Robert Stalnaker recently proposed a simple theory of propositions using the notion of a set of propositions being consistent, and conjectured that this theory is equivalent to the claim that propositions form a complete atomic Boolean algebra. This paper clarifies and confirms this conjecture. Stalnaker also noted that some of the principles of his theory may be given up, depending on the intended notion of proposition. This paper therefore also investigates weakened constraints on consistency and the corresponding classes of Boolean algebras.
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I would like to thank Robert Stalnaker for email correspondence out of which this paper developed. The proofs of Lemmas 5 and 24 are due to Stalnaker (p.c.), as is the suggestion to consider the principle of upward closure; I am grateful for being able to include them here. Thanks also to a reviewer for this journal, as well as to Andrew Bacon, Michael Caie, Cian Dorr and Jeremy Goodman for very helpful discussion.
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Fritz, P. On Stalnaker’s Simple Theory of Propositions. J Philos Logic (2020). https://doi.org/10.1007/s10992-020-09557-6
- Boolean algebras
- Robert Stalnaker