4 very easy arguments. Are they valid?

[ruby sparks]

Sorry, I'll rewrite that, preserving the OP labels where appropriate.

A5:

P1'": Joe is not an elephant
P2": Joe is an elephant
C'": Joe is an elephant and Joe is not an elephant

(My P1'" seems to be a summary of P1, P1' & P1" in A4?)

 

[Speakpigeon]

also, here you can get some basic info on the technical vs. non-technical distinction.

What Is Validity?
In everyday speech, the word “valid” is often used to mean “good”, “reasonable” or “correct”. Throw all that away.
In philosophy, “valid” is a technical term with a very specific meaning. And it doesn’t mean “good”, “reasonable” or “correct”.
In philosophy, “validity” is a property of (...)
https://systematicphilosophy.com/2011/05/20/what-is-validity/

Thanks for the link!

And I do agree with what the "experts" say here. Validity in the "technical" sense is definitely not reasonable and not correct.

Me, I prefer the reasonable and the correct sense of validity.
EB

 

[Treedbear]

...
And I do agree with what the "experts" say here. Validity in the "technical" sense is definitely not "reasonable" and not "correct".
...

FIFU

 

[fast]

Sorry, I'll rewrite that, preserving the OP labels where appropriate.

A5:

P1'": Joe is not an elephant
P2": Joe is an elephant
C'": Joe is an elephant and Joe is not an elephant

(My P1'" seems to be a summary of P1, P1' & P1" in A4?)

The only issue I have is with the “and” conjunction within the conclusion when it comes to validity.

Let Joe is an elephant be Y
Let Joe is not an elephant be N

C thus is: Y and N
There is a P that leads to Y
There is a P that leads to N

Therefore, we can draw two conclusions:
C1: Y
C2: N

Independently, there’s no issue.

There is more to Y and N when “AND” is added. Subtle, but there. Each thought is sensible when considered independent, but the coupling is more than two distinct thoughts: That something is the case AND not the case is contradictory when pitted up against each other, so C: Y;N taken as separate are validly derived through form. That “and” however, if scrutinized leads me to way overthink things that likely takes me beyond the scope of the intended convention.

The flow gives us “Y” and “N”, so valid, (p2 implies Y)(P1 implies N)
But, how do we get to “Y and N” as a single thought.

It’s not like there’s an “and” between premises that stands as the foundation for the structural integrity that gives rise to valid structural form.

P1: If the clock strikes two, it’s after one
P2: the clock strikes two.
Therefore C: it’s after one
What makes this argument valid besides having nothing to do with true premises is not because of some “and” between the premises.
I could have said, C: if the clock strikes two, it’s after one AND the clock strikes two. That’s true, but it’s not a function of form. There’s something deeper between the relationship between the premises that makes the argument valid, not the “and” conjunction.

So, am I overthinking things? Maybe, but what makes the conclusion valid is the fact two separate parts can be derived through proper form—not the conjunction.

Take this for instance:
P1: Joe is not an elephant
P2: joe is white
P3: Joe is not white
P4: Joe is an elephant
Therefore, C: Joe is not an elephant

That’s valid

 

[fromderinside]

AND is NAND '0'

 

[Angra Mainyu]

I get that A1, A2 & A3 are valid.

Okay.

So, it is the case that an argument (in this case A4) can still be valid even if the conclusion is a contradiction of itself, is that it?

Yes, but if you do not know that, it is not clear to me that you get that A1, A2 and A3 are valid, because it is not clear to me that you understand the meaning of "valid", at least in this context.
An easy valid argument is your example from this post.

If yes, I'm bound to ask, what is the point? I don't mean that rudely. I mean, so what if it's valid?

In the other thread, Speakpigeon made the claim that a conclusion and its negation cannot both follow, and she has not conceded that her claim was false. Several people have been confused in the other thread, perhaps as a result of her claim, or perhaps on their own, or a combination. I am trying to clear the matter a little bit, at least hopefully for some of the readers.

Or to put it another way, and using layman's language, if you fed this into a 'logic machine', and the machine was set up to detect logical flaws, and setting aside the issue of truth, a red button would flash on the machine to indicate that something was wrong with the argument, even if it was valid.

I do not think so, but I would need more information about what the logic machine does.

Could I even use a layman's term and say the argument is nonsensical (even if valid)?

You could, but I would disagree. It is not nonsensical. If it were, one would not understand it at all.
Incidentally, it is very common in mathematics to argue in a similar way: assume P is false, and then from there and a number of other hypotheses, derive a contradiction. Hence, P is true (well, under the other hypotheses, but some of them are often implicit). There are some debates among mathematicians regarding the scope in which that is doable; for example, intuitionists would not allow that kind of argument to conclude that P is true, but they would have no problem to use that kind of argument to derive ¬¬P, so the idea of deriving contradictions from some premises is still useful under intuitionism (even if to a lesser extent).

Also, quick query. Does P2’’ not contradict P1/P1'/P1" (as a set of premises)? I mean, can you even have (or use, or introduce) P2" along with (after having used) P1, P1' & P1" without invoking a contradiction in your set of premises? If that makes technical sense, which it may not, and if there is such a thing as contradictory premises (which it seems to me there should and can be).

I'm not sure what it means to "invoke a contradiction" in the set of premises, but yes, the set of premises is an inconsistent set of statements (equivalently, they can't all be true; equivalently, they entail a contradiction; equivalently, the conjunction of the premises is a contradiction).

In other words, if (emphasis if, because logic is not my area of any expertise) there is a contradiction there also, again, would a red button on the hypothetical logic machine not start to flash even before you finished inputting, ie before you even got to the (separately, of itself) contradictory conclusion C'"?

Whether the machine will flash depends on what it does. But having contradictory premises is not a reason to reject an argument. If it were, then a lot of math papers would have to be rejected. But despite Speakpigeon's attacks against mathematicians and mathematics, the fact is that there is nothing wrong with those papers, which have furthered knowledge in many areas of mathematics - and some of those have practical applications.

 

[Torin]

Whether the machine will flash depends on what it does. But having contradictory premises is not a reason to reject an argument. If it were, then a lot of math papers would have to be rejected. But despite Speakpigeon's attacks against mathematicians and mathematics, the fact is that there is nothing wrong with those papers, which have furthered knowledge in many areas of mathematics - and some of those have practical applications.

Logic simply abstracts from the method of thought a person uses (e.g., modus ponens is one method of thinking). Nothing, therefore, prevents a logic that abstracts from perverse methods of thought such as asserting a contradiction. These logics might even be "useful" in some situations. They just aren't based on reality or reason.

 

[Angra Mainyu]

Whether the machine will flash depends on what it does. But having contradictory premises is not a reason to reject an argument. If it were, then a lot of math papers would have to be rejected. But despite Speakpigeon's attacks against mathematicians and mathematics, the fact is that there is nothing wrong with those papers, which have furthered knowledge in many areas of mathematics - and some of those have practical applications.

Logic simply abstracts from the method of thought a person uses (e.g., modus ponens is one method of thinking). Nothing, therefore, prevents a logic that abstracts from perverse methods of thought such as asserting a contradiction. These logics might even be "useful" in some situations. They just aren't based on reality or reason.

The methods used in modern logic are based on reason, and they often are applicable to reality. Mathematics is full of arguments that, from some hypotheses, reach a contradiction. That is useful to disprove hypotheses.

 

[ruby sparks]

Yes, but if you do not know that, it is not clear to me that you get that A1, A2 and A3 are valid, because it is not clear to me that you understand the meaning of "valid", at least in this context.
An easy valid argument is your example from this post.

I might be wrong, but I think I understand what valid is. As I understand it, the conclusion must follow from the premises. Which as you say it does in both your A4 and my A5.

In the other thread, Speakpigeon made the claim that a conclusion and its negation cannot both follow, and she has not conceded that her claim was false. Several people have been confused in the other thread, perhaps as a result of her claim, or perhaps on their own, or a combination. I am trying to clear the matter a little bit, at least hopefully for some of the readers.

As I am starting to understand it now (thanks to this thread and your contributions, and those of some others) a conclusion and it's negation can follow, so long as both are in the premises.

I do not think so, but I would need more information about what the logic machine does.

Even without details, I do not see how you could not think so. Bear in mind that I am not saying that the machine is alerting us to invalidity.

ETA: For a better (I hope) answer, see my last point below. In short, I can see how do do not think so.

You could, but I would disagree. It is not nonsensical. If it were, one would not understand it at all.

That's true. The word nonsensical was the wrong word.

Incidentally, it is very common in mathematics to argue in a similar way: assume P is false, and then from there and a number of other hypotheses, derive a contradiction. Hence, P is true (well, under the other hypotheses, but some of them are often implicit). There are some debates among mathematicians regarding the scope in which that is doable; for example, intuitionists would not allow that kind of argument to conclude that P is true, but they would have no problem to use that kind of argument to derive ¬¬P, so the idea of deriving contradictions from some premises is still useful under intuitionism (even if to a lesser extent).

And that then, would seem to be an example of what I allowed for, that such things (valid arguments or processes which lead to or involve contradictions) might nonetheless be useful in some ways, probably ways that I don't understand.

I'm not sure what it means to "invoke a contradiction" in the set of premises, but yes, the set of premises is an inconsistent set of statements (equivalently, they can't all be true; equivalently, they entail a contradiction; equivalently, the conjunction of the premises is a contradiction).

My termonology might be wrong, but what I'm saying is that for instance, in my argument A5, the 'same issue or feature' which is in the conclusion (which I think we agree is a contradiction) is already present in the two preceding premises. In other words, the two premises seem to contradict each other in the same way the conclusion does.

Whether the machine will flash depends on what it does. But having contradictory premises is not a reason to reject an argument. If it were, then a lot of math papers would have to be rejected. But despite Speakpigeon's attacks against mathematicians and mathematics, the fact is that there is nothing wrong with those papers, which have furthered knowledge in many areas of mathematics - and some of those have practical applications.

Sure. See what I said above and in my own previous post. I am not surprised to hear that contradictions can be useful, albeit I am not familiar with the ways and might not even understand them.

Perhaps you might still wonder why I called a contradiction a 'wrong'. Is a contradiction a wrong? Perhaps it isn't, in logic or mathematics. Perhaps wrong is the wrong word. Or is it just a logically incompatible word? I am probably using the word 'wrong' wrongly, in that there is nothing wrong in mathematical or logical terms. I think I'm right in saying that contradiction might mean something isn't true, but of course we are only doing validity, not truth.

Perhaps my hypothetical logic machine should have not just one beeping light but a row of them. One for 'invalidity', one for 'contradictory' and so on. In the case of A4 & A5 only the latter would beep/flash, and there would be no need (and no reason) to reject the conclusion on grounds of validity, or indeed reject it at all, unless one was specifically seeking an argument or a conclusion that did not involve contradiction.

 

[ruby sparks]

Sorry, I'll rewrite that, preserving the OP labels where appropriate.

A5:

P1'": Joe is not an elephant
P2": Joe is an elephant
C'": Joe is an elephant and Joe is not an elephant

(My P1'" seems to be a summary of P1, P1' & P1" in A4?)

The only issue I have is with the “and” conjunction within the conclusion when it comes to validity.

Let Joe is an elephant be Y
Let Joe is not an elephant be N

C thus is: Y and N
There is a P that leads to Y
There is a P that leads to N

Therefore, we can draw two conclusions:
C1: Y
C2: N

Independently, there’s no issue.

There is more to Y and N when “AND” is added. Subtle, but there. Each thought is sensible when considered independent, but the coupling is more than two distinct thoughts: That something is the case AND not the case is contradictory when pitted up against each other, so C: Y;N taken as separate are validly derived through form. That “and” however, if scrutinized leads me to way overthink things that likely takes me beyond the scope of the intended convention.

The flow gives us “Y” and “N”, so valid, (p2 implies Y)(P1 implies N)
But, how do we get to “Y and N” as a single thought.

It’s not like there’s an “and” between premises that stands as the foundation for the structural integrity that gives rise to valid structural form.

P1: If the clock strikes two, it’s after one
P2: the clock strikes two.
Therefore C: it’s after one
What makes this argument valid besides having nothing to do with true premises is not because of some “and” between the premises.
I could have said, C: if the clock strikes two, it’s after one AND the clock strikes two. That’s true, but it’s not a function of form. There’s something deeper between the relationship between the premises that makes the argument valid, not the “and” conjunction.

So, am I overthinking things? Maybe, but what makes the conclusion valid is the fact two separate parts can be derived through proper form—not the conjunction.

I do not know if you are overthinking. We might even have to define the word.

But I think I agree with you.

Though as I see it, if both premises are used (in A5) there is, it seems, an implicit 'and' after the first one and before the second one. But I could be wrong, allowing that I'm now not sure about using that word. Lol.

Take this for instance:
P1: Joe is not an elephant
P2: joe is white
P3: Joe is not white
P4: Joe is an elephant
Therefore, C: Joe is not an elephant

That’s valid

Looks valid to me (now that I think I'm learning something).

 

[ruby sparks]

Picking up on (or perhaps merely rehashing) fast's last example, how about this?

A6

P1'": Joe is not an elephant
P2": Joe is an elephant
C'"': Joe is an elephant

A7

P1'": Joe is not an elephant
P2": Joe is an elephant
C'"'': Joe is not an elephant

Both valid? My (amateur) guess is....yes. Even though (a) in both cases the conclusion is only one of two things that must follow and (b) in both cases the conclusion is also a premise.

In other words, I'm thinking fast's last example (just above) would also have been valid if the conclusion had been any of the premises.

If (if) so, I'm curious to know what the correct term would be to describe something which is valid but which is 'incomplete' (layman's term again) if the conclusion is not the only conclusion that must follow from the premises? In other words if a certain premise (or premises) is/are not taken into account, not used, or ignored.

I'm thinking that if you were sitting a hypothetical logic exam and the question was 'what conclusion follows from the premises in argument A5 (or A4)' you might not get full marks if you only gave either of the answers in A6 or A7 above. Possibly in some ways, though not all, similar to how you would not get full marks in a GCSE Maths exam by giving 7 as the answer to the question, 'what is the square root of 49?' (Though we could be pedantic there and say the latter question, and possibly also the former one, was/were worded misleadingly or ambiguously in implying that there is 'a' or only one, answer).

 

[fast]

If the animal walks, it’s an elephant
The animal walks
The animal doesn’t walk
Therefore, it’s an elephant

The first two has skin in the game for arriving at the conclusion.

If the animal walks, it’s an elephant
The animal walks
The animal doesn’t walk
Therefore, its not an elephant

That’s a mess!!

On the one hand, it might be valid yet with a false conclusion, but then again, it might not be valid at all if P2 has no skin in the game:

If the animal walks, it’s an elephant
The animal doesn’t walk
Therefore, it’s an elephant

It could very well be that if ithe animal walks that it’s an elephant, but if theres an animal that doesn’t walk, it could still be an elephant—the poor thing

 

[fast]

If the animal walks, it’s an elephant
The animal walks
The animal does not walk
Therefore, it’s an elephant

That’s valid (1 and 2 entails C)

If the animal walks, it’s an elephant
The animals walks
The animal does not walk
Therefore, it’s not an elephant

1 and 3 does not infer C, so is it a valid argument with a false conclusion or an invalid argument?

See, there could be an elephant that doesn’t walk.

 

[OLDMAN]

A 3 is were it goes off the tracks.

 

[fast]

By the way, is a sentence a premise merely because we stipulate it as such? A sentence that is purportedly a premise is brought into question if has no attachment to the inference leading to a conclusion.

 

[Angra Mainyu]

I might be wrong, but I think I understand what valid is. As I understand it, the conclusion must follow from the premises. Which as you say it does in both your A4 and my A5.

Okay, you got it.

As I am starting to understand it now (thanks to this thread and your contributions, and those of some others) a conclusion and it's negation can follow, so long as both are in the premises.

Yes, and I would say - following nearly every logician, mathematician or philosopher - that even if they are not in the premises. For example, "Joe is an elephant" and its negation "It is not the case that Joe is an elephant" both follow from the premises of A4, but only one of them is one of the premises. In fact, even if neither of them were in the premises, both a conclusion and its negation could follow. For example, we can modify A4 as follows:

A8:

P1: Joe is either a squid or a giraffe.
P1''': Joe is either an elephant or a squid.
P1': A giraffe is not an elephant.
P2''': Joe is not a squid.
C'''': Joe is not an elephant, and Joe is an elephant.

In this case, we use P1''' and P2''' to derive "Joe is an elephant", and from P2''' and P1 we get "Joe is a giraffe", which together with P1' entail "Joe is not an elephant".
So, both "Joe is an elephant" and its negation follow from the premises of A6, even though neither statement is one of the premises.

My termonology might be wrong, but what I'm saying is that for instance, in my argument A5, the 'same issue or feature' which is in the conclusion (which I think we agree is a contradiction) is already present in the two preceding premises. In other words, the two premises seem to contradict each other in the same way the conclusion does.

Sure, though the full set of premises is inconsistent. It might be that any proper subset is consistent. In particular, there need not be two premises that contradict each other (e.g., ee A8 above, or even A4).

Picking up on (or perhaps merely rehashing) fast's last example, how about this?

A6

P1'": Joe is not an elephant
P2": Joe is an elephant
C'"': Joe is an elephant

A7

P1'": Joe is not an elephant
P2": Joe is an elephant
C'"'': Joe is not an elephant

Both valid? My (amateur) guess is....yes. Even though (a) in both cases the conclusion is only one of two things that must follow and (b) in both cases the conclusion is also a premise.

In other words, I'm thinking fast's last example (just above) would also have been valid if the conclusion had been any of the premises.

That is correct.

If (if) so, I'm curious to know what the correct term would be to describe something which is valid but which is 'incomplete' (layman's term again) if the conclusion is not the only conclusion that must follow from the premises? In other words if a certain premise (or premises) is/are not taken into account, not used, or ignored.

Those are two different things. That the (stated) conclusion isn't the only thing that follows from the premises is also a property of A4, A8, and a lot of other arguments that need all of the premises to derive the stated conclusion. I don't think there is a special term for that, other than saying that one or more of the premises are unnecessary, superfluous, etc.

I'm thinking that if you were sitting a hypothetical logic exam and the question was 'what conclusion follows from the premises in argument A5 (or A4)' you might not get full marks if you only gave either of the answers in A6 or A7 above.

Actually, the correct answer would be (in most such exams) that every statement follows from the premises of A4 (or A5, or A8). That is because every statement follows from a contradiction. That is the principle of explosion, which is accepted by nearly everyone. There are some philosophers who oppose it, though, and paraconsistent logics do not have it. But that is another matter, central in the other thread but not related to whether a conclusion and its negation can both follow, which as far as I know everyone accepts (well, everyone in the fields of philosophy, math and logic).

- - - Updated - - -

A 3 is were it goes off the tracks.

Why do you think that?
A3 is valid and nothing goes off the tracks, but I can't address your objection if I do not know why you object.

 

[Angra Mainyu]

If the animal walks, it’s an elephant
The animal walks
The animal doesn’t walk
Therefore, it’s an elephant

The first two has skin in the game for arriving at the conclusion.

That's valid. Whether a premise has skin in the game or not is not relevant when it comes to validity.

If the animal walks, it’s an elephant
The animal walks
The animal doesn’t walk
Therefore, its not an elephant

That’s a mess!!

That is valid, but for a very different reason, which I wanted to leave for the other thread so that in this one, I could focus on the fact that a conclusion and its negation can both follow. The reason the argument is valid is that from the second and third premise one can derive a contradiction, and everything follows from a contradiction (principle of explosion), though some philosophers (in my experience, very few) reject that (they do not reject the fact that a conclusion and its negation can both follow, though).

On the one hand, it might be valid yet with a false conclusion, but then again, it might not be valid at all if P2 has no skin in the game:

If the animal walks, it’s an elephant
The animal doesn’t walk
Therefore, it’s an elephant

It could very well be that if ithe animal walks that it’s an elephant, but if theres an animal that doesn’t walk, it could still be an elephant—the poor thing

That is invalid. But the matter is not about whether the premise has skin in the game.

If the animal walks, it’s an elephant
The animals walks
The animal does not walk
Therefore, it’s not an elephant

1 and 3 does not infer C, so is it a valid argument with a false conclusion or an invalid argument?

See, there could be an elephant that doesn’t walk.

1 and 3 do not imply C, but 2 and 3 imply everything (including C and its negation). So, it is valid. However, that is not what I wanted to focus on this thread, because I wanted to separate the matter of explosion from that of whether a conclusion and its negation can both follow.

By the way, is a sentence a premise merely because we stipulate it as such? A sentence that is purportedly a premise is brought into question if has no attachment to the inference leading to a conclusion.

Sure, we freely choose our premises.

 

[Torin]

The methods used in modern logic are based on reason, and they often are applicable to reality. Mathematics is full of arguments that, from some hypotheses, reach a contradiction. That is useful to disprove hypotheses.

If you're saying that there are true contradictions (which is how I read your post), then your ideas have nothing to do with reason or reality. If that's not what you're saying then okay.

 

[Angra Mainyu]

The methods used in modern logic are based on reason, and they often are applicable to reality. Mathematics is full of arguments that, from some hypotheses, reach a contradiction. That is useful to disprove hypotheses.

If you're saying that there are true contradictions (which is how I read your post), then your ideas have nothing to do with reason or reality. If that's not what you're saying then okay.

No, I'm not saying any of the sort. Contradictions are of course false. Reaching a contradiction with a valid argument implies at least one of the premises is false. When all of them but one are known to be true, from that one gets that the other premise is false. This is a common method for providing things in math (assume X, reach a contradiction from that and known statements, then conclude ¬X).

 

[Speakpigeon]

No, I'm not saying any of the sort. Contradictions are of course false. Reaching a contradiction with a valid argument implies at least one of the premises is false. When all of them but one are known to be true, from that one gets that the other premise is false. This is a common method for providing things in math (assume X, reach a contradiction from that and known statements, then conclude ¬X).

Good, excellent, repair has been completed if ever any damage was done.

So, I won't post anything else in this very interesting thread either.
EB