referential theory of meaning See MEANING, PHI -. LOSOPHY OF LANGUAG. reflection principles, two varieties of internal statements related to correctness in formal axiomatic systems. (1) Proof-theoretic reflection principles are formulated for effectively presented systems S that contain a modicum of elementary number theory sufficient to arithmetize their own syntactic notions, as done by Kurt Gödel in his 1931 work on incompleteness. Let ProvS(x) express that x is the Gödel number of a statement provable in S, and let nA be the number of A, for any statement A of S. The weakest reflection principle considered for S is the collection Rfn(S) of all statements of the form ProvS(nA) P A, which express that if A is provable from S then A (is true). The proposition ConS expressing the consistency of S is a consequence of Rfn(S) (obtained by taking A to be a disprovable statement). Thus, by Gödel’s second incompleteness theorem, Rfn(S) is stronger than S if S is consistent. Reflection principles are used in the construction of ordinal logics as a systematic means of overcoming incompleteness. (2) Set-theoretic reflection principles are formulated for systems S of axiomatic set theory, such as ZF (Zermelo-Fraenkel). In the simplest form they express that any property A in the language of S that holds of the universe of ‘all’ sets, already holds of a portion of that universe coextensive with some set x. This takes the form A P (Dx)A(x) where in A(x) all quantifiers of A are relativized to x. In contrast to proof-theoretic reflection principles, these may be established as theorems of ZF.
See also GÖDEL’S INCOMPLETENESS THEO- REMS , ORDINAL LOGIC , SET THEOR. S.Fe. reflective equilibrium, as usually conceived, a coherence method for justifying evaluative principles and theories. The method was first described by Goodman, who proposed it be used to justify deductive and inductive principles. According to Goodman (Fact, Fiction and Forecast, 1965), a particular deductive inference is justified by its conforming with deductive principles, but these principles are justified in their turn by conforming with accepted deductive practice. The idea, then, is that justified inferences and principles are those that emerge from a process of mutual adjustment, with principles being revised when they sanction inferences we cannot bring ourselves to accept, and particular inferences being rejected when they conflict with rules we are unwilling to revise. Thus, neither principles nor particular inferences are epistemically privileged. At least in principle, everything is liable to revision.
Rawls further articulated the method of reflective equilibrium and applied it in ethics. According to Rawls (A Theory of Justice, 1971), inquiry begins with considered moral judgments, i.e., judgments about which we are confident and which are free from common sources of error, e.g., ignorance of facts, insufficient reflection, or emotional agitation. According to narrow reflective equilibrium, ethical principles are justified by bringing them into coherence with our considered moral judgments through a process of mutual adjustment. Rawls, however, pursues a wide reflective equilibrium. Wide equilibrium is attained by proceeding to consider alternatives to the moral conception accepted in narrow equilibrium, along with philosophical arguments that might decide among these conceptions. The principles and considered judgments accepted in narrow equilibrium are then adjusted as seems appropriate. One way to conceive of wide reflective equilibrium is as an effort to construct a coherent system of belief by a process of mutual adjustment to considered moral judgments and moral principles (as in narrow equilibrium) along with the background philosophical, social scientific, and any other relevant beliefs that might figure in the arguments for and against alternative moral conceptions, e.g., metaphysical views regarding the nature of persons. As in Goodman’s original proposal, none of the judgments, principles, or theories involved is privileged: all are open to revision. See also COHERENTISM , RAWLS. M.R.D.