In reply to, "Does nature jump?" Mikhail Katz notes that:
There is a different idea in Leibniz called the Law of Continuity. One of its formulations is
the rules of the finite are found to succeed in the infinite and vice versa. (Leibniz to Varignon, 2 feb 1702).
As noted by Abraham Robinson, this is remarkably close to the transfer principle of infinitesimal analysis: if a formula holds for standard inputs, it will hold also for all inputs. For example, knowing that cos^2 x + sin^2 x =1 for all standard x, we would conclude that it holds for all x, including infinitesimal and infinite values. If anything, this represents a discontinuity: one jumps from finite to infinite values!
So I was wondering, at first: given a justification function j(S), then since I earlier declared that j(∃ω) = ω, would I not have to say that for infinitesimals e, j(∃e) = e? But then believing in some lone infinitesimal, at any given time, would be only infinitesimally justified, and how would we be done?
However, then, what if we moved to plural quantification, and put the function like j(∃ee)? This could be had to equal, say, 1/2, or 1, or 2, or ω, or whatever, on the supposition that the existences of the e's together are composed into the existences of the greater numbers. (There would either have to be uniform principles for composing token e's, or type-different e's per composition, I suppose.) Although Aristotelian continua are not divisible into "actual" proper parts, and hence seem to exhibit a fundamental unity and not plurality, yet for theories of continua that admit of intrinsic/internal plurality, are logics of plural quantification required to express/interpret such theories more adequately, more perspicuously? So far, I would like to try to be neutral about how far the e's in the term "∃ee" extend: obviously not only a finite number of times, since then they would correspond to normal rational numbers, but given the possibility of e.g. amorphous cardinals I would like to be ambivalent between choice-friendly and choice-inimical extensions of plural quantifiers.
Alternative formulation of the question: how do logics of plural quantification approach the structure of the Continuum? As outsiders making helpful comments, so to speak, or as insiders describing the essential structure of the Continuum? (Or as some other such faction...?)