universe/laws?

5.3 Subjectivism and Bayesian induction: de Finetti

Section 3 of the article Bayes' theorem should be read in conjunction with this section.
5.3.1 Subjectivism

Bruno de Finetti (1906–1985) is the founder of modern subjectivism in probability and induction. He was a mathematician by training and inclination, and he typically writes in a sophisticated mathematical idiom that can discourage the mathematically naïve reader. In fact, the deep and general principles of de Finetti's theory, and in particular the structure of the powerful representation theorem, can be expressed in largely non-technical language with the aid of a few simple arithmetical principles. De Finetti himself insists that “questions of principle relating to the significance and value of probability [should] cease to be isolated in a particular branch of mathematics and take on the importance of fundamental epistemological problems,” (de Finetti FLL, 99) and he begins the first chapter of the monumental “Foresight” by inviting the reader to “consider the notion of probability as it is conceived by us in everyday life” (de Finetti FLL, 100).

Subjectivism in probability identifies probability with strength of belief. Hume was in this respect a subjectivist: He held that strength of belief in a proposition was the proportion of assertive force that the mind devoted to the proposition. He illustrates this with the famous example of a six-sided die (Hume THN, 127–130), four faces of which bear one mark and the other two faces of which bear another mark. If we see the die in the air, he says, we can't avoid anticipating that it will land with some face upwards, nor can we anticipate any one face landing up. In consequence the mind divides its force of anticipation equally among the faces and conflates the force directed to faces with the same mark. This is what constitutes a belief of strength 2/3 that the die will land with one mark up, and 1/3 that it will land with the other mark up.

There are three evident difficulties with this account. First is the unsatisfactory identification of belief with mental force, whether divided or not. It is, outside of simple cases like the symmetrical die, not at all evident that strength of feeling is correlated with strength of belief; some of our strongest beliefs are, as Ramsey says (Ramsey 1931, 169), accompanied by little or no feeling. Second, even if it is assumed that strength of feeling entails strength of belief, it is a mystery why these strengths should be additive as Hume's example requires. Finally, the principle according to which belief is apportioned equally among exclusive and exhaustive alternatives is not easy to justify. This is known as the principle of indifference, and it leads to paradox if unrestricted. (See interpretations of probability, section 3.1.) The same situation may be partitioned into alternative outcomes in different ways, leading to distinct partial beliefs. Thus if a coin is to be tossed twice we may partition the outcomes as

2 Heads, 2 Tails, (Heads on 1 and Tails on 2), (Tails on 1 and Heads on 2)

which, applying the principle of indifference yields P(2 Heads) = 1/4

or as

Zero Heads, One Head, Two Heads

which yields P(2 Heads) = 1/3.

Carnap's c-functions c* and c†, mentioned in section 5.1 above, provide a more substantial example: c† counts the state descriptions as alternative outcomes and c* counts the structure descriptions as outcomes. They assign different probabilities. Indeed, the continuum of inductive methods can be seen as a continuum of different applications of the principle of indifference.

These difficulties with Hume's mentalistic view of strength of belief have led subjectivists to associate strength of belief not with feelings but with actions, in accordance with the pragmatic principle that the strength of a belief corresponds to the extent to which we are prepared to act upon it. Bruno de Finetti announced that “PROBABILITY DOES NOT EXIST!” in the beginning paragraphs of his Theory of Probability (de Finetti TOP). By this he meant to deny the existence of objective probability and to insist that probability be understood as a set of constraints on partial belief. In particular, strength of belief is taken to be expressed in betting odds: If you will put up p dollars (where, for example, p = 0.25) to receive one dollar if the event A occurs and nothing (forfeiting the p dollars) if A does not occur, then your strength of belief in A is p. If £ is a language like that sketched above, the sentences of which express events, then a belief system is given by a function b that gives betting odds for every sentence in £. Such a system is said to be coherent if there is no set of bets in accordance with it on which the believer must lose. It can be shown (this is the “Dutch Book Theorem”) that all and only coherent belief systems satisfy the laws of probability. (See interpretations of probability, section 3.5.2, and section 3 of the entry on Bayesian epistemology as well as the supplement to the latter on Dutch Book arguments for comprehensive discussions.) The Dutch Book Theorem provides a subjectivistic response to the question of what probability has to do with partial belief; namely that the laws of probability are minimal laws of calculative rationality. If your partial beliefs don't conform to them then there is a set of bets all of which you will accept and on which your gain is negative in every possible world.

As just cited the Dutch Book Theorem is unsatisfactory: It is clear, at least since Jacob Bernoulli's Ars Conjectandi in 1713 that the odds at which a reasonable person will bet vary with the size of the stake: A thaler is worth more to a pauper than to a rich man, as Bernoulli put it. This means that in fact betting systems are not determined by monetary odds. Subjectivists have in consequence taken strength of belief to be given by betting odds when the stakes are measured not in money but in utility. (See interpretations of probability, section 3.5.3.) Frank Ramsey was the first to do this in (Ramsey 1926, 156–198). Leonard J. Savage provided a more sophisticated axiomatization of choice in the face of uncertainty (Savage 1954). These, and later, accounts, such as that of Richard Jeffrey (Jeffrey LOD) still face critical difficulties, but the general principle that associates coherent strength of belief with probability remains a fundamental postulate of subjectivism.

7.2 David Armstrong on states of affairs, laws and induction.

D.M. Armstrong, like Williams and Stove, is a rationalist about induction. There is however a significant difference of emphasis and structure that marks Armstrong's approach off from that of Williams and Stove: The problem of induction was for the latter couple the topic and focus of their work on the question. Armstrong's major project on the other hand has for some three decades been the formulation and development of a theory of universals. (See the entry on properties where Armstrong's theory is discussed.) The problem of induction is treated in a brief paper (Armstrong 1991) and an eight-page section in (Armstrong 1983), which work is itself an application of the theory of universals. Armstrong's account of the problem of induction thus gains depth and richness, first in the light of his thesis that laws of nature are connections of universals, announced and defended in (Armstrong 1983) and secondly because it is a natural application of the elaborate theory of universals and states of affairs in which this thesis is developed. This theory yields a few essential metaphysical principles that underlie much of Armstrong's philosophy of science, including his views on induction, and that are usefully kept in mind:

Naturalism and physicalism:
Everything that exists is a physical entity in space / time.

Factualism:
Everything that exists is either (i) a state of affairs or (ii) a constituent of a state of affairs. These constituents include properties (including relations) and particulars.

Properties are of two sorts:
There are universals and ordinary, or second-class, properties. The difference between them is that second-class properties belong to particulars contingently, while this relation is always necessary in the case of universals.

About one-third of (Armstrong 1983) is devoted to stating and supporting three criticisms of what Armstrong calls the regularity theory of law. Put very generally, the various forms of the regularity theory all count laws, if they count them at all, as contingent generalizations or mere descriptions of the events to which they apply: “All there is in the world is a vast mosaic of local matters of fact, just one little thing and then another” as David Lewis put this view in (Lewis 1986, ix). One sort of regularity theory holds that laws of nature supervene on Lewis's vast mosaic. Armstrong argues against all forms of the regularity theory. Laws, on his view, are necessary connections of universals that neither depend nor supervene on the course of worldly events but determine, restrict, and govern those events. The law statement, a linguistic assertion, must in his view be distinguished from the law itself. The law itself is not linguistic, it is a state of affairs; “that state of affairs in the world which makes the law statement true” (Armstrong 1991, 505). A law of nature is represented as ‘N(F, G)’ where F and G are universals and N indicates necessitation: Necessitation is inexplicable, it is “a primitive, which we are forced to postulate” (Armstrong 1983, 92). That each F is a G, however, “does not entail that F-ness [the universal F] has N to G-ness” (Armstrong 1983, 85). That is to say that the extensional inclusion ‘all Fs are Gs ‘ may be an accidental generalization and does not imply a lawlike connection between Fs and Gs. In a “first formulation” of the theory of laws of nature (Armstrong 1983, 85), if N(F, G) is a law, “it entails the corresponding Humean or cosmic uniformity: (x)(Fx ⊃ Gx)”. In later reconsideration, (Armstrong 1983, 149) however, this claim is withdrawn: N(F, G) does not entail that all Fs are Gs, for some Fs may be “interfered with,” preventing the law's power from its work.

Armstrong's rationalism does not lead him, as it did Williams and Stove, to see the resolution of the problem of induction as a matter of demonstrating that induction is necessarily a rational procedure: “[O]rdinary inductive inference, ordinary inference from the observed to the unobserved , is, although invalid, nevertheless a rational form of inference. I add that not merely is it the case that induction is rational, but it is a necessary truth that it is so” (Armstrong 1983, 52). Armstrong does not argue for this principle; it is a premise of an argument to the conclusion that regularity views imply the inevitability of inductive skepticism; the view, attributed to Hume, that inferences from the observed to the unobserved are not rational (Armstrong 1983, 52). Armstrong seems to understand ‘rational’ not in Williams' stronger sense of entailing deductive proofs, but in the more standard sense of (as the OED defines it) “Exercising (or able to exercise) one's reason in a proper manner; having sound judgement; sensible, sane.” (Williams' “ordinary sagacity,” near enough.)

The problem of induction for Armstrong is to explain why the rationality of induction is a necessary truth. (Armstrong 1983, 52) Or, in a later formulation, to lay out “a structure of reasoning which will more fully reconcile us (the philosophers) to the rationality of induction” (Armstrong 1991, 505). His resolution of this problem has two “pillars” or fundamental principles. One of these is that laws of nature are objective natural necessities and, in particular, that they are necessary connections of universals. The second principle is that induction is a species of inference to the best explanation (IBE; see section 6.4 above), “[T]he core idea is very simple: observed regularities are best explained by hypotheses of strong laws of nature [i.e., objective natural necessities], hypotheses which in turn entail conclusions about the unobserved” (Armstrong 2001, 503).

An instantiation of a law is of the form

N(F, G) a's being F, a's being G

where a is an individual. Such instantiations are states of affairs in their own right.

As concerns the problem of induction, the need to explain why inductive inferences are necessarily rational, one part of Armstrong's resolution of the problem can be seen as a response to the challenge put sharply by Goodman: Which universal generalizations are supported by their instances? Armstrong holds that necessary connections of universals, like N(F, G), are lawlike, supported by their instances, and, if true, laws of nature. It remains to show how and why we come to believe these laws. Armstrong's proposal is that having observed many Fs that are G, and no contrary instances, IBE should lead us to accept the law N(F, G). “[T]he argument goes from the observed constant conjunction of characteristics to the existence of a strong law, and thence to a testable prediction that the conjunction will extend to all cases” (Armstrong 1991, 507).

A must see talk on TED:

Link : http://www.ted.com/talks/lee_cronin_making_matter_come_alive.html

Re TED
A philosophically naive speaker. He doesn't seem to realise that "evolution" requires "cognate life" to define it. i.e" order", "disorder" and "competition" require a cognate operation involving a "perception of change". That is the crux of objections to materialism - not vitalism or creationism.