260 research outputs found
Pointwise intersection in neighbourhood modal logic
We study the logic of neighbourhood models with pointwise intersection, as a
means to characterize multi-modal logics. Pointwise intersection takes us from
a set of neighbourhood sets (one for each member of a set
, used to interpret the modality ) to a new neighbourhood set
, which in turn allows us to interpret the operator .
Here, is in the neighbourhood for if and only if equals the
intersection of some . We show that the
notion of pointwise intersection has various applications in epistemic and
doxastic logic, deontic logic, coalition logic, and evidence logic. We then
establish sound and strongly complete axiomatizations for the weakest logic
characterized by pointwise intersection and for a number of variants, using a
new and generally applicable technique for canonical model construction.Comment: Submitted to Advances in Modal Logic 201
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
On a Straw Man in the Philosophy of Science - A Defense of the Received View
I defend the Received View on scientific theories as developed by Carnap, Hempel, and Feigl against a number of criticisms based on misconceptions. First, I dispute the claim that the Received View demands axiomatizations in first order logic, and the further claim that these axiomatizations must include axioms for the mathematics used in the scientific theories. Next, I contend that models are important according to the Received View. Finally, I argue against the claim that the Received View is intended to make the concept of a theory more precise. Rather, it is meant as a generalizable framework for explicating specific theories
Potential infinity, abstraction principles and arithmetic (Leniewski Style)
This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties
Characterising Testing Preorders for Finite Probabilistic Processes
In 1992 Wang & Larsen extended the may- and must preorders of De Nicola and
Hennessy to processes featuring probabilistic as well as nondeterministic
choice. They concluded with two problems that have remained open throughout the
years, namely to find complete axiomatisations and alternative
characterisations for these preorders. This paper solves both problems for
finite processes with silent moves. It characterises the may preorder in terms
of simulation, and the must preorder in terms of failure simulation. It also
gives a characterisation of both preorders using a modal logic. Finally it
axiomatises both preorders over a probabilistic version of CSP.Comment: 33 page
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