# Infinitesimals and plural quantification

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...?)

• Very flattering :-) Could you possibly clarify what j(∃ω) = ω means exactly? Incidentally, I should have added an ellipsis or two in my quotation from the letter from Leibniz to Varignon (the actual sentence is much longer). Commented Oct 18, 2023 at 12:12
• @MikhailKatz my idea is to axiomatize an abstract/general justification function for propositions/sentences (whether the ordered-pair account of functions is needed, I don't say, I just appeal to the generic sense of an input-output factoid), then look for fixed points of such a function, then replace the normal axiom of infinity with the claim that ω is the first such fixed point besides 1 (since I set j(∃1) = 1 as an axiom, too). I suggest that j-values > 1 are "hyperjustification," and then for a deontic-logic-of-belief these values express supererogatory belief. Commented Oct 18, 2023 at 12:32
• So finitism and ultrafinitism are acceptable as a "moral" baseline of belief (we ought to accept propositions about finite numbers) but the transfinitist set theorist is also permitted (via j-"supererogation") to "believe in" infinity, indeed whichever infinity they can ground in the appropriate j-outputs. Commented Oct 18, 2023 at 12:33
• " then look for fixed points of such a function" : Which function exactly? Commented Oct 18, 2023 at 12:34
• Since in Z2 AC cannot infer well ordering but the reverse is entailed it hints order may get lost in the vast higher order abstraction even assuming the already risky AC, thus plural quantification bearing more ontology would get lost too in such higher realms. But the justification (S4) logic with the KK principle and bismilarity embeddable as a decidable fragment of FOL is perhaps not such realm so either AC or plural quantification are ok at least in most of their conceivable applications, while if you need amorphous sets or their cardinals, AC and plural quantification need to be denied... Commented Oct 19, 2023 at 0:14