The Evolution of Reason: Logic as a Branch of Biology

By William S. Cooper | Go to book overview

Appendix
Formal Theory

This appendix provides a proof of Theorem 4.1. More generally, it offers some minimal mathematical foundations for an evolutionary derivation of classical decision and utility theory, subjective probability theory, and deductive logic. The definitions and theorems constitute the mathematical underpinnings of the theory outlined in the main text. The conditions of Model 1 are assumed.


DEFINING RATIONALITY

Let S be the set of states of nature for a decision problem. The members of S are denoted by variables s, s', etc. and its subsets are A, B, etc. Let F be the set of all potential consequences f, g, h, etc. involved in the problem. An act for the problem is a function f from S into F.

A choice function is a function C mapping every set of acts for the problem to one of the set's members. In the case of sets with just two members, for any acts f, g clearly either C ({f, g}) =f or C ({f, g}) =g. Choice functions are interpreted behaviorally. That is, C ({f, g}) is the act that an organism with the choice function C actually chooses or would choose if confronted with a choice between f and g. For brevity, expressions of form C ({f, g}) =f will be written f C g, which may be read “f is chosen over g.

A strict preference relation > between acts is definable in terms of the choice function. Clearly it should be required that if f>g then f C g. The converse, however, is unwanted because f C g can happen also in case of indifference. It suffices to require for f>g, beside f C g, that if the function f or the function g is modified slightly (i. e. changed on any sufficiently small set of states) then the choice between

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