New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold. The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X -> 1+1 carry such effect module structure, and can be used as predicates. Predicates are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays an important role. It appears here in the form of an adjunction, where we use maps 1 -> X as states. For such a state s and a predicate p, the validity probability s |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of `instruments', using L\"uders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C*-algebras, side-effect appear exclusively in the non-commutative case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems

A universe of processes and some of its guises

Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can then be converted into mathematical structure. With very little structural effort (i.e. in very abstract terms) and in a very short time span the categorical quantum mechanics (CQM) research program has reproduced a surprisingly large fragment of quantum theory. It also provides new insights both in quantum foundations and in quantum information, and has even resulted in automated reasoning software called `quantomatic' which exploits the deductive power of CQM. In this paper we complement the available material by not requiring prior knowledge of category theory, and by pointing at connections to previous and current developments in the foundations of physics. This research program is also in close synergy with developments elsewhere, for example in representation theory, quantum algebra, knot theory, topological quantum field theory and several other areas.Comment: Invited chapter in: "Deep Beauty: Understanding the Quantum World through Mathematical Innovation", H. Halvorson, ed., Cambridge University Press, forthcoming. (as usual, many pictures

Strong Complementarity and Non-locality in Categorical Quantum Mechanics

Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.Comment: 15 pages (incl. 5 appendix). To appear: LiCS 201

Higher Theory and the Three Problems of Physics

According to the Butterfield--Isham proposal, to understand quantum gravity we must revise the way we view the universe of mathematics. However, this paper demonstrates that the current elaborations of this programme neglect quantum interactions. The paper then introduces the Faddeev--Mickelsson anomaly which obstructs the renormalization of Yang--Mills theory, suggesting that to theorise on many-particle systems requires a many-topos view of mathematics itself: higher theory. As our main contribution, the topos theoretic framework is used to conceptualise the fact that there are principally three different quantisation problems, the differences of which have been ignored not just by topos physicists but by most philosophers of science. We further argue that if higher theory proves out to be necessary for understanding quantum gravity, its implications to philosophy will be foundational: higher theory challenges the propositional concept of truth and thus the very meaning of theorising in science.Comment: 23 pages, 1 table

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi