History and Commentary

The historical order in which the variety of semantic models for justification logics were created was quite different from the order they have been presented in this chapter.

The whole justification logic project has been semantically motivated from the beginning and grew from an attempt to find the right logical format for and the basic laws of reasoning about mathematical proofs.

In addition, Artemov (1995) introduced a constructive version of the canoni­cal model for LP, which was, in the current terminology, a special case of basic models and established completeness of LP. In a more general setting, the ab­stract logic semantics for LP was developed in Mkrtychev (1997) as basic and Mkrtychev models (the current terminology). Mkrtychev models for J, JT, and J4 were introduced in Kuznets (2000).

Fitting models for LP were introduced and developed in Fitting (2005). Fit­ting models for a variety of justification logics that were sublogics of LP, such as J, J4, JD, and JT were studied in Artemov (2008). Fitting models for JT45 were developed in Pacuit (2006) and Rubtsova (2006b). Extensions of Fitting models for combinations of modal logics with justifications were introduced in Artemov (2006) and Artemov and Nogina (2005). More recently, Fitting mod­els for an infinite family of justification logics were created in Fitting (2016a). This family will be discussed in Chapter 8.

Modular models for J were introduced in Artemov (2012) and for a number of other justification logics in Kuznets and Goetschi (2012). Basic models in their current general form are due to Artemov (2018).

Conceptually, basic models and modular models, supported by soundness and completeness theorems, provide an answer to the ontological question What kind of logical object is a justification? However, these models might be difficult to work with, and more convenient constructions, such as Mkrty- chev and Fitting models, are widely used in this area. Technically, basic models and Mkrtychev models may be regarded as special cases of Fitting models. On the other hand, Fitting models can be identified with modular models having the JYB property. This provides a natural hierarchy of the classes of models:

Even the smallest class, basic models, is already sufficient for mathematical completeness results for justification logics, and interesting consequences can be shown, as in Example 3.7. So, the main idea of going higher to Fitting and modular models is not a pursuit of completeness but rather a desire to offer nat­ural models for a variety of epistemic situations involving evidence, belief, and knowledge. There are also important technical advantages. Mkrtychev models have played an important role in proving results about the complexity of justi­fication logics, and Fitting models provide a broad and uniform, though non­constructive, treatment of realization. This will be discussed in Chapters 6 and 7.

Basic models reduce all the justification information to the syntactic eval­uation of justifications, i.e., sets of formulas t*. Conceptually, this approach reflects only one reason for not knowing F: there is no sufficient justification for F available. In contrast, Fitting models take into account two reasons for not knowing F: the Kripkean reason, F fails in some possible world, and the awareness reason, no justification for F is available. Modular models view ev­idence and belief as independent concepts that can help to analyze a broader class of epistemic situations.