November 01, 2004

Take that, logical positivists!

A propos of my brother’s last post and the comments regading it, here’s another example of a flaw in the application of logical principles to reality. Context: in my philosophy of science class I’m reading “The Philosophy of Natural Science” by Carl Hempel. Hempel was a member of the Vienna Circle, much devoted to Carnap, etc. Today he is probably most famous for formulating the so-called “paradox of confirmation” for logical statements, which as I understand goes as follows (I’m using his example): given two logically equivalent statements, such as for example (1)”all crows are black” and (2)”all non-black objects are not crows,” any evidence p which supports one of the statements supports the other as well. Hence, for example the statement p “object x is not black and is not a crow” supports statement (1) as well as statement (2), therefore any non-black non-crow, a fish, a book, a blueberry pie, all provide evidence that crows are black. The obvious response is that this is ludicrous, since p has nothing to do with crows and is therefore irrelevant to the question of whether crows are all black. But let us re-imagine the question. While it may seem that taking non-black non-crows at randomn would provide no evidence regarding the color of crows, if all of the non-black objects in the world were gathered and recorded and none of them were crows, would one not have to concede that the complementary point, “all crows are black,” would have been proved? Or take another example: say we had a box with 10 objects in it, of which an uncertain number werere crows and 4 of the objects were black, and say one decided to test the two statements “all crows in the box are black” and “all non-black objects in the box are not crows.” If an object were pulled out of the box at randomn and proved to be not black and not a crow, then we would know that more than 15% of the non-black objects were not crows, which definitely provides indicative evidence for both of the statements. And by the time 5 non-black non-crows had been pulled out, we would be all but certain that all the crows in the box were black. Therefore, I think that the seeming paradox is simply an illusion of scale. Of course, on a practical level, the paradox holds true, at least in this case: the category of non-black non-crows is so huge that finding examples of them probably won’t provide much evidence of anything. Therefore, the principle of logical equivalences does virtually nothing to advance the investigation. And in fact, I think it is likely that the problem would exist not only in empirical investigations but also in the investigation of certain mathematical properties, for example. Something to keep in mind for those who are convinced of the infinite power of logic to solve both abstract and practical problems.

Comments

Curt’s “Logic”:all non-black objects are not crows

Curt: the principle of logical equivalences does virtually nothing to advance the investigation … Something to keep in mind for those who are convinced of the infinite power of logic to solve both abstract and practical problems.

I’m convinced that the power of Logic can solve any problem (not just practical and abstract problems).

In fact, I would say that Logic is the only way to solve any problem.

But then again I am not blessed with magic (incomprehensible) “free will” powers as some who post here claim to possess.

Logical = Comprehensible, Consistent, Parsimonious

Curt: I think it is likely that the problem would exist not only in empirical investigations but also in the investigation of certain mathematical properties, for example.

For example?

(And if you really want to impress me, respond in an illogical/random (yet comprehensible) fashion.)

Since mathematics isn't a positivist pursuit, I'm not quite sure why you think a positivist "paradox" (which isn't, as you demonstrate, a paradox) would crop up in mathematics.

Since the issue is one of gathering evidence via examples taken from large categories, I was speculating that the problem might also be true in finding, for example, geometrical shapes with certain properties, but I'm not quite sure why I was even speculating about that since it's pretty much irrelevant to my point, which was how the "paradox" shows how a logically valid mode of confirmation breaks down in empirical investigations. Therefore, you needn't even bother to attack the mathematical point, because I'm not going to stand on it or defend it.

I’m convinced that the power of Logic can solve any problem (not just practical and abstract problems).

In fact, I would say that Logic is the only way to solve any problem.

It's all very well for you say that, but it doesn't exactly address the question of how the paradox of confirmation is not a problem or how it's surmounted by logic.