Conditionals and theory change: Revisions, expansions, and additions

Abstract

ABSTRACT. This paper dwells upon formal models of changes of beliefs, or theories, which are expressed in languages containing a binary conditional connective. After defining the basic concept of a (non-trivial) belief revision model. I present a simple proof of G~irdenfors's (1986) triviality theorem. I claim that on a proper understanding of this theorem we must give up the thesis that consistent revisions ('additions') are to be equated with logical expansions. If negated or 'might' conditionals are interpreted on the basis of 'autoepistemic omniscience', or if autoepistemic modalities (Moore) are admitted, even more severe triviality results ensue. It is argued that additions cannot be philosophically construed as ~parasitic' (Levi) on expansions. In conclusion I outline somed logical consequences of the fact that we must not expect 'monotonic' revisions in languages including conditionals. BELIEF REVISION MODELSBeliefs and theories are expressed by way of sentences of a certain language. In this paper I shall presuppose a language L that contains the operators of propositional logic (-, &, v, ~) and allows the formulation of conditionals of the form 'If A then B'. As in natural language, conditionals will not be given a special syntactical treatment in advance. In particular I do not exclude negations of conditionals.Belief revision models (BRMs) are formal models of the dynamics of epistemic states or theories. A belief set, or theory, is a set of sentences of a language L which is closed under a logic (consequence relation, derivability relation) ~-for L. The logic will be supposed to include classical propositional logic, in particular modus ponens, and the deduction theorem is required to hold for F-. At first we shall consider only one kind of theory change, viz., so-called 'revisions'. By the revision of a theory K we mean the assimilation of a sentence A into K. Theory revisions figure, for example, in processing a new piece of factual information or in following the lines of a hypothetical argument. We can distinguish two cases. If A is consistent witb~ K (under F-), the revision seems to be a matter of routine. But if A is inconsistent with K (under F-), then the affair will become tricky. It is for the sake of such belief-contravening changes If both K and A are F-consistent, so is K*.While (S) and (I) appear to be beyond controversy, (C) could be disputed.1 If the sentence A in (C) is hopelessly out of question then an epistemic or 'theoretical' breakdown of K into inconsistency might be regarded as possible. Although I do not think that (C) really is too strong, one might restrict this requirement to sentences A that are 'intuitively innocent'. The arguments to come would have to undergo a similar restriction at various places, but, so I believe, they would suffer no loss of their persuasive power. For this reason I will employ the simpler and still plausible requirement (C).Notice that (S), (I), and (C) are criteria for BRMs. I have omitted the quantificational prefixes VKE K VA ~ Sentences(L...

A reason maintenance perspective on relevant Ramsey conditionals

“…Rott [37] argues convincingly that (M ) is indeed true, but only trivially so. Infor-mally, once we admit conditionals into belief sets (as per (A K 2)), no belief set can be a proper subset of another.…”

“…In addition, even with closure, success, and consistency reinstated, we will show below that the system avoids the result on several grounds. Notably, it encapsulates variants of the solutions proposed in [37,14,28]. In addition, the triviality trap is also avoided as a result of adopting relevance logic and only one direction of the Ramsey test.…”

“…We shall consider six proposals [37,14,4,28,13,12], each identifying a different set of background assumptions as the culprit, or a different way out of the triviality trap. Except for [4], all proposals attempt to invalidate the use of (M ) in the proof.…”

“…Inspecting Table 1, [37,14,12] take issue with (A * 3) and (A * 4). Most probably [37,14,12] would identify the main glitch in the triviality proof with step 7. 6 Though they agree on the culprit, each of these authors proposes a different way out of the triviality trap.…”

“…6 Though they agree on the culprit, each of these authors proposes a different way out of the triviality trap. Rott [37] informally considers employing a non-monotonic logic instead of Cn 0 . Hansson [14] presents a detailed theory founded on belief base revision.…”