One of the main takeaways from this discussion, for me, is that while some formal arguments may be valid, they are not necessarily valid in an informal setting.

To wit,

B
Therefore A→B
Formally valid.

Water was added to the lake.
Therefore,
If it is cloudy out, then water was added to the lake.
Informally not valid.

as well as -

A ^ B
Therefore, (A→B).
Formally valid.

Kangaroos are marsupials and Paris is the capital of France.
Therefore,
If kangaroos are marsupials, then Paris is the capital of France.
Informally not valid.

Hawt damn. Looks like it was worth all the posts after all :D

The "following" of a rule versus it's being merely "present" can be illustrated by the following example:
A->B
B^C
Therefore, C.
In this example, the rule A-> B does not do any work — NotAristotle

This issue is directly parallel to the earlier discussion about the nature of validity:

A: "If the premises are inconsistent then the argument is valid."
B: "Validity has to do with the conclusion following from the premises, and inconsistency is not evidence that the conclusion follows from the premises."

If one cannot recognize that something can follow from something else in different ways, then they will not be able to recognize the difference between a material conditional and a disjunction; and this is similar to the way that if one cannot recognize the essence of validity, then they will not be able to exclude or even recognize degenerative cases.

"Validity has to do with the conclusion following from the premises, and inconsistency is not evidence that the conclusion follows from the premises." — Leontiskos

That ((P→Q)∧Q), therefore P is not valid, whereas ((A∧¬A)∧(P→Q)∧Q), therefore P is valid, does seem strange to me. Inconsistent premises don't seem to have anything to do with whether the argument "follows." Although I have a feeling that Tones will have something to say about that.

That ((P→Q)∧Q), therefore P is not valid, whereas ((A∧¬A)∧(P→Q)∧Q), therefore P is valid, does seem strange to me. Inconsistent premises don't seem to have anything to do with whether the argument "follows." Although I have a feeling that Tones will have something to say about that. — NotAristotle

I am a man and I am not a man. Therefore I am rich.

The argument is valid; the conclusion follows from the premise. We can show this in four parts:

1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.

- The difference between an argument from the definition of validity and an argument from explosion has been explained multiple times throughout this thread. Tones himself recognized it. You continue to conflate the two.

("I can give a valid argument moving from A to B," is not the same as, "The argument that was given is valid.")

I don't know what you're talking about.

It seems that that argument would be valid, but only if one accepts that an argument is valid iff there is no interpretation s.t. all premises are true and the conclusion is false per Tones' definition.

If it turned out that validity required more than what that definition suggests (I think it does), then the argument you stated may well turn out to not be valid, as I think is the case.

Maybe another way of coming at this is as follows - the conclusion is true. Period. Under that understanding, "there is no interpretation where the conclusion is false" ergo there is no interpretation s.t. all the premises are true and the conclusion is false. But the conclusion being true does not seem to guarantee that the argument is valid. But with Tones' definition, it would. Similarly, inconsistent premises also guarantee the validity of the argument according to Tones' definition, but that also seems problematic.

It seems that that argument would be valid, but only if one accepts that an argument is valid iff there is no interpretation s.t. all premises are true and the conclusion is false per Tones' definition.

If it turned out that validity required more than what that definition suggests (I think it does), then the argument you stated may well turn out to not be valid, as I think is the case. — NotAristotle

You seem to be putting the cart before the horse.

It's not the case that the word "valid" means something and then we try to give a proper description of this meaning, and that we disagree on the proposed definition.

Rather, a bunch of logicians got in a room together and decided that if an argument's conclusion follows from its premises using the rules of inference then they will name this type of argument "valid". And that if the premises are also in fact true then they will name this type of argument "sound".

Besides, if someone gave the argument you gave -- "I am a man and I am not a man. Therefore I am rich" that is a nonsensical argument; the conclusion just has nothing to do with the premises, you might as well argue "I am a human and it might snow this week, therefore I live in Antartica." Even if conclusion and premise are all true i.e. the argument is sound, what kind of argument is that?

you might as well argue "I am a human and it might snow this week, therefore I live in Antartica." — NotAristotle

But that argument isn't valid.

Even if conclusion and premise are all true i.e. the argument is sound, — NotAristotle

That is not what "sound" means.

Your argument is that: If logicians have defined validity, then that definition is correct. Logicians have defined validity. Therefore, that definition is correct. This is a valid argument as far as I can tell. It is, however, unsound, as premise 1 is faulty.

If I did live in Antartica it would have to be valid wouldn't it?

as premise 1 is faulty — NotAristotle

It's not. You might as well claim that mathematicians are wrong to define the "=" sign as meaning "is equal to".

If I did live in Antartica it would have to be valid wouldn't it? — NotAristotle

No.

Why not? It satisfies the definition, does it not?

Why not? It satisfies the definition, does it not? — NotAristotle

No, it doesn't.

This is a valid argument:

P1. If I am a man then I am mortal
P2. I am a man
C1. Therefore, I am mortal

This is an invalid argument:

P1. If I am a man then I am mortal
P2. I am a man
C1. Therefore, I am English

Both premises and both conclusions are true, but the second conclusion doesn't follow from the premises (whereas the first conclusion does).

1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true. — Michael

I am a man and I am not a man.
Therefore, I am rich.

These are two different arguments, and the validity of the first does not ensure the validity of the second.

It's one argument:

P1. "I am a man and I am not a man" is true
P2. If "I am a man and I am not a man" is true then "I am a man" is true.
P3. If "I am a man" is true then "I am a man or I am rich" is true.
P4. If "I am a man and I am not a man" is true then "I am not a man" is true.
P5. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
C1. Therefore, "I am rich" is true

P2 through P5 can be kept implicit as they simply express the commonly understood rules of inference.

Okay I agree with you that only one of those two arguments is valid. Now, in a non-circular way, explain why the one follows but the other does not.

It's one argument: — Michael

No, they are two different arguments. One involves inferential reasoning and the other does not.

In simple terms, given these two premises:

P1. If I am a man then I am mortal
P2. I am a man

You can use the rules of inference to derive the conclusion "I am mortal" using a priori reasoning, but you cannot use the rules of inference to derive the conclusion "I am English" using a priori reasoning.

No, they are two different arguments. One involves inferential reasoning and the other does not. — Leontiskos

It's one argument that uses deductive reasoning to derive the conclusion from the premise.

P2 - P5 simply make explicit the rules of inference and can normally be left unsaid.

- This shouldn't be so hard.

Argument 1:

1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true. — Michael

Argument 2:

I am a man and I am not a man.
Therefore, I am rich.

Two different arguments.

You want to claim that argument 2 is an enthymeme of argument 1. But it need not be. And the question at hand is whether argument 2 is valid independent of argument 1.

"You can use the rules of inference to derive the conclusion "I am mortal" using a priori reasoning, but you cannot use the rules of inference to derive the conclusion "I am English" using a priori reasoning"

That is well said.

Perhaps we we disagree about what may be considered a rule of inference. Unless you think an argument that is invalid only coincidentally doesn't follow? Or is it invalid because it does not follow?

Perhaps we we disagree about what may be considered a rule of inference. Unless you think an argument that is invalid only coincidentally doesn't follow? Or is it invalid because it does not follow? — NotAristotle

Logicians coined the term "valid argument" as a shorthand for "an argument with a conclusion that can be derived from the premises using a priori reasoning".

Logicians coined the term "sound argument" as a shorthand for "a valid argument with true premises".

The "argument 1" sits in between the premise and the conclusion of "argument 2" to make a single argument:

P1. "I am a man and I am not a man" is true
P2. If "I am a man and I am not a man" is true then "I am a man" is true.
P3. If "I am a man" is true then "I am a man or I am rich" is true.
P4. If "I am a man and I am not a man" is true then "I am not a man" is true.
P5. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
C1. Therefore, "I am rich" is true

P2 through P5 are normally left implicit as they are simply stating the commonly accepted rules of inference.

Ah, I see, then we will say as a shorthand "invalid" as a way of saying it does not follow, that is, that the conclusion cannot be derived using a priori reasoning.

My question is, if I use a priori reasoning, how can I conclude that "I live in Antartica" (assuming that is true) based on the premise "Pluto is a planet and Pluto is not a planet". How does the conclusion "follow?" I saw your reasoning from the earlier argument, I'm just wondering what rule of inference leads to this conclusion.

To be more specific, it seems to me that in the argument you stated, P1, P5, and C1 cannot all be true. That is, if C1 is true then P1 cannot be true. And if P1 is true then C1 cannot be.

- So you think it is literally impossible to give argument 2 without implying argument 1?

This is dumb beyond belief.