valid. In fact, the only two who claimed the argument valid, Angra Mainuy and Bomb#20, both have a training in mathematical logic, which explains their claim and invalidates it as biased for the purpose of the poll.

However, both Bomb#20 and Angra Mainuy at one point started to play teacher (repeated below) by authoritatively asserting validity to posters who had commented that the argument was not valid, and this on the basis that validity was a consequence of the definition of validity. This showed clearly that both thought the word "validity" means mathematical logic validity. Yet, my question was about validity without any reference to mathematical logic. Thus, the debate turned on to whether there was such a thing as a notion of logical validity outside mathematical logic. Given the answers by most posters, clearly there is, but our specialists of mathematical logic here disputed this. So, somewhere, I asked my question about the justification of the mathematical definition of logical validity/ Is there a justification that it is correct? Because if not, then people should abstain from pretending there's just the mathematical logic's notion of validity.
EB

This is a poll on the logical validity of the following argument:

A squid is not a giraffe
A giraffe is not an elephant
An elephant is not a squid
Joe is either a squid or a giraffe
Joe is an elephant
Therefore, Joe is a squid

Is this argument logically valid?
Either way, why?
EB

No because the stated premises say nothing about the relationship between squids or elephants as categories; therefore the conclusion that Joe is an elephant cannot be logically arrived at. It is like A human is not an animal; An animal is not a soul., Joe is either a human or an animal. Joe is a human. Therefore Joe is a soul. (the unstated premise is A Human is a soul--which most posters on this forum would debate. In the case of the elephant Joe; one would have to prove that, given the accepted meanings of the words, squids are all elephants, for the logic to hold.

Bomb#20 already proved the conclusion from the premises, so it is valid.

 

[A Toy Windmill]

I had started a thread on the validity of an argument (repeated below). Nearly all posters have voted either not valid or nonsensical, which amounts to not valid. So, I take this as a demonstration that the empirical evidence is that the argument is seen by most people as not valid. In fact, the only two who claimed the argument valid, Angra Mainuy and Bomb#20, both have a training in mathematical logic, which explains their claim and invalidates it as biased for the purpose of the poll.

One might suggest that diagnosing validity requires training, and there are simple problems where people's untrained intuitions are wrong. I would say that understanding just the difference between validity and soundness requires some training, as does understanding the formalism developed by Aristotle.

However, both Bomb#20 and Angra Mainuy at one point started to play teacher (repeated below) by authoritatively asserting validity to posters who had commented that the argument was not valid, and this on the basis that validity was a consequence of the definition of validity. This showed clearly that both thought the word "validity" means mathematical logic validity. Yet, my question was about validity without any reference to mathematical logic. Thus, the debate turned on to whether there was such a thing as a notion of logical validity outside mathematical logic. Given the answers by most posters, clearly there is, but our specialists of mathematical logic here disputed this. So, somewhere, I asked my question about the justification of the mathematical definition of logical validity/ Is there a justification that it is correct? Because if not, then people should abstain from pretending there's just the mathematical logic's notion of validity.

I'm a specialist in mathematical logic, and like other specialists, I would tell you that there isn't one notion of validity in mathematical logic.

You are right that the material implication has always been controversial. It trips up students all the time. Alternative logics have been developed to try to model implication in a way that better fits our pre-formal understanding. You have mentioned one before: relevant logic.

This is a poll on the logical validity of the following argument:

A squid is not a giraffe
A giraffe is not an elephant
An elephant is not a squid
Joe is either a squid or a giraffe
Joe is an elephant
Therefore, Joe is a squid

Is this argument logically valid?
Either way, why?
EB

If you'll indulge me: here's the same argument with two of the premises deleted:

A giraffe is not an elephant
Joe is either a squid or a giraffe
Joe is an elephant
Therefore, Joe is a squid

Is this invalid?

 

[Speakpigeon]

One might suggest that diagnosing validity requires training, and there are simple problems where people's untrained intuitions are wrong. I would say that understanding just the difference between validity and soundness requires some training, as does understanding the formalism developed by Aristotle.

Since all mathematicians abiding by the definition I provided of validity as used in mathematical logic will be systematically wrong about the validity of my squid argument, I think common sense validity wins hands down.

And you're just wrong about the necessity of training.

I'm a specialist in mathematical logic, and like other specialists, I would tell you that there isn't one notion of validity in mathematical logic.

I know that, but I can't possibly cater for all tastes. The situation in mathematical logic is just to complicated to follow in details. Do you even know exactly how many theories of logic there are? I've never seen anyone venturing a figure.

You are right that the material implication has always been controversial. It trips up students all the time. Alternative logics have been developed to try to model implication in a way that better fits our pre-formal understanding. You have mentioned one before: relevant logic.

In other words, alternative logics try to better model our intuitive notion of validity, which straightforwardly contradict your initial claim about the necessity of training by implying that ordinary folks get it more often right than mathematicians abiding by the definition of validity consistent with material implication (i.e. most mathematicians, I think).

As demonstrated here and elsewhere with my squid argument: Systematically, all untrained posters get it right saying the argument is not valid. Trained posters get it always wrong. QED, training makes you wrong. .

This is a poll on the logical validity of the following argument:

A squid is not a giraffe
A giraffe is not an elephant
An elephant is not a squid
Joe is either a squid or a giraffe
Joe is an elephant
Therefore, Joe is a squid

Is this argument logically valid?
Either way, why?
EB

If you'll indulge me: here's the same argument with two of the premises deleted:

Why do you not answer my question first? Is my squid argument valid or not?

A giraffe is not an elephant
Joe is either a squid or a giraffe
Joe is an elephant
Therefore, Joe is a squid

Is this invalid?

There's no difficulty answering this one but I'll wait that you answer mine first.
EB

 

[A Toy Windmill]

There's no difficulty answering this one but I'll wait that you answer mine first.

EB

The argument is classically valid.

 

[fast]

Joe is a squid
Joe is not a squid
Therefore, Joe is a walrus.

Is that valid?

 

[fast]

I ask the previous question to rule something out.

If Joe is a person, Joe can walk.
Joe is a person.
If joe is a tree, then Joe cannot walk.
Joe is a tree.
Therefore, Joe can walk.

That’s what I guess he would call classically valid.

To me, it’s like mixing two puzzles together. So long as there’s enough ingredients in the premises to get to the conclusion, the argument is valid — contradictions and discrepancies be damned.

But, my earlier question lacked the necessary ingredients but contained a contradiction.

I’m trying to hone in on the thrust that gives rise to this “everything follows” mind set. It kind of reminds me of the notion that we have no rights if our rights can be altered—or something like that.

 

[Angra Mainyu]

In fact, the only two who claimed the argument valid, Angra Mainuy and Bomb#20, both have a training in mathematical logic, which explains their claim and invalidates it as biased for the purpose of the poll.

Obviously, my claim is not "invalidated" by my training in mathematical logic. But that aside, if I had been asked about this argument before I had any training in mathematical logic, of course I would have found it valid. It's intuitively clear to me that it is valid.

 

[Angra Mainyu]

As demonstrated here and elsewhere with my squid argument: Systematically, all untrained posters get it right saying the argument is not valid. Trained posters get it always wrong. QED, training makes you wrong.

Again, there are plenty of cases where the intuitions of people with no previous training, in nearly all cases, go wrong (purely for example, the Monty Hall Problem, where the folk probabilistic intuitions nearly always fail). On the other hand, after learning mathematics (more specifically in the example, probability theory), humans are considerably less likely to go wrong.

Training in mathematics in general - with or without studying mathematical logic - makes you better at logic. But some of us are just better than logic than most, regardless of the reason. (but seriously, for example Bomb#20 essentially destroys the arguments of their opponents in nearly every debate on this forum).

 

[Angra Mainyu]

]If you'll indulge me: here's the same argument with two of the premises deleted:

A giraffe is not an elephant
Joe is either a squid or a giraffe
Joe is an elephant
Therefore, Joe is a squid

Is this invalid?

It's valid of course. But I'd like to use your modification for an intuitive argument for interested readers.

Here's an alternative:

A giraffe is not a mollusc.
Tom is either a squid or a giraffe.
Tom is a mollusc.
Therefore, Tom is a squid.
​

I'm not going to ask whether it is valid. I'm going to ask about what they should or should not reckon, on the basis of the given information.

S1: Let us suppose that Bob and Alice live in an isolated rural community, and have never heard of giraffes, moluscs, squids, elephants, or any of those things. But they meet Daniel, who is not from their community, and we have the following exchange:

Daniel: A giraffe is not a mollusc.
Daniel: Tom is either a squid or a giraffe.
Daniel: Tom is a mollusc.
Alice: I get it. Tom is a squid.
Bob: You don't know that.
Alice: Well, I'm saying that on the basis of what Daniel said, Tom is a squid. I do not know whether Daniel is mistaken or pulling our legs - no offense, Daniel -, but on the basis of what he said, Tom is a squid.
Bob: No, you shouldn't make that assessment on the basis of the information given by Daniel.


Who's right, Alice or Bob?
Obviously, Alice is right. It's intuitively obvious, right?

Now let us consider the second scenario:

S2: Let us suppose that Bob and Alice live in an isolated rural community, and have never heard of giraffes, moluscs, squids, elephants, or any of those things. But they meet Daniel, who is not from their community, and we have the following exchange:

Daniel: A giraffe is not a elephant.
Daniel: Tom is either a squid or a giraffe.
Daniel: Tom is an elephant.
Alice: I get it. Tom is a squid.
Bob: You don't know that.
Alice: Well, I'm saying that on the basis of what Daniel said, Tom is a squid. I do not know whether Daniel is mistaken or pulling our legs - no offense, Daniel -, but on the basis of what he said, Tom is a squid.
Bob: No, you shouldn't make that assessment on the basis of the information given by Daniel.

Who's right, Alice or Bob?
Obviously, Alice is right. It's intuitively obvious, right?

 

[A Toy Windmill]

I ask the previous question to rule something out.

If Joe is a person, Joe can walk.
Joe is a person.
If joe is a tree, then Joe cannot walk.
Joe is a tree.
Therefore, Joe can walk.

That’s what I guess he would call classically valid.

To me, it’s like mixing two puzzles together. So long as there’s enough ingredients in the premises to get to the conclusion, the argument is valid — contradictions and discrepancies be damned.

Yes. The observation here is that adding additional premises to a valid argument cannot make it invalid. It's something that most logics accept, and goes by the name "weakening", since the more assumptions your argument has, the fewer scenarios it applies to, and so the weaker it is.

With weakening, the following argument is valid:

1) Assume p.
2) Assume q.
3) Therefore p (from 1)

And so we generally have the implication

K) p → (q → p).

This is sometimes regarded as a paradox of material implication, but it's really about weakening. If we combine this implication with modus tollens:

MT) (~p → ~q) → (q → p)

We have:

1) Assume ~p
2) ~q → ~p (from K)
3) Therefore p → q (from MT)

And so ~p → (p → q)

So weakening and modus tollens commit us to anything following from a contradiction.

We're presumably happy with modus tollens, but weakening is held in doubt in a few logics. In these logics, all assumptions must be used in an argument, and dangling assumptions, like the q in the proof that p → (q → p) are invalid.

 

[A Toy Windmill]

Again, there are plenty of cases where the intuitions of people with no previous training, in nearly all cases, go wrong (purely for example, the Monty Hall Problem, where the folk probabilistic intuitions nearly always fail).

I'd volunteer the Wason Selection Task. It's really basic, uncontroversial logic concerning implication (not material), which most people get wrong.

 

[fast]

Joe is a squid
Joe is not a squid
Therefore, Joe is a walrus.

Is that valid?

This one. Is this one valid?

 

[A Toy Windmill]

Joe is a squid
Joe is not a squid
Therefore, Joe is a walrus.

Is that valid?

This one. Is this one valid?

Yes, if you allow me previous argument.

1) Joe is a squid.
2) Joe is not a squid.
3) If Joe is not a walrus, then Joe is not a squid (from 2 by K)
4) If Joe is a squid then Joe is a walrus (from 3 by MT)
5) Joe is a walrus (MP from 1 and 4)

 

[Angra Mainyu]

Again, there are plenty of cases where the intuitions of people with no previous training, in nearly all cases, go wrong (purely for example, the Monty Hall Problem, where the folk probabilistic intuitions nearly always fail).

I'd volunteer the Wason Selection Task. It's really basic, uncontroversial logic concerning implication (not material), which most people get wrong.

Isn't that the material conditional? (https://en.wikipedia.org/wiki/Wason_selection_task)

But in any case, it's interesting that subjects agree with the correct solution once it has been explained to them.
Still, I guess Speakpigeon will just take it as evidence against the material conditional, and insist on an incorrect solution.

But I wonder how will Speakpigeon reject the Monty Hall Problem as an example of systematic failure of nearly everyone without training (granted, it's a probability puzzle; nonetheless, it's an example of systematic failure of human untrained intuitions).

 

[fast]

Joe is a squid
Joe is not a squid
Therefore, Joe is a walrus.

Is that valid?

This one. Is this one valid?

Yes, if you allow me previous argument.

1) Joe is a squid.
2) Joe is not a squid.
3) If Joe is not a walrus, then Joe is not a squid (from 2 by K)
4) If Joe is a squid then Joe is a walrus (from 3 by MT)
5) Joe is a walrus (MP from 1 and 4)

The “yes” part confuses me because the conditional “if” is tacked on.

There were originally two issues and one of them was adequately explained. But, it was also said that a contradiction implies anything follows but for a different reason. That reason was in question.

My first premise contradicts the second, so either the conclusion follows because of some questionable reason or it doesn’t.

 

[Speakpigeon]

There's no difficulty answering this one but I'll wait that you answer mine first.

EB

The argument is classically valid.

Since you're not actually saying "This argument is valid", I have to assume you accept the argument is not valid (just kidding).

You talk like you are a government official, using code words. My question was whether the argument was valid, not whether it is "classically" valid (who cares?). I know it is "classically" valid but that doesn't imply (oops) that it is valid, just like a dead man is not a man meaning you haven't replied to my question.

So you can't answer a simple question with a straight answer.

Second point, "classical". For posters here who likely won't know, "classical" here doesn't mean Aristotelian classical (just as classical logic doesn't mean logic). "Classical" here is just used to contrast one brand of mathematical logic contrasted with other, more exotic brands, which assume three or more truth values, whatever that may mean. However, the so-called "classical logic" isn't in fact classical since it is effectively contradictory with Aristotelian logic. I think posters here should be told.

So, not only are you using code words most people won't understand, but these words already mean something, and something different, to most people. Words like validity, truth, proof, etc. mean what they mean to most people since Aristotle. Mathematical logic redacted the definition of validity to make it fit the truth-table definition of the material implication, while pretending to toe the line of Aristotelian logic.
EB