Handling Inconsistencies in the Early Calculus

The early calculus is a popular example of an inconsistent but fruitful scientific theory. This paper is concerned with the formalisation of reasoning processes based on this inconsistent theory. First it is shown how a formal reconstruction in terms of a sub-classical negation leads to triviality. This is followed by the evaluation of the chunk and permeate mechanism (C&P) proposed by Brown and Priest in (Journal of Philosophical Logic, 33(4), 379–388, 2004) to obtain a non-trivial formalisation of the early infinitesimal calculus. Different shortcomings of this application of C&P as an explication of inconsistency tolerant reasoning are pointed out, both conceptual and technical. To remedy these shortcomings, an adaptive logic is proposed that allows for conditional permeations of formulas under the assumption of consistency preservation. First the adaptive logic is defined and explained and thereafter it is demonstrated how this adaptive logic remedies the defects C&P suffered from.