From the very beginning, justification logics had important connections with modal logics.
Indeed, this was the main reason for studying them. For instance, as we have noted several times, the first justification logic LP connects with the modal logic S4, and this plays an important role in giving an arithmetic interpretation for intuitionistic logic.
We will examine the details of this LP/S4 connection in Chapter 9. Both basic models and Mkrtychev models lack specific machinery to help bring the justification/modal connection out. Fitting semantics, introduced in Fitting (2003, 2005), is a possible world semantics. Because of this, justification/modal connections are consequently much easier to grasp. Modular models extend this use of possible world machinery to a still more general setting because JYB, justification yields belief, is not required though it is essentially built into Fitting models. As we will see in Chapter 7, Fitting models turned out to be the appropriate tools for establishing deep general results connecting modal and justification logics, but we can't go into details at this point.In this chapter we first present modal semantics. This is familiar and standard, and we have been making some use of it all along. To do this we must first establish notation and set up basic machinery. Then we introduce Fitting semantics and prove general completeness results.
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