3 research outputs found
On an Intuitionistic Logic for Pragmatics
We reconsider the pragmatic interpretation of intuitionistic logic [21]
regarded as a logic of assertions and their justications and its relations with classical
logic. We recall an extension of this approach to a logic dealing with assertions
and obligations, related by a notion of causal implication [14, 45]. We focus on
the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on
polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the
S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic
that correctly represents the duality between intuitionistic and co-intuitionistic logic,
correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism
as a distributed calculus of coroutines is then used to give an operational
interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear
calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation
of linear co-intuitionism is given as in [9]. Also we remark that by extending the
language of intuitionistic logic we can express the notion of expectation, an assertion
that in all situations the truth of p is possible and that in a logic of expectations
the law of double negation holds. Similarly, extending co-intuitionistic logic, we can
express the notion of conjecture that p, dened as a hypothesis that in some situation
the truth of p is epistemically necessary
Quantum geometry, logic and probability
Quantum geometry on a discrete set means a directed graph with a weight
associated to each arrow defining the quantum metric. However, these `lattice
spacing' weights do not have to be independent of the direction of the arrow.
We use this greater freedom to give a quantum geometric interpretation of
discrete Markov processes with transition probabilities as arrow weights,
namely taking the diffusion form for the
graph Laplacian , potential functions built from the
probabilities, and finite difference in the time direction.
Motivated by this new point of view, we introduce a `discrete Schroedinger
process' as for the Laplacian associated
to a bimodule connection such that the discrete evolution is unitary. We solve
this explicitly for the 2-state graph, finding a 1-parameter family of such
connections and an induced `generalised Markov process' for in
which there is an additional source current built from . We also discuss
our recent work on the quantum geometry of logic in `digital' form over the
field , including de Morgan duality and its possible
generalisations.Comment: 23 pages, 5 figure
INTERMEDIATE LOGICS AND POLYHEDRA
Polyhedra enjoy a peculiar property: every geometric shape with a certain \u201cregularity\u201d \u2013 in specific terms, certain classes of (closed) topological manifolds \u2013 can be captured by a polyhedron via triangulation, that is, by subdividing the geometric shapes into appropriate \u201ctriangles\u201d, called simplices (which, in the 1- and 0-dimensional case, are simply edges and vertices, respectively). Therefore, one might well wonder: what is the intermediate logic of the class of triangulable topological manifolds of a given dimension d? The main result of the present work is to give the answer to this question in the case of 1-dimensional manifolds, that is, the circle and the closed interval