Visualization in propositional logic

Abstract

We are visually exploring a current problem in propositional logic related to information processing, specifically n-traces. Traces represent subsets of possible consequences which can be inferred classically from partitions of the set of inputs. We are interested in the relationship between a given set of Boolean inputs and its respective trace(s). Let /spl Sigma/ be a set of sentences or data. In particular, suppose that /spl Sigma/ represents information received by a central processor from N distinct channels. Each channel is self-consistent, but the distinct channels may supply conflicting data. Then /spl Sigma/ will contain no sentence of the form /spl alpha//spl and//spl sim//spl alpha/, but it may contain both the sentences /spl alpha/ and /spl sim//spl alpha/. If the processor must draw inferences from /spl Sigma/ using methods of standard logic, then when separate channels supply conflicting information, the processor is justified in inferring every sentence of the language. To avoid this, we must give the processor some comparatively conservative inference strategy. One such strategy redeploys standard inferential methods in a way that introduces a relative measure of incoherence and provides a formulation for incoherence-tolerant inference. We define l(/spl Sigma/), the incoherence level of /spl Sigma/, as the cardinal of the least partition of /spl Sigma/ into consistent subsets. Then any sentence can be inferred from /spl Sigma/ if it is inferrable by standard methods from at least one element of every l(/spl Sigma/)-partition of /spl Sigma/. We define a set T/sup l/ that is an l(/spl Sigma/)-trace over /spl Sigma/ for which the processor will infer /spl beta/ if /spl beta/ is a standard consequence of every element of T/sup l/.