Monday, May 13, 2019

What Paradox?

I feel like the arguments I advanced against the Raven Paradox yesterday are valid, and yet not completely satisfying, because they do not address its strictly logical sense.  In other words, I think statements like "all ravens are black" are not categorical and thus can't be proven in the sense that mathematical theorems are proven; nevertheless, I feel that there is a deeper logical inadequacy in the raven paradox that would invalidate it even if it did not have this deficiency.

The basis of the raven paradox is that the statement "all ravens are black" should be logically equivalent to its "contrapositive," "all non-black objects are not ravens."  Nevertheless, there seems to be a difference between the two, because the first statement asserts something about ravens and the second does not.  To illustrate, let's consider unicorns.  Suppose I say that "all unicorns are white."  The contrapositive is that "all non-white objects are not unicorns."  In this case, the contrapositive is certainly true, but the statement itself is either false or at least meaningless.  Can anyone disagree that all non-red objects are not unicorns?  No, because no objects are unicorns, and so all objects, by any characteristic we choose, are not unicorns.  All non-red objects are not unicorns; all non-white objects are not unicorns; all non-black objects are not unicorns.  And since, by premise, the contrapositive must be able to stand in place of the positive statement, we can also say that all unicorns are red, all unicorns are white, and all unicorns are black.  (We don't need to deal with what it means to be "red" or "black," i.e. can a unicorn be red and black, because we are using this as an absolute statement.  We could just as easily pick traits that are more self-contradictory, such as "all unicorns have exactly four legs," "all unicorns have exactly three legs," and "all unicorns have exactly two legs.")

There may be some branch of logic where it makes sense to say both "all unicorns are black" and "all unicorns are white," since unicorns don't exist.  But I don't think it passes the intuitive test.  By this way of thinking, every non-existent object by definition is characterized by having every property.  I cannot even invent an imaginary object and give it a single colour, since any non-existent object will fulfill every contrapositive of the form "all non-[colour] objects are non-[imaginary objects]."

It looks to me that the contrapositive is logically consistent with its positive statement, but it is not logically identical.  Saying "all non-black objects are not ravens" says nothing positive about ravens, which is irrelevant as long as at least one raven exists; if none does, however, there is a great difference between the statement and its contrapositive.  A quick glance at the Wikipedia article on contrapositives shows some examples that may not fit this paradigm; in other words, there may be cases where a statement and its contrapositive are literally stating the same thing.  I need to investigate it more.  But I feel certain that, in this case, they are not logically equivalent.

How can this be so?  As a software developer, this makes me think of database nulls.  Nulls are fields in a table that have no meaningful value.  They aren't empty, or blank, they are literally meaningless.  An important property of nulls is that they are not equal to anything, even other nulls.  If you try to match tables by comparing null fields with each other, you will get nothing.  Field A is null, and Field B is null, but Field A is not equal to Field B.

The set of objects that are unicorns is null.  You could say it is empty, but I think in this case it makes more sense to think of it as null, because there is no such thing as a unicorn.  Because it is null, every contrapositive whatever applies to it, but none of them says anything about the set, because the set is null and nothing can be said about it.  Therefore, you cannot substitute the contrapositive "all non-black objects are not ravens" for the positive statement "all ravens are black."  They are not logically equivalent in every case, and substitution requires that they mean literally and absolutely the same thing.  Because we can imagine a circumstance where one is true and the other is false, they do not always mean the same thing, and therefore we cannot consider a white shoe as evidence that all ravens are black.