In the history of classical justification logic there have been two conceptual breakthroughs.

The first was the origin itself: Artemov’s logic of proofs, the discovery of explicit counterparts for modal logics and proof semantics. The second was the Mkrtychev-Fitting semantics, at which point justification logic departed from arithmetical semantics and embraced general epistemic models, adding justification components to Kripke models for modalities.

A third conceptual breakthrough appears to be happening now: going past (explicit counterparts of) modal logic. We must take into consideration very general evidence tracking, and this naturally leads us beyond a purely modal framework because some epistemic scenarios include reasoning about reasons that is basically not of a modal nature. For such scenarios, a modeling that directly uses justification logic tools without intermediate modal logic steps could be appropriate.

There have already been significant developments in this direction. The pa­pers (Krupski, 1997, 2001) described the logic of proofs for single-conclusion proof predicates (note Example 9.2). This departs from modal logic but is still within the arithmetical provability paradigm. Fitting's semantical realization methods, Chapter 7, actually produce quasi-realizations of modal logics in “+’’-free versions of justification logics, which themselves are not counter­parts of standard modal logics. Modular models (Artemov, 2012; Kuznets and Studer, 2012; cf. also Chapter 3) offer semantics of justifications using possi­ble worlds constructions with justifications and closure conditions, in which modalities can be defined a posteriori. Artemov (2018) drops “+” and consid­ers single-conclusion justification epistemic logics and models, that is, where for every justification term t there is at most one F such that = t:F.

This chapter offers a refined version of Artemov (2018). It analyzes a well- known epistemic scenario, the Russell Prime Minister Example, using justifi­cation logic with additional predicates representing “t is accepted as a justifi­cation for F” and “t is a knowledge-producing justification for F.” We argue that these predicates are not closed under sum on justifications and hence do not correspond to modalities.

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