Rationality in Fragmented Belief Systems

This chapter deals with the question of which notion of rationality best fits with a fragmentation picture of belief that holds that we are mostly rational. According to this picture, coherence is not a requirement of rationality for the entire belief system. Coherence is only rationally required within belief fragments. The chapter argues, however, that fragmentation still needs to offer a different rationality criterion across belief fragments to account for a variety of cases in which we would intuitively ascribe irrationality to the subject. It proposes that the requirement of evidence responsiveness is a good candidate for making sense of the idea that there are certain normative relations in place among beliefs from different belief fragments.

A note on Ohm's rationality criterion for conics

An irreducible quadratic polynomial P(X, Y) in two variables over a field k is called a conic over k. It is called rational if its function field is simple transcendental over k (equivalently if P is parameterisable by rational functions). Ohm's rationality criterion states that P is rational if and only if (i) the locus of P is non-empty and (ii) k is algebraically closed in the function field of P. To show the irredundancy of (ii), Ohm gives an example of a non-rational conic with a non-empty locus. That locus, however, consists of a single point.In this note, we show that a better example cannot exist by showing that if the locus of a conic contains more than one point then it is rational. We also show that the only rational conic whose locus consists of one point is the conic XY + 1 over the field of two elements.