Sat Jan 21, 2012
60 Evans Hall, 12:10–2 AM
Sherrilyn Roush (University of California, Berkeley)
Reasoning and the Growth of Error
I address several questions about controlling the growth of error introduced by reasoning. For example, if every step of reasoning introduces a new source of error, how can we possibly improve our reliability – as we think we do – by proof-checking, filling in gaps in proofs, or consulting an expert, all of which add steps to our reasoning? I will devote most attention to the topic of closure of knowledge under known implication. How can knowledge be closed if in one step of valid deductive reasoning I can go from knowledge that my car is parked on Main Street to a belief that it has not been stolen since I parked it, which it seems I do not know? I argue that this closure problem is entirely a matter of how fast false positive error in the conclusion grows over deductive reasoning from the premise, and develop a graded model with upper bounds on this growth. It turns out that problem examples like the car theft case, and more radical skeptical cases, can be generated only by committing an error of reasoning.