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On the Logic of Belief and Propositional Quantification
We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as âeverything I believe is trueâ or âthere is some-thing that I neither believe nor disbelieve.â Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss
Semigroups with if-then-else and halting programs
The "ifâthenâelse" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using ifâthenâelse. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions â modeling halting programs â we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included
Independences and Partial -Transforms in Bi-Free Probability
In this paper, we examine how various notions of independence in
non-commutative probability theory arise in bi-free probability. We exhibit how
Boolean and monotone independence occur from bi-free pairs of faces and
establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove
that bi-freeness is preserved under tensoring with matrices. Finally, via
combinatorial arguments, we construct partial -transforms in two settings
relating the moments and cumulants of a left-right pair of operators
Constructive version of Boolean algebra
The notion of overlap algebra introduced by G. Sambin provides a constructive
version of complete Boolean algebra. Here we first show some properties
concerning overlap algebras: we prove that the notion of overlap morphism
corresponds classically to that of map preserving arbitrary joins; we provide a
description of atomic set-based overlap algebras in the language of formal
topology, thus giving a predicative characterization of discrete locales; we
show that the power-collection of a set is the free overlap algebra
join-generated from the set. Then, we generalize the concept of overlap algebra
and overlap morphism in various ways to provide constructive versions of the
category of Boolean algebras with maps preserving arbitrary existing joins.Comment: 22 page
Partial orderings with the weak Freese-Nation property
A partial ordering P is said to have the weak Freese-Nation property (WFN) if
there is a mapping f:P ---> [P]^{<= aleph_0} such that, for any a, b in P, if a
<= b then there exists c in f(a) cap f(b) such that a <= c <= b. In this note,
we study the WFN and some of its generalizations. Some features of the class of
BAs with the WFN seem to be quite sensitive to additional axioms of set theory:
e.g., under CH, every ccc cBA has this property while, under b >= aleph_2,
there exists no cBA with the WFN
Quantifiers on languages and codensity monads
This paper contributes to the techniques of topo-algebraic recognition for
languages beyond the regular setting as they relate to logic on words. In
particular, we provide a general construction on recognisers corresponding to
adding one layer of various kinds of quantifiers and prove a corresponding
Reutenauer-type theorem. Our main tools are codensity monads and duality
theory. Our construction hinges on a measure-theoretic characterisation of the
profinite monad of the free S-semimodule monad for finite and commutative
semirings S, which generalises our earlier insight that the Vietoris monad on
Boolean spaces is the codensity monad of the finite powerset functor.Comment: 30 pages. Presentation improved and details of several proofs added.
The main results are unchange
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