Thursday, May 14, 2009

Propositional Functions

I do not claim to be a Frege Scholar, but I am interested in this distinction between propositions and propositional functions that Drew and Mark Textor talked about last night. My interest is of course epistemological and probabilistic. Take the assertion that:
Fido smokes.
Lets stipulate that there is a particular dog named Fido. Let's also disambiguate "smokes" so that it means inhales tobacco with the nicotine delivery mechanism known as "fags", not some wierdly tensed expression for the early signs of catching fire.
Now I gather that according to Frege there is a fundamental difference between "Fido" and "smokes", and we can pretty easily get a grip on what this difference is. One way of saying what the difference is is that Fido names an object, whereas "smokes" doesn't name anything but predicates of something. Therefore, in a sense, "Fido" stands alone, whereas "smokes" doesn't. Fido is complete, whereas smokes is incomplete. We can use the ontology of propositions to make this difference clearer. "Fido" names a dog, "Fido smokes" names a proposition (or a truth value) but "smokes" names nothing.
Drew made an interesting distinction between a variable and a gap. We can complete "smokes" with a variable easily. Someone smokes. Who smokes? Who smokes dies. If you smoke please do so outside. Smoking causes cancer. We can also names those who smoke "smokers". The thought begins to emmerge that "smokes" is little different from a collective name for all those who smoke. We can think of smokes as a set or class, constituted by its members. The only difference between "Fido" and "smokes" is that Fido names one thing, whereas "smokes" names many things.
Now we enter into epistemology. We have a grammatical trick for converting predicates into collective names. "Ravens are black", is no different in structure from "black things are ravens".
The switch involves a difference in meaning, but this is just because "are" is directional. Since there are more than one Raven and more than one black thing, we can see that these two statements fail to satisfy the law of excluded middle until we quantify the first term with "all" "some" "no" or any proportion or range. So "75% of Ravens are black" is true or false and this has a clear empirical meaning. The meaning is very different from "75% of black things are Ravens". A probabilistic account presents itself. "Ravens are black" is the probability X is black given x is a raven. We then can complete it by a number or a range. P(B R) = 0.75. for 75% of ravens are black. P(B R) = 1, for all ravens are black. P(B R) > 0 for some ravens are black, and P(B R) = 0 for no ravens are black.
Now I don't believe that "Ravens" or "black things" name sets, or are constituted by their extension. The reason I don't believe this is because we can understand and act upon P(B R) = 0.75 without being acquainted with all the ravens that every have been or will be, and without being acquainted with all the black things. All we need are two independent criteria, one for verifying that x is a raven, and one for verifying that x is black. Our belief P(B R) = 0.75 is justified by its success, without the need to be true or false. 0.75 is the success rate of inferences from x is a raven to x is black.
So we still have "x", we still have these objects that underlie everything. This is because Fido smokes does not work like this at all. Fido smokes is a stand alone proposition that automatically obeys the law of excluded middle. We might be able to wonder what P(smokes Fido). But it would be a mistake to think there is any objective answer to this other than 0 or 1. If we were forced to bet, we might consider the classes to which Fido belongs and derive a probability this way. P(smokes dog) is probably very low, I don't know the exact figure. But if we knew that Fido was a circus dog, then we might be better off using P(smokes circus dog) which could be a lot higher. However, both these probabilities would be informed by our coming to know propositions of the form Fido smokes. Finding out that Fido is a circus dog and that Fido smokes would inform our P(smokes circus dog). If our prior belief was 0, we could not thereby reject the testimony of our own eyes when we see Fido lighting up. Our experience of particulars are the foundations on which the whole edifice rests. It is through dogs like Fido that we learn about dogs, and philosophers like Socrates that we learn about Philosophers. But it is not through "smokes" that we learn about Fido, or through "is mortal" that we learn about Socrates.