Notes on Mathematical Modal Logic
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چکیده
These notes were written by Wang Yafeng following a course of three intensive lectures on classical themes in mathematical modal logic given by Johan van Benthem in the Berkeley-Stanford Logic Circle, San Francisco, May 2015. Some of this material was collected in the monograph van Benthem [13], [16], other parts come from later publications. We will provide some references to further relevant work in this document, but our bibliography is not self-contained. Our text starts with a model-based perspective on modal logic. From this perspective, modal logic is just a special fragment of first order logic with certain syntactic restrictions. More precisely, modal logic can be translated into first order logic via the standard translation ST: M, s φ iff M, s ST (φ). For instance, p can be translated as ∀y(Rxy → Ry). We then move on to a frame-based perspective on modal logic. A frame is simply a model stripped of its valuation, and the validity of a modal formula in a frame can be translated into second order logic: F, s φ iff ∀~ PST (φ). From this perspective, modal logic is a special fragment of monadic second order logic with all the quantifiers out in front (called Π1 formulas). Interestingly, some modal formulas also correspond to first-order conditions of independent interest, and we would like to understand why they do so. Many important results in the classical era of modal logic have to do with either of these perspectives. In the first half of these notes, we will mainly prove two results from the model-based perspective: the modal invariance theorem and the modal Lindström theorem. As a highlight from the frame based perspective, we will prove the so-called Sahlqvist Theorem that covers a wide range of modal axioms from the literature, and we will also present some key examples of modal formulas that lack first-order correspondents.
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