It should be clearly understood that it is not meant that every invalid syllogism will offend directly against one of these two rules. As a direct test for the detection of invalid syllogisms we must still fall back upon the four rules given in section [201].[316] All that we have succeeded in shewing is that ultimately these four rules are not independent of one another.
[316] If, for example, for our rule of distribution we select the rule relating to the distribution of the middle term, then the invalid syllogism,
| All M is P, | |
| No S is M, | |
| therefore, | No S is P, |
does not directly involve a breach of either of our two independent rules. But if this syllogism is valid, then must also the following syllogism be valid:
| All M is P (original major), | |
| Some S is P (contradictory of original conclusion), | |
| therefore | Some S is M (contradictory of original minor); |
and here we have undistributed middle. Hence the rule relating to the distribution of the middle term establishes indirectly the invalidity of the syllogism in question. The principle involved is the same as that on which we shall find the process of indirect reduction to be based.
Take, again, the syllogism: PaM, SeM, ∴ SaP. This does not directly offend against the rules given above; but the reader will find that its validity involves the validity of another syllogism in which a direct transgression of these rules occurs.
204. Proof of the Rule of Quality.—For the following very interesting and ingenious proof of the Rule of Quality (as stated in the preceding section) I am indebted to Mr R. A. P. Rogers, of Trinity College, Dublin. In this proof the symbol fn( ) is used to denote the form of a proposition, the terms which the 295 proposition contains in any given case being inserted within the brackets. Thus, if fx(P, M) symbolises All M is P, then fx(B, A) will symbolise All A is B: or, again, if fy(S, M) symbolises Some S is not M, then fy(B, A) will symbolise Some B is not A. It will be observed that the order in which the terms are given does not necessarily correspond with the order of subject and predicate.
Let f1( ), f2( ), f3( ) be propositions belonging to the traditional schedule. Then “f1(P, M), f2(S, M), ∴ f3(S, P)” will be the expression of a syllogism; and, since the syllogism is a process of formal reasoning, if the above syllogism is valid in any case, it will hold good if other terms are substituted for S, M, P (or any of them). Thus, substituting S for M, and S for P, if “f1(P, M), f2(S, M), ∴ f3(S, P)” is a valid syllogism, then “f1(S, S), f2(S, S), ∴ f3(S, S)” will be a valid syllogism.
It follows, by contraposition, that if “f1(S, S), f2(S, S), ∴ f3(S, S)” is an invalid syllogism, then “f1(P, M), f2(S, M), ∴ f3(S, P)” will be an invalid syllogism.