Wlodzimierz Rabinowicz

How to Model Relational Belief Revision

Some years ago, we proposed a generalization of the well-known approach to belief revision due to Peter Gardenfors (cf. Gardenfors 1988). According to him, for each theory G (i.e., each set of propositions closed under logical consequence) and each proposition A, there is a unique theory, G*A, which would be the result of revising G with A as new piece of information. There is a unique theory which would constitute the revision of G with A. Thus, belief revision is seen as a function. Our proposal was to view belief revision as a relation rather than as a function on theories. The idea was to allow for there being several equally reasonable revisions of a theory with a given proposition. If G and H are theories, and A is a proposition, then GRAH is to be read as: H is an admissible revision of G with A. (Cf. Lind-strom and Rabinowicz, 1989 and 1990.)

Actual truth, possible knowledge

F.B. Fitch is credited with a simple argument purporting to show that the verificationist claim: (Ver) Truth implies Knowability, leads to the unacceptable conclusion: Truth implies Knowledge.

Intensions and Extensions

Consider two sentences: (a) Every father is a parent (b) Either Socrates is a philosopher or it is not the case that Socrates is a philosopher.

Universality and Universalizability

While the relevance closure of a proposition, X, is the set of all w such that R+⊆X, we shall define the deontic closure of X as the set of all w such that D(w⊆X. (D(w)—{v∊W:wDv}.) Thus, the deontic closure of X is the proposition true in exactly those worlds in which X is obligatory. In other words, the deontic closure of X is nothing other than the proposition that X is obligatory.

Leibnizianism Once Again

What about Leibnizians? Do individuals matter in their view? In a sense, they do matter very much. According to Leibnizians, the individuals involved in a situation can never be replaced without changing some of the universal aspects of the situation. Leibnizianism amounts to the claim that individual differences between situations always imply some universal differences. To put it formally, for any w, v∊W, and any individual X⊆W such that w∊X and \([v \notin X\), there is some universal Y⊆W such that w∊Y and \([v \notin Y\).(It is easy to ascertain that this condition is equivalent to the assumption that C is an identity relation.)

Morality Without Purity

Let us recall our formulation of the Principle of Universalizability in terms of the concept of automorphism and the relation D: D stays invariant under automorphisms.

Extensions of Leibnizianism

Leibnizianism reduces exact similarity to identity. In section 8.3, we suggested that there may exist stronger variants of Leibnizianism, which reduce R to R+. Such a reduction must consist in the derivation of the condition (RR+), according to which R+ includes R. This is sufficient, because the converse of this condition trivially follows from any variant of D 10.3. Before we start searching for such extensions of Leibnizianism, let us recall that Leibnizianism, taken by itself, is fully compatible with the negation of (RR+). This is shown by our Example E in section 10.3.

Universality and Relevance

In section 8.2, we suggested that R+ and the concept of a universal aspect are sufficient for the definition of the relation R. Now it is time to construct such a definition.

Universality and Intensions

Consider two groups of intensional propositions: 1. (a) Every father is a parent; (b) The discovery of radioactivity was made by a woman; (c) There are people who are professional politicians. 2. (a) Either Socrates is a philosopher or it is not the case that Socrates is a philosopher; (b) Marie Curie-Sklodowska is a woman; (c) Jimmy Carter is a professional politician.

Universalizability in Morals and Elsewhere

By an n-place operation on propositions we shall understand a function which transforms n-tuples of propositions into propositions. In particular, deontic closure is a 1-place operation on propositions. The same applies to the operations of complement and relevance closure. As additional illustrations, let us consider the following: Unavoidability. If X⊆W then Y shall be said to bethe unavoidability closure of X iff Y is the set of worlds in whichX is unavoidable, i.e., iff Y is the set of worlds in which X is the case and in which nothing can be done—no actions can be performed—which would prevent X’s being the case. In section 3.2, we introduced the alternative-relation A on W. D was thought to be included in A and perhaps even to be definable in terms of this relation. Clearly, the precise nature of A may be fixed in a number of different ways, and different interpretations of A will result in different specifications of D. However, the following interpretation of A seems to be especially natural: wAv iff everything that is unavoidable in w obtains in v. Given this interpretation, the unavoidability closure of X turns out to be identical with {w∊W: ∀v∊W(wAv → v∊X)}—just as the deontic closure of X={w∊W:∀v(wDv→v∊X)}.