The Great Art of Modeling
Johan van Benthem
Abstract:
‘Possible worlds semantics’ for modal logic is a widely used
term, sometimes with ominous metaphysical connotations, but
what does this style of modeling involve today? We discuss
three main issues, using epistemic logic as a running example,
and drawing upon both mathematical results and practices in
the expertise of working researchers. Our first question is
a foundational one: how does one associate a type of model
with a language, and what considerations affect that choice?
Our focus is on invariance and definability results, familiar
from the mathematical and computational tradition, though less
so in philosophy. The second question is less deep, but maybe
even more challenging in practice: once we have chosen a type
of models for a language, how does one select and then maintain
models appropriate to concrete scenarios of application? While
there is a lot of ‘art’ to this in the literature, there is
very little ‘science’ of model construction for modal logics.
We show how this works in dynamic epistemic logics, and
identify some current challenges for a true ‘modeling theory’
as opposed to the more abstract usual ‘model theory’. Finally,
we discuss the pervasive tension between ‘thin’ and ‘thick’
worlds in modal logic, using examples from game theory, and
pointing out how the contrast can be made fruitful.