# Thread: 4 very easy arguments. Are they valid?

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).

2. 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

3. 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.

4. A 3 is were it goes off the tracks.

5. 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.

6. Originally Posted by ruby sparks
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.

Originally Posted by ruby sparks
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.

Originally Posted by ruby sparks
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).

Originally Posted by ruby sparks

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.

Originally Posted by ruby sparks
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.

Originally Posted by ruby sparks
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 - - -

Originally Posted by OLDMAN
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.

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

Originally Posted by fast
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).

Originally Posted by fast
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.

Originally Posted by fast

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.

Originally Posted by 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.
Sure, we freely choose our premises.

8. Originally Posted by 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.

9. Originally Posted by Torin
Originally Posted by 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).

10. Originally Posted by Angra Mainyu
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

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