Ramsey test

This article is about the test for conditionals. For a branch of combinatorics named after Frank P. Ramsey, see Ramsey theory.

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. Roughly, the idea is that one should add the antecedent p hypothetically to one's beliefs, minimally revise one's belief state to accommodate this supposition, and then see whether the consequent q would be accepted in the revised state. If so, the conditional if p, q is acceptable.

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 largely independent traditions: in belief revision theory, 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.

Informal statement

In Ramsey's own words, when two people are arguing about whether if p, q, while both are uncertain about p, they "are adding p hypothetically to their stock of knowledge" and reasoning on that basis about q, thereby "fixing their degrees of belief in q given p". The Ramsey test abstracts from this remark the following guiding idea:

Ramsey test (informal). To decide whether to accept a conditional if p, q in a given belief state K, (i) hypothetically add p to K and revise K in a minimal and rational way to accommodate p; (ii) accept if p, q iff 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.

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. In this footnote he characterises conditional judgements pragmatically, in terms of how an agent's beliefs and expectations would change under the supposition of the antecedent.

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. On this picture, believing a conditional is not merely believing a truth-functional proposition, but adopting a rule 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, writers such as Robert Stalnaker and David Lewis model the hypothetical addition of p by selecting a suitable "closest" or most plausible p-world to evaluate q at, thus embedding a Ramsey-style test in a semantic theory of counterfactuals and other conditionals.
• In probabilistic approaches (notably the work of Ernest Adams and Vann McGee), Ramsey's suggestion that agents "fix their degrees of belief in q given p" is read as a recipe for tying the acceptability of conditionals to conditional probabilities.
• In belief revision theory, especially within the AGM framework, Peter Gärdenfors and others formulate the Ramsey test using explicit belief-change operations on deductively closed belief sets, and study the constraints that such a test imposes on rational revision functions.

Subsequent work has elaborated Ramsey-style tests in dynamic epistemic logic, non-monotonic reasoning, game theory, and artificial intelligence, often under the heading of "epistemic" or "subjunctive" conditionals.

Formal formulations

Belief-revision version

In AGM-style belief revision theory, 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.

In symbols, this is often written as:

$p>q\in K\quad {\text{iff}}\quad q\in K*p.$

This formulation connects a logic of conditionals with a logic of belief change: principles about > can be translated into postulates on *, and conversely.

Probabilistic versions

A different family of formulations reads Ramsey's 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.

One influential proposal, sometimes called the Ramsey–Adams thesis, holds that, for suitable indicative conditionals, the acceptability or assertability of if p, q is governed by the conditional probability $P(q\mid p)$ rather than by the unconditional probability of a material implication. While Ramsey himself did not explicitly state an equation between the probability of a conditional and a conditional probability, his footnote is widely cited as the motivating idea for such accounts.

David Lewis famously proved that, under plausible assumptions, the probability of a conditional cannot in general be equal to the corresponding conditional probability, a result often referred to as Lewis's triviality theorem. This has led probabilistic Ramsey-style theorists to weaken or reinterpret the connection between acceptability and conditional probability, or to restrict attention to certain classes of conditionals.

Possible-worlds semantics

In conditional logics with possible-worlds semantics, an idea inspired by the Ramsey test is implemented via selection functions or ordering semantics: a model assigns to each world w and antecedent p a set of "closest" or most plausible p-worlds, and the conditional "if p, q" is true at w when all of these selected worlds satisfy q. Robert Stalnaker's landmark paper "A Theory of Conditionals", which began the field of conditional logic, was explicitly inspired by the Ramsey test.

On one influential reading, the Ramsey test guides what counts as a "closest" world: the selected p-worlds represent the outcome of minimally revising one's beliefs to accommodate p, so that checking whether q holds at those worlds corresponds to seeing whether q would be believed after the supposition of p.

Triviality results and limitations

Gärdenfors' triviality theorem

When the AGM form of the Ramsey test is combined with the standard AGM postulates for rational belief revision, one obtains strong constraints on the interaction between conditionals and revision. Gärdenfors showed that, if conditionals built with > belong to the same language as non-conditional propositions and can themselves serve as inputs to revision, then there is no non-trivial revision operator that simultaneously satisfies the AGM postulates and the Ramsey test.

Very roughly, the Ramsey test plus AGM implies that any belief state containing p and q must also contain the conditional p > q, and that iterated applications of the test collapse the space of belief states in an undesirable way. This is sometimes called a triviality or impossibility result, because the only revision functions that satisfy all the assumptions leave no room for a sensible logic of conditionals.

In subsequent work, Gärdenfors proposed weakened versions of the test and explored further triviality theorems, including results for negative conditionals ("the negative Ramsey test").

Responses and modifications

A substantial literature explores ways of avoiding triviality while retaining some Ramsey-style insight. Strategies include:

• Restricting the language so that conditionals cannot appear as ordinary beliefs or inputs to revision.
• Weakening the preservation or minimal-change postulates governing revision, while keeping a strong Ramsey test.
• Weakening the Ramsey test itself or limiting it to certain classes of conditionals (for example, epistemic rather than metaphysical conditionals).
• Moving from belief sets to richer epistemic states, such as ranking functions or plausibility orderings, and letting the Ramsey test operate at this more fine-grained level.

Some authors argue that the Ramsey test should be seen not as a single, rigid principle, but as a schema that can be instantiated in different ways for different kinds of conditionals and different models of belief change.

Variants and generalisations

Epistemic and subjunctive conditionals

Ramsey's footnote is often read as targeting epistemic or indicative conditionals—claims about what follows from what we know or believe. More recent reconstructions of Ramsey's theory, however, treat his "variable hypotheticals" as underwriting both indicative and counterfactual readings, thereby extending the Ramsey test to subjunctive conditionals that concern what would be the case under a hypothetical change in circumstances.

On such views, the same basic pattern—suppose p, adjust one's system of laws and background beliefs appropriately, then see whether q would be accepted—can be implemented with different kinds of suppositions and different belief-revision mechanisms for different sorts of conditionals (counterfactual, normative, counterlegal, and so on).

Negative and iterated Ramsey tests

Besides the positive Ramsey test for accepting if p, q, there are proposals for "negative Ramsey tests" that aim to characterise when a conditional of the form if p, not q should be rejected or accepted. These have been shown to generate further triviality results when combined with standard belief-revision postulates, prompting more fine-grained treatments of denial and of conditionals with negative consequents.

A further extension concerns iterated Ramsey tests, where conditionals are evaluated relative to sequences of revisions rather than single steps. This connects the Ramsey test with work on iterated belief revision and dynamic epistemic logics, in which the evolution of an agent's plausibility ordering under multiple pieces of information is explicitly modelled.

Applications

Because it links conditionals with hypothetical belief change, the Ramsey test has applications across several areas:

• In the semantics of natural language, it underlies many accounts of indicative and counterfactual conditionals, often in combination with possible-worlds or dynamic semantic tools.
• In formal epistemology and non-monotonic reasoning, it motivates representation theorems relating conditional logics, default rules, and belief-revision policies, and helps explain why conditionals often fail to validate monotonic inference patterns.
• In artificial intelligence and belief management systems, Ramsey-style tests are implemented via AGM-like belief-revision operators, ranking functions, or conditional knowledge bases, and used to drive reasoning about actions, observations, and hypothetical scenarios.

Recent work continues to explore the Ramsey test's role in modelling learning, decision-making under uncertainty, and the social aggregation of conditional opinions.

See also

• Belief revision
• AGM postulates
• Conditional (logic)
• Indicative conditional
• Counterfactual conditional
• Conditional probability
• Non-monotonic logic
• Frank P. Ramsey