Valid, but unsound.

1. is a conditional contradiction.
2. fulfils the condition'

And from a contradiction, anything and everything follows. This is the principle of explosion.

That is to say, "1. A -> not-A" is impossible; when the impossible can happen, anything can happen.

Let's put something in for A, say "Sue is sitting."

1. That Sue is sitting implies that Sue is not sitting.
2.Sue is sitting.
Therefore, Sue is not sitting.

A truth table will tell you this (the whole statement) is true if Sue is sitting or if she isn't sitting.

I just quoted Priest so I have him on hand:

The notion of validity that comes out of the orthodox account is a strangely perverse one according to which any rule whose conclusion is a logical truth Is valid and, conversely, any rule whose premises contain a contradiction is valid. By a process that does not fall far short of indoctrination most logicians have now had their sensibilities dulled to these glaring anomalies. However, this is possible only because logicians have also forgotten that logic isa normative subject: it is supposed to provide an account of correct reasoning. When seen in this light the full force of these absurdities can be appreciated. Anyone who actually reasoned from an arbitrary premise to, e.g., the infinity of prime numbers, would not last long in an undergraduate mathematics course.

Lewis wrote a lot about this too.

I mean, you can always just laugh at these and ignore them too, there is always a judgement call element in logic anyhow.

no way that the conclusion can follow from the premises. — NotAristotle

Define "follows from".

In ordinary logic, a conclusion follows from a set of premises if and only if there is no interpretation in which all the premises are true and the conclusion is false; and P -> Q is true if and only if at least one of these: (1) P is false, (2) Q is true; and ~P is true if and only if P is true. Thereby:

There is no interpretation in which
A -> ~A
and
A
are both true
and
~A is false

Indeed
A -> ~A
A
therefore ~A
is an instance of modus ponens:

P -> Q
P
therefore Q

The fact that {A -> ~A, A} is inconsistent doesn't contradict that there are no interpretations in which both A -> ~ A and A are true but ~A is false, as indeed there are no such interpretations since there are no interpretations in which A -> ~A and A are both true, even without consulting modus ponens.

If you propose a context with different definitions of "follows from" or different definitions for the truth or falsehood of '->' or '~', then you're welcome to state your definitions.

A truth table will tell you this is true is Sue is sitting or if she isn't sitting. — Count Timothy von Icarus

I can't parse that.

This is the principle of explosion. — unenlightened

That is correct, but it is not necessary to appeal to explosion, since the argument is valid as it is an instance of modus ponens.

In this case we don't need to appeal to the fact that the premises are inconsistent. If the logic includes modus ponens, then the example is valid, even if the logic does not include explosion.

1. A -> not-A
2. A
Therefore,
3. not-A. — NotAristotle

#1 is a contradiction, reducible to ~ A or ~A. Since it concludes A cannot be true, the antecedent (if A) is always false.
#2 is false and contradicts #1 that establishes ~A.
#3 is not a conclusion, but is a restatement of #1.

As given, 3 is a conclusion.

3 follows from 1 and 2 by modus ponens.

[EDIT:

A -> ~A is not a contradiction.

A is false or not depending on an interpretation.]

I agree that A -> not-A seems like a questionable premise; perhaps that is the premise you think is untrue. But what makes " A -> not-A " a premise that is not true? Does it have something to do with truth tables?

I think adding content to the logical propositions definitely demonstrates the absurdity of the argument; but it seems to me that the absurdity is implicit in the structure of the argument itself - we don't really need the content to see that.

The argument seems a bit less problematic if the second premise were changed to: "Sue is not sitting" because then it seems to me that the argument can at least be true in some sense.

I can't parse that.

Understandable, there is a typo there. I mean the conclusion column is true regardless of the truth value of A.

In this case we don't need to appeal to the fact that the premises are inconsistent. If the logic includes modus ponens, then the example is valid, even if the logic does not include explosion.

Indeed.

3 follows from 1 and 2 by modus ponens. — TonesInDeepFreeze

1 means "If A is true, A is false." This means A can never be true, despite it being true. It's a walking contradiction. This in itself can be taken to mean A is false because, as noted A -> ~A is logically equivalent to ~A or ~A as a disjunction of the conditional ( A --> B = ~A or B). 1 therefore means ~A.

This can be reduced to:

1. ~ A
2. A
Therefore ~A.

The conclusion is a restatement of #1. 2 is a contradiction of 1..

Am I understanding you to be saying, similarly to unenlightened, that one of the premises must be false given that they are "inconsistent?" The argument is valid but unsound you are saying?

If so, can you say which premise is false and why?

If so, can you say which premise is false and why? — NotAristotle

1 is false. "If A is true, then A is false" is a necessarily false statement.
"If A is true, then A is false" is logically equivalent to "A is false or A is false." This means that A is false.

By truth table,
if A is true, then 1 is false. Then (1^2) is false, and ((1^2)=>3) is true.
If A is false, then 1 is true but 2 is false, and (1^2) is false, then ((1^2)=>3) is true.
Not to be confused with 3 being true.

A --> B = ~ A v B.
A --> ~A = ~A v ~A
~A v ~ A = ~A

You missed my point.

Of course, there are different ways to show the validity of the argument. But my point is that one of the ways doesn't require appealing to explosion or even contradiction since the argument is in the form of modus ponens.

That's correct. [EDIT: except for use of '=' instead of '<->']

So is:

A -> B
A
therefore B

Well, in the intuitive natural language context I think people would simply want to reject the entailment. E.g., "But that my dog is alive doesn't entail that he is dead."

It could be more interesting in an instance of self-reference. E.g. A is the proposition "this proposition is false." This would, as far as I can see, be a case where intuition would actually tell us that if A is true it entails that A is false.

I cannot think of any concrete examples where we wouldn't simply dismiss it as gibberish though.

what makes " A -> not-A " a premise that is not true? Does it have something to do with truth tables? — NotAristotle

A -> ~A is true when A is false and it is false when A is true.

But my point is that one of the ways doesn't require appealing to explosion or even contradiction since the argument is in the form of modus ponens. — TonesInDeepFreeze

I'd argue A --> ~ A is not of the form A --> B as required as a first premise of modus ponens.

The generic modus ponens syntax requires that the antecedent and consequent be different, meaning that A --> A is not logically equivalent to A -->B because the latter is not reducible to a contradiction.

one of the premises must be false given that they are "inconsistent?" — NotAristotle

There is no interpretation in which both premises are true.

If the interpretation has A as true, then A -> ~A is false.

If the interpretation has A -> ~A as true, then A is false.

And no interpretation has both A and A -> ~A as false.

The argument is valid but unsound you are saying? — NotAristotle

The argument is valid, and there is no interpretation in which the argument is sound.

My understanding of a valid argument is that it is one such that if the premises are true, the conclusion must be true. Sounds like you are saying the initial argument is at least inconsistent; does that prevent it from being a valid argument? Or is it valid, just unsound?

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

An argument is not "inconsistent". What are inconsistent are sets of formulas. What is inconsistent here is the set of formulas that is the set of premises.

It's a valid argument with two premises that cannot both be true and so is necessarily unsound.

I mean the conclusion is true regardless of the truth value of A. — Count Timothy von Icarus

It's a valid argument, so the conclusion is true in any interpretation in which all the premises are true. There are no interpretations in which all the premises are true. The conclusion is true in some interpretations and false in other interpretations.

It's a valid argument with a necessarily false premise and so is necessarily unsound. — Michael

It's a valid argument only if you allow that A --> ~A is of the form A-->~B.

I don't think it follows proper modus ponens syntax. The antecdent and consequent cannot be the same because if they are then it is reducible to simply ~A.

I'd argue A --> ~ A is not of the form A --> B as required as a first premise of modus ponens. — Hanover

Then you'd argue incorrectly

The generic modus ponens syntax requires that the antecedent and consequent be different, meaning that A --> A is not logically equivalent to A -->B because the latter is not reducible to a contradiction. — Hanover

Modus ponens is any argument of this form:

P -> Q
P
therefore Q

There is no restriction on what P and Q can be.

That includes taking P to be A, and taking Q to be ~A.

A -> ~A
A
therefore ~A

is most certainly an instance of modus ponens.

"If A is true, then A is false" is a necessarily false statement. — Hanover

That is incorrect.

If A is false then "If A is true then A is false" is true.

"If A is true, then A is false" is logically equivalent to "A is false or A is false." This means that A is false. — Hanover

That is correct.

That's wrong.

If A is false then "If A is true then A is false" is true. — TonesInDeepFreeze

If my dog does not have have fleas, then "if my dog has fleas, then my dog does not have fleas" is false.