I finally got my internet connection back up at home now. Turns out I had a bad cable out on the side of the appartment building. Now all I have to do is get to the backlog of super interesting comments! I hope I’ll have some time to do that this weekend.

In the meantime here is something I was puzzeling about today. Consider the following argument

If P then P
Not P
So, not P

Is this a valid argument? That would depend on whether it is an instance of modus tollens or denying the antecedent; but how can we tell which one it is an instance of? We have the same problem for modus ponens and asserting the consequent.

If P then P
P
Therefore, P

So what are we supposed to say about this? I suppose one could deny ‘p –> p’ since it is equivelent to the law of the excluded middle (~p v p) and there are those who would deny that it is true but that doesn’t seem to solve the problem. We still won’t be able to tell what argument form this is an instance of and so can’t know if it is valid or not. But if that is the case then we may be commiting a fallacy when we infer that a sentence must be true because its negation can’t be true and that would mean that reductio arguments have a deep problem.

So, any thgoughts on whether these are valid arguments or not?

## 11 thoughts on “A Puzzle About Reductios”

1. Premise: Not P
Therefore: Not P

(forget If P then P)

Knowing and recognizing argument forms is nice, but it is not as important as understanding that logic is more fundamentally about showing what can be proven given certain assumptions. There is no traditional ‘argument form’ here because there is no argument: your conclusion is in the premises, and if you assume your premises are true, then this is valid by definition. It is not necessary to perform any operation for an argument to be valid.

If you want to you can use any argument form available, but this has to do with whatever way you think is best to arrive at the conclusion: as long as one strategy exists, then you can show the conclusion is necessitated by the premises. It matters not which method you choose.

2. *I meant to say ‘valid and sound’ at the end of the second sentence of the first paragraph

3. Is this a valid argument form? That would depend on whether it is an instance of modus tollens or denying the antecedent

I find the question confusing. If this is a form of an argument, then it is not an instance of any form. Suppose, then, that this is not a a form, but an instance of a form of argument. In that case, it is definitely valid. Every argument instantiates infinitely many invalid argument forms. That does not make every argument invalid. All that is needed to establish that an argument is valid is that it is an instance of a single valid form (no matter how many invalid forms it happens to instantiate).

4. Hi Mike, yeah that was a bit confusing. I meant to be asking ‘what form is this argument an instance of?’ There doesn’t seem to be any way to answer that.

5. Yeah, like Mike says, the argument instantiates several forms. So it’s an ill-formed question to ask which is “the” form it takes, as if there were only one.

6. RB,

It is not unusual to find questions about “the” form of an argument. I think people mean by this the most obvious form. So logic texts will talk about the “final form”. By that they mean what form articulates all of the logical connectives. But take any two premise argument, it will have this form:

1. p
2. q
3. /:. r

Pretty clearly invalid, and analogously for other arguments. But as I mentioned, if you find one valid form, the argument is valid. This is why people still discuss the ontological argument. The fact that a valid form has not been found for it, and lots of invalid one’s have, it does not follow that there’s not some valid form for this argument. And to badly paraphrase the Beatles, all you need is one. Apologies for that.

7. Noah,

‘p therefore p’ is a valid argument, and it doesn’t beg the question, or so I think. The paper that convinced me of this was Roy Sorenson’s “P therefore P, without Circularity’. Have you read it?

Mike,

Yeah I wasn’t thinking about that when I wrote the post, and I see the point. But in this case the argument instantiates two final forms. It is both modus ponens and affirming the consequent (or modus tollens and denying the antecedent). Or to put in another way; the way it instantiates modus ponens is also the way that it instantiates affirming the consequent. In the other kinds of cases you point out there is a principled way to show how the argument instantiates the valid argument form in a way that it doesn’t instantiate the invalid form but in these cases we can’t do that.

8. Hi Richard,
Right, to return to the argument, letting A be a propositional constant and reserving p, q, r, etc. for prop. variables, it looks like this.

(I)
If A then A
Not A
So, not A

But then it’s final form is this (it has only one),

(II)
If P then P
Not P
So, not P

And that is plainly valid, as any truth-table will show. But (II) is not the form of denying the antecedent. The following is the form of denying the antecedent.

(III)
If p then q
Not p
So, not q

And (III) and (II) are not the same form since there are instances of (III) that are not instances of (II). Here’s one,

(IV)
If A then B
Not A
So, not B

(IV) is not an instance of (II), but it is an instance of (III). It is (III) that is the invalid form of denying the antecedent, not (II). That’s why we would likely conclude that (IV) is an invalid argument. The argument in (I) is an instance of both the invalid form in (III) and the valid form in (II). That’s why (I) is valid.

9. Thanks for the comment Mike, I am finding this very helpful!

So, I guess I wasn’t really seeing your strategy before since I wasn’t thinking of (ii) as a seperate (and valid) argument form and was thinking (i) must be an instance of modus tollens or denying the antecedent. But OK, so now (i) is an instance of two valid forms. The converse of my question then comes up, in virtue of which form is (i) valid?

When you say that (i) is valid because it instantiates a valid form no matter how many invalid forms it instantiates do you mean that it is common for an argument to be interpreted as an instance of a valid form and an invalid form even when one is restricted to the same syntax and semantics? So, take the following argument

(v)
Either A, or B, or both
Not B
So, A

Can (v) be an instance of an invalid form with the same level of expressiviness as the language used to state (v) does? Or what about (vi)?

(vi)
A or A
A
So, not A

(vi) is invalid. Can it be seen as an instance of a valid argument form? I may be missing something, but I don’t see how it could. And if this is right, then it seems to me that (i) is in trouble.

10. Richard, by (vi)

(vi)
A or A
A
So, not A

do you mean (vii)?

(vii)
A or A
not A
So, not A

Suppose you don’t mean (vii). In that case (vi) is an instance of no valid forms at all. On the other hand, (vii) has both valid and invalid forms. Here is an invalid form for (vii)

p
q
/:. r

I’m not sure what you mean by the following: “Can (v) be an instance of an invalid form with the same level of expressiviness as the language used to state (v) does?”

11. No, I actually meant (vi) (as an instance of asserting the alternative). So, you agree that (vi) cannot be considered an instance of a valid argument form. Are there ANY that can?

When you say that (v) instantiates the invalid argument form

p
q
/:. r

you ignore the more fine-grained structure of the argument (for instance, the propositional connectives). Once you make those connective explicit we can see that the argument has the form of disjunctive syllogism, which is a valid argument form. Is there a way of see (v) as also instantiating an invalid argument form using the language of the propositional calculus? That is, keeping the same propositional connective and constants can we see this argument (v) as instantiating an invalid argument form in this sense? It seems to me that this would be highly unusual if it were true! But this is the case with (i), so isn’t that strange?