## Friday, July 11, 2008

### Hempel's paradox and bounded predicates

If a naturalist decides to inductively prove the sentence

All ravens are black

she is expected to go out to the country and seek for ravens, checking the color of every raven sighted. After numerous individual confirmations the naturalist feels confident to inductively generalize and declare the sentence "all ravens are black" true. But this sentence is logically equivalent to

Everything that is not black is not a raven.

Now, to obtain confirmation of this sentence the naturalist need not even leave home! All she has to do is look for things around the room which are not black and check out that they are not ravens. There is a myriad of valid instances outside the realm of ornitology, so it seems one can take the inductive step without even having sighted a single raven.

One of the most usual resolutions of Hempel's Paradox follows a statistical or Bayesian approach: sighting a black raven provides evidence that all ravens are black in a much higher degree than sighting a non-black thing which is not a raven (vg. a green apple) because there are many fewer ravens than non-black things. Without going into discussing this argument, I would like to point out that the observation "there are fewer ravens than non-black things" can be interpreted in an ontological sense that is far stronger than the mere fact that the cardinal of the set of ravens is less than the cardinal of the set of non-black things. Furthemore, I contend that it is this ontological interpretation that gives Hempel's puzzle its paradoxical flavor.

Consider the formulation of "all ravens are black" in first order logic:

for all x (RxBx),

where R is the predicate for "raven" and B for "black". This sentence is equivalent to:

for all x (−Bx → −Rx),

as we already knew. Both formulations are instances of the general schema:

for all x (ΦxΨx)

where Φ = R in the first case and Φ = −B in the second case. This first order logic formulation is not explicit about the universe of discourse considered, that is, the class U of entities over which x is meant to range. It is naturally expected that U is large enough that it contains all the objects for which Φ is true, while being small enough so that the elements considered are relevant to the sentence. So, for the case Φ = R the following are natural universes of discourse:

• The class of all corvids
• The class of all birds
• The class of all macroscopic animals

Now, what is a natural universe of discourse for the formulation in which Φ = −B? The chosen class U must contain all things non-black, but not being black is a property held by ontologically remote entities:

• Kingfishers are not black
• Daffodils are not black
• Rubies are not black
• Quarks are not black
• Emotions are not black
• π is not black
• Sets are not black

It seems that, short of taking U to be the class of everything, we cannot choose a universe of discourse comprehensive enough: almost anything is not black.

And this is, in my opinion, the ontological qualm behind Hempel's paradox: When applying induction to prove "for all x (ΦxΨx)" we expect that Φ is a bounded predicate, i.e. one for which a natural universe of discourse exists. "Raven" is a bounded predicate, whereas "non-black" is not. The Bayesian explanation of the paradox assumes that a universe of discussion has already been established, but it is the very choice of universe that poses the problem.

In a later entry we will try to give some formalization to the notion of a predicate being bounded using some elementary set theory machinery.