Abstract
Traditional epistemic and doxastic logics cannot deal with inconsistent beliefs nor do they represent the evidence an agent possesses. So-called ‘evidence logics’ have been introduced to deal with both of those issues. The semantics of these logics are based on neighbourhood or hypergraph frames. The neighbourhoods of a world represent the basic evidence available to an agent. On one view, beliefs supported by evidence are propositions derived from all maximally consistent collections evidence. An alternative concept of beliefs takes them to be propositions derivable from consistent partitions of one’s inconsistent evidence; this is known as Schotch-Jennings Forcing. This paper develops a modal logic based on the hypergraph semantics to represent Schotch-Jennings Forcing. The modal language includes an operator \(U(\varphi _1, \ldots , \varphi _n;\psi )\) which is similar to one introduced in Instantial Neighbourhood Logic. It is of variable arity and the input formulas enjoy distinct roles. The U operator expresses that all evidence at a particular world that supports \(\psi \) can be supported by at least one of the \(\varphi _i\)s. U can then be used to express that all the evidence available can be unified by the finite set of formulas \(\varphi _1,\ldots , \varphi _n\) if \(\psi \) is taken to be \(\top \). Future developments will then use that semantics as the basis for a doxastic logic akin to evidence logics.
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Notes
- 1.
Usually, they are presented with Es (\(\square \)s) in the place of all the \(\left\langle E\right\rangle \) (\(\lozenge \)s) which makes the connection to the modal logic K clearer in which \(K_1=K\). But we are choosing to remain consistent with the notation in the literature on evidence logic.
- 2.
This is the name of the axiom as presented in [6].
- 3.
A more appropriate name would be ‘fixed-width forcing’.
- 4.
We could define a cover of \({\mathcal X}\) as a subset of \(\mathcal {P}(\mathcal {P}(W))\), \(\varPi \) such that for each \(\pi \in \varPi \), \(\cap \pi \ne \varnothing \) and for each \(X\in {\mathcal X}\) there is \(\pi \in \varPi \) such that \(\cap \pi \subseteq X\) and if \(\varPi \) is a partition of \({\mathcal X}\) we say that \(\varPi \) would be a partition cover. However, the definition on offer is slightly more economical.
- 5.
Of course, it is also a property that can be trivially satisfied by taking \(\varGamma \) to be large enough—assuming that each \(X\in {\mathcal E}(x)\) can be represented by a formula. Of course, if \(\varGamma \) has other properties, e.g., finiteness, that makes a better case for a non-trivial unification.
- 6.
\(X^c\) is the relative complement of X with respect to W.
- 7.
The operator in [3] is \(\square (\varphi _1,\ldots ,\varphi _n;\psi )\) which is true at \({\mathcal M},x\) iff there is \(X\in {\mathcal E}(x)\) such that \(X\subseteq \llbracket \psi \rrbracket \) and \(X\cap \llbracket \varphi _i\rrbracket \ne \varnothing \) for each \(i\le n\). Its dual would be true, then, iff for all \(X\in {\mathcal E}(x)\) if \(X\subseteq \llbracket \psi \rrbracket \), then \(X\subseteq \llbracket \varphi _i\rrbracket \) for some \(i\le n\). Whereas that operator says that all of the evidence is sufficient for at least one of \(\varphi _i\)s—when it is sufficient for \(\psi \), our U operator says that any piece of evidence is necessary for at least one of the \(\varphi _i\)s, when it is sufficient for \(\psi \).
- 8.
As abbreviations, we will write \(\overline{\varphi }\) to mean \(\varphi _1,\ldots , \varphi _n\), and \((\overline{\varphi }/\psi )_i\) to mean
$$\varphi _1, \ldots , \varphi _{i-1}, \psi , \varphi _{i+1}, \ldots , \varphi _n$$for \(i\le n\).
- 9.
We are using a ‘hypergraph model’ in the sense found in [9] rather than in [6]. Our hypergraph models are what they call neighbourhood models and what [5] calls ‘Minimal Models’. Topologically speaking, it would make more sense to call neighbourhood models those minimal models \(\left\langle W,{\mathcal E}\right\rangle \) in which for each \(x\in W\), \(x\in \bigcap {\mathcal E}(x)\) since a neighbourhood of x would usually contain x.
- 10.
A syntactic derivation of this equivalence proceeds as follows: Suppose \(U(\lnot \varphi ;\varphi )\wedge E\varphi \). An instance of UV is \((U(\lnot \varphi ;\varphi )\wedge E\varphi )\rightarrow \square (\lnot \varphi \rightarrow \varphi )\), so we can infer \(\square (\lnot \varphi \rightarrow \varphi )\) which is equivalent to \(\square \varphi \) in any normal modal logic. Conversely, suppose \(\square \varphi \). Thus, in any normal modal logic \(\square \varphi \rightarrow \square (\lnot \varphi \rightarrow \bot )\) is a theorem. By U\(\bot \), \(U(\bot ;\varphi )\) is a theorem of U, and by U\(\square \)L, \(\square (\lnot \varphi \rightarrow \bot )\rightarrow (U(\bot ;\varphi )\rightarrow U(\lnot \varphi ;\varphi ))\) is a theorem and thus, \(U(\lnot \varphi ;\varphi )\) follows. Since \(\square \varphi \rightarrow \square (\top \rightarrow \varphi )\) is a theorem of any normal modal logic, using N, and E\(\square \), we can derive \(E\varphi \).
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Brunet, T.D.P., Payette, G. (2023). An Evidence Logic Perspective on Schotch-Jennings Forcing. In: Hansen, H.H., Scedrov, A., de Queiroz, R.J. (eds) Logic, Language, Information, and Computation. WoLLIC 2023. Lecture Notes in Computer Science, vol 13923. Springer, Cham. https://doi.org/10.1007/978-3-031-39784-4_9
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