A three-valued doxastic logic based on Kleene’s and Bochvar’s ideas

Janusz Wesserling

doi:10.21697/spch.2019.55.4.04

Published: 2019-12-30

Vol. 55 No. 4 (2019)

A three-valued doxastic logic based on Kleene’s and Bochvar’s ideas

Janusz Wesserling

Studia Philosophiae Christianae

Section: Papers

https://doi.org/10.21697/spch.2019.55.4.04

Abstract

In this paper, I shall propose the construction of a three-valued logic of beliefs, which I call: LSB3_1 (short for: Three-valued Logic for a type of Strong Belief). I shall also state and prove the completeness of LSB3_1 with respect to a given semantics. LSB3_1 is based on preformal assumptions and intuitions, which are stated in section 1. Section 2 includes the syntax and division of LSB3_1 statements into internal and external. Section 3 presents the semantics of LSB3_1, as well as a number of tautologies and non-tautological formulae in LSB3_1 with their intuitive interpretation. The axiomatic system for LSB3_1 and its comparison to Kleene’s strong logic are provided in section 4. The completeness theorem for LSB3_1 is presented in section 5. I shall define the term conjunctive normal form and provide lemmas which lead to proving the reduction of the LSB3_1 language formulae before proving the completeness theorem.

Keywords:

doxastic logic, strong belief, Kleene’s strong three-valued logic, internal connectives, external connectives, completeness theorem

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Wesserling, J. (2019). A three-valued doxastic logic based on Kleene’s and Bochvar’s ideas. Studia Philosophiae Christianae, 55(4), 89–113. https://doi.org/10.21697/spch.2019.55.4.04

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