Quantified Temporal Alethic Boulesic Doxastic Logic
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Logica Universalis
Quantified Temporal Alethic Boulesic Doxastic Logic Daniel R¨onnedal Abstract. The paper develops a set of quantified temporal alethic boulesic doxastic systems. Every system in this set consists of five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. There are no systems in the literature that combine all of these branches of logic. Hence, all systems in this paper are new. Every system is defined both semantically and proof-theoretically. The semantic apparatus consists of a kind of T × W models, and the proof-theoretical apparatus of semantic tableaux. The ‘quantified part’ of the systems includes relational predicates and the identity symbol. The quantifiers are, in effect, a kind of possibilist quantifiers that vary over every object in the domain. The tableaux rules are classical. The alethic part contains two types of modal operators for absolute and historical necessity and possibility. According to ‘boulesic logic’ (the logic of the will), ‘willing’ (‘consenting’, ‘rejecting’, ‘indifference’ and ‘non-indifference’) is a kind of modal operator. Doxastic logic is the logic of beliefs; it treats ‘believing’ (and ‘conceiving’) as a kind of modal operator. I will explore some possible relationships between these different parts, and investigate some principles that include more than one type of logical expression. I will show that every tableau system in the paper is sound and complete with respect to its semantics. Finally, I consider an example of a valid argument and an example of an invalid sentence. I show how one can use semantic tableaux to establish validity and invalidity and read off countermodels. These examples illustrate the philosophical usefulness of the systems that are introduced in this paper. Mathematics Subject Classification. Primary 03B45, Secondary 03B42, 03B44. Keywords. Quantified modal logic, Modal logic, Temporal logic, Boulesic logic, Doxastic logic, Semantic tableaux.
The first version of this paper was finished in 2018. I would like to thank everyone who has commented on the text since then. Thanks also to the editor Jean-Yves B´eziau for helpful guidance.
D. R¨ onnedal
Log. Univers.
1. Introduction The purpose of this paper is to develop a set of quantified temporal alethic boulesic doxastic systems. Every system is semantically defined by some class of T × W models. According to a T × W model, both worlds and times are basic and truth is relativised to world-moment pairs. Hence, a sentence may be true at some world-moment pairs and false at others. For more on the T × W approach, see, for example [113,115]. Our proof theory is built on semantic tableaux. I will introduce a set of semantic tableau systems and prove that they are sound and complete with respect to their semantics. Every quantified temporal alethic boulesic doxastic system includes five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. The quantified, temporal and modal parts are well-known. The doxas
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