Two Treatments of Definite Descriptions in Intuitionist Negative Free Logic

Sentences containing definite descriptions, expressions of the form `The F', can be formalised using a binary quantier that forms a formula out of two predicates, where ℩x[F;G] is read as `The F is G'. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INF℩ of intuitionist negative free logic extended by such a quantier, which was presented in [4], INF℩ is first compared to a system of Tennant's and an axiomatic treatment of a term forming ℩ operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INF℩ in which the G of ℩x[F;G] is restricted to identity. INF℩ is then compared to an intuitionist version of a system of Lambert's which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion.

Cohomology Operations

This chapter concerns cohomology operations. Although motivic cohomology was originally defined for smooth varieties over the perfect field 𝑘, it is more useful to view it as a functor defined on the pointed 𝔸1-homotopy category Ho ·, discussed previously in chapter 12. After defining cohomology operations and giving a few examples, the chapter turns to an axiomatic treatment of the motivic Steenrod operations. The motivic Milnor operations are then presented. Thereafter, this chapter demonstrates that the sequence of Milnor operations 𝑄𝑖 is exact on the reduced cohomology of the suspension Σ‎𝔛 attached to a Rost variety 𝑋, using the degree map 𝑡𝒩. It concludes with Voevodsky's motivic degree theorem.

Universal properties for universal types in bifibrational parametricity

In the 1980s, John Reynolds postulated that a parametrically polymorphic function is an ad-hoc polymorphic function satisfying a uniformity principle. This allowed him to prove that his set-theoretic semantics has a relational lifting which satisfies the Identity Extension Lemma and the Abstraction Theorem. However, his definition (and subsequent variants) has only been given for specific models. In contrast, we give a model-independent axiomatic treatment by characterising Reynolds’ definition via a universal property, and show that the above results follow from this universal property in the axiomatic setting.

Two Sides of the Same Coin: Belief Revision and Enforcing Arguments

We study a type of change on knowledge bases inspired by the dynamics of formal argumentation systems, where the goal is to enforce acceptance of certain arguments. We put forward that enforcing acceptance of arguments can be viewed as a member of the wider family of belief change operations, and that an axiomatic treatment of it is therefore desirable. In our case, laying down axioms enables a precise account of the close connection between enforcing arguments and belief revision. Our analysis of enforcing arguments proceeds by (i) axiomatizing it as an operation in propositional logic and providing a representation result in terms of rankings on sets of interpretations, (ii) showing that it stands in close relationship to belief revision, and (iii) using it as a gateway towards a principled treatment of enforcement in abstract argumentation.

Origami folds in higher-dimension

We present a generalization of mathematical origami to higher dimensions. We briefly explain Huzita- Justin’s axiomatic treatment of mathematical origami. Then, for concreteness, we apply it to origami on 3-dimensional Euclidean space in which the fold operation consists of selecting a half-plane and reflect- ing one half-plane across it. We finally revisit the subject from an n-dimensional point of view.

Inferring Rationales from Choice: Identification for Rational Shortlist Methods

A wide variety of choice behavior inconsistent with preference maximization can be explained by Manzini and Mariotti's Rational Shortlist Methods. Choices are made by sequentially applying a pair of asymmetric binary relations (rationales) to eliminate inferior alternatives. Manzini and Mariotti's axiomatic treatment elegantly describes which behavior can be explained by this model. However, it leaves unanswered what can be inferred, from observed behavior, about the underlying rationales. Establishing this connection is fundamental not only for applied and empirical work but also for meaningful welfare analysis. Our results tightly characterize the surprisingly rich relationship between behavior and the underlying rationales. (JEL D11, D12, D83, M37)

Comparative uncertainty: theory and automation

In recent decades, qualitative approaches to probabilistic uncertainty have received more and more attention. We propose a characterisation of partial preference orders through a uniform axiomatic treatment of a variety of qualitative uncertainty notions. To this end, we prove a representation result that connects qualitative notions of partial uncertainty to their numerical counterparts. We describe an executable specification, in the declarative framework of Answer Set Programming, that constitutes the core engine for qualitative management of uncertainty. Some basic reasoning tasks are also identified.

Frobenius n-homomorphisms, transfers and branched coverings

The main purpose is to characterise continuous maps that are n-branched coverings in terms of induced maps on the rings of functions. The special properties of Frobenius n-homomorphisms between two function spaces that correspond to n-branched coverings are determined completely. Several equivalent definitions of a Frobenius n-homomorphism are compared and some of their properties are proved. An axiomatic treatment of n-transfers is given in general and properties of n-branched coverings are studied and compared with those of regular coverings.