Decision LogicGuy HoweDepartment of PhilosophyAustralian National [email protected] 4452SummaryIn this paper, I present a summary of my work to date on an improved formalism for
decision-making. This formalism, which I call Decision Logic, adds binary ?preference
relations? to a logic, generating a partial ordering over sentences from the logic which is
sufficient to represent the kind of ?reasoning? employed in mainstream decision theory.
Beyond this, the formalism can also be used to represent the reasoning behind a given
decision. Introduction
?I will/should do y because X? and ?X thus y? have a similar syntactic and, one feels,
semantic structure. They express a relationship between a set of sentences and a single
sentence, roughly that one is willing to accept the latter whenever one accepts the former.
The natural question at this point is whether decision making should not be formalised as a
logic.In any such formalism, one might hope to draw on the dominant existing formalism, decision
theory, and its adjuncts such as game theory, microeconomics and public choice theory.
These formalisms represent a decision system as a well-ordering of equivalence classes (a
preference or preference relation) on a decision set. The choice any such system makes from
any subset of the decision set is then defined to be the union of all maxima in the contraction
of the preference to the subset (in fact, since a complete ordering is usually insisted on, there
will be only one such maximum set in the traditional theory; however it is part of my intent
here to relax the completeness assumption). I am for simplicity ignoring the possibility of
infinitely ascending chains in the preference.Decision theory and its adjuncts have proceeded by analysing the results of adding extra
constraints to this model, eg probabilistic constraints or constraints obtained from
simultaneous or competing decision systems. It would seem obvious to add logical
constraints to this list. One would expect that most decision sets would have logical
relationships such as entailment between their elements.
Decision theory, if it is to represent logical constraints, should let me argue about orderings
and also to order arguments. If we model an argument with a proof from some logic L, and if
L is truth-functional, then my willingness to accept the conclusion of an argument is entirely
dependent on my willingness to accept its assumptions. Choosing between the consequences
of two or more arguments is thus equivalent to choosing between their respective sets of
assumptions. In such a scheme, reasoned, consistent decision-making (suitably defined) is
equivalent to finding an ordering between subsets of a set of sentences, given an ordering
over that set?s elements.