Boole does not make use of the operations here termed logical addition and subtraction. The advantages obtained by the introduction of them are three, viz. they give unity to the system; they greatly abbreviate the labor of working with it; and they enable us to express particular propositions. This last point requires illustration. Let i be a class only determined to be such that only some one individual of the class a comes under it. Then
2d. Boole uses the ordinary sign of multiplication for logical multiplication. This debars him from converting every logical identity into an equality of probabilities. Before the transformation can be made the equation has to be brought into a particular form, and much labor is wasted in bringing it to that form.
3d. Boole has no such function as ab. This involves him in two difficulties. When the probability of such a function is required, he can only obtain it by a departure from the strictness of his system. And on account of the absence of that symbol, he is led to declare that, without adopting the principle that simple, unconditioned events whose probabilities are given are independent, a calculus of logic applicable to probabilities would be impossible.
The question as to the adoption of this principle is certainly not one of words merely. The manner in which it is answered, however, partly determines the sense in which the term “probability” is taken.
In the propriety of language, the probability of a fact either is, or solely depends upon, the strength of the argument in its favor, supposing all relevant relations of all known facts to constitute that argument. Now, the strength of an argument is only the frequency with which such an argument will yield a true conclusion when its premises are true. Hence probability depends solely upon the relative frequency of a specific event (namely, that a certain kind of argument yields a true conclusion from true premises) to a generic event (namely, that that kind of argument occurs with true premises). Thus, when an ordinary man says that it is highly probable that it will rain, he has reference to certain indications of rain,—that is, to a certain kind of argument that it will rain,—and means to say that there is an argument that it will rain, which is of a kind of which but a small proportion fail. “Probability,” in the untechnical sense, is therefore a vague word, in as much as it does not indicate what one, of the numerous subordinated and co-ordinated genera to which every argument belongs, is the one the relative frequency of the truth of which is expressed. It is usually the case, that there is a tacit understanding upon this point, based perhaps on the notion of an infima species of argument. But an infima species is a mere fiction in logic. And very often the reference is to a very wide genus.
The sense in which the term should be made a technical one is that which will best subserve the purposes of the calculus in question. Now, the only possible use of a calculation of a probability is security in the long run. But there can be no question that an insurance company, for example, which assumed that events were independent without any reason to think that they really were so, would be subjected to great hazard. Suppose, says Mr. Venn, that an insurance company knew that nine-tenths of the Englishmen who go to Madeira die, and that nine-tenths of the consumptives who go there get well. How should they treat a consumptive Englishman? Mr. Venn has made an error in answering the question, but the illustration puts in a clear light the advantage of ceasing to speak of probability, and of speaking only of the relative frequency of this event to that.3
1. So that, for example, ā denotes not-a.
2. a;b,c must always be taken as (a;b),c, not as a;(b,c).
3. See a notice, Venn’s Logic of Chance, in the North American Review for July, 1867.
On the Natural Classification of Arguments
P 31: Presented 9 April 1867
PART I. §1. Essential Parts of an Argument
In this paper, the term “argument” will denote a body of premises considered as such. The term “premise” will refer exclusively to something laid down (whether in any enduring and communicable form of expression, or only in some imagined sign), and not to anything only virtually contained in what is said or thought, and also exclusively to that part of what is laid down which is (or is supposed to be) relevant to the conclusion.
Every inference involves the judgment that, if such propositions as the premises are are true, then a proposition related to them, as the conclusion is, must be, or is likely to be, true. The principle implied in this judgment, respecting a genus of argument, is termed the leading principle of the argument.
A valid argument is one whose leading principle is true.
In order that an argument should determine the necessary or probable truth of its conclusion, both the premises and leading principle must be true.
§2. Relations between the Premises and Leading Principle
The leading principle contains, by definition, whatever is considered requisite besides the premises to determine the necessary or probable truth of the conclusion. And as it does not contain in itself the subsumption of anything under it, each premise must, in fact, be equivalent to a subsumption under the leading principle.
The leading principle can contain nothing irrelevant or superfluous.
No fact, not superfluous, can be omitted from the premises without being thereby added to the leading principle, and nothing can be eliminated from the leading principle except by being expressed in the premises. Matter may thus be transferred from the premises to the leading principle, and vice versa.
There is no argument without premises, nor is there any without a leading principle.
It can be shown that there are arguments no part of whose leading principle can be transferred to the premises, and that every argument can be reduced to such an argument by addition to its premises. For, let the premises of any argument be denoted by P, the conclusion by C, and the leading principle by L. Then, if the whole of the leading principle be expressed as a premise, the argument will become
But this new argument must also have its leading principle, which may be denoted by L’. Now, as L and P (supposing them to be true) contain all that is requisite to determine the probable or necessary truth of C, they contain L’. Thus L’ must be contained in the leading principle, whether expressed in the premise or not. Hence every argument has, as portion of its leading principle, a certain principle which cannot be eliminated from its leading principle. Such a principle may be termed a logical principle.
An argument whose leading principle contains nothing which can be eliminated is termed a complete, in opposition to an incomplete, rhetorical, or enthymematic argument.1
Since it can never be requisite that a fact stated should also be implied in order to justify a conclusion, every logical principle considered as a proposition will be found to be quite empty. Considered as regulating the procedure of inference, it is determinate; but considered as expressing truth, it is nothing. It is on this account that that method of investigating logic which works upon syllogistic forms is preferable to that other, which is too often confounded with it, which undertakes to enunciate logical principles.
§3. Decomposition of Argument
Since a statement is not an argument for itself, no fact concluded can be stated in any one premise. Thus it is no argument to say All A is B; ergo Some A is B.
If one fact has such a relation to another that, if the former is true, the latter is necessarily or probably true, this relation constitutes a determinate fact; and therefore, since the leading principle of a complete argument involves no matter of fact, every complete argument has at least