Semantics out of context: nominal absolute denotations for first-order logic and computation

Call a semantics for a language with variables absolute when variables map to fixed entities in the denotation. That is, a semantics is absolute when the denotation of a variable a is a copy of itself in the denotation. We give a trio of lattice-based, sets-based, and algebraic absolute semantics to first-order logic. Possibly open predicates are directly interpreted as lattice elements / sets / algebra elements, subject to suitable interpretations of the connectives and quantifiers. In particular, universal quantification "forall a.phi" is interpreted using a new notion of "fresh-finite" limit and using a novel dual to substitution. The interest of this semantics is partly in the non-trivial and beautiful technical details, which also offer certain advantages over existing semantics---but also the fact that such semantics exist at all suggests a new way of looking at variables and the foundations of logic and computation, which may be well-suited to the demands of modern computer science

Nominal techniques

This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2893582.2893594 Programming languages abound with features making use of names in various ways. There is a mathematical foundation for the semantics of such features which uses groups of permutations of names and the notion of the support of an object with respect to the action of such a group. The relevance of this kind of mathematics for the semantics of names is perhaps not immediately obvious. That it is relevant and useful has emerged over the last 15 years or so in a body of work that has acquired its own name: nominal techniques. At the same time, the application of these techniques has broadened from semantics to computation theory in general. This article introduces the subject and is based upon a tutorial at LICS-ICALP 2015 [Pitts 2015a]. </jats:p

From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions

Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments. It takes a semantics in (permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction. Then we have higher-order logic (HOL) and its models in ordinary (i.e. Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full or partial function spaces. This raises the following question: how are these two models of binding connected? What translation is possible between PNL and HOL, and between nominal sets and functions? We exhibit a translation of PNL into HOL, and from models of PNL to certain models of HOL. It is natural, but also partial: we translate a restricted subsystem of full PNL to HOL. The extra part which does not translate is the symmetry properties of nominal sets with respect to permutations. To use a little nominal jargon: we can translate names and binding, but not their nominal equivariance properties. This seems reasonable since HOL---and ordinary sets---are not equivariant. Thus viewed through this translation, PNL and HOL and their models do different things, but they enjoy non-trivial and rich subsystems which are isomorphic

Consistency of Quine's New Foundations using nominal techniques

We build a model in nominal sets for TST+; typed set theory with typical ambiguity. It is known that this is equivalent to the consistency of Quine's New Foundations. Nominal techniques are used to constrain the size of powersets and thus model typical ambiguity

A new foundational crisis in mathematics, is it really happening?

The article reconsiders the position of the foundations of mathematics after the discovery of HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.Comment: Final versio

Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness

We give a semantics for the lambda-calculus based on a topological duality theorem in nominal sets. A novel interpretation of lambda is given in terms of adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is necessary)