Truthmaker semantics for justification logic: Subset approaches

Daniele Sansoni

Justification logic may be thought of as an extension of propositional logic by the means of expressions of the form “t : ϕ”, where ϕ is a formula and t a justification term: the resulting formula may be read as “t justifies ϕ”, “t is evidence for ϕ”, “t is why ϕ”, and the likes. Justification terms can be simple or complex, the latter construed through the application of function symbols. Ultimately, justification logic may be considered an explicit counterpart to plain modal logic: in fact, justification terms can be thought of as tagged and internally structured versions of the □ operator.

Justification logic has been noted to provide a hyperintensional framework: specifically, logical equivalence does not warrant equivalence of justifications, as JL ⊢ ϕ ↔ ψ̸ ⇒ JL ⊢ t : ϕ ↔ t : ψ. In the epistemic field, hyperintensional features of justification logic have been linked by Artemov and Fitting to a syntactical approach to knowledge: this syntactical approach is reflected in one of the most common semantics for justification logic, the Fitting relational semantics. In Fitting models, formulae are interpreted in a standard semantic way (as truth values), while justifications terms are interpreted syntactically (as the sets of formulae for which they provide justification, at an evaluation point).

An alternative route to Fitting semantics has been presented by Lehman through subset semantics, which interprets terms as sets of possible worlds and operations on terms as operations on sets of possible worlds. In this approach, the formula “t : ϕ” holds iff the interpretation of t is a subset of the set of worlds where ϕ holds. In subset semantics, non-standard possible worlds are required in the model for maintaining hyperintensionality: while hyperintensionality is a given in Fitting relational semantics, it needs to be earned in the subset approach. In spite of its hyperintensional features, truthmaker semantics for justification logic is still in its infancy: building on a first proposal made by Faroldi adapting the relational framework of Fitting semantics, we will advance a similar approach lifting ideas for a truthmaker semantics for justification logic from subset semantics.

In general, we aim to pursue different, interrelated objectives: the first amounts to providing a bilateral truthmaker semantics for justification logics based on a language L⋆ CS. The proposed exact truthmaking clause for “t : ϕ” will state that the formula is verified iff the set of states assigned to t is a subset of the verifiers of ϕ. Moreover, terms shall be interpreted as sets of states, while functors on terms as operations between sets of states. This first main route will follow as closely as possible the original subset approach, construing possible worlds through classically worldly state space: we will approach completeness and validity for the class of L⋆ CS justification systems. A survey of topics of interest of the work in progress (regarding matters such as flat and structured versions of the proposed exact truthmaker clause, subject matter of justification terms, and a bottom-up truthmaker inspired justification logic) shall be also undertaken.