Boole’s Laws of Thought

George Boole An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities monograph, 1854.

Boole’s ‘Laws of Thought’ are well known as having led to Boolean algebra and to informed developments in probability theory, for example those by Keynes in his treatise.

Boolean algebra

The key to Boole’s algebra is:

“We may in fact lay aside the logical interpretation of the symbols in the given equation; convert them into quantitative symbols, susceptible only of the values 0 and 1; perform upon them as such all the requisite processes of solution; and finally restore to them their logical interpretation.” [Ch V]

Note that in Boole’s system one can have a ‘quantitative symbol’ that stands for either 0 or 1 (representing unknown), unlike modern Boolean algebra where the symbols must stand for one or the other.

Probability

Imprecision

Boole’s treatment of probability extends his algebra by allowing ‘truth values’ between 0.0 and 1.0. As with his algebra, it allows for gaps and unknowns:

“As the method is independent of the number and the nature of the data, it continues to be applicable when the latter are insufficient to render determinate the value sought. When such is the case, … there will correspond terms in the final logical equation, the interpretation of which will inform us what new data are requisite in order to determine the values of those constants, and thus render the numerical solution complete. If such data are not to be obtained, we can still, by giving to the constants their limiting values 0 and 1, determine the limits within which the probability sought must lie independently of all further experience. When the event whose probability is sought is quite independent of those whose probabilities are given, the limits thus obtained for its value will be 0 and 1, as it is evident that they ought to be, and the interpretation of the constants will only lead to a restatement of the original problem.” [Ch. I]

For example:

“Proposition I.

From the mere records of the decisions of a court or deliberative assembly, it is not possible to deduce any definite conclusion respecting the correctness of the individual judgments of its members.” [Ch. XXI]

Thus Boole’s methods sometimes yields an imprecise probability [a,b], with [0.0,1.0] being complete ignorance. In much modern probability theory the insufficiency of the data is supplemented by a subjective assessment, to yield a definite (subjective) value. In contrast, Boole is aiming to establish normative limits that depend objectively on the data.  Boole’s approach is more precise than much modern imprecise probability theory, as via the use of his ‘constants’ he can capture correlations, etc.

For example, suppose a coin has an arbitrary bias to heads. Its probability of Heads is [0,0.5] and its probability of Tails is [0.5,1], so the probability of Heads or Tails is [0.5,1.5], which is unnecessarily imprecise. If we follow Boole in supposing the probability of Heads to be v then the probability of ‘Heads or Tails’ is precisely 1.0.

The scientific method: hypotheses

Boole also describes the use of hypotheses in the scientific method:

“It has been observed that a solution may consist entirely of terms affected by arbitrary constant coefficients,—may, in fact, be wholly indefinite. … To obtain a definite solution it is necessary, in such cases, to have recourse to hypotheses possessing more or less of independent probability, but incapable of exact verification. Generally speaking, such hypotheses will differ from the immediate results of experience in partaking of a logical rather than of a numerical character; in prescribing the conditions under which phænomena occur, rather than assigning the relative frequency of their occurrence. This circumstance is, however, unimportant. Whatever their nature may be, the hypotheses assumed must thenceforth be regarded as belonging to the actual data, although tending, as is obvious, to give to the solution itself somewhat of a hypothetical character.

….

It will be manifest that the ulterior value of the theory of Probabilities must depend very much upon the correct formation of such mediate hypotheses, where the purely experimental data are insufficient for definite solution, and where that further experience indicated by the interpretation of the final logical equation is unattainable. “

Thus where the context is insufficient to determine a precise probability, one can consider probabilities condition on hypotheses that provide the missing data or ‘logic’.

The impossibility of unconditional empirical results

“These results only illustrate the fact, that when the defect of data is supplied by hypothesis, the solutions will, in general, vary with the nature of the hypotheses assumed ; so that the question still remains, only more definite in form, whether the principles of the theory of probabilities serve to guide us in the election of such hypotheses. I have already expressed my conviction that they do not, a conviction strengthened by other reasons than those above stated. … Still it is with diffidence that I express my dissent on these points from mathematicians generally ,… and I venture to hope, that a question, second to none other in the Theory of Probabilities in importance, will receive the careful attention which it deserves.” [Ch XX]

“One of the first conclusions to which it leads is that of the necessary insufficiency of any data that experience alone can furnish, for the accomplishment of the most important object of the inquiry. But in setting clearly before us the necessity of hypotheses as supplementary to the data of experience, and in enabling us to deduce with rigour the consequences of any hypothesis which may be assumed, the method accomplishes all that properly lies within its scope. ” [Ch. XXI]

This is key to understanding the limitations of knowledge, even science.

Regularity

Probability estimates depend on assumptions of regularity that may not always be valid:

“We learn that we are not to expect, under the dominion of necessity, an order perceptible to human observation, unless the play of its producing causes is sufficiently simple; nor, on the other hand, to deem that free agency in the individual is inconsistent with regularity in the motions of the system of which he forms a component unit.”

Boole’s examples seem somewhat quaint, but the point remains that continuing regularity is not to be taken for granted. For example, there can be no unconditional scientific proof that revolutions or crises will not happen, since the regularity of the observed phenomena is a default assumption of the method.

Criticisms

Despite what has been said above, Boole sometimes goes too far in an attempt to get a precise answer. For example:

“To meet a possible objection, I here remark, that the above reasoning does not require that the drawings of a white and a marble ball should be independent,in virtue of the physical constitution of the balls. The assumption of their independence … is founded upon our total ignorance … . Probability always has reference to the state of our actual knowledge, and its numerical value varies with varying information.”

The assertion is that if the dependence is unknown one can assume independence, rather than treat it as a variable. But as Keynes showed, this can be misleading.

Dave Marsay

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