If something like my weak inferentialism is to be taken seriously, it should be able to account for common structures of mathematical reasoning. In particular, let's take two examples: mathematical induction (used for reasoning about the natural numbers and other discrete structures) and the continuum (used for reasoning about real numbers and other continuous structures).
My favorite way of thinking about mathematical induction is as a "nothing more" operator-- the principle of mathematical induction essentially asserts that there aren't any numbers except the ones whose existence is assured by the other axioms.
Viewed as a second-order axiom, this is the only axiom of number theory whose logical consequences are not computably enumerable. If we view it instead as a first order infinite axiom schema, the consequences are computably enumerable, but are an incomplete characterization of the natural numbers. (This is as a result of applying the classical semantics to 1st-order and 2nd-order logics.)
In a weak inferentialist system like the one I described, it's quite possible to list the consequences of the first-order axiom schema or other stronger versions which approach the 2nd-order axiom. However, the full 2nd-order axiom causes some trouble. The result depends on how paradoxes are resolved. In my description, I said that Kripke's theory of truth would provide a way of avoiding paradoxes. Kripke's theory, however, leaves open some questions.
Basically, Kripke says that some sentences must have definite truth values, and some cannot, but leaves some sentences between the two which we are allowed to assign values or not based on preference. The least fixed-point is the theory resulting from leaving all these middle sentences undefined as well; it's generally the preferred theory. However, there are others, such as those based on supervaluation. (Note: This is not an entirely complete way of spelling out what's going on here. There are actually two different things which Kripke leaves open: the fixed-point and the valuation scheme. When I say "least fixed point" I'll actually mean the least fixed point with a Kleene evaluation scheme. However, I won't go into those details here.)
The full inferentialist 2nd-order induction axiom would say that for any predicate, we can conclude that if it is true of 0 and is true of n+1 whenever it is true of n, it is true of all numbers.
According to the least-fixed-point, this assertion should always be undefined. This is because the quantification over all predicates necessarily includes undefined predicates, for which the whole statement will come out undefined.
I believe the supervaluation version will instead allow the assertion to be well-defined. As I understand it, supervaluation says that a statement is true if it is true for any consistent assignment of truth values to the undefined sentences. This means that, conceptually, for the instances of induction which are handed ill-defined predicates, we ask "what would be true of well-defined extensions of this predicate?"
This is hopeful, but at the same time it seems unfortunate that the properties of the system would depend so much on which version of Kripke's theory is used. The minimal fixed-point seems (at least to some) like the nicest one, so relying on a different version needs some explanation. Can the choice be motivated in a way more strongly tied to the thesis of weak inferentialism? I'll leave that question for later.
Now, for the continuum. By continuum, I mean to refer to any entity with the cardinality of the powerset of the natural numbers. That powerset (ie, the set of all sets of natural numbers) is one such entity; the real numbers are another. These objects have the essential feature that not every element can be described-- there are far more elements than there can be formulas in any symbolic language. It is a larger sort of infinity.
The main question here is, does this sort of system allow for a classical continuum, or is it more like a constructive continuum (in which only the describable elements exist)?
The answer is, again, dependent on the version of Kripke's truth that we choose. Supervaluation seems to do just what classical mathematics wants: "all possible valuations" will include an uncountable number of possibilities if the setup is right. This will cause universal generalizations about sets if natural numbers (such as the induction axiom!) to take the correct truth values. I could be wrong here, though-- I am not sufficiently familiar with supervaluation.
Least fixed point will again seem to fail us, refusing to make many generalizations which are taken to be sensible in classical mathematics.
