Friday, February 20, 2009

An Improved Deductive Argument

An extraordinarily astute observer -- an anonymous Cardozo colleague (who very probably wishes to avoid the embarrassment of being associated with my blogs) -- points out that the proposed inference in the below argument is valid only if it is assumed that Hs must prefer either L or ~L; Hs cannot be in the position of having no preferences about L.

For your convenience, the argument in issue is:

Stipulation: All Hs prefer P or ~P [Hs prefer P or ~P but not both]
Premise 1: if H prefers P --> H prefers L
Premise 2: if H prefers ~P --> H prefers ~L
Premise 3: (H prefers L) is True
[Therefore]: The inference, or conclusion, [(H prefers P) is True] is valid
So let's restate the argument this way:
Stipulation 1: All Hs prefer P or ~P [Hs prefer P or ~P but not both]
Stpulation 2: All Hs prefer L or ~L [Hs prefer L or ~L but not both]
Premise 1: if H prefers P --> H prefers L
Premise 2: if H prefers ~P --> H prefers ~L
Premise 3: (H prefers L) is True
[Therefore]: The inference, or conclusion, [(H prefers P) is True] is valid
So now we have a better deductive argument, apparently an ironclad one. But this will just go to prove that a perfectly good deductive argument can get you into a lot of trouble.


Flash!: My astute colleague makes the further point:

If you are going to add the assumption that either H prefers L or H prefers -L but not both, I think you can drop premise 1. You just need H prefers -P to imply H prefers -L.
I think my colleague is correct. But at the moment I will proceed on the assumption that logical overkill is not always a bad thing. Besides, since I want to get his basic point out in a hurry and since I am slow on the uptake, I will leave the modified argument (above) unchanged for the moment.

the dynamic evidence page

coming soon: the law of evidence on Spindle Law


Anonymous said...

I think that it's still an example of affirming the consequent...

Premise 1 sets out the conditional. Premise 3 asserts the consequent. But unless the conditional in Premise 1 is interpreted as a biconditional, then the argument likely won't be valid.
(abundant examples, e.g., "If Homer prefers peas, then Homer prefers legumes. Homer prefers legumes. Therefore, Homer prefers peas" is not valid unless the "if" means "if and only if".)

Unknown said...

Dear Philosoraptor, I will respond informally -- since that's really all I can do. The argument is this: if Hs must either prefer P or not-P and if H prefers L, H must prefer P -- because H cannot prefer not-P because if H does so H must prefer not-L (see Premise 2). If you think that a biconditional has to go into this argument to make it work formally speaking, please feel free to add it (and I won't be in a position to object. [I am tempted to say that the argument above generates a biconditional, but I speak loosely].) My basic point: I have a hard time seeing how the informally-stated deductive argument fails to force the inference that H prefers P. Of course, one or more of the premises or assumptions of the argument may be false. But that point takes us elsewhere. My main objective at the moment is to give either-you're-with-me-or-against-me reasoning the maximum possible force it can have. (Since I think most such reasoning is defective when it is viewed from a broader perspective, I wouldn't be unhappy to discover that it fails as a deductive argument even when the assumptions and premises most favorable to the argument are granted.)

Anonymous said...

Hello, and thanks for taking the time. My concern is not about whether P and not-P are the only alternatives for H to choose between, but instead with whether P is the only way to get to L. (And likewise, whether not-P is the only way to get to not-L.)

In specific dialectical contexts, as you said, it might be perfectly clear to everyone evaluating the argument that P is the only way to get to L. But I had in mind contexts where that information needs to be an explicit part of the argument in order for the argument to succeed.

Anonymous said...

1. I suggest that a preferable way of setting it out is:

Premise: if H prefers ~P, then H prefers ~L.
Premise : If H prefers L, then H does not prefer ~L.
Premise : H prefers L.
Conclusion : H prefers P.

2. It’s confused to say:
if H prefers P --> H prefers L

You could say (H prefers P)  (H prefers L)
If H prefers P, then H prefers L

but you can’t use “If” together an implication sign.

3. Why say “(H prefers L) is True”?

Why not just say “H prefers L”?

4. I suggest it’s preferable to avoid “The inference, or conclusion, [(H prefers P) is True] is valid”.

This is because arguments are valid, whereas assertions, conclusions etc are true.

5. I also suggest that there is an ambiguity in your premise:
All Hs prefer P or ~P

It could mean, “Either (All Hs prefer P) or (All Hs prefer ~P)
or “For every H, it is the case that H prefers or that H prefers ~P”

Anonymous said...

gulp, sorry, but my first point missed a premise:

either H prefers P or H prefers ~P