The Church–Fitch knowability paradox in the light of structural proof theory

Synthese 190 (14):2677-2716 (2013)
Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox is presented following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism.
Keywords Church–Fitch’s paradox  Knowability principle  Structural proof theory  Proof analysis  Intuitionistic bimodal logic  Labelled sequent calculus
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DOI 10.1007/s11229-012-0061-7
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Jc Beall & Greg Restall (2000). Logical Pluralism. Australasian Journal of Philosophy 78 (4):475 – 493.

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Your work is technically very interesting neverthless I have some remarks.

1/ I don't always understand the claim that the Fitch Paradox threatens Anti-Realist philosophy.
If everybody accepts the Knowability Principle restricted to basic propositions,
it sounds more like a victory than a defeat for the knowability advocates.
It seems that what is threatened is more the capacity of modal logic to represent the knowability.

2/ In your intuitionistic frame, as you say in proposition 5.8, it is impossible to have 'A' and 'not K A' in the same world.
So the Fitch Paradox is avoided but the result is a very poor epistemic logic where you cannot express that some truths are unknown.

3/ more technically in the figure below the proposition 5.7
I don't understand what happens in the world y.
You have      y Rk y ;    y Rk z ;    y: p   z: not p
but you have not    'y: not K p' .
Does it stand that 'y: not not K p' ?
It is very counterintuitive.

Else I wrote a dissertation on these points.
I have published a - maybe insufi ... (read more)