Arguments can be:
1. Valid, consistent, and sound
2. Valid, consistent, and unsound
3. Valid, inconsistent, and unsound
4. Invalid

OP's argument is (3), and is an example of the principle of explosion.

If P then not P
P
Not P

This is valid and not sound, but also not coherent. — Hanover

P → ¬P
∴ ¬P ∨ ¬P
∴ ¬P
P
∴ ¬P

Or more simply:

¬P
P
∴ ¬P

It's not raining and it's raining therefore it's not raining.. So yeah, it's "incoherent" in that its premises are inconsistent.

The majority respond to populist, easy answers. They're not going to understand or want to hear complicated proposals that aren't going to give them everything they want. So the side that gives them what they want is the side that is going to win.

A bit of pragmatism over principle shouldn't be ignored.

Fighting fire with fire needs to stop. There has to be a movement that rejects post-truth ideologies. — Christoffer

"They go low, we go high" just ain't working. The problem is that Democrats haven't been fighting fire with fire.

Populism is fine, just so long as it's good policies.

They need a liberal populist "news" channel to compete with Fox News. Bit of competitive propaganda and empty soundbites will do wonders.

I think Trump may have won on policy. — bert1

This policy?



https://x.com/MattWalshBlog/status/1854164649433858119

Now that the election is over I think we can finally say that yeah actually Project 2025 is the agenda. Lol — Matt Walsh

What names do you have in mind to pin our hopes behind? — Mr Bee

I'll do it. I mean, I'm not American, and don't know anything, but then that's probably a good thing.

I won't golf, because golf sucks, but I will sleep in till at least 10am. I'm not a morning person.

If the brain gives out it just means the brain gives out. Just a body before and just a body after. — NOS4A2

There is a moral difference between a living body with a functioning brain and a living body without a functioning brain.

Brain death is death of the person.

And if the brain could be removed but kept alive then even though it's a single organ it's also a person.

I listen to Max Richter at least once a week. — frank

I love his soundtrack for The Leftovers.

The one on the left is what the one on the right looked like about 9 months earlier. In those 9 months, what changed for you? — NOS4A2

The one on the right has a functioning brain. That’s an important difference because it’s an especially important organ. If my heart gives out and I’m on life support then I’m still a person but if the brain gives out and I’m on life support then there’s no person anymore, just a body.

Awww. Do you feel bad now Michael? — Srap Tasmaner

I took it to refer to the word count. :smirk:

There is a reason we don't need an additional implication operator ― that is, one that might appear in a premise, say, and another for when we make an inference. — Srap Tasmaner

I mean your post does use two different operators?

In fact there are a few that come to mind:

1. A → ¬A
2. A ⊢ ¬A
3. A ⊨ ¬A
4. A ∴ ¬A

As a specific example:

1. I am a penguin → My name is Michael
2. I am a penguin ⊢ My name is Michael

(1) is true and (2) is false.

You might want to double-check that. — Srap Tasmaner

Good catch. Trying to translate English into proportional logic is hard.

Checking the validity of one argument using another is done all the time. — Hanover

Checking the soundness of one argument using another is done all the time.

Here are two arguments:

P1. If my name is Michael then I am 36 years old
P2. My name is Michael
C1. Therefore I am 36 years old

P1. If my name is Michael then I am not 36 years old
P2. My name is Michael
C1. Therefore I am not 36 years old

Both arguments are valid, but only one is sound.

The premises don't have to be inconsistent for that. They're just never both true. — frank

If they are consistent then they can both be true. If they can never both be true then they are inconsistent.

If you have an argument in which there is an interpretation where both premises are false, but there are no cases where both premises are true, then the argument is valid. That wouldn't be a case of explosion. — frank

The reason that there is no interpretation where both premises are true is because the premises are inconsistent, i.e. that their conjunction is a contradiction. As such the argument is valid whatever the conclusion (i.e. anything follows).

Explosion is that any proposition can be proven from a contradiction. What Tones is explaining is that if you have an argument in which there is never a case where both premises are true, the argument is valid. — frank

That's the same thing.

What I was trying clarify is that he's not talking about explosion. It's simply that if there is no interpretation in which all the premises are true, the argument is valid. — frank

That is explosion.

I agree, but this was the specific exchange:

He's just using the definition of validity:

An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.
- TonesInDeepFreeze

There is no interpretation in which all the premises are true. Therefore, the argument is valid. — frank

What he says certainly follows from what you said, but it isn't what you (literally) said (at least not in the quote he posted), and isn't the definition of validity.

You and I don't have different definitions of validity.

That's what I was trying to clarify.

You say that because you're not linking your first argument to your second. — Hanover

Why would I? Every argument is its own thing. If the conclusion deductively follows from the premises then the argument is valid.

The fact that two contradictory premises entail two contradictory conclusions does not mean that neither argument is valid. It says it right there in the Wikipedia article:

this arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all.

Right, so you're talking about the principle of explosion?

Given that frank and I were talking about the definition of "valid", I (mis)understood him as claiming that you were saying "an argument is valid if and only if there is no interpretation in which all the premises are true".

He claimed that you and I were giving different reasons for why the argument in the OP is valid.

this argument is not valid becasue all the premises are true and that conclusion is false.. — Hanover

They're not all true. One of them is false. Either it is raining or it is not raining.

But if it were the case that both "it is raining" and "it is not raining" were true then it would be the case that "George Washington is made of rakes" is true (and that "George Washington is not made of rakes" is true).

The inferences in OP's argument are right in line with the idea that if A is true then it is also false. — Count Timothy von Icarus

This is the misunderstanding.

A → ¬A does not mean "if A is true then A is also false".

As I said above, these mean two different things:

1. A → ¬A
2. A → (A ∧ ¬A)

"if ... then ..." in propositional logic does not mean what it means in English.

I really don't understand what you're trying to say. Have a look at this.

The following argument is valid:

It is raining
It is not raining
George Washington is made of rakes

And the following argument is valid:

It is raining
It is not raining
George Washington is not made of rakes

And the following argument is valid:

It is raining
It is not raining
It is raining

And the following argument is valid:

It is raining
It is not raining
It is not raining

As the article says, "this arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical because although the above is a logically valid argument, it is not sound (not all of its premises are true)."

This is just what the word "valid" means. I think you think it means something else.

I don't quite understand what you're trying to say here. I'm just explaining very basic terminology.

If the conclusion follows from the premises then the argument is valid. If the argument is valid and the premises are true then the argument is sound.

The argument in the OP is valid because the conclusion follows from the premises, but it's unsound because one of its premises is false.

There aren't any interpretations where all the premises are true. So it's valid. — frank

That's not what he's saying. I don't know how to explain this to you in an even simpler way.

He's not saying what you think he's saying. These are two different claims:

1. An argument is valid if there is no interpretation in which all the premises are true
2. An argument is valid if there is no interpretation in which all the premises are true and the conclusion is false.

You are claiming that he is asserting (1), when in fact he is asserting (2), as am I.

There is no interpretation in which all the premises are true. Therefore, the argument is valid. — frank

That's not what he's saying.

if the premises are both true then the conclusion is true

And as previously mentioned, P → Q ↔ ¬P ∨ Q. So the above can be rephrased as:

a. One of the premises is false or the conclusion is true.

And (a) is true because one of the premises is false.

I can't see that we are.

We both agree that the argument is valid because the conclusion deductively follows from the premises, i.e. that if the premises are both true then the conclusion is true.

Yes, the argument is valid as I said. But it isn't sound because one of the premises is false.

and a zygote merely has the potential to develop into 1 or more human beings. — Relativist

And 2 zygotes have the potential to develop into 1 human being (a chimera).

Much like a sperm and an ovum have the potential to develop into 1 human being.

Ok. I see. But then, what about the second premise? If A is false, wouldn't the second premise actually be not-A? — frank

I'll rephrase it into English for you.

1. If Socrates is mortal then Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal

Given that P → Q ↔ ¬P ∨ Q, this can be rephrased as:

1. Socrates is not mortal or Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal

This can be simplified to:

1. Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal

The argument is valid (as per the principle of explosion) but is unsound because (1) and (2) cannot both be true.

But in this case, they're the same variable. They're both A. — frank

No, P is A. Q is ¬A.

I read it, thanks. It just looks like that if the A in the antecedent is false, the A in the consequent should be false too. — frank

No. It doesn't say that Q being true depends on P being true. Q can be true whether P is true or false.

It just can't be that P is true and Q is false.

The logic is explained in that link I posted.

These mean the same thing:

1. P → Q
2. ¬Q → ¬P

These do not mean the same thing:
1. P → Q
3. P → (P ∧ Q)

The misunderstanding is that many here are misinterpreting (1) as (3). "if ... then ..." in propositional logic does not mean what it means in ordinary English.

https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference)

P → Q ↔ ¬P ∨ Q

In this case, P = A and Q = ¬A.

A → ¬A ↔ ¬A ∨ ¬A

However, the reductio shows that the first premise is unsound but why is it unsound? It's unsound because it's logically contradictory. If A then not-A necessarily implies A and not-A, which tells me the argument must be invalid. — Benkei

1. "Sound" in this context means "the premises are true and the conclusion follows". It doesn't make sense to say that premises are sound or unsound; it is arguments that are sound or unsound.

2. "Valid" in this context means "if the premises are true then the conclusion follows". An argument can be valid even if one or more of its premises are false (and even if one or more of its premises are necessarily false).

3. A → ¬A does not mean A ∧ ¬A. It means ¬A ∨ ¬A. The argument in the OP is equivalent to:

¬A ∨ ¬A
A
∴ ¬A

To offer a more meaningful example:

I am not 36 years old or I am not 36 years old
I am 36 years old
Therefore, I am not 36 years old

It's possible for the first premise to be true (it's true if I am not 36 years old) and it's possible for the second premise to be true (it's true if I am 36 years old), but it's not possible for both premises to be true.

The argument is an example of the principle of explosion; from a contradiction anything follows.