Studies and Exercises in Formal Logic

by John Neville Keynes

These rules exclude the combinations of premisses marked respectively (a), (b), (c) above.

Assigning the valid unweakened conclusion in the case of each of the twelve combinations which remain, we have the following; AYA, AII, AOη, YAY, YEE, YηO, IAI, IEη, EYE, EIO, OAO, ηYη. From this, the table of valid (unstrengthened and unweakened) moods for all four figures may be expanded as before.

330. Formal Inferences not reducible to ordinary Syllogisms.[415]—The following is an example of what is usually called the argument à fortiori: 385

	B is greater than C,
	A is greater than B,
therefore,	A is greater than C.

As this stands, it is clearly not in the ordinary syllogistic form since it contains four terms; an attempt is, however, sometimes made to reduce it to ordinary syllogistic form as follows:

	B is greater than C,
therefore,	Whatever is greater than B is greater than C,
but	A is greater than B,
therefore,	A is greater than C.

[⁴¹⁵] Attempts to reduce immediate inferences to syllogistic form have been already considered in section [110]. In the present section, non-syllogistic mediate inferences will be considered.

With De Morgan, we may treat this as a mere evasion, or as a petitio principii. The principle of the argument à fortiori is really assumed in passing from B is greater than C to Whatever is greater than B is greater than C. It may indeed be admitted that by the above reduction the argument à fortiori is resolved into a syllogism together with an immediate inference. But this immediate inference is not one that can be justified so long as we recognise only such relations between terms or classes as are implied by the ordinary copula; and if anyone declined to admit the validity of the argument à fortiori he would decline to admit the validity of the step represented by the immediate inference.

The following attempted resolution[416] must be disposed of similarly:

Whatever is greater than a greater than C is greater than C,
A is greater than a greater than C,
therefore, A is greater than C.