Bohrification

New foundations for quantum logic and quantum spaces are constructed by merging algebraic quantum theory and topos theory. Interpreting Bohr's "doctrine of classical concepts" mathematically, given a quantum theory described by a noncommutative C*-algebra A, we construct a topos T(A), which contains the "Bohrification" B of A as an internal commutative C*-algebra. Then B has a spectrum, a locale internal to T(A), the external description S(A) of which we interpret as the "Bohrified" phase space of the physical system. As in classical physics, the open subsets of S(A) correspond to (atomic) propositions, so that the "Bohrified" quantum logic of A is given by the Heyting algebra structure of S(A). The key difference between this logic and its classical counterpart is that the former does not satisfy the law of the excluded middle, and hence is intuitionistic. When A contains sufficiently many projections (e.g. when A is a von Neumann algebra, or, more generally, a Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be compared with the traditional quantum logic, i.e. the orthomodular lattice of projections in A. This time, the main difference is that the former is distributive (even when A is noncommutative), while the latter is not. This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in "Deep Beauty" (ed. H. Halvorson

Lifting Coalgebra Modalities and IMELL Model Structure to Eilenberg-Moore Categories

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic (IMELL), known as a linear category, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by R. Blute and P. Scott\u27s work on categories of modules of Hopf algebras as models of linear logic, we study Eilenberg-Moore categories of monads as models of IMELL. We define an IMELL lifting monad on a linear category as a Hopf monad - in the Bruguieres, Lack, and Virelizier sense - with a mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to Eilenberg-Moore categories of IMELL lifting monads. We explain how monoids in the Eilenberg-Moore category of the monoidal coalgebra modality can induce IMELL lifting monads and provide sources for such monoids. Along the way, we also define mixed distributive laws of bimonads over coalgebra modalities and lifting differential category structure to Eilenberg-Moore categories of exponential lifting monads

A Distributive Theory of Criminal Law

In criminal law circles, the accepted wisdom is that there are two and only two true justifications of punishment-retributivism and utilitarianism. The multitude of moral claims about punishment may thus be reduced to two propositions: (1) punishment should be imposed because defendants deserve it, and (2) punishment should be imposed because it makes society safer. At the same time, most penal scholars notice the trend in criminal law to de-emphasize intent, centralize harm, and focus on victims, but they largely write off this trend as an irrational return to antiquated notions of vengeance. This Article asserts that there is in fact a distributive logic to the changes in current criminal law. The distributive theory of criminal law holds that an offender ought to be punished, not because he is culpable or because punishment increases net security, but because punishment appropriately distributes pleasure and pain between the offender and victim. Criminal laws are accordingly distributive when they mete out punishment for the purpose of ensuring victim welfare. This Article demonstrates how distribution both explains the traditionally troubling criminal law doctrines of felony murder and the attempt-crime divide, and makes sense of current victim-centered reforms. Understanding much of modern criminal law as distribution highlights an interesting political contradiction. For the past few decades, one, if not the most, dominant political message has emphasized rigorous individualism and has held that the state is devoid of power to deprive a faultless person of goods (or rights\u27) in order to ensure the welfare of another. But many who condemn distribution through the civil law or tax system embrace punishment of faultless defendants to distribute satisfaction to crime victims. Exposing criminal law as distributionist undermines these individuals\u27 claimed pre-political commitment against government distribution

Bialgebraic Semantics for Logic Programming

Bialgebrae provide an abstract framework encompassing the semantics of different kinds of computational models. In this paper we propose a bialgebraic approach to the semantics of logic programming. Our methodology is to study logic programs as reactive systems and exploit abstract techniques developed in that setting. First we use saturation to model the operational semantics of logic programs as coalgebrae on presheaves. Then, we make explicit the underlying algebraic structure by using bialgebrae on presheaves. The resulting semantics turns out to be compositional with respect to conjunction and term substitution. Also, it encodes a parallel model of computation, whose soundness is guaranteed by a built-in notion of synchronisation between different threads