The Ramsey test (also called the Ramsey test for conditionals) is a proposal for how to evaluate and accept conditional sentences of the form if p (then) q. Informally, the guiding idea is that, to decide whether to accept a conditional if-p-then-q in a given belief state K, one must hypothetically add p to K and revise K in a minimal and rational way to accommodate p; then, one must accept if-p-then-q if, and only if, q would be accepted in the resulting belief state. In contemporary formulations, the "minimal" change of belief is made precise by a belief revision operation or by a rule for changing probabilities or possible-world rankings, depending on the framework. The test originates in a brief footnote in Frank P. Ramsey's 1929 essay General Propositions and Causality, but has since been developed in several overlapping research traditions, namely, in the theory of § Belief revision, in § Probabilistic approaches to conditionals, in § Possible-worlds semantics, and in dynamic and non-monotonic logics. The Ramsey test has also been shown to generate various "triviality" results when combined with seemingly natural constraints on rational belief change, leading to an extensive literature on how it should be formulated and where its limits lie.
Historical background
Ramsey's footnote Ramsey's discussion of conditionals appears in a single footnote to General Propositions and Causality (1929), posthumously published in The Foundations of Mathematics and Other Logical Essays:If two people are arguing "If p will q?" and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q; so that in a sense "If p, q" and "If p, ¬q" are contradictories. We can say they are fixing their degrees of belief in q given p. If p turns out false, these degrees of belief are rendered void. If either party believes ¬p for certain, the question ceases to mean anything to him except as a question about what follows from certain laws or hypotheses.Ramsey connects conditionals with what he calls "variable hypotheticals", general rules of the form "if anything is F it is G", which, on his view, function as laws or policies guiding future judgement and action. To Ramsey, believing a conditional is adopting a rule such that, under the supposition of p together with one's background information and laws, one is disposed to accept q. Although Ramsey's writings on conditionals attracted relatively little attention at first, the footnote was rediscovered in the later twentieth century, where it came to be seen as a key inspiration for non-material accounts of conditionals in philosophy of language, epistemology, and formal logic.
Later developments From the 1960s onwards, Ramsey's idea was taken up in at least three overlapping research programmes:
In possible-worlds semantics, philosophers Robert Stalnaker and David Lewis model the hypothetical addition of p by selecting a "closest" or most plausible p-world to evaluate q at, which is an approach to conditionals inspired by the Ramsey test. In probabilistic approaches, the Ramsey test is read as a recipe for tying the acceptability of conditionals to conditional probabilities. In belief revision theory, especially the AGM framework, the Ramsey test is the inspiration for an equivalence between a belief set's acceptance of a conditional and the consequent's membership in belief set after it is revised by the antecedent. Subsequent work has elaborated versions of the Ramsey test in dynamic epistemic logic, non-monotonic reasoning, game theory, and artificial intelligence, often under the heading of "epistemic" or "subjunctive" conditionals. The precise formulations vary; Bradley argues that the Ramsey test should be seen as a schema that can be instantiated in different ways for different kinds of conditionals and different models of belief change.
Formalizations
Belief revision In AGM belief revision, a belief state is represented by a deductively closed set of sentences K, and revision by a new sentence p is represented by an operator * that outputs a new belief set K* p. The Ramsey test is then commonly expressed as a definitional equivalence for an epistemic conditional connective >:
Ramsey test (AGM form). K accepts the conditional p > q if and only if q is a member of the revised belief set K * p. That is, p > q ∈ K iff q ∈ K ∗ p . {\displaystyle p>q\in K\quad {\text{iff}}\quad q\in K*p.}
This formulation connects a logic of conditionals with a belief revision theory, so that principles about the conditional connective > can be translated into postulates on the revision operation *, and conversely.
Probabilistic approaches to conditionals Some formulations read the Ramsey test as inherently probabilistic. On this view, to evaluate if p, q one should adjust one's credence function as if p were known and then look at the resulting probability of q. An influential proposal, called "the Ramsey–Adams thesis" or "Adams's thesis", holds that, for suitable indicative conditionals, the acceptability or assertability of if p, q is governed by the conditional probability P ( q ∣ p ) {\displaystyle P(q\mid p)} . Some proposals, such as Robert Stalnaker's, go further and identify the probability of the conditional with the corresponding conditional probability, e.g. P ( if p , q ) = P ( q ∣ p ) {\displaystyle P({\text{if }}p,q)=P(q\mid p)} . However, in 1976, philosopher David Lewis proved a famous series of triviality results which show that, under plausible assumptions, such equations cannot hold generally for non-trivial probability functions, motivating weaker connections between conditional probability and conditional acceptability. A related line of work studies how probabilities for compounds and embeddings of conditionals can be assigned while avoiding triviality, for example by restricting the language or weakening closure assumptions.
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