Saturday, December 04, 2010

A Treatise on Probability, Chapter 2

We know many things simply by contemplating the objects of direct acquaintance (experience, understanding, and perception). We can can leverage this direct knowledge to arrive at indirect knowledge about other propositions. For example, perhaps we know the proposition “3 horses—red, green, and blue--crossed the finish line” and we wish to know something about the proposition “The red horse crossed the finish line.” What moves must we make to arrive at knowledge about the red horse crossing the finish line? We must know a secondary proposition that expresses the logical relationship between the two propositions. In this case, that proposition would be “p bears probability-relationship .33 with respect to proposition h.” Yet how did we come to know this secondary proposition? By directly perceiving a logical relation: in this case certain axioms of probability. I’m not sure if that example is perfect, but it’s the best I can do.

Justified True Belief
Chapter 2 begins by defining knowledge in the traditional tripartite sense: justified true belief. One might have a certain (i.e., 100%) degree of rational belief in some proposition (and thus knowledge).  However, probable degrees of rational belief cannot be considered knowledge, because they might turn out to be false (“x is probable” and “x is false” are compatible propositions...well, that might be disputed but not today). For instance, perhaps I find it very probable that my wife was at home during lunch yesterday. However, there is a small chance that she ran out to shop for Christmas presents and didn’t tell me. After all, it’s the beginning of December. Therefore, even though my degree of rational belief would be very high, the belief may turn out to be false. So I don’t properly “know” that my wife was home during lunch yesterday in the sense of having a justified true belief.  In fact, Keynes would also say that even if, at some later point in time, I was able to have proper knowledge of my wife's location on that day: the probability relationship with respect to the evidence (at time t1) remains the same, regardless of the new evidence at time t2.  As a sidenot: Bayesian probability theory operates on the basis of acquiring new evidence and updating probabilities as new evidence comes to light.  So you're constantly re-evaluating the probability relationship at tx, where x is some time when new evidence came to bear on some conclusion.

Degrees of Belief, Primary and Secondary Propositions
So far, one of the more interesting claims in this book is that “belief, whether rational or not, is capable of degree.” I’m not familiar with the literature on this topic, but suspect that such a view has been (or still is) contested. Perhaps some would say that belief is like an on or off switch, or that doubting is really just holding another belief in mind in addition to the belief in question. Speculations aside, when we say that some proposition is probable (to some degree) with respect to a body of evidence, do we actually know anything? Yes. We know something about the proposition: a second proposition that expresses the “purely logical” relationship.
”The proposition (say, q), that we know in this case is not the same as the proposition (say, p) in which we have a probable degree of rational belief. If the evidence upon which we base our belief is h, then what we know, namely q, is that the proposition p bears the probability-relation of degree α to the set of propositions h; and this k knowledge of ours justifies us in a rational belief of degree α in the proposition p. It will be convenient to call propositions such as p, which do not contain assertions about probability-relations, “primary propositions”; and propositions such as q, which assert the existence of a probability-relation, “secondary propositions.”
So in the case of certainty, we have both knowledge about the proposition (“p bears the probability-relation of degree α to the set of propositions h”), and also knowledge of the proposition itself—we know both the primary and secondary propositions.  In other cases, only the secondary proposition is known--we only know about the proposition. If evidence doesn't confer 100% certainty on the proposition in question, then we don't know the proposition in question.  We only know about the proposition in question, since it might turn out to be false. Regardless, “rational belief of whatever degree can only arise out of knowledge…”  And thus Keynes has solved his problem.  Since rational belief must arise out of knowledge (I've seen no argument for this but it seems reasonable to lay aside objections for now), then he must annex secondary propositions about which we can have knowledge.  Voila, we know secondary propositions about primary propositions and we have a rational degree of belief.

Direct and Indirect Knowledge
Keynes began the book by making the direct/indirect knowledge distinction, so I’m glad he came back to discuss it a bit in this chapter. We all start (temporally or logically, not sure?) with a direct acquaintance of things.  Keynes says, “the important class of things which we have direct acquaintance are our own sensations, which we may be said to experience, the ideas or meanings, about which we may be said to understand, and factors or characteristics or relations of sense-data or meanings, which we may be said to perceive..” So experience, understanding, and perception are the three major forms of direct acquaintance.  But how do does direct acquaintance give rise to knowledge, since the objects of knowledge and belief are propositions, and not experiences, perceptions, or understandings (I'm a bit unclear on exactly what each of those refers to)?  Nevertheless, Keynes says that by contemplating the objects of acquaintance, we obtain direct knowledge of the propositions that express them.
Let us take examples of direct knowledge. From acquaintance with a sensation of yellow I can pass directly to a knowledge of the proposition “I have a sensation of yellow.” From acquaintance with a sensation of yellow and with the meaning of “yellow,” “colour,” “existence,” I may be able to pass to a direct knowledge of the propositions “I understand the meaning of yellow,” “my sensation of yellow exists,” yellow is a colour.” Thus, by some mental process of which it is difficult to give an account, we are able to pass from direct acquaintance with things to a knowledge of propositions about the things of which we have sensations or understand the meaning.”
The 4 step program to rational belief
Step 1: Acquire direct knowledge by contemplating the objects of acquaintance
Step 2: Take knowledge from step 1 and put it beside some other unknown proposition
Step 3: Perceive the logical relation between the two sets of propositions.
Step 4: Now you know a secondary proposition, and thus can have a degree of rational belief in the unknown proposition to the same degree that your secondary proposition expresses a probability-relationship between the two sets of propositions.

Step 3 is yet again the one that has me scratching my head...what is this logical relation and how do we know it?  To reiterate: we pass from a knowledge of h to a knowledge about p, and we do so by perceiving a logical relation between them.  Keynes says that we must know every proposition that constitutes the set of propositions h.  We must know them mind you.  Well, all the scientists just got up and walked out...a few logicians stayed behind...but we'll continue.  "The logic of knowledge is mainly occupied with a study of the logical relations, direct acquaintance with which permits direct knowledge of the secondary proposition asserting the probability-relation, and so to indirect knowledge about, and in some cases of, the primary propositions."

Knowledge Proper
Keynes admits that we might not always be able to put our finger on the what logical relation is being perceived, but he says that all legitimate transitions from knowledge of h to knowledge about p are the result of some kind of logical relation being perceived. If we can’t put our finger on the exact logical relation then Keynes calls it “uncompleted knowledge,” otherwise if we can distinctly put our finger on the logical relation, then we have “knowledge proper.”

Memory and Knowledge
What if we remember some known proposition, but we can’t remember how we acquired it as knowledge? What if we can’t tell the difference between remembered knowledge and present knowledge (or I'd add, what if the neural processes involved turn out to be indistinguishable with respect to tense)? Memory poses a problem for any theory of knowledge, and Keynes doesn’t want to get involved in that debate so he sets the issue aside.

Using precise language
I enjoyed this little footnote at the end of the chapter, where Keynes defends the reason that he will not attempt to be overly precise in his use of "probability" in the book.
This question, which faces all contemporary writers on logical and philosophical subjects, is in my opinion much more a question of style-and therefore to be settled on the same sort of considerations as other such questions-than is generally supposed. There are occasions for very exact methods of statement, such as are employed in Mr Russell's Principia Mathematica. But there are advantages also in writing the English of Hume. Mr Moore has developed in Principia Ethica an intermediate style which in his hands has force and beauty. But those writers, who strain after exaggerated precision without going the whole hog with Mr Russell, are sometimes merely pedantic. They lose the reader's attention, and the repetitious complication of their phrases eludes his comprehension, without their really attaining, to compensate, a complete precision. Confusion of thought is not always best avoided by technical and unaccustomed expressions, to which the mind has no immediate reaction of understanding; it is possible, under cover of a careful formalism, to make statements, which, if expressed in plain language, the mind would immediately repudiate. There is much to be said, therefore, in favour of understanding the substance of what you are saying all the time, and of never reducing the substantives of your argument to the mental status of an x or y.


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