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3. No knowledge is certain.

3.1 All statements about the world are only more or less probable.

3.2 Any proposition of reason may turn out to be false due to confusion of ideas or to errors of calculation or logic.

3.2.1 That is, any chain of reasoning may turn out to be invalid.

3.2.2 The above statement may appear to be too strong. Although for consistency's sake I may be forced to assume that any proposition is only probably true, I am not seriously suggesting that there is some non-zero probability that Pythagoras' theorem may turn out to be false.

3.3 Any apparent fact of experience may turn out to be false due to delusion, carelessness or, where at second hand, to misunderstanding in transmission or even to mischievous intent.

3.4 However, there are many propositions, both of reason and experience, that, for practical purposes are indubitable.

3.5 That all things are to some degree doubtful does not imply that they are all equally doubtful, nor does it recommend a thoroughgoing scepticism or a limp agnosticism.

3.6 There are, however, many areas in which it is more candid to profess ignorance than knowledge and others in which there is very considerable doubt as to the truth.

3.6.1 There will usually be, however, even in areas of considerable doubt, reasons provisionally to hold one view or another.

3.7 Because we hold that no knowledge is certain, we must clarify what we mean by truth.

3.7.1 A statement about the world is true if the statement correctly mirrors the world. We might denote this kind of truth by the term absolute truth.

3.7.2 A deductive statement in a model (whether or not the model correctly pictures the world) is true if it correctly follows from the premises & rules of the model, logically applied. This second, more limited, form of truth we will refer to as relative truth.

3.7.3 It is of fundamental importance to distinguish between absolute & relative truth. When asserting any proposition we must make clear to others and to ourselves whether we intend the proposition to be taken as true of the world, ie. true absolutely, or only true within a model or argument, ie. true relatively. Much confusion and self deceit will be avoided by obeying this injunction.

3.7.4 Since we are profoundly uncertain of our knowledge of the world, we may only say of any of any statement that it is true with a greater or lesser degree of probability. The same remark applies to the relative truth of difficult statements within complex models. For instance, certain extended mathematical proofs may be no more than probably true.

3.8 A requirement of great importance, therefore, is the ability to assign a probability to any statement about the world (or within a model), in accord with the likelihood of it being a true statement.

3.8.1 Ideally, we would like to assign a mathematical probability to any statement, ie. a real number in the range 0 to 1, with 0 representing impossibility & 1 representing certainty. As in the frequency theory of probability, the assigned number should represent the proportion of situations in which the statement is expected to turn out to be true. However, in most cases this approach has a spurious aura of precision, because most situations are not regular or repeatable events. It must be noted that where the possible outcomes of an experiment form a continuum or other infinite set, the a priori probability of any particular outcome is 0. Nonetheless, we would not describe any particular outcome, a posteriori, as unlikely. Rather, we would divide the domain of possible outcomes into equivalence classes (whose total probability may be obtained by summation or integration) and judge the outcome of the experiment by the probability of its class.

3.8.2 In practical life, where it is unreasonable to assign a numerical probability to an event, we do assign non-mathematical probabilities to statements and base our actions on them. for instance, a jury may decide that a defendant is guilty "beyond reasonable doubt". That is, using the mathematical model, be is probably guilty with a probability that approaches 1. however, the ascription of a number (say 0.95) to this probability does not have as precise a meaning as the accuracy of the number suggests. A possibility is to apply a tolerance factor (eg. 0.95 +/- 0.05), which would indicate the degree of uncertainty.

3.8.3 It also makes sense to say that certain statements are more probable than others, even when they do not refer to the same domain of experience. For instance, one could take odds on the contention that a colony will be founded on the moon before 100m is run in under 9.0s. By taking odds one might be able to assign a probability to any statement, though, because of the marginal utility of goods, definition would be lost at the ends of the spectrum (unless comparative probabilities such as those in the previous example are adopted). However, because the set of all possible statements about possible experience is not closed, we cannot easily assign mathematical probabilities to them without slipping into non-linearity. That is, the accumulation of new statements might require a renormalisation of the whole scale of probabilities to smooth out anomalies brought about by bunching.

3.8.4 It would seem to be possible to assign a priori probabilities to statements about the world, the probability being assigned a priori to that particular potential experience, by reference to other actual experiences, though not a priori to all experience. This assumes that there is a regularity in physical & even human behaviour that may be assumed to apply unless there is strong evidence to the contrary.

3.8.5 A statement with a low a priori probability may yet have a higher a posteriori probability because of the strength of actual testimony or experimental evidence.

3.9 Anything of a miraculous nature should be accorded a very low a priori probability, otherwise it would not be categorised as a miracle.

3.9.1 For example, it is a priori very improbable that a statue bled, shed tears or did anything else that is not normally associated with a statue. Hence, we might assign such statements a probability close to 0.

3.9.2 The alleged view of David Hume, that no testimony is sufficient to establish a miracle, would appear to be tautological and, therefore, to say nothing about the world. Hume is alleged to define a miracle as an impossible event, rather than as one that is merely very improbable, because according to Hume no amount of evidence is sufficient to establish a miracle. I suspect that Hume has been much maligned in this. He states that a miracle is a violation of the laws of nature; and as firm and unalterable experience has established these laws, the proof against miracle, from the very nature of the case, is as entire as any argument from experience can possibly be imagined (Enquiry, X). The objection easily seized on is that Hume was wrong to assume the laws of nature as established. however, true though this is (and especially so in Hume's day), it is not the key issue. The question is : presented with a phenomenon that I do not understand, how am I to come to the conclusion that it is a miracle ? The question divides into two, depending upon whether I have witnessed the event personally or have heard it from others. If I had witnessed the supposed miracle myself, I would have to ask myself whether it was more probable that I had been hallucinating or had been deceived than that a law of nature had been violated. If I had received the account from others, I would have to ask whether it is more probable that the testimony is unreliable than that the miracle is genuine. The former must always be more probable. Alleged miracles tend not to be random portents, but evidence for some contentious proposition (eg. for the immaculate conception). Those that accept miracles incline towards those miracles that support their own ideas, especially those of which they are unsure. The key question to ask ourselves when tempted to believe the report of a miracle is "would I believe this report (or its analogue) if it were reported to me by my enemies in support of a proposition I find obnoxious" ? In short, while we may not have sufficient reason to say that miracles are impossible, we never have sufficient reason actually to believe that one has occurred.

3.10 We must note that beliefs are not held in isolation, but form a network of interconnected beliefs commonly called a world view.

3.10.1 Any world view must be self-consistent. That is, all its statements must be simultaneously relatively true. For a world view to be absolutely true, all its component beliefs must simultaneously be true of the world.

3.10.2 Any world view may be condensed into an irreducible set of propositions, none of which duplicates any of the contents of another & which collectively cover the world view. Since, however, there may be difficulty in ensuring the independence of the propositions, we may have to be satisfied with a non-disjoint covering set (as in topology).

3.10.3 Let us suppose that a world view is composed of a set of irreducible propositions {pi} enumerated by the index set I & let each of these propositions have probability P(pi). Then, the probability of the world view is (or is closely related to) the product, over I, of these probabilities, ie. PieI(P(pi)).

3.10.4 Naively, we can deduce two consequences from the above, given below. Firstly, no sophisticated world view can have a high probability of being true in all its parts, because the number of irreducible propositions it contains will be be large while their probabilities will often be low. To take a trivial example, let us suppose our world view consists of an irreducible set of 20 propositions, each of which we take to be 90% certain; then, we have only the right to be 12% certain of the truth of our world view. Secondly, the greater the number of irreducible propositions in a world view, the lower the probability of that world view. Hence the force and importance of Occam's Razor, of which more will be said later.

© Theo Todman 1992 - 2000.
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