Dave S. Henley

Our online interaction with information-systems may well provide the largest arena of formal logical reasoning in the world today. Presented here is a critique of the foundations of Logic, in which the metaphysical assumptions of such 'closed world' reasoning are contrasted with those of traditional logic. Closed worlds mostly employ a syntactic alternative to formal language namely, recording data in files. Whilst this may be unfamiliar as logical syntax, it is argued here that propositions are expressed by data stored in files which are essentially non-linguistic and so cannot be expressed by simple formulae F(a), with the inference-rules normally used in Logic. Hence, the syntax of data may be said to define a fundamentally new kind of logical form for simple propositions. In this way, the logic of closed systems is shown to be non-classical, differing from traditional logic in its truth-conditions, inferences and metaphysics. This paper will be concerned mainly with how the reference and certain inferences in such a closed system differ metaphysically from classical logic.

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures.

This work proposes a logical analysis for the kind of knowledge or insight provided by Buddhist enlightenment, which is often presented only in the form of contradictions and riddles. The comprehension of contradictions is perplexing to most western logic, and yet developed here is a theory demonstrating how a non truth-functional interpretation can be attached to certain naturalistic contradictions. In this way, the logical and psychological status of Enlightenment can be analysed in a manner consistent with the claims of much Eastern thought. For example, it is argued that by breaking the logical principle of identity, it is possible to understand the contradiction "Everybody's friend is nobody's friend" in such a way as to learn something new about friendship, thereby causing us to modify our concept of friendship. In this way our grasp of contradictions can lead to conceptual change. It is further noted that conflicts in all areas of life can be expressed by contradictions, so that our capacity in general to grasp naturalistic contradictions is also a basis for conflict-resolution -thus justifying the term 'Enlightenment'. This pragmatic comprehension of contradictions is thus shown to formalise, not only the Middle Way of Buddhism, but also Socratic dialectic. Finally, the analysis explains why gnomic utterances in every religion and Zen in particular, are sometimes expected to trigger spiritual enlightenment.