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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?

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I don't see how it matters, given that Fitch's Paradox doesn't rely on any sort of complete set, but only on specialization. It goes from the Knowability Principle

(KP)  ∀p(p→◊Kp)

to the paradoxical claim

   (p∧¬Kp)→◊K(p∧¬Kp)

Just by assuming that there is something that is not known:

   p∧¬Kp

The only thing that is required is substitution for the free instances of p in a well-formed formula Ψ in ∀p(Ψ) in order to get a particular instance. You haven't said how you expect "any" to differ from "for all", but surely you have to allow substitution, otherwise what does "any" mean?

In fact, I don't see how the logic would change at all if you read ∀ as "any" rather than "for all". The only difference I see is that with your characterization of the differences, the "for all" interpretation justifies a Fregean comprehension principle that says any WFF with substitutions forms a set, but that's not logic; it's a theory on top of the logic.

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    Is, "Any truth is known," less invidious that, "All truths are known," though? Dec 14, 2023 at 19:55
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    I don't see how it could be. Deal a card face down, say it is the Jack of Spades. No one knows what the value of the card is before it is turned over, so "the card is the Jack of Spades" is a truth that is not known. This example falsifies "any truth is known". Dec 14, 2023 at 20:28
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The fallacy in Fitch's paradox:

  • a limited time
  • selfish
  • lying pig
  • using "eternal tools"
  • ..inconsistently

enables such "unrealistic" premises as:

And suppose that collectively we are non-omniscient, that there is an unknown truth: (NonO) ∃p(p ∧ ¬Kp)

An "infinte unselfish truthful (+realistic) pig", would rather "suppose":

...that collectively "we" are omniscient, that there is no unknown truth...

false premise -> (skip) blahblablahablah (~3000 words)

🍻

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  • basically all "fallacies" in "this world"(!) base on the (wrong?(!)) premise now==eternity
    – xerx593
    Jan 14 at 12:20
  • just down vote without comment .anonymously ..like a "confused guy" would do it !(?;)
    – xerx593
    Jan 14 at 13:27

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