From the Knowability Paradox to the existence of proofs

Abstract

The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if ϕ is true, then there exists a proof of ϕ. Building on the work of Artemov (Bull Symb Log 7(1): 1-36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence.

Intuitionistic Epistemic Logic

“…IEL ⊢ K¬A → ¬A. 24 24 It is easy to check that K¬A → ¬A could be used instead of KA → ¬¬A to axiomatize IEL.…”

“…We are not, of course, the first to outline arguments that an intuitionistic conception of truth supports coreflection, see for instance Dean & Kurakawa (2009); Hart (1979); Khlentzos (2004); Murzi (2010); Percival (1990); Martino & Usberti (1994); Williamson (1982Williamson ( , 1988; Wright (1993a). 11 Our contention is that coreflection, when properly understood in line with the intended BHK semantics, is a fairly immediate consequence of uncontroversially intuitionistic views about truth, and should therefore be endorsed as foundational for a properly intuitionistic epistemology.…”

Truth, Demonstration and Knowledge. A Classical Solution to the Paradox of Knowability

“…While sharing some features of the view developed in this paper, the Stockholm approach is subject to the problem raised in the text (Dummett [1982] offers an early statement of the problem when considering the hypothesis that there are feasibly unknowable constructive demonstrations of mathematical sentences, although he does not bring the point to bear on the paradox of knowability). Something like the Stockholm approach has recently been adopted by a few other authors, in my view without significant improvements as far as the problem raised in the text is concerned (for example, Hand [2010] imposes a ban on any consideration that is not meaning-theoretic -without hinting at which finely discriminating meaning-theoretic argument is supposed to establish the knowability of choice functions but not of constructive demonstrations-while Dean and Kurokawa [2010] go for a language in which the contents of Church-Fitch sentences are inexpressible -thus making one wonder what is so bad about the (TFPK)-version of unrestricted anti-realism). A glaring gap in all these works is that they simply assume their favoured version of (TD) without explaining what argument is supposed to yield it in the first place (a point pressed e.g.…”

The logic of justified belief, explicit knowledge, and conclusive evidence

“…For implicit belief, w |= Bϕ means ∃v w.∀u v : u |= ϕ. (25) Since c ϕ ϕ and c θ ∈sub(c ϕ ) θ = ϕ, we have (25) iff…”

“…Justification Logic, due to Sergei Artemov and originally conceived as a solution to a long-standing open problem concerning the intended semantics of Gödel's provability logic [3], has since developed into a wide-ranging study of the notions of evidence and justification; see, e.g., [1,2,[4][5][6][7][8][9][10]13,[18][19][20][21][23][24][25][29][30][31]33,[42][43][44][45]50,53,54,[59][60][61][62]. By making explicit the "evidence" supporting a given assertion, this formalism can capture one of the main ingredients in the epistemological analysis of knowledge: the epistemic justification underlying the knowledge or belief possessed by an agent endowed with only limited logical resources and bounded rationality.…”