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So I have been looking into Modal Logic as well as epistemic logic (and its dynamic versions) with the hopes of studying their applications to Fitch's paradox.

Fitches paradox refers to the proof that from the supposition that everything true is knowable, we get that everything true is already known, which seems absurd.

My main question is that this seems to specify a very particular language. So my understanding has been that:

Modal logic: Necessitation = Box (for all related worlds) Possibility = Diamond (there exists related world)

Epistemic Logic: Necessitation = K_a (for all related states), Possibility = K^ (there exists related state)

However most proofs of Fitch's paradox seems to model 'knowable' as (Diamond K p), which seems to combine both languages together. I find it difficult to interpret what this specific language means, and what Diamond here really refers to. For example, if I wanted to formally model the language that includes both K and Diamond, what would be the semantics?

Thanks so much

  • See Fitch's Paradox of Knowability : "Let K be the epistemic operator ‘it is known by someone at some time that.’ Let ◊ be the modal operator ‘it is possible that’." And see there the "logics" involved, i.e. the modal axioms needed (reagarding ◊) as well as the epistemic axioms needed (regarding the epistemic operator K). – Mauro ALLEGRANZA Jul 9 '18 at 12:00
  • Thanks for the response. I read the stanford encyclopedia article but I was more interested in a semantics for possibility here. That is, suppose one wanted to model this in a given epistemic logic, how abouts would this work? For example would it include both necessity and Knowledge, and how would one differentiate between states and worlds? Especially because knowledge here is more broad than usual, 'known by someone at some time' instead of just 'known by agent a currently' Thanks also for the link, I will take a look. – Kevin Jul 9 '18 at 12:12
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See : Jonathan Kvanvig, The Knowability Paradox (2006 Oxford University Press), page 8 :

The theorem proved by Fitch on the way to investigating the logic of certain value concepts and from which the paradox arises is:

⊢ ∼α(p & ∼αp),

where ‘p’ is some sentence in a formal language and ‘α’ is an operator of that language meeting certain restrictions. It is sufficient for meeting these restrictions that a is at least as strong as a truth operator, and that it distributes over conjunction. If we let a be the truth operator itself, then the theorem implies the unremarkably obvious idea that the following conjunction is provably untrue: p and it is not true that p. If, however, we let K be the value for a in the above theorem, where ‘K’ is interpreted as ‘‘it is known by someone at sometime that’’, we have the material for paradox.

See also Epistemic Logic : Multi-Modalities, with ref to : Ronald Fagin et alii, Reasoning About Knowledge (1995, The MIT Press).

And see also : Walter Carnielli & Claudio Pizzi, Modalities and Multimodalities (2008, Springer).

  • Thank you. From a bit more reading, would it be correct to say that we operate under a basic modal logic, and then add an additional operator 'a' that has certain properties. When we then decide to formalise the paradox, we can substitute 'a' for the knowledge operator K, which we assume to be closed under conjunction. This latter fact being true in general in epistemic logic, but we taken more generally in the derivation. – Kevin Jul 9 '18 at 13:03
  • @Kevin - correct; as you can see in SEP and the book, the "ingredients" needed are some simple modal principles (like Necessitation rule) as well as some "natural" axioms regarding the K operator (also in this case, quite straightforward if we read K as a knowledge modality). – Mauro ALLEGRANZA Jul 9 '18 at 13:08

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