Justification of the mathematical definition of logical validity?

[Speakpigeon]

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

The Wason selection task can be explained with the classical material conditional, but that doesn't mean it requires it. I think wiki is being sloppy here.

There are good reasons that people get it wrong.

Mostly, because they usually right but then the researchers themselves don't understand that they do.
EB

 

[fast]

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

EB

A Toy Windmill has already answered. Will you now answer his question about his argument?

No, he didn't. You guys have a habit of pretending you've answered when you haven't.

Well, I guess we get used to it.
EB

I once asked admissions if I owed them any money. The lady RESPONDED, “the computer doesn’t show that I do.” I wasn’t in the best of moods and said, “you didn’t answer my question.” Needless to say she disagreed. I explained that I wasn’t interested in solely what the computer showed and inquired as to if there was a lingering bill that hadn’t been put in the system. She said she wouldn’t have any way of knowing. I suggested asking someone that would. Come to find out, I did; it just hadn’t been entered yet.

It pays to be watchful when people respond, as not every response is an answer. How many apples are on the table? Well, let’s see, there were originally 12 and sally took four. That might sound like an answer, but it’s a damn math problem; besides, it doesn’t even account for the one that rolled onto the floor not expressed in the response.

Then, years later, and on this forum, ambiguity reared it’s ugly head. It would have seemed to me that the typical politician’s so-called answers are not in fact answers but mere responses, but I had to come to the unfortunate realization that the term, “answer” has more than one meaning.

So while you may very well be correct that he did not answer your question, it would not be a contradiction to say he did. It depends on which meaning of the term is being used and the truth of the matter.

Further, in analogy, concerning behavior X:

Tom says, “X is a crime.”
Ann says, X is not a crime.”

That is not necessarily contradictory if ann’s use of “crime” is different than Tom’s use. Ann may in fact be correct.

However, if Ann had instead said, “Tom, you are incorrect,” then even if X is not a crime in how she thinks it is not, her actual statement is incorrect since it hinges on his use, not hers.

Now, did he answer your question? Well, it seems you are using “answer” in how I used “answer” at admissions. Angra is apparently using the other, but he didn’t explicitly deny that you were incorrect (although he likely thinks you are incorrect—since he did answer your question in the second more broad sense of the term).

However, I’m with ya and grasp why you say he didn’t, as he didn’t (if he didn’t) give more than a mere response.

 

[Angra Mainyu]

We know that Joe is not a squid. But, that doesn’t tell me what he is—just what he’s not. Now, if I want to validly conclude that Joe is a walrus, I’m going to need a bit more than “Joe is a squid.” However, what more is there? Sure, we can add premises, but then it would be a different argument.

That’s what Toy Windmill did. He added more premises. Where’d they come from? If he derives them from what was given, great, but then, how? If he added them on his own, then no fair—different argument.

I could just as easily have put “Joe is a raccoon” as the conclusion, but the best I can get is Joe is possibly a raccoon or possibly a walrus or possibly a blue jay with a broken wing. The conclusion in this instance isn’t about what Joe possibly is but what he actually is. Neither premise taken together or apart provides the ingredients necessary to hone in on what should follow other than either Joe is a squid (because it follows from itself) or Joe is not a squid (from P2).

I've been thinking of ways of trying to convince you. I don't know whether it will work, but let me try something.

Let's consider the following two arguments:


Argument 1:

Premise 1: Either Meteoro is a car, or Meteoro is a truck.
Premise 2: Meteoro is not a truck.
Conclusion: Meteoro is a car.


Argument 2:

Premise 1: Chita is a chimpanzee.
Conclusion: Either Chita is a chimpanzee, or Chita is a dog.


What do you make of those arguments? In other words, would you say that they both are instances of proper reasoning? Or just one of them? (if so, which one?). Or none of them? Similarly, do you think they are valid M(both, one of them, or none), in the sense in which you understand the word "valid"?

Argument 1 is valid.
Argument 2 almost had me, but argument 2 is invalid.

That's too bad, I almost succeeded but no luck.
But why do you think that argument 2 is invalid?

Regardless, let us say it is invalid, in whatever sense you understand the word "invalid". I also want to know whether you think it is proper reasoning. After all, this sort of argument can never lead you from truth to falsehood, right? I mean, if it is true that Chita is a chimpanzee, it is also true that either Chita is a chimpanzee, or Chita is a dog (if that were false, then Chita would be neither a chimpanzee nor a dog). Or do you think it is possible that Chita is a chimpanzee, but false that (either Chita is a chimpanzee, or Chita is a dog)?

 

[Angra Mainyu]

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

EB

A Toy Windmill has already answered. Will you now answer his question about his argument?

No, he didn't. You guys have a habit of pretending you've answered when you haven't.

Well, I guess we get used to it.
EB

Now you accuse us of lying. While you are not lying because you believe what you say, you make false and disparaging claims, with reckless disregard from the truth. You even ignore my reply to this thread, and my arguments that utterly destroyed your position, showing its irrationality, in this thread among others.

 

[fast]

That's too bad, I almost succeeded but no luck.
But why do you think that argument 2 is invalid?

Well, it’s not a lost cause yet. If the presence of “Chita is a chimpanzee” being contained in the conclusion is sufficient, fine, but my hesitation isn’t a product of thinking it’s poor reasoning; my issue is that there is reasoning at all. See, even though the premises and conclusion both contain propositions, my idea of validity separates the premises from the conclusion such that the premises must contain enough information without extra reasoning to get to the destination (conclusion).

The conclusion (the cake) in this instance has ingredients (premises) that are not in the kitchen (argument).

Let’s imagine a classroom setting where there are two tables. Table one has a bunch of index cards, each containing a premise. On table two is a red card containing a conclusion.

P1) If Tom is a duck, then Tom quacks
P2) if Tom is a duck, then Tom doesn’t quack
P3) Tom is a duck
P4) Tom is not a duck.

Conclusion (in red) is on another table, and it reads, “Tom doesn’t quack.”

Is the argument valid? Yes, because there are enough cards on the table to get me to the conclusion. All I have to do is find out which ones that ARE PRESENT IN THE ARGUMENT that get me to the conclusion I want to be at. I can go over and pick up cards P2 and P3 and there I have the necessary ingredients to get me to precisely what the conclusion is.

In your argument, the conclusion might very well be true, but truth is a distraction. The issue is validity. There needs to be actual cards ON THE TABLE — and enough to pave a trail to the conclusion for it to be valid. The only premise you have is Chita is a chimpanzee. I have nothing else to work with (on the table) to get me to what you wrote as a conclusion.

I don’t mind reasoning through the premises I have (to work them out) and see if there is a way for the conclusion to follow, but if I inject reasoning to come up with what you did, I have to add premises—and that requires more cards, and if we’re fresh out of cards, well, hopefully you at least see where I’m coming from.

 

[Angra Mainyu]

Well, it’s not a lost cause yet. If the presence of “Chita is a chimpanzee” being contained in the conclusion is sufficient, fine, but my hesitation isn’t a product of thinking it’s poor reasoning; my issue is that there is reasoning at all. See, even though the premises and conclusion both contain propositions, my idea of validity separates the premises from the conclusion such that the premises must contain enough information without extra reasoning to get to the destination (conclusion).

I'm not sure I follow. The premises must contain information without extra reasoning to get to the conclusion? But then, how do you reason from your premises to the conclusion?

At any rate, I would say this: The premise "Chita is a chimpanzee" contains enough information to get "Either Chita is a chimpanzee, or the Moon is made of cheese". No extra information is required.

The conclusion (the cake) in this instance has ingredients (premises) that are not in the kitchen (argument).

But the conclusion does not have extra premises. There is no assumption in the conclusion. For this reason, you cannot get a false conclusion by adding a disjunction.

Still, let us say that valid arguments are like you say, so it is invalid to get:

Premise 1: P.
Conclusion: P or Q.

Even though invalid (not from my perspective, but let's grant it) wouldn't you say that this is proper reasoning? Maybe you say it's not reasoning at all. I don't know what else to say.

How about this: don't you think that deriving (P or Q) from P is a method that does not involve any faulty reasoning, or logic, or any fault whatsoever? Or do you think there is some sort of fault, epistemic impropriety, or whatever, in deriving (P or Q) from P?

I don’t mind reasoning through the premises I have (to work them out) and see if there is a way for the conclusion to follow, but if I inject reasoning to come up with what you did, I have to add premises—and that requires more cards, and if we’re fresh out of cards, well, hopefully you at least see where I’m coming from.

I'm afraid I do not , which is why I say I failed. I was close (I almost had you!), but no luck. I do not see myself as adding any further information (no premise at all, if you like) to get (P or Q) from P.

 

[Speakpigeon]

Hey, today, July 9th, at 5:51pm, walking back from borrowing music from my local library I understood why mathematicians won't likely ever get it right about logic (unless they change job).

Walking is probably the best way to understand things. Maybe it has to do with letting go. Letting your brain do the real stuff and then let you know the result.

Some people round here could give it a try. Just a thought. Still, that's one question solved, although, sorry, I ain't sharing that.
EB

 

[fromderinside]

Happy birthday to me. Is there anything less meaningful than a bloke in Paris demonstrating poverty by sharing he's picked up music at a public library and sharing his current local time in a post when his post in our local time is a bit different, perhaps a lot different. And I am sharing that.

Walking in Paris. Can't get out of my head the image of a little boy in a bow and jacket being walked to school by mummy and daddy at nine am in Macron's neighborhood.

This whole collage brings me around to thinking whether it's even relevant to be discussing inane aspects of philosophical argument technique.

My current POV,

 

[fast]

I'm not sure I follow. The premises must contain information without extra reasoning to get to the conclusion? But then, how do you reason from your premises to the conclusion?

At any rate, I would say this: The premise "Chita is a chimpanzee" contains enough information to get "Either Chita is a chimpanzee, or the Moon is made of cheese". No extra information is required.


But the conclusion does not have extra premises. There is no assumption in the conclusion. For this reason, you cannot get a false conclusion by adding a disjunction.

Still, let us say that valid arguments are like you say, so it is invalid to get:

Premise 1: P.
Conclusion: P or Q.

Even though invalid (not from my perspective, but let's grant it) wouldn't you say that this is proper reasoning? Maybe you say it's not reasoning at all. I don't know what else to say.

How about this: don't you think that deriving (P or Q) from P is a method that does not involve any faulty reasoning, or logic, or any fault whatsoever? Or do you think there is some sort of fault, epistemic impropriety, or whatever, in deriving (P or Q) from P?

I don’t mind reasoning through the premises I have (to work them out) and see if there is a way for the conclusion to follow, but if I inject reasoning to come up with what you did, I have to add premises—and that requires more cards, and if we’re fresh out of cards, well, hopefully you at least see where I’m coming from.

I'm afraid I do not , which is why I say I failed. I was close (I almost had you!), but no luck. I do not see myself as adding any further information (no premise at all, if you like) to get (P or Q) from P.

It’s like pouring some premises from one bucket into another bucket. Nothing can go in the second bucket unless it came from the first.

P1) P
P2) Q

From that, you can have valid conclusions so long as the ingredients are there:

C1) P
C2) Q
C3) P and Q
C4) P or Q
C5) Q but not P
C6) P but not Q
C7) not P and not Q
C8) not P or not Q

The thing is, there needs to be a P and Q to arrive at a conclusion with both in there.

Your argument:
P1) P
C) P or Q

It makes so much sense when you put it like that, and it does makes sense to me (intuitively true even) but if all I have in my bucket is a P, that’s the only thing that can be dumped out of it. No “or Q’s” are in the premises to find it’s way to the conclusion; that’s why it’s invalid. Not because the reasoning isn’t there for the grasping but because there’s no trace of it in the premises listed.

Add, if P, not Q, then I’m good. Heck, just let Q show up for the fun in the premises; he can be apart of the show

 

[fromderinside]

What if P is P but at a different time or place? Oh sure P'. Pete and Pete" were in a boat ..... is different from Pete and RePete were in a boat how? Don't you hate it when an adult messes up a nice third grader's ploy.

 

[Angra Mainyu]

Hey, today, July 9th, at 5:51pm, walking back from borrowing music from my local library I understood why mathematicians won't likely ever get it right about logic (unless they change job).

Walking is probably the best way to understand things. Maybe it has to do with letting go. Letting your brain do the real stuff and then let you know the result.

Some people round here could give it a try. Just a thought. Still, that's one question solved, although, sorry, I ain't sharing that.
EB

Actually, I do a lot of walking. It's a good way of working on difficult problems. But you also need to put your ill-placed contempt aside, and try to think.

 

[Angra Mainyu]

I'm not sure I follow. The premises must contain information without extra reasoning to get to the conclusion? But then, how do you reason from your premises to the conclusion?

At any rate, I would say this: The premise "Chita is a chimpanzee" contains enough information to get "Either Chita is a chimpanzee, or the Moon is made of cheese". No extra information is required.


But the conclusion does not have extra premises. There is no assumption in the conclusion. For this reason, you cannot get a false conclusion by adding a disjunction.

Still, let us say that valid arguments are like you say, so it is invalid to get:

Premise 1: P.
Conclusion: P or Q.

Even though invalid (not from my perspective, but let's grant it) wouldn't you say that this is proper reasoning? Maybe you say it's not reasoning at all. I don't know what else to say.

How about this: don't you think that deriving (P or Q) from P is a method that does not involve any faulty reasoning, or logic, or any fault whatsoever? Or do you think there is some sort of fault, epistemic impropriety, or whatever, in deriving (P or Q) from P?

I don’t mind reasoning through the premises I have (to work them out) and see if there is a way for the conclusion to follow, but if I inject reasoning to come up with what you did, I have to add premises—and that requires more cards, and if we’re fresh out of cards, well, hopefully you at least see where I’m coming from.

I'm afraid I do not , which is why I say I failed. I was close (I almost had you!), but no luck. I do not see myself as adding any further information (no premise at all, if you like) to get (P or Q) from P.

It’s like pouring some premises from one bucket into another bucket. Nothing can go in the second bucket unless it came from the first.

P1) P
P2) Q

From that, you can have valid conclusions so long as the ingredients are there:

C1) P
C2) Q
C3) P and Q
C4) P or Q
C5) Q but not P
C6) P but not Q
C7) not P and not Q
C8) not P or not Q

The thing is, there needs to be a P and Q to arrive at a conclusion with both in there.

Your argument:
P1) P
C) P or Q

It makes so much sense when you put it like that, and it does makes sense to me (intuitively true even) but if all I have in my bucket is a P, that’s the only thing that can be dumped out of it. No “or Q’s” are in the premises to find it’s way to the conclusion; that’s why it’s invalid. Not because the reasoning isn’t there for the grasping but because there’s no trace of it in the premises listed.

Add, if P, not Q, then I’m good. Heck, just let Q show up for the fun in the premises; he can be apart of the show

Sorry, it was crucial not to add it on the premises. With that, I have to give up, because if I can't get (P or Q) from P, no way I'm going to make the case that everything follows from a contradiction in a way you can find persuasive.

 

[A Toy Windmill]

I think there's something quite aberrant about deducing (P or Q) from P. You have P, why water things down by throwing in a random Q? It's an utterly absurd
thing to do, and not even those lying idiot mathematicians would engage in such behaviour.

But what if your friend Alice says: roll up! Roll up! If you have either a P or a Q, I have a million quid for you!

Now you're there with a P. And you say, hang on, I have P, which means I can fulfill Alice's requirements and get me my million quid. Those requirements say I need either a P or a Q, and indeed, I have a P.

I have P. Therefore, I have either P or Q. And so I'm destined to be a millionaire.

The lesson here is that the rules of logic are often less about the conclusions you can make, and more about the requirements you can fulfill. Always check which side of the turnstile you're standing.

 

[fast]

I accept that if P, then P or Q, but what I grasp is that the conclusion (P or Q) is true if the premise (P) is true, and while I have this foggy notion that it probably follows (is valid), what bothers me is that the reasoning feels hidden. It’s like a senior multi-variable calculus professor working out a problem on the board but redacting the parts he feels we should already know.

There’s probably some explicit argument somewhere that demonstrates what you’re espousing.

Either way, the fact P is contained in the conclusion will work for now.

Sooo:

P1. it’s the case that cats are birds
P2. it’s not the case that cats are birds
Therefore, C. Donkey’s are trees

Accepting that P or Q follows from P is one thing, but with that, you have a way to convince me (or pursuade me) that it follows that donkey’s are trees (just follows, not true, I get that) from the mere fact the argument that has that conclusion contains contradictory premises? Good luck with that

 

[A Toy Windmill]

Sooo:

P1. it’s the case that cats are birds
P2. it’s not the case that cats are birds
Therefore, C. Donkey’s are trees

Accepting that P or Q follows from P is one thing, but with that, you have a way to convince me (or pursuade me) that it follows that donkey’s are trees (just follows, not true, I get that) from the mere fact the argument that has that conclusion contains contradictory premises? Good luck with that

1) It's the case that cats are birds.
2) So it's the case that cats are birds or donkeys are trees (you accept this)
3) It's not the case that cats are birds.
4) So donkeys are trees. (it's the remaining possibility left in 2 given 3)

I'm mostly at a loss as to where you and Speakpigeon are unhappy, but can keep throwing out hypotheses: logic doesn't care about context. It wants universal rules that can be used in any situation, and demands that you accept all the consequences, no matter what other premises are around. So if you accept P entailing P or Q, and you accept disjunctive syllogism (what got me to 4), then you're forced to accept the above argument. The context of contradictory premises is irrelevant. Similarly, if Speakpigeon claims to accept weakening and modus tollens, they are forced to accept the argument I gave previously.

That's the theory, anyway. Speakpigeon is demonstrably free to say "nuh-uh."

 

[Angra Mainyu]

I accept that if P, then P or Q, but what I grasp is that the conclusion (P or Q) is true if the premise (P) is true, and while I have this foggy notion that it probably follows (is valid), what bothers me is that the reasoning feels hidden. It’s like a senior multi-variable calculus professor working out a problem on the board but redacting the parts he feels we should already know.

There’s probably some explicit argument somewhere that demonstrates what you’re espousing.

Either way, the fact P is contained in the conclusion will work for now.

Sooo:

P1. it’s the case that cats are birds
P2. it’s not the case that cats are birds
Therefore, C. Donkey’s are trees

Accepting that P or Q follows from P is one thing, but with that, you have a way to convince me (or pursuade me) that it follows that donkey’s are trees (just follows, not true, I get that) from the mere fact the argument that has that conclusion contains contradictory premises? Good luck with that

No problem!

Premise 1: P
Premise 2: ¬P.

From Premise 1, we get

Conclusion 1: (P v Q)

Now, from Premise 2 and Conclusion 1, we get

Conclusion 2: Q.

Note that you already accepted the fact that from (P v Q) and ¬P, we get Q, in this post (Argument 1).

Also note that Q is anything you like to say!

 

[Speakpigeon]

Similarly, if Speakpigeon claims to accept weakening and modus tollens, they are forced to accept the argument I gave previously.

That's the theory, anyway. Speakpigeon is demonstrably free to say "nuh-uh."

Nuh-uh.

You can't prove your theory using logic. I'm sure you understand that. So, please explain that to AM. Me, I have tired of it.
EB

 

[fast]

Trickery!

And to think I almost bought it!

I accepted it as part of a conclusion—not a premise.

Cats meow
They don’t
So, you’re both sentenced to a long ass marriage with a woman you despise!!

Logic!!

But yeah, I’m starting to catch on.

 

[Angra Mainyu]

Trickery!

And to think I almost bought it!

I accepted it as part of a conclusion—not a premise.

Cats meow
They don’t
So, you’re both sentenced to a long ass marriage with a woman you despise!!

Logic!!

But yeah, I’m starting to catch on.

Lol!
But you are starting to catch on, so I guess you do get that if you get C1 as a conclusion from some premises P1, ..., Pn (n is a positive integer), and you also get C2 from (C1,P1,..., Pn), then that implies that you get C2 from P1, ..., Pn, right?

 

[fast]

I’m catching on to what’s being done. There just seems to be a mismatch between what’s being done and what’s being said.

You had to utilize the conclusion in my argument and incorporate it as a premise before turning around to arrive at the conclusion, which means a premise was added in order to show validity. Strangly enough, it doesn’t matter what my conclusion is, as you can see beforehand that ultimately the OLD argument CAN BE shown to be valid —BUT ONLY IF you razzle dazzle my argument with premises never given; that’s why you can’t tell me specifically what the unstated conclusion is to an argument with an undisclosed conclusion.

However, that it’s a two-step process isn’t a major concern for me. No wonder you refused to accept the hidden “or Q” in the argument:

P: P
C: P or Q

I’m a rat
Or, something else
Therefore, I’m a rat or something else

I’m not a rat
Therefore, I’m something else

That’s apparently how it’s going down.

Cool!