Could Dummettians use indefinite extendibility to get around Fitch's paradox? I was reading over my answer to "Metaphysical indeterminacy and necessity" here on the PhilosophySE, and I quoted the following there:
One expression of the open-endedness thought is the distinction between generalizations made with ‘any’ and those made with ‘all’. This distinction was first explicitly drawn by Russell. (Russell in fact attributes the idea to Frege, but this attribution is dubious given Frege’s absolutism about the domain of quantification.) Intuitively, ‘any’ expresses schematic generality, whereas ‘all’ expresses the logical product of its instances. The meaning of ‘all’ is therefore dependent on a determinate range of instances. The use to which Russell put the distinction was to generalize over domains which are not determinate: indenitely extensible domains, such as the domains of propositions and properties.
Then the SEP article on quantification cites Dummett to the following effect:
At the core of the problem lies the assumption that the set-theoretic paradoxes cast doubt upon the existence of a comprehensive domain of all objects. What they reveal, according to Dummett (1991, 1993), is the existence of indefinitely extensible concepts like set, ordinal, and object. For Dummett, the indefinite extensibility of set is incompatible with the existence of a comprehensive domain of all sets, since no matter what putative domain of all sets we isolate, we find that we can employ Russell’s reasoning to characterize further sets that lie beyond the putative domain of all sets with which we began. The set of all non-self-membered sets in the initial domain cannot, on pain of contradiction, be in that domain, which means that it must lie in a more comprehensive domain of all sets. If there is no domain of all sets, there is, the thought continues, no hope for a domain of all objects.
And the objectionable disjunct with which Fitch leaves us is, "If all truths are knowable, all truths are known" (the less objectionable disjunct, from a normal point of view, is just, "Not all truths are knowable"). But if Dummett (or constructivists or intuitionists or predicativists or whoever) were to say, "No, what we would have said is that any truth is knowable, from among the indefinite extension of truth and knowledge thus far," would this then get around the objectionable disjunct? In other words, does the anti-realist have to contend that there are "all truths" to be knowable or known in the first place? Rather than reinterpreting one of the other steps in Fitch's deduction, can we reinterpret the initial step, by reinterpreting the initial quantifier, to get a different result?
