Justification of the mathematical definition of logical validity?

[Speakpigeon]

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

Is that valid?

No. Ask anyone not trained in mathematical logic and they will just say the argument is meaningless.
EB

 

[Speakpigeon]

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).

Yes, I couldn't care less. It's a derail. You should try instead to come up with a proper justification that the definition of validity you use is somehow correct.

But, I guess you can't produce that which doesn't exist. Even with faulty logic.
EB

 

[Speakpigeon]

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.

Yeah, so different mathematical logic notions of validity inevitably contradict each other.
EB

 

[Angra Mainyu]

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).

Yes, I couldn't care less. It's a derail. You should try instead to come up with a proper justification that the definition of validity you use is somehow correct.

But, I guess you can't produce that which doesn't exist. Even with faulty logic.
EB

First, it is not a derail. It addresses your claim that because the vast majority of people without training in logic say it's invalid, they're right and mathematicians are wrong. There are plenty of examples in which human untrained intuitions go wrong, and expertise helps.

Second, I already provided a justification. Now, that justification is under the hypothesis that all mathematical statements are either true or false. Many mathematicians disagree, and without that hypothesis, I could have a very different conversation with them. However, given that you said when I asked that you believe all statements are either true or false (which implies that all mathematical statements are either true or false), then my rationale holds. If you want to challenge it, go for it. But you already tried in other thread, with no success (even if you will never realize that).

 

[fast]

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

Is that valid?

No. Ask anyone not trained in mathematical logic and they will just say the argument is meaningless.
EB

I got that. I just don’t know what kind of yes it was that he gave. Is it the kind of yes meaning no unless I accepted his additional premises or was it the kind of yes which remains yes where the additional premises explains why it’s yes?

 

[Angra Mainyu]

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

Is that valid?

No. Ask anyone not trained in mathematical logic and they will just say the argument is meaningless.
EB

I got that. I just don’t know what kind of yes it was that he gave. Is it the kind of yes meaning no unless I accepted his additional premises or was it the kind of yes which remains yes where the additional premises explains why it’s yes?

Before I got any training in mathematical logic, I would have said readily that the argument was valid, or more precisely, probably I would not have used the word "valid" (or synonyms), but I would have said the conclusion followed from the premises.
Then again, I did have the basic logic course (not mathematical logic, though) I took in high school. Before that, who knows? It's sort of ancient. Maybe it would have taken me a while to figure it out, though I think I would have probably figured it out. But if not, well, that only is another example of someone without training failing to figure that out.

 

[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?

 

[fast]

I got that. I just don’t know what kind of yes it was that he gave. Is it the kind of yes meaning no unless I accepted his additional premises or was it the kind of yes which remains yes where the additional premises explains why it’s yes?

Before I got any training in mathematical logic, I would have said readily that the argument was valid, or more precisely, I would not have used the word "valid", but I would have said the conclusion followed from the premises.

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).

 

[Angra Mainyu]

I got that. I just don’t know what kind of yes it was that he gave. Is it the kind of yes meaning no unless I accepted his additional premises or was it the kind of yes which remains yes where the additional premises explains why it’s yes?

Before I got any training in mathematical logic, I would have said readily that the argument was valid, or more precisely, I would not have used the word "valid", but I would have said the conclusion followed from the premises.

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).

You don't know that Joe is a squid. It's a premise, sure, but if you were to know the premises, you would have to say that you also know that Joe is not a squid, and that is not possible. But let's simplify a little. Would you say the following does follow?

Joe is a squid
Joe is not a squid
Therefore, (Joe is a squid and Joe is not a squid).

 

[Angra Mainyu]

I got that. I just don’t know what kind of yes it was that he gave. Is it the kind of yes meaning no unless I accepted his additional premises or was it the kind of yes which remains yes where the additional premises explains why it’s yes?

Before I got any training in mathematical logic, I would have said readily that the argument was valid, or more precisely, I would not have used the word "valid", but I would have said the conclusion followed from the premises.

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"?

 

[fast]

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).

You don't know that Joe is a squid. It's a premise, sure, but if you were to know the premises, you would have to say that you also know that Joe is not a squid, and that is not possible. But let's simplify a little. Would you say the following does follow?

Joe is a squid
Joe is not a squid
Therefore, (Joe is a squid and Joe is not a squid).

Sorry about the “know” part.

Yes, the argument is valid.

The following three arguments are valid:

Argument #1:
Joe is a squid
Joe is not a squid
Therefore, Joe is a squid


Argument #2:
Joe is a squid
Joe is not a squid
Therefore, Joe is not a squid


Argument #3:
Joe is a squid
Joe is not a squid
Therefore, Joe is a squid and Joe is not a squid

The reason argument number 1 is valid is because there are enough premises in the argument to reach the conclusion. Premise 1 is sufficient to do that. Same reasoning for argument 2 except it’s the second premise that works.

The third argument requires both, and together, the conclusion is reached.

 

[fast]

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.

 

[fast]

I’m going to go out on a limb and say argument 2 has proper reasoning but nevertheless invalid. The conclusion may be true, but that doesn’t make THAT argument valid. It takes a different hidden argument to arrive at the conclusion.

 

[A Toy Windmill]

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

My apologies. But I'm mostly avoiding absolute statements on this.

If you accept weakening and modus tollens, then contradictions imply everything. If you don't (and I personally think there are good reasons to reject weakening) then contradictions may not imply everything.

 

[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.

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.

 

[fast]

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

My apologies. But I'm mostly avoiding absolute statements on this.

If you accept weakening and modus tollens, then contradictions imply everything. If you don't (and I personally think there are good reasons to reject weakening) then contradictions may not imply everything.

Oh, okay! That’s helpful. I’ve not developed a good intuitive sense of modus tollens. Maybe that’s where I should focus on ... and perhaps in another thread with basics.

 

[Speakpigeon]

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

 

[Speakpigeon]

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).

That's it in a nutshell.

Thanks for demonstrating that people untrained in logic are perfectly capable of articulating what validity means to them.

And, this is exactly Aristotelian logic. All good!
EB

 

[Speakpigeon]

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

My apologies. But I'm mostly avoiding absolute statements on this.

If you accept weakening and modus tollens, then contradictions imply everything. If you don't (and I personally think there are good reasons to reject weakening) then contradictions may not imply everything.

I accept Modus Tollens and weakening. But, no, that doesn't imply that contradiction implies everything.
EB

 

[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