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doi:10.1111/j.17552567.2011.01119.x Everything is Knowable – How to Get to Know Whether a Proposition is Truetheo_1119 1..22
, 2012
"... Abstract: Fitch showed that not every true proposition can be known in due time; in other words, that not every proposition is knowable. Moore showed that certain propositions cannot be consistently believed. A more recent dynamic phrasing of Mooresentences is that not all propositions are known af ..."
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Abstract: Fitch showed that not every true proposition can be known in due time; in other words, that not every proposition is knowable. Moore showed that certain propositions cannot be consistently believed. A more recent dynamic phrasing of Mooresentences is that not all propositions are known after their announcement, i.e., not every proposition is successful. Fitch’s and Moore’s results are related, as they equally apply to standard notions of knowledge and belief (S 5 and KD45, respectively). If we interpret ‘successful ’ as ‘known after its announcement ’ and ‘knowable ’ as ‘known after some announcement’, successful implies knowable. Knowable does not imply successful: there is a proposition j that is not known after its announcement but there is another announcement after which j is known. We show that all propositions are knowable in the more general sense that for each proposition, it can become known or its negation can become known. We can get to know whether it is true: �(Kj ⁄ K¬j). This result comes at a price. We cannot get to know whether the proposition was true. This restricts the philosophical relevance of interpreting ‘knowable ’ as ‘known after an announcement’. Keywords: modal logic, knowability, Fitch’s paradox, dynamic epistemics, public announcements 1. Successful – the Historical Record
Section A.2 Philosophical and applied logics The Dynamic Capabilities of Public Announcements
"... Fitch addressed the issue for which formulas ϕ it holds that ϕ → ♦Kϕ, i.e., which truths are knowable or in other words can be learnt [2]. He observed that this requirement is inconsistent with the existence of an unknown truth, i.e., with ψ∧¬Kψ. For example, given a propositional variable p, one ca ..."
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Fitch addressed the issue for which formulas ϕ it holds that ϕ → ♦Kϕ, i.e., which truths are knowable or in other words can be learnt [2]. He observed that this requirement is inconsistent with the existence of an unknown truth, i.e., with ψ∧¬Kψ. For example, given a propositional variable p, one can easily compute that the sentence p∧¬Kp cannot be made known, as from this and (p∧¬Kp) → ♦K(p∧¬Kp) we derive that it is conceivable for K(p∧¬Kp) to be (become) true. But for knowledge, and also for introspective belief, this is an inconsistency. Clearly, this issue relates to the phenomenon of Mooresentences. These are sentences such that Kϕ is inconsistent. The prime example is again p ∧ ¬Kp. In his recent paper What one may come to know [3] Van Benthem proposed to interpret the ♦ operator in Fitch’s ϕ → ♦Kϕ, that stands for ‘at some future stage, ’ as: ‘for some restriction of the information state, resulting from an announcement. ’ We have taken up this proposal. Public announcement logic is an extension of multiagent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. Consider an extension of public announcement logic wherein we can express what may become true, whether known or not, without explicit reference to announcements realising that. Let us work our way upwards from a concrete announcement. When p is true, it becomes known by announcing it. Formally, in public announcement