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|Title:||A Study of Berntsonian Modal Logics|
|Abstract:||In [Ber20], Daniel Berntson argues for accepting a conditional logic whose models are not transitive but instead have an alternative model property. In this paper, we will consider a variety of model properties similar to those considered in [Ber20] as possible alternatives for transitivity. However, we restrict our attention purely to modal logic rather than conditional logic, as was considered by Berntson. For each of the Berntsonian model properties we consider, we will first derive a characterizing formula for the property, with the additional requirement that the resulting logic from adding the characterizing formula as an additional axiom to the minimal modal logic, K, is complete for the set of frames it characterizes. However, for two properties being considered, we will see that there is no characterizing formula. For one of these, strict connectedness, we introduce the weaker notion of a quasi-characterizing formula and show that over the class of generated frames, strict connectedness is equivalent to a combination of other properties under consideration. In the remaining sections, we explore the connections between the various properties, both syntactically and semantically. We will give a complete characterization of the implications that exist among the properties, giving semantic and syntactic proofs for each implication that holds and demonstrating semantically that no further implications can hold.|
|Type of Material:||Princeton University Senior Theses|
|Appears in Collections:||Mathematics, 1934-2021|
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