6. Preserving What?

In this essay Gillman Payette and Peter Schotch present an account of the key notions of level and forcing in much greater generality than has been managed in any of the early publications. In terms of this level of generality the hoary notion that correct inference is truth-preserving is carefully examined and found wanting. The authors suggest that consistency preservation is a far more natural approach, and one that can, furthermore, characterize an inference relation. But an examination of the usual account of consistency reveals problems that, in general, can be corrected by means of an auxiliary notion of inference (forcing) which relies upon a kind of generalization of consistency, called level. Preservation of the latter is shown to be another of the properties which characterize a logic and forcing is shown to preserve it. The essay ends with a sketch of a result which locates forcing among all possible level-preserving inference relations.