And the next U.K. Prime Minister will be?

[Speakpigeon]

You still don't falsify theorems. You show that a purported theorem was never a theorem, and falsify the conjecture.

You're obviously wrong here.

You can obviously falsify a conjecture but you can also falsify a theorem, exactly as I said, by showing that the proof is wrong.

And of course, you couldn't falsify a theorem unless it had been stated as a theorem to being with. Assuming it has a proof, that proof may not be valid. You falsify such a theorem by showing the proof not valid. Easy as pie.

You clearly never had to think too much about that kind of niceties, I guess it's more philosophy that mathematics, but mathematicians might do well doing a bit of philosophy now and then. All great mathematicians did.

Such as that all theorems are statements and as such each of them is either true or false. If you show a statement to be false, you falsify it. There's nothing else to understand to it, it is obvious, and yet you just said "You still don't falsify theorems". Well, sorry, love, me, I do.

You should also stop your idiot disparaging suggestion that I'm likely a "crackpot". I've been on this board for years and I always argue my point and I'm rather good at it. I definitely don't look like I'm a crackpot. Insisting on this ad hom just make yourself look off kilter. The lady protests too much, that sort of thing. Still, keep at it, I really don't mind as for myself.

I'll respond to you again once you've address Angra Mainyu's excellent defence of the definition of logical validity.

I already replied. AM's justification is not what I asked and I also already made that clear.

But, please, don't reply. It's clear now you don't have anything interesting to say about logic and likely very little on mathematical logic. And generally speaking, you don't seem to understand much at all.
EB

 

[Speakpigeon]

It's like someone sending a paper to a serious geology journal claiming they have conclusive evidence the Bible is right and the Earth is less than 15000 years old. It would be rejected without even reading it, and with pretty good reason.

Your logic really sucks. I wouldn't have any reason to send a paper about the Bible being right or the Earth being less than 15000 years old.
EB

Your failure to understand what others say is a constant annoyance. It's an analogy, of course.

You think I didn't understand that?!

And you think analogy are not meant to prove a point?!

Whoa. Man. You really have serious shortcomings.
EB

 

[Speakpigeon]

P1: P and Q
C: P or Q

Is that valid?
I’ve been wrestling with that for two days

ETA: I want to say yes, but certain examples get tricky.
I need either some apples or some oranges
You have some apples and oranges
Seems easy enough

But
I need either some salt or pepper
You have both, but they’re mixed
Maybe what was really meant was I need either pure salt or pure pepper

Any pitfalls to consider?

Does the conclusion need to be an "or" to highlight the issue?

I need some salt. You have salt and pepper, but they're mixed.

What's your point? Remember, you said logical validity is arbitrary and different mathematicians use different definitions.
EB

 

[Speakpigeon]

P1: P and Q
C: P or Q

Is that valid?
I’ve been wrestling with that for two days

Yes, it is valid.

If your premise is that both P and Q are true, then surely it is also true that either P and Q are both true, or P is but not Q, or Q is but not P.

This is the case because by definition A or B is true either because both A and B are true, or A is true, or B is true. Exactly what we have here since P and Q are both true as per your premise.

P1: P and Q
ETA: I want to say yes, but certain examples get tricky.
I need either some apples or some oranges
You have some apples and oranges
Seems easy enough

But
I need either some salt or pepper
You have both, but they’re mixed
Maybe what was really meant was I need either pure salt or pure pepper

Any pitfalls to consider?

If that can help, the following is definitely not valid:

P and Q
C: Either P or Q

If you are somewhat confused about these two different "or", that may explain why you're "wresting" with P and Q, therefore P or Q.
EB

 

[fast]

I’ve composed two posts, one that touches on it and one that explains in great detail. I felt one wasn’t enough and the other over the top.

I’m just happy to know it’s valid.

 

[Speakpigeon]

I’ve composed two posts, one that touches on it and one that explains in great detail. I felt one wasn’t enough and the other over the top.

I’m just happy to know it’s valid.

Good.

But, "P and not P, therefore Q" is still not valid and never will be.
EB

 

[fast]

I’ve composed two posts, one that touches on it and one that explains in great detail. I felt one wasn’t enough and the other over the top.

I’m just happy to know it’s valid.

Good.

But, "P and not P, therefore Q" is still not valid and never will be.
EB

It’s not valid. Obviously. But, it’s valid, and though not obvious, it is. It’s been shown how.

It’s valid and it’s not valid would ordinarily be considered contradictory, but then again, that’s because we’re accustomed to contrasting words spelled the same as being equivalent, yet because words are ambiguous, it’s not necessarily the case that what appears to be the same word is.

 

[A Toy Windmill]

What distinguishes modern logic more than anything is that, like mathematics, it's highly recursive and systematic. The syntax is recursive, as when we say

for any proposition P and Q, we can form the proposition P → Q

Recursion now commits us to admitting complex propositions that are beyond the ability for a human to parse in their heads, like this:

(((((S → S) → (Q → S)) → ((S → S) → (P → Q))) → (((R → P) → (S → S)) → ((Q → R) → (Q → S)))) → ((((S → R) → (R → S)) → ((R → Q) → (Q → S))) → (((R → Q) → (Q → Q)) → ((R → S) → (R → R)))))

Next, you agree to some recursive rules of deduction, as with:

given a theorem or axiom P, and a theorem or axiom P → Q, we take Q to be a theorem.

And now the mathematical logician can start cranking a handle, applying the recursive deduction rules to the recursive syntax to show an infinity of deductions to which you're committed. The commitments are sometimes surprising. They're sometimes upsetting. Sometimes you shrug your shoulders at them and say "them's my dues". Other times you are so upset that you force yourself to go back and modify your starting rules.

I've said, in another thread that, that if you commit yourself to just a few simple rules then you are committed to all of classical propositional logic. This does not mean we have to accept all of classical propositional logic. It just means that, if we don't accept all of classical propositional logic, we've got some simple culprit rules to single out as being problematic.

I gave three axioms in that thread:

1) P → (Q → P)

This axiom is related to weakening. It's regarded as problematic by relevant and linear logicians, who object to the fact that it ignores or discards a Q.

2) [P → (Q → R)] → [(P → Q) → (P → R)]

This axiom is related to a complementary notion of weakening called "contraction." Linear logicians object to it because it says that P can be used twice.

3) (¬P → ¬Q) → (Q → P)

This axiom is the classical axiom. Intuitionistic logicians object to it because it conjures up a positive P.

I personally find it interesting to talk about all these objections, and to discuss what happens when we change the rules.

 

[Speakpigeon]

I’ve composed two posts, one that touches on it and one that explains in great detail. I felt one wasn’t enough and the other over the top.

I’m just happy to know it’s valid.

Good.

But, "P and not P, therefore Q" is still not valid and never will be.
EB

It’s not valid. Obviously. But, it’s valid, and though not obvious, it is. It’s been shown how.

It’s valid and it’s not valid would ordinarily be considered contradictory, but then again, that’s because we’re accustomed to contrasting words spelled the same as being equivalent, yet because words are ambiguous, it’s not necessarily the case that what appears to be the same word is.

There is just one word "valid" and it is defined by how it is spelt. What words mean can only be decided on the context. Thus, it is incombent on each of us to make sure we use words in a way which isn't ambiguous. The term "valid" applied to logical arguments was already used in the 17th century. An argument was valid in the same sense as an argument was a syllogism, as defined by Aristotle. In the 20th century, mathematicians started to use the word "valid" with a different definition, yet without making clear that their notion of validity was different from that of Aristotle. This was just plain equivocation on their part.

So, to say it's valid and it's not valid is just a contradiction and you can't argue anything from a contradiction because contradictions don't imply anything because arguments with contradictory premises are not valid.

Still, this is a free country and it is probably still legal to equivocate.

Legal but you won't ever win an argument in court by assuming contradictory premises. You would probably by charged with contempt of court.
EB

 

[Speakpigeon]

I personally find it interesting to talk about all these objections, and to discuss what happens when we change the rules.

Why objections? You said logical rules in mathematical logic were arbitrary. Why object at all once this is admitted?

My position is different. There is just one logic and it is the logic of the human brain. It should be investigated by science, not by mathematicians, who clearly haven't a clue what logic is.

In this context, it makes sense to object to particular rules or definitions. It makes sense to say that mathematical logic is not correct.

However, in the context of mathematical logic, methods of logic are not correct because they are arbitrary. But then, why are there "objections"? Not only objections, but mathematicians writing books containing ad hominens against other mathematicians.

So, as far as I can tell, there is no coherence even in terms of the nature of the various method in use in mathematical logic. Some mathematicians believe their method is correct in that it would represent what the human mind does (they are all wrong), other develop completely arbitrary methods without regard for human logic (and that's fine as long as they don't make claims about about validity as used in human logic). And there are probably all sorts in between, including mathematicians who never think about what they do. They just do it. And other who are just terminally confused. Most can't even explain themselves. The few who write books just obfuscate the various issues facing mathematical logic. There won't be any solution any time soon.
EB

 

[A Toy Windmill]

You said logical rules in mathematical logic were arbitrary.

I didn't. I said that some mathematicians argue that definitions are neither correct nor incorrect, only useful or adequate. I am not one of those.

My broader point was that mathematicians rarely give a justification for any definitions. They may motivate a definition, but they mostly let the mathematics speak for itself.

There's no reason to single out the mathematician's attitude towards the term "validity" as a particular problem. Mathematicians also have definitions for colloquial notions such as "number", "straight line", "continuous" and "collection", and their usages of these terms may well diverge from colloquial usage, leading to some odd poll results on forums such as these. But so what?

 

[fast]

Words denote meaning.

We use words to stand in for lexical meaning, and as such, it’s perfectly acceptable to say that words have meaning.

The meaning of words is a function of usage. However, lexical usage (as opposed to stipulative usage) is a function of collective usage (and not individual usage). So, meaning (lexical meaning, that is) is a function of collective usage; however, there is more.

The lexical meaning of words is a function of collective usage of fluent speakers of a given language. That said, the source of meaning although predicated on not only usage but collective usage, it’s not necessarily by how any one small group of people might use the term.

The dictionary is an authoritative source for the lexical meaning of words. What we will find upon examination, however, are definitions (including both the definiendum and definiens). From that, we can glean what a word means.

A definition is (in short) an explanation of meaning. When you consult your dictionary (oh, let’s say like The American Heritage Dictionary of the English Language), you could find what A Toy Windmill may call a colloquial use of the word. There are other sources. For example, a teacher or parent could explain what a word means.

We are capable of using words in an alternative or unusual manner and communicate a meaning that differs from lexical usage. Example, after a few drinks, your neighbor’s husband might be found at a bar trying to cat around with a kitty. Stipulative usage is common. In fact, even many fields of study may collectively use terms particular to their field. Even legal terms are stipulative and have stipulative definitions written into law.

You are using the term, “valid” in a lexical manner. The definition cited (somewhere around one of these threads) is a clear example of a stipulative usage of the term. In fact, it’s even referred to as a technical term.

For more insight about something that goes beyond meaning, one can do an analysis of the referent of the word. Referent is of course different than meaning. All words have meaning (well, almost all), but whether a term is a referring term or a nonreferring term is a different matter; moreover, some referring terms are successful while some are not. Some nonreferring terms (like “of” and “although”) are often confused with referring terms that fail to refer (e.g. unicorn).

Both the stipulative and technical term “valid” has evolved to mean what it does today. No term’s meaning is immune from change or evolution. It’s an etymological fallacy to think that the original meaning of a term is the current meaning of a term.

The lexical definitions are what they are just as the stipulative definitions are what they are.

Justification. I suppose neither kind of meaning needs a justification, and though I can speculate why the stipulative term “valid” has evolved to be more precise, I too am curious why a construct of propositional or deductive logic would include a variety of validity that would diverge to such an extent as to conflict with common ordinary lexical usage as it does.

 

[A Toy Windmill]

I enjoyed that post.

Justification. I suppose neither kind of meaning needs a justification, and though I can speculate why the stipulative term “valid” has evolved to be more precise, I too am curious why a construct of propositional or deductive logic would include a variety of validity that would diverge to such an extent as to conflict with common ordinary lexical usage as it does.

I made a suggestion above. Mathematical logicians, being mathematicians, aimed to be systematic. And when you are systematic with a bunch of intuitive rules, you sometimes unearth pathologies. This phenomenon happens with other mathematical concepts, such as the mathematician's definition of "continuous", which, from its inception, has admitted pathologies (the term pathology is used broadly by mathematicians to refer to this phenomenon).

In this specific disagreement about validity, my attempts from the outset have been to correct our divergent understanding by pointing to general rules such as weakening, showing that, if we commit to them systematically, we will be forced to accept pathologies that offend our intuitive sense. This might only show that general rules such as weakening should be questioned. It might only show that our intuitive notion of validity isn't systematic. Both are fine by me.

 

[Speakpigeon]

You said logical rules in mathematical logic were arbitrary.

I didn't. I said that some mathematicians argue that definitions are neither correct nor incorrect, only useful or adequate. I am not one of those.

My broader point was that mathematicians rarely give a justification for any definitions. They may motivate a definition, but they mostly let the mathematics speak for itself.

There's no reason to single out the mathematician's attitude towards the term "validity" as a particular problem. Mathematicians also have definitions for colloquial notions such as "number", "straight line", "continuous" and "collection", and their usages of these terms may well diverge from colloquial usage, leading to some odd poll results on forums such as these. But so what?

Obviously not the same situation at all. People don't take the mathematical equation of a line to be the line itself. They don't take equations to be the real stuff of the world.

The problem with mathematical logic is that most people, as indeed demonstrated again and again on this and other forums, have become increasingly confused about logic. They have come to believe that logic is literally what mathematicians call "mathematical logic". Philosophers, at least analytic philosophers, themselves are gradually forgetting the expertise they had on logic, nearly all deferring to mathematicians to tell them what is logic. And the reason is simple. It is the impressive complexity of mathematical logic, the use of a formal language seen as cryptic by outsiders and making it impossible for the outsiders to produce an effective critique of mathematical logic, it is the might of the mathematical academy, and also the usefulness of mathematics which people naively extend to mathematical logic. Mathematical logic has become a de facto dogma served by a plethoric army of mathematicians.

How could any definition of validity used in mathematical logic be correct when mathematicians themselves think it's not something which could be either correct or incorrect? All definitions used in mathematical logic are de facto incorrect representations of human logic. That they are not meant as representation or model doesn't alter this fact. And yet, there are a large majority of computer scientists, philosophers and mathematicians themselves who believe it is correct, even more "correct" than human logic! We're deep down into the rabbit hole of Alice in Wonderland complete with the Mad Hatter and the Queen of Hearts.
EB

 

[Speakpigeon]

I enjoyed that post.

Justification. I suppose neither kind of meaning needs a justification, and though I can speculate why the stipulative term “valid” has evolved to be more precise, I too am curious why a construct of propositional or deductive logic would include a variety of validity that would diverge to such an extent as to conflict with common ordinary lexical usage as it does.

I made a suggestion above. Mathematical logicians, being mathematicians, aimed to be systematic. And when you are systematic with a bunch of intuitive rules, you sometimes unearth pathologies. This phenomenon happens with other mathematical concepts, such as the mathematician's definition of "continuous", which, from its inception, has admitted pathologies (the term pathology is used broadly by mathematicians to refer to this phenomenon).

In this specific disagreement about validity, my attempts from the outset have been to correct our divergent understanding by pointing to general rules such as weakening, showing that, if we commit to them systematically, we will be forced to accept pathologies that offend our intuitive sense. This might only show that general rules such as weakening should be questioned. It might only show that our intuitive notion of validity isn't systematic. Both are fine by me.

There are no "pathologies" in our ordinary intuitive notion of set. Many mathematicians have come to believe their is a problem with intuition, that it is unreliable. Yet, there is no problem with our ordinary intuitive notion of set. The pathology is entirely in the mathematical formalisations of our intuitive notions.
EB

 

[A Toy Windmill]

Let's talk again when you've gone back to school and got your degree.

 

[Speakpigeon]

Let's talk again when you've gone back to school and got your degree.

LOL. I think I got it this morning at 9 h 48.

Talking of pathology, you should know that your behaviour looks pathological. You stall whenever you are required to justify yourself and this without any good reason. You are unable to engage except on the basis that you would be the teacher and people are prepared to play the part of the good pupils. This is pathological.

This is a forum. We're here to debate in a rational way as much as possible. Sometimes we may have good reasons for not engaging but you don't have any. This suggests that suffisance is your motivation. Yeah, a pathological trait.

But, fine, you're making my point for me. Evil academic.
EB

 

[Speakpigeon]

I personally find it interesting to talk about all these objections, and to discuss what happens when we change the rules.

LOL. You find it "personally interesting to talk about all these objections"!

Apparently, not all objections and not with all those who raise objections.

And obviously, by "talking" here, you don't mean having anything like a debate.

It is clear you are not interested at all "talking" about objections. You are only interested giving lectures.

I guess it's a psychological constant in evil academics.
EB

 

[Angra Mainyu]

I personally find it interesting to talk about all these objections, and to discuss what happens when we change the rules.

LOL. You find it "personally interesting to talk about all these objections"!

Apparently, not all objections and not with all those who raise objections.

And obviously, by "talking" here, you don't mean having anything like a debate.

It is clear you are not interested at all "talking" about objections. You are only interested giving lectures.

I guess it's a psychological constant in evil academics.
EB

Actually, you only give lectures, albeit vastly confused ones. For example, you keep saying things like

LOL. Yes, I'm sure my comment here makes no sense to you. Still, I read what I wrote and I can still sign up to it. Of course it is valid. Trivially valid. Obviously valid.

The clue is given, intentionally, in my comment, but you sure don't want to see it.

Now that is a confused - and evil - lecture. It is confused because you fail to realize I just debunked some of your claims again. It is evil because you accuse me - falsely and with no epistemic justification at all - that I do not want to see an alleged clue, but also evil because of the way you talk about "the clue is given...", etc. You systematically refuse to clarify your position, speak in riddles, and when your position is clear enough to be debunked and is debunked, you come to believe that this is one of the times when you spoke in riddles, claim to have spoken in riddles, and mock those who debunked your claims.

Let me make it clear to you again: Your constant "clues" and deliberate refusal to clarify your claims is not an acceptable way of debating. You do that to pretty much all of your debate opponents.

 

[Speakpigeon]

Actually, you only give lectures, albeit vastly confused ones. For example, you keep saying things like

LOL. Yes, I'm sure my comment here makes no sense to you. Still, I read what I wrote and I can still sign up to it. Of course it is valid. Trivially valid. Obviously valid.

The clue is given, intentionally, in my comment, but you sure don't want to see it.

Now that is a confused - and evil - lecture. It is confused because you fail to realize I just debunked some of your claims again. It is evil because you accuse me - falsely and with no epistemic justification at all - that I do not want to see an alleged clue, but also evil because of the way you talk about "the clue is given...", etc. You systematically refuse to clarify your position, speak in riddles, and when your position is clear enough to be debunked and is debunked, you come to believe that this is one of the times when you spoke in riddles, claim to have spoken in riddles, and mock those who debunked your claims.

Let me make it clear to you again: Your constant "clues" and deliberate refusal to clarify your claims is not an acceptable way of debating. You do that to pretty much all of your debate opponents.

I've just been going for a walk and thought about my day and it's been tremendously productive. You can see on this forum my willingness to debate and I think I do a good job in this respect. But I'm not here to teach you logic. First, you're not even interested. Second, I won't be able to publish anything. Your new academic imaginary friend here just said it very elegantly. Crackpots can't be right.
EB