Reapproaching Ramsey: Conditionals and Iterated Belief Change in the Spirit of AGM

According to the Ramsey Test, conditionals reflect changes of beliefs: α?>?β is accepted in a belief state iff β is accepted in the minimal revision of it that is necessary to accommodate α. Since Gärdenfors’s seminal paper of 1986, a series of impossibility theorems (“triviality theorems”) has seemed to show that the Ramsey test is not a viable analysis of conditionals if it is combined with AGM-type belief revision models. I argue that it is possible to endorse that Ramsey test for conditionals while staying true to the spirit of AGM. A main focus lies on AGM’s condition of Preservation according to which the original belief set should be fully retained after a revision by information that is consistent with it. I use concrete representations of belief states and (iterated) revisions of belief states as semantic models for (nested) conditionals. Among the four most natural qualitative models for iterated belief change, two are identified that indeed allow us to combine the Ramsey test with Preservation in the language containing only flat conditionals of the form α?>?β. It is shown, however, that Preservation for this simple language enforces a violation of Preservation for nested conditionals of the form α?>?(β?>?γ). In such languages, no two belief sets are ordered by strict subset inclusion. I argue that it has been wrong right from the start to expect that Preservation holds in languages containing nested conditionals.     

Belief revision,epistemic conditionals and the Ramsey test

Epistemic conditionals have often been thought to satisfy the Ramsey test (RT): If A, then B is acceptable in a belief state G if and only if B should be accepted upon revising G with A. But as Peter Gärdenfors has shown, RT conflicts with the intuitively plausible condition of Preservation on belief revision. We investigate what happens if (a) RT is retained while Preservation is weakened, or (b) vice versa. We also generalize Gärdenfors' approach by treating belief revision as a relation rather than as a function.In our semantic approach, the same relation is used to model belief revision and to give truth-conditions for conditionals. The approach validates a weak version of the Ramsey Test (WRR) — essentially, a restriction of RT to maximally consistent belief states.We prove that alternatives (a) and (b) are both consistent, but argue that (b) is philosophically more promising. Gärdenfors' belief-revision axioms are compatible with WRR together with RT from left to right: the only direction of the test that is defensible on intuitive grounds.An earlier version of this paper was presented at the conference on the dynamics of knowledge and belief at Lund University, 24–26 August 1989. We wish to thank Sven Danielsson, Peter Gärdenfors, Sören Halldén, David Makinson, Hugh Mellor, Michael Morreau, Nils-Eric Sahlin and Brian Skyrms for their very helpful suggestions and remarks. We are also grateful for thought-provoking criticism and comments from two anonymous referees.     

Iterated revision and minimal change of conditional beliefs

We describe a model of iterated belief revision that extends the AGM theory of revision to account for the effect of a revision on the conditional beliefs of an agent. In particular, this model ensures that an agent makes as few changes as possible to the conditional component of its belief set. Adopting the Ramsey test, minimal conditional revision provides acceptance conditions for arbitrary right-nested conditionals. We show that problem of determining acceptance of any such nested conditional can be reduced to acceptance tests for unnested conditionals. Thus, iterated revision can be accomplished in a virtual manner, using uniterated revision.     

Minimal Belief Change and the Pareto Principle

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decision-theoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory – a deductively closed set of sentences or propositions – and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of constraints than the standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. The Ramsey test is a well-known way of establishing connections between belief revision postulates and axioms for conditionals (“if p, then q”). Pareto-minimal theory change corresponds exactly to three characteristic axioms of counterfactual systems: a theory revision operator that satisfies the Ramsey test validates these axioms if and only if the revision operator is Pareto-minimal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Two notions of epistemic validity

How to accept a conditional? F. P. Ramsey proposed the following test in (Ramsey 1990). (RT) ‘If A, then B’ must be accepted with respect to the current epistemic state iff the minimal hypothetical change of it needed to accept A also requires accepting B. In this article we propose a formulation of (RT), which unlike some of its predecessors, is compatible with our best theory of belief revision, the so-called AGM theory (see (Gärdenfors 1988), chapters 1–5 for a survey). The new test, which, we claim, encodes some of the crucial insights defended by F. P. Ramsey in (Ramsey 1990), is used to study the conditionals epistemically validated by the AGM postulates. Our notion of validity (PV) is compared with the notion of negative validity (NV) used by Gärdenfors in (Gärdenfors 1988). It is observed that the notions of PV and NV will in general differ and that when these differences arise it is the notion of PV that is preferable. Finally we compare our formulation of the Ramsey test with a previous formulation offered by Gärdenfors (GRT). We show that any attempt to interpret (GRT) as delivering acceptance conditions for Ramsey's conditionals is doomed to failure.     

Undercutting and the Ramsey test for conditionals

There is an important class of conditionals whose assertibility conditions are not given by the Ramsey test but by an inductive extension of that test. Such inductive Ramsey conditionals fail to satisfy some of the core properties of plain conditionals. Associated principles of nonmonotonic inference should not be assumed to hold generally if interpretations in terms of induction or appeals to total evidence are not to be ruled out.     

God − Moore = Ramsey (A Reply to Chalmers and Hájek)

Famously, Frank P. Ramsey suggested a test for the acceptability of conditionals. Recently, David Chalmers and Alan Hájek (2007) have criticized a qualitative variant of the Ramsey test for indicative conditionals. In this paper we argue for the following three claims: (i) Chalmers and Hájek are right that the variant of the Ramsey test that they attack is not the correct way of spelling out an acceptability test for indicative conditionals. But there is a suppositional variant of the Ramsey test which is still stated in purely qualitative terms, which avoids the problems, and which looks correct. (ii) While the variant of the Ramsey test that Chalmers and Hájek criticize is not correct, it is still a good approximation of a correct formulation of the Ramsey test which may be usefully employed in various contexts. (iii) The variant of the Ramsey test that Chalmers and Hájek suggest as a substitute for the deficient version of the Ramsey test is itself subject to worries similar to those raised by Chalmers and Hájek, if it is given a non-suppositional interpretation.     

The Ramsey Test and Conditional Semantics

Proponents of the projection strategy take an epistemic rule for the evaluation of English conditionals, the Ramsey test, as clue to the truth-conditional semantics of conditionals. They also construe English conditionals as stronger than the material conditional. Given plausible assumptions, however, the Ramsey test induces the semantics of the material conditional. The alleged link between Ramsey test and truth conditions stronger than those of the material conditional can be saved by construing conditionals as ternary, rather than binary, propositional functions with a hidden contextual parameter. But such a ternary construal raises problems of its own.     

Moore-paradoxical belief, conscious belief and the epistemic Ramsey test

Chalmers and Hájek argue that on an epistemic reading of Ramsey??s test for the rational acceptability of conditionals, it is faulty. They claim that applying the test to each of a certain pair of conditionals requires one to think that one is omniscient or infallible, unless one forms irrational Moore-paradoxical beliefs. I show that this claim is false. The epistemic Ramsey test is indeed faulty. Applying it requires that one think of anyone as all-believing and if one is rational, to think of anyone as infallible-if-rational. But this is not because of Moore-paradoxical beliefs. Rather it is because applying the test requires a certain supposition about conscious belief. It is important to understand the nature of this supposition.     

Conditional Ranking Revision

In the context of a general framework for belief dynamics which interprets revision as doxastic constraint satisfaction, we discuss a proposal for revising quasi-probabilistic belief measures with finite sets of graded conditionals. The belief states are ranking measures with divisible values (generalizing Spohn’s epistemology), and the conditionals are interpreted as ranking constraints. The approach is inspired by the minimal information paradigm and based on the principle-guided canonical construction of a ranking model of the input conditionals. This is achieved by extending techniques known from conditional default reasoning. We give an overview of how it handles different principles for conditional and parallel revision and compare it with similar accounts.     

New surprises for the Ramsey Test

In contemporary discussions of the Ramsey Test for conditionals, it is commonly held that (i) supposing the antecedent of a conditional is adopting a potential state of full belief, and (ii) Modus Ponens is a valid rule of inference. I argue on the basis of Thomason Conditionals (such as ‘If Sally is deceiving, I do not believe it’) and Moore’s Paradox that both claims are wrong. I then develop a double-indexed Update Semantics for conditionals which takes these two results into account while doing justice to the key intuitions underlying the Ramsey Test. The semantics is extended to cover some further phenomena, including the recent observation that epistemic modal operators give rise to something very like, but also very unlike, Moore’s Paradox.     

Partial monotonicity and a new version of the Ramsey test

We introduce two new belief revision axioms: partial monotonicity and consequence correctness. We show that partial monotonicity is consistent with but independent of the full set of axioms for a Gärdenfors belief revision sytem. In contrast to the Gärdenfors inconsistency results for certain monotonicity principles, we use partial monotonicity to inform a consistent formalization of the Ramsey test within a belief revision system extended by a conditional operator. We take this to be a technical dissolution of the well-known Gärdenfors dilemma.In addition, we present the consequential correctness axiom as a new measure of minimal revision in terms of the deductive core of a proposition whose support we wish to excise. We survey several syntactic and semantic belief revision systems and evaluate them according to both the Gärdenfors axioms and our new axioms. Furthermore, our algebraic characterization of semantic revision systems provides a useful technical device for analysis and comparison, which we illustrate with several new proofs.Finally, we have a new inconsistency result, which is dual to the Gärdenfors inconsistency results. Any elementary belief revision system that is consequentially correct must violate the Gärdenfors axiom of strong boundedness (K^*8), which we characterize as yet another monotonicity condition.This work was supported by the McDonnell Douglas Independent Research and Development program.     

The Suppositional Ramsey Test and Decision-Instability

I analyse the relationship between the Ramsey Test (RT) for the acceptance of indicative conditionals and the so-called problem of decision-instability. In particular, I argue that the situations which allegedly bring about this problem are troublesome just in case the relevant conditionals are evaluated by non-suppositional versions, e.g. causal/evidential, of the test. In contrast, a suppositional RT, by highlighting the metacognitive nature of the evaluation of indicative conditionals, allows an agent to run a simulation of such evaluation, without yet committing her to the acceptance of such conditionals. I conclude that a suppositional interpretation of RT is superior to its non-suppositional counterparts and by briefly showing that a suppositional RT is compatible with a deliberational decision theory.     

The probability of causal conditionals

Conditionals in natural language are central to reasoning and decision making. A theoretical proposal called the Ramsey test implies the conditional probability hypothesis: that the subjective probability of a natural language conditional, P(if p then q), is the conditional subjective probability, P(q/p). We report three experiments on causal indicative conditionals and related counterfactuals that support this hypothesis. We measured the probabilities people assigned to truth table cases, P(pq), P(p notq), P( notpq) and P( notp notq). From these ratings, we computed three independent predictors, P(p), P(q/p) and P(q/ notp), that we then entered into a regression equation with judged P(if p then q) as the dependent variable. In line with the conditional probability hypothesis, P(q/p) was by far the strongest predictor in our experiments. This result is inconsistent with the claim that causal conditionals are the material conditionals of elementary logic. Instead, it supports the Ramsey test hypothesis, implying that common processes underlie the use of conditionals in reasoning and judgments of conditional probability in decision making.     

On some limits of hypothetical thinking

Faced with extreme demands, hypothetical thinking runs the danger of total failure. Paradoxical propositions such as the Liar (“I am lying”) provide an opportunity to test it to its limits, while the Liar's nonparadoxical counterpart, the Truthteller (“I am telling the truth”), provides a useful comparison. Two experiments are reported, one with abstract materials (“If I am a knave then I live in Emerald City”) and one with belief-laden materials (a judge says: “If I am a knave then I enjoy pop music”). In both experiments, conditionals with Truthteller-type antecedents were “collapsed” to responses of conditional probability closely resembling estimates of control items. Liar-type antecedents, in contrast, dramatically weakened belief in conditionals in which they were embedded. The results are discussed in the framework of the theory of hypothetical thinking.     

自省主体对Moore-型信息的处理与融合

The main task is to discuss the issue in belief dynamics in which philosophical beliefs and rational introspective agents incorporate Moorean type new information. First, a brief survey is conducted on Moore’s Paradox, and one of its solutions is introduced with the help of Update Semantics. Then, we present a Dynamic Doxastic Logic (DDL) which revises the belief of introspective agents put forward by Lindström &; Rabinowicz. Next, we attempt to incorporate Moorean type new information within the DEL (DDL) framework, as advised by van Benthem, Segerberg et al. Though we maintain the principle of “the primacy of new information” from the literature on traditional belief revision theory, several unsuccessful ways are also presented. We then conclude that some special kind of success (weak success) can still be found in those revision processes although absolute success does not hold. At last, the relevant problem of “learnability” is re-considered through weak success.     

Suppositions, extensionality, and conditionals: a critique of the mental model theory of Johnson-Laird And Byrne (2002)

P. N. Johnson-Laird and R. M. J. Byrne proposed an influential theory of conditionals in which mental models represent logical possibilities and inferences are drawn from the extensions of possibilities that are used to represent conditionals. In this article, the authors argue that the extensional semantics underlying this theory is equivalent to that of the material, truth-functional conditional, at least for what they term "basic" conditionals, concerning arbitrary problem content. On the basis of both logical argument and psychological evidence, the authors propose that this approach is fundamentally mistaken and that conditionals must be viewed within a suppositional theory based on what philosophical logicians call the Ramsey test. The Johnson-Laird and Byrne theory is critically examined with respect to its account of basic conditionals, nonbasic conditionals, and counterfactuals.     

Rational belief change,popper functions and counterfactuals

This paper uses Popper's treatment of probability and an epistemic constraint on probability assignments to conditionals to extend the Bayesian representation of rational belief so that revision of previously accepted evidence is allowed for. Results of this extension include an epistemic semantics for Lewis' theory of counterfactual conditionals and a representation for one kind of conceptual change.