Introduction to Traces

The philosophical interest in traces arises from the problem posed by the instance of inconsistencies or contradictions between sources of information. Consider the case in which $\Sigma$ is a subset of the set of sentences of some language. In particular, suppose that $\Sigma$ records information received by a central processor from n distinct channels. Each channel is self-consistent but the distinct channels may supply conflicting data. Then $\Sigma$ will contain no sentence of the form $\alpha \land \lnot\alpha$, but it may contain both the sentences $\alpha, \lnot\alpha$. Now suppose that the processor is required to draw (or test) inferences from $\Sigma$. If the processor adopts the methods of standard logic, then when separate channels have supplied conflicting information, it can justifiably infer every sentence of the language. To avoid this, it must be given some comparatively conservative inference strategy. One such strategy described fully in [2] and [4] re-deploys standard inferential methods in a way which introduces a relative measure of incoherence and provides a formulation for incoherence-tolerant inference.

The incoherence level of $\Sigma$, $l(\Sigma)$, is defined as the cardinal of the least partition of $\Sigma$ into consistent (non-contradictory) subsets. The strategy permits any sentence $\alpha$ to be inferred from $\Sigma$ that is inferrable by standard methods from at least one cell of every $l(\Sigma)$-partition of $\Sigma$.The utility of traces consists in their providing an efficient implementation of that strategy. In the implementation, a set $T^{\it l}\subseteq \wp(\Sigma)$ is an $l(\Sigma)$-trace over $\Sigma$,then the processor will infer $\beta$, if $\beta$ is a standard consequence of every element of $T^{\it l}$.The goal of this paper is to provide an effective procedure by which to distinguish patterns that arise from the process of partitioning a set into consistent cells. If cycles in the production of traces are predictable, a visual solution can be found that would further research towards a completeness proof that relies on a visual approach.

Section 1 describes traces. Section 2 describes patterns that arise in the construction of truth tables for traces and their theoretical interest for the completeness problem. Section 3 introduces L-systems and the relevance to the trace problem. The visualization approach and observations are described in section 4. Section 5 concludes with a synthesis of the different approach in this paper. Section 6 concludes with a discussion about future goals and how they may be achieved.