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It may therefore be concluded, as already asserted, that all inference consists in substituting, for terms of propositions, other terms of equivalent ratiocinative value.
$ 81. FORMAL AND MATERIAL SUBSTITUTIONS.-Substitution of terms may be either formal or material. The former includes all cases where the substituted term is the original term in a modified form,-as, where the elements of a complex term are arranged in a different order, as, e. g., where YX is substituted for XY; or, as where the original term is qualified by some other word or words expressing a formal relation existing between the substituted term and the original, -as, e. g., where in the proposition “ Y is X,” we substitute for “Y”“ some Y," or for “X”. not Not-X”; or, as in the example given above, where we substitute for“ negro” and “ fellow-creature" the terms “ negro in suffering" and " fellow
“ . creature in suffering.” Material substitutions are those where a new term is substituted, as, e. g., where we substitute for a term a synonym, or species for genus, or genus for species.
OF MATHEMATICAL REASONING
§ 82. MATHEMATICS THE TYPE OF ALL RATIOCINATION. – Hence it would seem that
the most perfect type of ratiocination is presented by the mathematical, and especially by the algebraic methods of demonstration; and this is, in fact, the case, as may be illustrated by two familiar examples:
1st Example. Thesis.—The angles of a plain triangle are together equal to two right angles; or, referring to the figure, a +b+c=d (Euclid, Book I., Prop. XXXII.).
a + b' + '= a (16., Prop. XXIX.). But
b' = b
Hence, substituting equivalents,
a+b' + c = a +b+c=d. Q. E. D. 2d Example. Thesis.—The formula for compound interest, i. e., S=p (1 + r)", in which р principal, n= number of years, r=rate of interest, and S = the amount.
At end of first year
S=p+ pr =p (1 + r). At end of second year
S=p (1 + r) + pr (1 + r) = p (1 + r)'.
At end of third year
S=p (1 + r)' + pr (1 + r)' = p (1 + r)'.
At the end of n years
S=p (1 + r)". .
$ 83. A CURRENT ERROR ON THIS POINT. It is indeed asserted by recent logicians that there is an essential difference between ordinary and mathematical, or, as it is otherwise expressed, between qualitative and quantitative reasoning. But this opinion arises from the failure to reflect that the comparison of magnitudes can be effected only by means of units of measurement that can be applied equally to the inagnitudes compared, and that these constitute the significates denoted by mathematical terms. Hence mathematical reasoning consists not in directly comparing the magnitudes considered, but in comparing the units that represent them; and mathematical terms must therefore be regarded as denoting - like other terms-collections or classes of individuals, i. e., of the units expressed.
AN OPINION OF MR. BAIN.—On this point we have the following from Mr. Bain: “ Logicians are aware that the form ' A equals B, B equals C, therefore A equals C' is not reducible to the syllogism. So with relation toʻgreater than' in the argument a fortiori; yet to the ordinary mind these inferences are as natural, as forcible, and as prompt as the syllogistic inference. But the first expression is a perfect syllogism differing from the ordinary form only in the different interpretation given to the copula; and this is true also of the argument a fortiori, if we give it the form, “ A < B, B < C.. A < C.” It is strange this is not recognized by the author; or, rather, would be strange were not the error common. What is meant, therefore, is that the mathematical cannot be reduced to the ordinary form of the syllogism. But this is not the case, for mathematical reasoning can readily be expressed in the ordinary logical forms, as, e. g., the equational syllogism in the two syllogisms following: a is b
b is a b is c
c is b ..a is c
..c is a;
and the argument a fortiori in the following: "a is b, b is c, ..a is c,"— meaning that the class of units denoted by a is contained in the class denoted by b, etc.
Or the inequalities may be converted into equations, as, e.g.,"a<b”into “a+x=b,” and the argument then be expressed in two syllogisms as above.
§ 84. REDUCTION OF Euclid's FIFTH PROPOSITION TO SYLLOGISMS.-Recognizing the mathematical form of the syllogism, there is no need of the cumbersome method usually adopted for the reduction of mathematical reasoning to syllogistic form, as, e. g., in the ancient example of the reduction of Euclid's Fifth Proposition given by Mansel in his notes to Aldrich; or the reduction of the same proposition by Mill (Logic, p. 142).
In fact, Euclid's demonstration is itself in syllogistic form, and needs only a slight variation in the statement of it to make this apparent, as, e. g., as follows:
Prop. V. The angles at the base of an isosceles triangle are equal to one another.
Or, referring to the figure, in the isosceles triangle A B C the angles a and c are equal.
The figure is constructed by producing the equal sides A B and AC to D and E, making the lines A D and A E equal, and by drawing the lines B E and D C
Major Premise. — Prop. IV.