its trustworthiness entirely on the trustworthiness of the data. If the latter be true, the former must be so. The premisses of a syllogism, though they may be immediately the conclusions of prior syllogisms, are ultimately the results of Induction, Observation, Perception, or Intuition; but whatever their origin may be, Deductive Logic has nothing to do with it. All that it is concerned with is, the legitimacy of the conclusion or conclusions that are drawn from the premiss or premisses. To its student Deductive Logic offers the following wholesome advice:-" If you wish to live happily in my domain, obey my Laws. If you desire to enjoy the peace of certitude, conform to the rules and conditions I have laid down. I take no account of your prejudices, passions, instincts, habits, associations, interests, and tendencies, which may induce you to infer any thing from any thing else you must, under all circumstances, implicitly or explicitly obey my Laws, if you desire to attain your object. If you reason from particulars to particulars, you reason against my express Law, and though your conclusions may in some cases be accidentally true, the means you employ to attain your end are none the less unlawful. If you reason from some to all, you do this at your own risk and responsibility. The Law which I lay down is that you infer the particular from the general, or the less general from the more general, and not conversely."

CHAPTER IX.

PROBABLE REASONING AND PROBABILITY.

or

§ 1. If both the premisses of a syllogism are necessary, assertory, or probable, the conclusion is necessary, assertory, or probable. If the modality of one premiss be different from that of the other, the conclusion has the less certain modality. For example, 'B must be A, C is B: .. C is A'; 'B is A, C is probably B: .. C is probably A.' Now what is the meaning of the propositions 'C is probably B' and 'C is probably A'? From the two premisses 'A is probably B' and 'B is probably C,' we may infer 'A is probably C. Is this inference always legitimate? Is the meaning of probably, or rather is the degree of probability, the same in the conclusion as in either of the premisses? Under what conditions is the conclusion valid? In order to answer these questions, we must first of all state the meaning of a Probable Proposition.

§ 2. The Meaning of a Probable Proposition.

'It will probably rain to-morrow,' or 'He will probably die,' means, subjectively, that my belief in the event in question is not full or complete, is of a degree less than the highest; and objectively, that the evidence for the happening of the event in question is not of such a nature as to make it a certainty. That this is the meaning of the proposition will be evident if we consider the meaning in the assertory form. 'It rains,' 'He is dead,' 'The sun rises,' 'Fire is burning': in each of these my belief is of the highest degree, and the event in ques

tion is quite certain: subjectively, there is no trace of doubt, and objectively, there is not the least uncertainty about the event. When the word probably is added to the copula, the proposition means, subjectively, that the state of my mind in regard to the event is a mixture of belief and doubt, partial belief caused by certain evidence for, and partial doubt caused by certain evidence against, the event, that is, a state of incomplete belief caused by incomplete evidence for the event; and it means, objectively, that there is some evidence for, and some against, the event, or at any rate that all the evidence attainable is not such as to make the event a certainty. For example, 'He will probably die' means that there are certain appearances that are symptoms of death, and that there are others which are not : that there are certain signs or marks from which we may infer that death will result, and that there are others from which we may infer the contrary; so that altogether the evidence is conflicting, and the state of mind resulting may be said to be a state of partial belief, or a mixture of belief and doubt.

In this sense the words 'probably,' 'probable,' 'probability' mean any degree of belief less than the highest, and any evidence for the event less than certainty. If we represent full belief and highest certainty by 1, we may represent different degrees of 'probability' by fractions such as 1, 3, 1, 1, 1, &c. In ordinary language the word 'probable' means 'more likely than not,' and in this sense 'probability' would always be represented by fractions greater than . But, in the widest sense in which it is used here, it may be represented by any fraction however small or large, and corresponds exactly to the mathematical word 'chance.'

The probability of a proposition may, then, be represented by a fraction. But what is the exact meaning of the fraction, and how do we get it? The meaning of the proposition 'It will probably rain to-morrow' is, we may say, that the probability of its raining to-morrow is ; or the meaning of the propo

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sition He will probably die this year' is that the probability

* *

*

of his dying is 3, or 3, or any other fraction. Now, how is this fraction obtained, and what is its real meaning? We cannot discuss this question here. We shall adopt the view held by Dr Venn, which appears to be the best and most reasonable. "I consider," says he, "that these terms (probability, chance) presuppose a series; within the indefinitely numerous class which composes a series, a smaller class is distinguished by the presence or absence of some attribute or attributes. * These larger and smaller classes respectively are commonly spoken of as instances of the 'event,' and of 'its happening in a given particular way.' Adopting this phraseology, which, with proper explanations, is suitable enough, we may define the probability or chance (the terms are here regarded as synonymous) of the event happening in that particular way as the numerical fraction which represents the proportion between the two different classes in the long run. Thus, for example, let the probability be that of a given infant living to be 80 years of age. The larger series will compose all men, the smaller all who live to 80. Let the proportion of the former to the latter be 9 to 1; in other words, suppose that 1 infant in 10 lives to 80. Then the chance or probability that any given infant will live to 80 is the numerical fraction" Conversely, if the probability of a man living to 80 be, this implies that in every 10 persons one only lives to that age. Similarly, if the probability of its raining to-morrow be, this implies that in every three cases like the present, rain happens in two cases on the following day. If the probability of a man's dying of a certain disease be, it means that in every three cases of that disease one dies. The two classes, one larger and the other smaller, the proportions between which constitute the probability are, in the last example, (1) the class of persons who have had that disease, and (2) the special class within the other of persons who have died of it; and the proportion of the second to the first is represented by the fraction 3.

1 Venn's Logic of Chance, 2nd ed., p. 145.

§ 3. The Rules of Immediate Inference.

Every probable proposition is thus connected with what Dr Venn aptly calls a Proportional proposition of the form 'm A's in n are B.' It can be shown that every probable proposition must ultimately be traced to a proportional proposition of that form, and that, without tracing it to such a proposition, we can give no rational account of its meaning, when the probability is represented by a fraction. A proportional proposition is to be distinguished from a universal of the form 'All A is B.' From the latter we may infer that 'Any A or sub-class of A is B.' From the former we may infer that 'Any A is probably B,' the probability being represented by the fraction Given that 9

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men in 10 of any assigned age live to 40, we may immediately infer that the probability of a man of that age living to 40 is. Given that 3 in 4 men in India are Hindus, we may immediately infer that the probability of a man in India being a Hindu is 2. Given that 2 in 4 candidates will pass at the examination, we may immediately infer that the probability of a candidate's passing is. Thus, from every proportional proposition, we may infer a probable one, the probability of which is represented by a fraction. Conversely, from a probable proposition we may infer a proportional one. Given the probable proposition 'A is probably B,' the probability of which is represented by the fraction, we may infer the proportional proposition 2 in 3 A's are B.' Given that the probability of a man under certain circumstances becoming rich is, we can immediately infer that 1 man in 10 under the same circumstances becomes rich. Given that the probability of an event happening is, we can infer that 3 events in 5 of that nature do usually happen.

any

Examples.

'Most A's are B': from this we can infer that the probability of A being B is greater than .

' of A are B' or '3 A's in 4 are B': from this we can infer that the probability of any A being B is 2.