We present and show topological completeness for LB, the logic of the topological border. LB is also a logic of epistemic ignorance. Also, we present and show completeness for LUT, the logic of unknown truths. A simple topological completeness proof for S4 is also presented using a T1 space.
We introduce an operator to represent the simple notion of being wrong. Read Wp to mean: the agent is wrong about p . Being wrong about p means believing p though p is false. We add this operator to the language of propositional logic and study it. We introduce a canonical model for logics of being wrong, show completeness for the minimal logic of being wrong and various other systems. En route we examine the expressiveness of the language. In conclusion, (...) we discuss an open question regarding K4. (shrink)
In this highly original text, Christopher Steinsvold explores an alternative semantics for logics of rational belief. Topologies, as mathematical objects, are typically interpreted in terms of space; here topologies are re-interpreted in terms of an agent with rational beliefs. The topological semantics tells us that the agent can never, in principle, know everything; that the agent's beliefs can never be complete. -/- A number of completeness proofs are given for a variety of logics of rational belief. Beyond this, the author (...) explores the philosophical question of why our beliefs can never be complete, and considers the possibility that a totality of truths is a dialethia. -/- This work will be of interest to all philosophers interested in epistemology, and modal logicians as well. (shrink)
We show the Boxdot Conjecture holds for a limited but familiar range of Lemmon-Scott axioms. We re-introduce the language of essence and accident, first introduced by J. Marcos, and show how it aids our strategy.
Patrick Grim has presented arguments supporting the intuition that any notion of a totality of truths is incoherent. We suggest a natural semantics for various logics of belief which reflect Grim’s intuition. The semantics is a topological semantics, and we suggest that the condition can be interpreted to reflect Grim’s intuition. Beyond this, we present a natural canonical topological model for K4 and KD4.
A message appears on the moon. It is legible from Earth, and almost no one knows how it was created. Markus West leads the government’s investigation to find the creator. -/- The message is simple and familiar. But those three words, written in blazing crimson letters on the lunar surface, will foster the strangest revolution humankind has ever endured and make Markus West wish he was never involved. -/- The message is ‘Drink Diet Coke.’ -/- When Coca-Cola denies responsibility, global (...) annoyance becomes indignation. And when his investigation confirms Coca-Cola’s innocence, Markus West becomes one of the most hated men on Earth. -/- Later, five miles above the White House, a cylinder is discovered floating in the night. It is 400 feet tall, 250 feet in diameter, and exactly resembles a can of Campbell’s Chicken Noodle Soup. Nearly everyone thinks the cylinder is a promotional stunt gone wrong, just like the lunar advertisement. And this is exactly what the alien in the cylinder wants people to think. -/- Ralph, an eccentric extraterrestrial who’s been hiding on the moon, needs Markus’s help to personally deliver a dark warning to the White House. Ralph has a big heart, a fetish for Andy Warhol, and a dangerous plan to save the world. (shrink)
Given the frequency of human error, it seems rational to believe that some of our own rational beliefs are false. This is the axiom of epistemic modesty. Unfortunately, using standard propositional quantification, and the usual relational semantics, this axiom is semantically inconsistent with a common logic for rational belief, namely KD45. Here we explore two alternative semantics for KD45 and the axiom of epistemic modesty. The first uses the usual relational semantics and bisimulation quantifiers. The second uses a topological semantics (...) and standard propositional quantification. We show the two different semantics validate many of the same formulas, though we do not know whether they validate exactly the same formulas. Along the way we address various philosophical concerns. (shrink)
Interpreting the diamond of modal logic as the derivative, we present a topological canonical model for extensions of K4 and show completeness for various logics. We also show that if a logic is topologically canonical, then it is relationally canonical.
The Boxdot Conjecture is shown to hold for a novel class of modal systems. Each system in this class is K plus an instance of a natural generalization of the McKinsey axiom. [Note from the editors: This paper was accepted for publication in 2011. It should have been published in 2014. The lateness of the appearance of the article is due entirely to an editorial oversight.].