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

[fast]

At risk of having a brick thrown at me, I do maintain my position that the argument is valid.

However, I am biased in my use of the term, “valid.”

I can’t speak for mathematicians, but if I were among logicians trained in deductive logic or educated philosophers with an analytical bent, I feel my answer would accord with theirs.

That’s not to say I cannot appreciate why some may not feel the argument is valid. I think there can be a number of reasons that would explain why some would not regard the argument as valid. A lack of understanding, a substantively different approach, lack of education on the subject matter, a subtle but substantively different definition of “valid.” I’m not gonna throw eggs for voting contrary.

Who’s the philosopher thats says, “Think with the learned. Speak with the vulgar[?]”

I still think language is the ultimate culprit behind the issue.

 

[Speakpigeon]

The term "valid", etymologically, comes from the idea of being healthy.

Different terms have been used throughout history since Aristotle, who only talked of "syllogism" but effectively meant straightforwardly a valid argument. But the idea most people have in mind without ever having to take formal lesson on logic, is that some arguments are, on the face of them, "healthy", while others are just plain sick.

Now, it is a fact that, given the premises assumed in this thread, most people will regard the conclusion that Boris Johnson will be the next Prime Minister as obviously and definitely sick.

As the most articulate people put it here, the premises don't make this particular conclusion necessary, ergo it is not valid.

And, as again noted by some posters here, given that the premises are self-contradictory, no conclusion could possibly be made necessary by the premises.

And this is what flatly contradicts the 166-year old mathematical logic view of validity, advocated here by Angra Mainyu.

I would really appreciate if more people could cast a vote. There are four options, including "Don't know" and "Doesn't make sense", so there's really no good reason that anyone could not find at least one option to vote for. Thanks for your cooperation.
EB

 

[Angra Mainyu]

Just to be clear. That the premises are contradictory does not imply that there is no conclusion that can be validly inferred from them. On the contrary, any conclusion follows validly from contradictory premises. Consider the following:

Premise 1: P.
Conclusion: (P or Q).

That conclusion (i.e., P or Q) follows from P. If P, then surely at least (P or Q).

So, here's another one:

Premise 1: P.
Premise 2: Not P.
Conclusion 1: P or Q (this follows from P, as in the previous argument).
Conclusion 2: Q (this follows from 'P or Q' (which is Conclusion 1) and 'Not P' (which is Premise 2).

Note that Q can be anything you want. From contradictory premises, everything follows.

Premise 1: Boris Johnson is the Prime Minister of the UK.
Premise 2: It is not the case that Boris Johnson is the Prime Minister of the UK.
Conclusion: The Moon is made of cheese, and the Earth is flat, all created by Yahweh less than 10000 years ago.

While the conclusion follows from the premises, obviously it would be a bad idea to believe it is true. But then again, the fact that a conclusion follows from some premises does not provide a good reason to believe the conclusion is true, independently of the premises. For that matter, there are (properly made, valid or whatever you call arguments in which the conclusion follows from the premises) arguments without contradictory premises and whose conclusion is " The Moon is made of cheese, and the Earth is flat, all created by Yahweh less than 10000 years ago". All of this is unproblematic.

I do recommend that before you agree with Speakpigeon's absurd claims, you take a closer look at the arguments, or - better yet - you take a logic course if you have not done so yet.

 

[Speakpigeon]

I do recommend that before you agree with Speakpigeon's absurd claims, you take a closer look at the arguments, or - better yet - you take a logic course if you have not done so yet.

I'm still waiting for you to provide a link or reference to recently published material by well-known logicians, be they mathematicians or philosophers, that would provide an articulate justification that any of the definitions of validity used in mathematical logic is correct.

Scientists are not reticent supporters of the claim that the scientific method is best to investigate nature. We are free to accept the justification they offer but it is easy to find such a justification to begin with. No so with the notion of validity used in mathematical logic. The last time it happened I'm aware of it was with Boole and Frege, who both thought mathematical logic described "the laws of thought" or the logic of how mathematicians reasoned. Since then, well, mathematical logic has evolved many various and very different notions of "logic" and it's just no longer possible for any mathematician to justify the kind of validity they use without vexing their colleagues. They all think theirs is best. They will even criticise each other, including using sometimes ad hominens. But not one of them is able to explain why we should favour the notion of validity they promote rather than our own innate logical sense. I think posters here, as indeed elsewhere, have shown they have in some cases a clear notion of what is valid and what isn't. What is the justification that mathematicians would be correct and we would be wrong? I still don't know. I've asked this question for perhaps two years now, here and elsewhere and no one ever produced any such justification. I also spent time at the top two libraries here in Paris reading books on logic by renown logicians, Quine, Russell, Tarski etc. I went there 45 times in 2016, just trying to find such a justification and I didn't find it. I did find an explanation of why mathematicians opted for the material implication, which also explains why they opted for their definition of validity and it all comes down to... well, they think there's no other possibility! However, their reasoning is based on a premise we don't know whether it's true at all, hence, we don't know whether the conclusion is true or not. So, no justification of any sort. Only the quiet suffisance and pressing suggestion you'll find implicit in absolutely all logic textbooks that there is no logic outside mathematical logic.

The result of that is impressive. You cannot find today anyone arguing anything about logic outside the dogmas of mathematical logic. Most philosophers interested in logic will abide by one of the dogmas of mathematical logic. Philosophers used to be free thinkers. Now they are intellectual workers paid by the big business of universities and academic publications. They don't have time to spend on critical thinking. They prefer to repeat the dogmas, much like the Scholastics did for so long, which earned them to be dismissed by our bright thinkers of the Enlightenment. Isn't that ironic that we should still sing the praise of people like Descartes, Copernicus, Galileo, Newton, Bacon etc. and yet we're doing exactly what the Scholastics are doing. There's something no a little worrying about this. You just have to look at my critics here. Not one could bring himself to really look at the history of it. They are just please with repeating the mantra of their suffisance. How could I be right when so many bright people say otherwise? Wait, what kind of argument is that?
EB

 

[A Toy Windmill]

I will be the first to argue that mathematical logic has nothing to say about laws of thought or the psychology of justification. I will be the first to argue that mathematical logic wasn't a quest to define validity in a way which lines up with polls on internet fora. Boole and Frege may have had different ambitions, but mathematicians don't judge mathematical logic by Boole and Frege's ambitions. Frege's work in logic was almost entirely unread until the middle of the 20th century, long after mathematical logic had grown into an established mathematical field.

I have no problem with mathematical logic being told to stay within the purview of mathematics, where logics are mathematical objects, some of which are accepted by mathematicians as adequate vehicles for formalizing and delimiting their subject. And so long as they are staying in their lane, you will struggle to find definitions of validity that will satisfy any philosopher. Mathematicians do not argue much over whether their definitions are correct. Many do not even consider such arguments coherent: definitions can be described as useful or adequate. They can never be described as correct or incorrect.

Mathematicians are far more interested in their theorems.

 

[fast]

Which is better?

Argument 1:
P1: P
P2: if not P, then Q
C: P or Q


If P and Q are not collectively exhaustive
Then P nor Q is possible, therefore,

Argument 2:

P1: P
P2: if not P, then possibly Q
C: P or Q

The following argument (argument M):
P1: P
C: P or Q

Has an unstated hidden premise or else I can’t get to the truth of P or Q

What about argument N:

P1: P
C: not P

The conclusion may be true, and it’s a logical possibility; after all If P is not true, the contradiction of it makes sense. Even a contrary is possible, but to go full blown “it follows” and thus “it’s valid” without the aid of incorporating the nonexplicit supporting thoughts hinders the entailment.

 

[Angra Mainyu]

Which is better?

Argument 1:
P1: P
P2: if not P, then Q
C: P or Q


If P and Q are not collectively exhaustive
Then P nor Q is possible, therefore,

Argument 2:

P1: P
P2: if not P, then possibly Q
C: P or Q

The following argument (argument M):
P1: P
C: P or Q

Has an unstated hidden premise or else I can’t get to the truth of P or Q

What about argument N:

P1: P
C: not P

The conclusion may be true, and it’s a logical possibility; after all If P is not true, the contradiction of it makes sense. Even a contrary is possible, but to go full blown “it follows” and thus “it’s valid” without the aid of incorporating the nonexplicit supporting thoughts hinders the entailment.

Argument 1 is valid, though it has an unnecessary premise (either one of them can be set aside). Argument 2 is also valid, though the second premise is not required. Argument 3 is valid, and I'm not sure what you mean by a hidden premise. Surely, if P is true, so is (P or Q). Argument 4 is not valid.

 

[fast]

I messed up argument N and kinda glad I did.

P
P or not P

That’s valid

So long as P is retained, validity holds.

 

[fast]

I’m following. But, I have two trains of thought. First, an acknowledgement of understanding:

P1: P
C: P or Q

Example
P1: I’m holding the salt shaker
C: I’m holding the salt shaker OR I’m holding a pepper shaker that someone mistakenly put oregano in

P1 is retained in the conclusion. Q could have been darn near anything. Validity would be unaffected.

The second train of thought:

It’s the spooky action of Q getting slipped in the conclusion that I find troublesome. I see how it’s valid, but it’s creepy. The presence of P takes care of the validity issue (I’ve accepted that). It’s that something you say isn’t a hidden premise making its way for its debut in the conclusion that is spectacular.

P1: P
C: P or Q

I’m not saying it’s invalid but I have this sense that nothing should make its way to the end of the road without traveling the road. It shouldn’t show up explicitly if there’s no origin.

This is what I’m doing to make it more tolerable:

P1: P
P2: Q
C: P or Q

You say that P2 isn’t necessary, but without it, why should I accept Q? You’re reasoning it out; I get that, but you’re not showing your work. I am. I made it known that the second premise is necessary. It doesn’t have to be explicit, but it has to be real or else there’s no thrust behind accepting the added “or Q” to the conclusion.

 

[fast]

P is to salt as Q is to pepper.

P1: P
C: P or Q

I see how the salt got in the conclusion. I mean, I literally see it. It’s in the territory where premises reside. It’s address is P1. Q, well, how pepper got in the pot is an issue. An illegal ingredient?

 

[A Toy Windmill]

Here's an idea. Try working backwards. Don't think of the premises as the start and the conclusion the end. Think of the conclusion as a goal and the premises the basis for the solution.

Thinking backwards, it's me saying "I need either salt or pepper." The argument shows that my needs are met on the basis of you having salt.

This way, the pepper hasn't come out of nowhere. It was there from the start, because we started at the goal.

Everything we want to say of logic and validity can be done this way. And there's a lesson there when it comes to getting computers to do logic. If you start at premises, and you work with inferences such as "P; therefore P or Q", you'll find yourself having to invent spurious Qs, or otherwise reaching for some strategy to synthesize them. If you work backwards, and stay goal-oriented, you'll keep things relevant.

 

[fast]

Here's an idea. Try working backwards. Don't think of the premises as the start and the conclusion the end. Think of the conclusion as a goal and the premises the basis for the solution.

Thinking backwards, it's me saying "I need either salt or pepper." The argument shows that my needs are met on the basis of you having salt.

This way, the pepper hasn't come out of nowhere. It was there from the start, because we started at the goal.

Everything we want to say of logic and validity can be done this way. And there's a lesson there when it comes to getting computers to do logic. If you start at premises, and you work with inferences such as "P; therefore P or Q", you'll find yourself having to invent spurious Qs, or otherwise reaching for some strategy to synthesize them. If you work backwards, and stay goal-oriented, you'll keep things relevant.

 

[Angra Mainyu]

Here's an idea. Try working backwards. Don't think of the premises as the start and the conclusion the end. Think of the conclusion as a goal and the premises the basis for the solution.

Thinking backwards, it's me saying "I need either salt or pepper." The argument shows that my needs are met on the basis of you having salt.

This way, the pepper hasn't come out of nowhere. It was there from the start, because we started at the goal.

Everything we want to say of logic and validity can be done this way. And there's a lesson there when it comes to getting computers to do logic. If you start at premises, and you work with inferences such as "P; therefore P or Q", you'll find yourself having to invent spurious Qs, or otherwise reaching for some strategy to synthesize them. If you work backwards, and stay goal-oriented, you'll keep things relevant.

I second A Toy Windmill's excellent reply.

 

[Speakpigeon]

I will be the first to argue that mathematical logic has nothing to say about laws of thought or the psychology of justification. I will be the first to argue that mathematical logic wasn't a quest to define validity in a way which lines up with polls on internet fora. Boole and Frege may have had different ambitions, but mathematicians don't judge mathematical logic by Boole and Frege's ambitions. Frege's work in logic was almost entirely unread until the middle of the 20th century, long after mathematical logic had grown into an established mathematical field.

I have no problem with mathematical logic being told to stay within the purview of mathematics, where logics are mathematical objects, some of which are accepted by mathematicians as adequate vehicles for formalizing and delimiting their subject. And so long as they are staying in their lane, you will struggle to find definitions of validity that will satisfy any philosopher. Mathematicians do not argue much over whether their definitions are correct. Many do not even consider such arguments coherent: definitions can be described as useful or adequate. They can never be described as correct or incorrect.

Mathematicians are far more interested in their theorems.

Sure. I could have said most of this.

Please explain to AM. Explain in particular that the various definitions of validity used in mathematical logic have absolutely zero value and zero authority when it comes to any human being considering or discussing the validity of logical arguments. This is a point he still doesn't get.

Further, there is the question of the use of the theories developed in mathematical logic. Mathematicians are human beings and they reason logically when they want to, and presumably they have to to prove theorems. As human beings, their logic must be the same as that of any other human being. So, either mathematical logic has no role whatsoever in how mathematicians prove theorems or it does. If no role, mathematical logic is literally useless, which would contradict what you say here. So, I will assume that you think it has a role. Now, i asked several times, without result, AM to provide examples of important mathematical theorems whose demonstration relied on mathematical logic. Can you yourself provide examples?

Further, it seems reasonable for non-mathematicians to assume mathematical theorems are logically proven. Yet, if mathematical logic is not logic but is nonetheless used to prove mathematical theorems, then non-mathematicians are wrong in their reasonable that mathematical theorems are logically proven. Isn't that a problem? Shouldn't people be told?
EB

 

[Speakpigeon]

Here's an idea. Try working backwards. Don't think of the premises as the start and the conclusion the end. Think of the conclusion as a goal and the premises the basis for the solution.

Thinking backwards, it's me saying "I need either salt or pepper." The argument shows that my needs are met on the basis of you having salt.

This way, the pepper hasn't come out of nowhere. It was there from the start, because we started at the goal.

Everything we want to say of logic and validity can be done this way. And there's a lesson there when it comes to getting computers to do logic. If you start at premises, and you work with inferences such as "P; therefore P or Q", you'll find yourself having to invent spurious Qs, or otherwise reaching for some strategy to synthesize them. If you work backwards, and stay goal-oriented, you'll keep things relevant.

I second A Toy Windmill's excellent reply.

Reply?

LOL. Who is he replying to?!
EB

 

[Speakpigeon]

I’m following. But, I have two trains of thought. First, an acknowledgement of understanding:

P1: P
C: P or Q

Example
P1: I’m holding the salt shaker
C: I’m holding the salt shaker OR I’m holding a pepper shaker that someone mistakenly put oregano in

P1 is retained in the conclusion. Q could have been darn near anything. Validity would be unaffected.

The second train of thought:

It’s the spooky action of Q getting slipped in the conclusion that I find troublesome. I see how it’s valid, but it’s creepy. The presence of P takes care of the validity issue (I’ve accepted that). It’s that something you say isn’t a hidden premise making its way for its debut in the conclusion that is spectacular.

P1: P
C: P or Q

I’m not saying it’s invalid but I have this sense that nothing should make its way to the end of the road without traveling the road. It shouldn’t show up explicitly if there’s no origin.

This is what I’m doing to make it more tolerable:

P1: P
P2: Q
C: P or Q

You say that P2 isn’t necessary, but without it, why should I accept Q? You’re reasoning it out; I get that, but you’re not showing your work. I am. I made it known that the second premise is necessary. It doesn’t have to be explicit, but it has to be real or else there’s no thrust behind accepting the added “or Q” to the conclusion.

It's no use. Mathematicians don't understand logic at all. Your questions will be met, invariably, with either fudge, argument from authority, and most often falsehood.
EB

 

[fast]

I’m following. But, I have two trains of thought. First, an acknowledgement of understanding:

P1: P
C: P or Q

Example
P1: I’m holding the salt shaker
C: I’m holding the salt shaker OR I’m holding a pepper shaker that someone mistakenly put oregano in

P1 is retained in the conclusion. Q could have been darn near anything. Validity would be unaffected.

The second train of thought:

It’s the spooky action of Q getting slipped in the conclusion that I find troublesome. I see how it’s valid, but it’s creepy. The presence of P takes care of the validity issue (I’ve accepted that). It’s that something you say isn’t a hidden premise making its way for its debut in the conclusion that is spectacular.

P1: P
C: P or Q

I’m not saying it’s invalid but I have this sense that nothing should make its way to the end of the road without traveling the road. It shouldn’t show up explicitly if there’s no origin.

This is what I’m doing to make it more tolerable:

P1: P
P2: Q
C: P or Q

You say that P2 isn’t necessary, but without it, why should I accept Q? You’re reasoning it out; I get that, but you’re not showing your work. I am. I made it known that the second premise is necessary. It doesn’t have to be explicit, but it has to be real or else there’s no thrust behind accepting the added “or Q” to the conclusion.

It's no use. Mathematicians don't understand logic at all. Your questions will be met, invariably, with either fudge, argument from authority, and most often falsehood.
EB

It’s been brought to my attention that what is being discussed falls under the heading of what is being called “deductive logic.” It’s also my understanding that deductive logic is a type of logic. Given that, a discussion regarding deductive logic is a discussion regarding logic. Of course, even that simple notion is fraught with dangers.

For instance, we can discuss astrology and CALL IT a discussion about astronomy, but no matter what we CALL IT, it remains what it is. So, if we’re discussing something that is neither regarding deductive logic nor logic, then CALLING IT either is a mistake.

However, I don’t get the sense that we’re not discussing deductive logic but rather having a discussion where not everyone is meeting eye to eye.

A second danger (that I don’t think is apt here) would be a misdirection of what’s being discussed by the wordage of the label itself. In this instance “deductive logic.” Even when used as a noun phrase (or technical term to describe the referent), there doesn’t seem to be a disconnect even should the phrase be interpreted as a type of logic that described as deductive. At any rate, this doesn’t seem to be one of those times where the sheer label itself is causing any confusion.

A third issue would be the ever-persistent ambiguity that words bring. Words have meaning, but of course, they often have more than one. Not only that, they can be somewhat imprecise and people will take it upon themselves to alter their scope in whatever articulate fashion they feel helps them to communicate. The potential for misunderstanding is no small deal.

I find it difficult to think mathematicians don’t understand logic at all. How it ought to be used?

Either way, why are mathematicians being brought up? Even if we aren’t talking about deductive logic but some deviant non-logical subject masquerading as such, what do mathematicians have to do with this?

 

[Speakpigeon]

I find it difficult to think mathematicians don’t understand logic at all. How it ought to be used?

Me, too. This is indeed an incredible fact.

Initially, I thought the basic problem there is with mathematical logic might sort of disappear through some subtlety in the proof process. Well, it doesn't.

Second, I initially thought of mathematicians understood what they were talking about. Well, turns out, they don't. Most of them aren't even capable of articulating anything rational in respect to logic. They don't even know it exists. They incredibly naively think that logic is just what they do, i.e. mathematical logic.

Which would make mathematicians the same as all of us. We know logic through Aristotle's notion of syllogisms, and most of us are only dimly aware of logic itself. What people actually say makes my point for me.

However, mathematicians are taught mathematical logic, which is in fact contradictory not only to Aristotle's logic, but to logic itself. Thus, they managed to sort of unlearn the little we intuitively understand about logic. In effect, the know even less than any idiot knows, which is already very little.

QED.

Either way, why are mathematicians being brought up? Even if we aren’t talking about deductive logic but some deviant non-logical subject masquerading as such, what do mathematicians have to do with this?

Very simple. Over the last few months, I happened to propose a number of topics and polls on various logical arguments, as you may have noticed. Indeed, this very thread is just one of those.

At no point did I ever even mention anything to do with mathematics or mathematical logic in my introductory posts. I would invariably ask whether people thought the argument was valid. Yet, invariably, some posters would come up with the idiotic argument from authority that the argument was valid just because mathematical logic said so.
EB

 

[Angra Mainyu]

I second A Toy Windmill's excellent reply.

Reply?

LOL. Who is he replying to?!
EB

He is replying to fast. Obviously.

 

[Angra Mainyu]

Sure. I could have said most of this.

Please explain to AM. Explain in particular that the various definitions of validity used in mathematical logic have absolutely zero value and zero authority when it comes to any human being considering or discussing the validity of logical arguments. This is a point he still doesn't get.

Stop grossly misrepresenting my position.
First, I never said or suggested that the various definitions of validity used in mathematical logic had value when it comes to the colloquial meanings of 'valid'. In fact, I said that in natural languages, like English, usually people do not need a fine-grained distinction that separates arguments with false premises with arguments in which the conclusion does not follow from the premises, so a colloquial use of "invalid" may include arguments of the former kind.
Second, I was never discussing the different usages of 'valid' in different logics in mathematics. Rather, I was talking about the specific definition that you called "the definition of logical validity used in mathematical logic since the beginning of the 20th century" (not "a", but "the"), and which is as follows:

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/

Note that this definition is not limited to mathematics. It's a standard definition used in philosophy (indeed, you quoted from philosophy Encyclopedia), and it is in that context used in the study of human logic. I would say it as a proper definition of an argument in which the conclusion follows from the premises (whether it is called it 'valid' or not), intuitively. But you have different intuitions. It does not matter, because that is not one of the points I intended to make.

Third, I did not claim or suggested that the definition in question matched common usage. But I showed that under assumptions you made, that definition provided our best method for finding mathematical truth. I showed that in several posts, for example this one.

Fourth, I showed that under your own assumptions, the rational assessment is that human logic is a disaster, as it fails to preserve truth, so we should ditch it and choose another, better definition of validity, such as what you call "the" definition of mathematical logic. Of course, I am not suggesting either that the definition in question fails to match human logic, or that human logic fails to preserve truth. I just debunked your position once again

Fifth, I did further debunking on your position, showing its untenability in several threads, such as this one, this one, this one,

Further, there is the question of the use of the theories developed in mathematical logic. Mathematicians are human beings and they reason logically when they want to, and presumably they have to to prove theorems. As human beings, their logic must be the same as that of any other human being. So, either mathematical logic has no role whatsoever in how mathematicians prove theorems or it does. If no role, mathematical logic is literally useless, which would contradict what you say here. So, I will assume that you think it has a role. Now, i asked several times, without result, AM to provide examples of important mathematical theorems whose demonstration relied on mathematical logic. Can you yourself provide examples?

First, again by "mathematical logic", you meant what you defined as such, namely the definition of validity that says "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."
I already told you: that is our intuitive sense of logic. We reason that way all the time. But I went further. I posted the relevant link to the SEP article in which it is explained that "Unfortunately, as Meyer and Friedman have shown, relevant arithmetic does not contain all of the theorems of classical Peano arithmetic." (I also provided more information about what happens when you limit logic in different ways).

Second, you misunderstand Aristotelian logic, and as a result, you say things follow in Aristotelian logic when they do not, which makes it hard to give examples, since you can simply say that that too follows in Aristotelian logic. In fact, you contradict of course your own position. For example, you claim

One derives a contradiction.

The right word is indeed "inconsistent". The premises are inconsistent, i.e. one premise implies the negation of the other premise.

Contradictory premises would be p and not p and that's not what we have here.

The premises here are not contradictory.

You need to make sure you know the basics before posting silly arguments.
EB

So, you claim that it is not a problem when "one premise implies the negation of the other premise.", but rather, the problem is with premises like "p and not p". Of course, that already debunks your own claim that the "Improved Squid Argument" (and several similar arguments you asks about) is invalid, since that is precisely a case in which no premise contradicts itself, but one or more premises imply the negation of another.

Remember: I do not want to talk to you anymore, but whenever you reply to a thread and misrepresent what I said (in that thread or any other(s)), I will reply by debunking some of what you claim. Either you will stop misrepresenting what I said, or I will keep debunking your position. (of course, that will not motivate you to stop if you fail to realize I'm debunking your position, but no matter, the debunking will continue in that case).