So far we have seen justification logics axiomatically presented—it was with axiomatic LP that the subject began.
But proof machinery is not enough—one also needs semantics. Technically, without a semantics it is difficult to show something is not provable. But more deeply, a semantics gives us some precise idea of what our proofs are about—what our ontological assumptions are.
Over the years, apart from the original provability semantics for the logic of proofs, LP, and related systems, three semantics have been introduced for justification logics. All three are very general and are discussed in this chapter and the next.Historically, the first nonprovability semantics for justification logic (then the logic of proofs LP) is from Mkrtychev (1997). It is called Mkrtychev semantics here. This was followed by Fitting semantics, which combined Mkr- tychev's machinery with that of Kripke's possible world semantics (Fitting, 2003, 2005). Chapter 4 is devoted to it. The third semantics is due to Artemov, and is called modular semantics (which includes the so-called basic semantics), see Artemov (2012), Kuznets and Studer (2012), and Artemov (2018). In many ways it is the simplest and most basic of the three, though it was the last to be created. Mkrtychev and modular semantics are covered in this chapter.
3.1