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On the weak Kleene scheme in Kripke's theory of truth
Journal of Symbolic Logic 56 (4):14521468 (1991)
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We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ℕcategoricity and discuss its usefulness and limits. 

The paper contains a survey on the complexity of various truth hierarchies arising in Kripke’s theory. I present some new arguments, and use them to obtain a number of interesting generalisations of known results. These arguments are both relatively simple, involving only the basic machinery of constructive ordinals, and very general. 

In this paper we study several translations that map models and formulae of the language of secondorder arithmetic to models and formulae of the language of truth. These translations are useful because they allow us to exploit results from the extensive literature on arithmetic to study the notion of truth. Our purpose is to present these connections in a systematic way, generalize some wellknown results in this area, and to provide a number of new results. Sections 3 and 4 contain (...) 