On 4/18/2013 3:19 PM, Virgil wrote: > In article > <firstname.lastname@example.org>, > WM <email@example.com> wrote: > >> Matheology § 254 >> >> 1. Finite cannot comprehend, contain, the Infinite. - Yet an inch >> or minute, say, are finites, and are divisible ad infinitum, that is, >> their terminated division incogitable. >> 2. Infinite cannot be terminated or begun. - Yet eternity ab ante >> ends now; and eternity a post begins now. So apply to Space. >> 3. There cannot be two infinite maxima. - Yet eternity ab ante and >> a post are two infinite maxima of time. >> 4. Infinite maximum if cut in two, the halves cannot be each >> infinite, for nothing can be greater than infinite, and thus they >> could not be parts; nor finite, for thus two finite halves would make >> an infinite whole. >> 5. What contains infinite quantities (extensions, protensions, >> intensions) cannot be passed through, - come to an end. An inch, a >> minute, a degree contains these; ergo, &c. Take a minute. This >> contains an infinitude of protended quantities, which must follow one >> after another; but an infinite series of successive protensions can, >> ex termino, never be ended; ergo, &c. >> 6. An infinite maximum cannot but be all-inclusive. Time ab ante >> and a post infinite and exclusive of each other; ergo, &c. >> 7. An infinite number of quantities must make up either an infinite >> or a finite whole. I. The former. - But an inch, a minute, a degree, >> contain each an infinite number of quantities; therefore an inch, a >> minute, a degree, are each infinite wholes; which is absurd. II. The >> latter. - An infinite number of quantities would thus make up a finite >> quantity, which is equally absurd. >> John Stuart Mill > > Clearly JSM is no more of a mathematician than WM. >
Well, most authors prior to Dedekind and Cantor would speak this way.
Although Mill's logic is not frequently mentioned in any modern context, it appears to have been influential. Frege certainly took the time to criticize some of Mill's ideas in the promotion of his own. And, in the context of description theory, the investigations of both Frege and Russell are understood in contrast to Mill's theories involving names. In particular, names for Russell are quantifiers that require instantiation.
Modern first-order logic and its model theory treat constant terms of the language signature in this fashion.