4. Three elements in four are conductors of electricity; four in five are conductors of heat: draw the conclusions which follow if the total number of elements be sixty.

5. The population of London is 4,000,000; and there are 500 distinguished men in all professions, suppose 100 in the medical, 150 in the literary, 100 in the army and navy, 50 in the church, 25 great orators, 25 engineers, 50 scientific men; what is the probability of a Londoner being a distinguished man, and belonging to any of these professions?

6. Test the following:-(1) Most A's are B; most B's are C: therefore some A is C. (2) Most A's are B; most A's are C: therefore some B is C.

7. The probability of a man's dying within two years of fever is , of cholera, of consumption. What are the probabilities of his dying of fever or consumption, of cholera or consumption, and of fever or cholera?

8. Thirty per cent. of fishes are edible, and twenty per cent. are freshwater. Calculate the probabilities of a fish being edible, freshwater, edible and freshwater, edible and not-freshwater, freshwater and not-edible, and not-edible and not-freshwater.

APPENDIX.

A. CANONS OR AXIOMS OF THE SYLLOGISM ACCORDING
TO LOGICIANS.

§1. Lambert's Canons for the so-called Imperfect Figures.In opposition to the view that all the figures except the first are imperfect, because they have no canons of their own like the 'Dictum de Omni et Nullo' for the first or perfect figure, and that, therefore, syllogisms in those figures must be reduced to the first, Lambert (in his Neues Organon, Leipzig, 1764) enunciates a distinct canon for each figure, and thus places them all on an equality. For the first figure Lambert recognizes the 'dictum de omni et nullo' as usual. For the second figure he lays down a canon called 'Dictum de Diverso,' which is as follows:-" If one term be contained in, and another excluded from, a third term, they are mutually excluded." This dictum is as self-evident as the 'dictum de omni et nullo.' On applying it to the sixteen possible combinations of premisses it will be found that the same valid moods are obtained as on any other method. It holds good in the moods Cesare, Camestres, Festino, and Baroko. In Cesare the term 'C' (taking A, B, and C as standing for the major, middle, and minor terms respectively) is included in ‘B' in the minor premiss, and in the major premiss the term 'A' is excluded from 'B'; therefore, according to the 'dictum de diverso,' 'C' and 'A' are excluded from each other, that is, the conclusion is 'No C is A.' In Baroko the term 'A' is included

in 'B' in the major premiss, and the term 'some C' is excluded from 'B' in the minor premiss; therefore, according to the same dictum, 'Some C' and 'A' are excluded from each other, that is, the conclusion is 'Some C is not A.' The 'dictum de diverso' is similarly applicable to Camestres and Festino, and thus distinguishes the valid from the invalid moods in the second figure.

For the third figure Lambert enunciates the following canon, which is called 'Dictum de Exemplo:-"Two terms which contain a common part, partly agree, or if one term contains a part which the other does not, they partly differ." This is also self-evident, and may be easily applied to syllogisms in the third figure. In the valid mood Darapti of this figure 'B' is a part of 'A' in the major premiss, and also a part of 'C' in the minor premiss, that is, 'A' and 'C' have a common part 'B'; therefore they partly agree, that is, 'Some C is A,' according to the first part of the 'dictum de exemplo.' In the mood Felapton of the same figure the term 'C' contains B' in the minor premiss, while 'B' is not contained in 'A,' according to the major premiss; therefore 'C' and 'A' partly differ, that is, 'Some C is not A,' according to the second part of the same dictum. The first part of the 'dictum de exemplo' is similarly applicable to the other affirmative moods, and the second part to the other negative moods; and thus it distinguishes the valid from the invalid moods in the third figure.

For the fourth figure Lambert gives a canon called 'Dictum de Reciproco,' which is stated as follows1:-"If no M is B, no B is this or that M; if C is or is not this or that B, there are B's which are or are not C." But it may be more clearly stated thus: If a term be included in a second term which is excluded from a third, then the third is excluded from the first; if a term be included in (or excluded from) a second term which is included in a third, then a part of the third is included in (or

1 Vide Mansel's Aldrich (1849), p. 80; Hamilton's Lectures, Vol. IV. p. 441; and Ueberweg's Logic, p. 372,

excluded from) the first. The first part is applicable to the mood Camenes, while the second part is applicable to the moods Bramantip, Dimaris, Fesapo, and Fresison in the fourth figure. Both parts of the dictum are self-evident, and require no explanation.

Lambert not only abolishes Reduction, and gives a canon for each of the so-called imperfect figures, but he also establishes their independence of the first figure and their equality with it, by showing that each figure is by its nature especially adapted for a particular kind of argument, and that we naturally think and express our thoughts in certain cases in one figure rather than in another. "For example, the proposition, Some stones attract iron, everyone will admit, because The magnet is a stone and attracts iron. This syllogism is in the third figure. In the first, by conversion of one of its premisses, it would run thus:All magnets attract iron (major premiss),

(A)

(I)

Some stones are magnets

(I) .. Some stones attract iron

...

"Here we are unaccustomed to the minor proposition, while it appears as if we must have all stones under review, in order to pick out magnets from among them. On the other hand, that the magnet is a stone is a proposition which far more naturally suggests itself, and demands no consideration. In like manner:-A circle is no square; for the circle is round,— the square not. This proof (in the second figure) is as follows, when cast in the first :

What is not round is no circle,
A square is not round,
Consequently, &c.

"Here the major proposition is converted by means of a terminus infinitus (i. e. contraposed), and its truth is manifested to us only through the consciousness that all circles are round. For, independently of this proposition, should we not hesitate,— there being innumerable things which are not round,—whether the circle were one of those which belonged to this category?

The

We think not; because we are aware. It is thus apparent that we use every syllogistic figure there, where the propositions, as each figure requires them, are more familiar and more current. difference of the figures rests, therefore, not only on their form, but extends itself, by relation to their employment, also to things themselves, so that we use each figure where its use is more natural: The First for finding out or proving the Attributes of a thing; the Second for finding out or proving the Difference of things; the Third for finding out and proving Examples and Exceptions; the Fourth for finding out and excluding Species of a Genus1."

Mill has the following lines on Lambert and his work: "A German philosopher, Lambert, whose Neues Organon (published in the year 1764) contains among other things one of the most elaborate and complete expositions which have ever been made of the syllogistic doctrine, has expressly examined which sorts of arguments fall most naturally and suitably into each of the four figures; and his investigation is characterized by great ingenuity and clearness of thought. His conclusions are: 'The first figure is suited to the discovery or proof of the properties of a thing; the second to the discovery and proof of the distinction between things; the third to the discovery or proof of instances and exceptions; the fourth to the discovery or exclusion of the different species of a genus.' The reference of syllogism in the last three figures to the 'dictum de omni et nullo' is, in Lambert's opinion, strained and unnatural; to each of the three belongs, according to him, a separate axiom, co-ordinate and of equal authority, with that dictum, and to which he gives the names of 'dictum de diverso' for the 2nd figure, ‘dictum de exemplo' for the 3rd, and 'dictum de reciproco' for the 4th. Mr Bailey (Theory of Reasoning, 2nd edition, pp. 70-74) takes a similar view of the subject2." A similar view is also taken by Archbishop Thomson and by Dr Martineau.

1 Hamilton's Lectures, Vol. IV. p. 439.
2 Mill's Logic, Vol. 1. pp. 194-5.