Nominal Terms and Nominal Logics: From Foundations to Meta-mathematics
Handbook of Philosophical Logic
Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in meta-mathematics. More specifically, we survey the application of nominal techniques to languages for unification, rewriting, algebra, and first-order logic. What characterises the languages of this chapter is that they are first-order in character, and yet they can specify and reason on names. In the languages we develop, it
... will be fairly straightforward to give first-order 'nominal' axiomatisations of namerelated things like alpha-equivalence, capture-avoiding substitution, beta-and etaequivalence, first-order logic with its quantifiers-and as we shall see, also arithmetic. The formal axiomatisations we arrive at will closely resemble 'natural behaviour'; the specifications we see typically written out in normal mathematical usage. This is possible because of a novel name-carrying semantics in nominal sets, through which our languages will have name-permutations and term-formers that can bind as primitive built-in features. This chapter draws together material from several papers to deliver a coherent account of a journey from the foundations of a mathematics with names, via logical systems based on those foundations, to concrete applications in axiomatising systems with binding. Definitions and proofs have been improved, generalised, and shortened, and placed into an overall narrative. On the way we touch on a variety of definitions and results. These include: the nominal unification algorithm; nominal rewriting and its confluence proofs; nominal algebra, its soundness, completeness, and an HSP theorem; permissive-nominal logic and its soundness and completeness; various axiomatisations with pointers to proofs of their correctness; and we conclude with a case study stating and proving correct a finite first-order axiomatisation of arithmetic in permissive-nominal logic.