Online research in philosophy

The possible-worlds semantics for modality says that a sentence is possibly true if it is true in some possible world. Given classical prepositional logic, one can easily prove that every consistent set of propositions can be embedded in a ‘maximal consistent set’, which in a sense represents a possible world. However the construction depends on the fact that standard modal logics are finitary, and it seems false that an infinite collection of sets of sentences each finite subset of which is intuitively ‘possible’ in natural language has the property that the whole set is possible. The argument of the paper is that the principles needed to shew that natural language possibility sentences involve quantification over worlds are analogous to those used in infinitary modal logic.

Let’s fix some terminology at the start. A world (or possible world – for me, the ‘possible’ is redundant) is, first, an individual, not a set or class; second, a particular, not a property or universal; third, concrete in this sense: it is fully determinate in all qualitative respects; and, fourth, a maximal interrelated whole: each world is internally unified, and isolated from every other world.1 There is at least one world, the world we are part of. It is an actual world, the actual world if there are no “island universes.”2 Worlds that are not actual (if any) are merely possible. A realist about possible worlds believes that there is a plenitudinous plurality of worlds: whenever something is possible – for example, that donkeys talk, or that pigs fly – there is a world in which it is true.

Some accounts of mental content represent the objects of belief as structured, using entities that formally resemble the sentences used to express and report attitudes in natural language; others adopt a relatively unstructured approach, typically using sets or functions. Currently popular variants of the latter include classical and neo-classical propositionalism, which represent belief contents as sets of possible worlds and sets of centered possible worlds, respectively; and property self-ascriptionism, which employs sets of possible individuals. I argue against their contemporary proponents that all three views are ineluctably plagued by generation gaps: they either overgenerate beliefs, undergenerate them, or both.

Robert Stalnaker’s formal semantics for his indicative conditional (which his 1975 paper takes over from his 1968 paper and Stalnaker and Thomason 1968) validate modus ponens, as one might expect. But they do so at the cost of a tension between his philosophical remarks in his 1975 paper and his formal constraints. Stalnaker commits himself to the following: he defines a “context set” as “the possible worlds not ruled out by the presupposed background information” (Stalnaker 1975 p 142). He later states a “pragmatic principle” that “normally a speaker is concerned only with possible worlds within the context set, since this set is defined as the set of possible worlds among which the speaker wishes to distinguish. So it is at least a normal expectation that the selection function should turn first to these worlds before considering counterfactual worlds—those presupposed to be non-actual” (p 144). Then two paragraphs later, in apparent reference to this principle he says “I would expect that the pragmatic principle stated above should hold without exception for indicative conditionals”. Yet when the actual world is not one in which the presuppositions all hold, from his definition of the “context set” it is not among the worlds of the context set, and elsewhere in his 1975 as well as his 1968 he stipulates that the selection function given the actual world and an antecedent true at the actual world yields the actual world (p 144 of Stalnaker 1975, condition 3 on p 104 of Stalnaker 1968). These remarks on the face of it lead to inconsistency if it is possible to presuppose falsehoods: for then the presuppositions create a context set which does not include the actual world (but may perfectly well nevertheless contain some possible worlds in which the antecedent of a given conditional holds when that antecedent is also actually true). In evaluating a conditional with a true antecedent which also holds in some world in the context set, Stalnaker enjoins us to employ a selection function which selects the actual world, and to (“without exception”) employ a selection function which selects some world in the context set in preference to any world outside it..

It is commonly held, plausibly, that many true beliefs are true only contingently, that is, are actually true (or true with respect to the actual world) but would be false were the world in some relevant ways otherwise (i.e. are false with respect to some other possible worlds). However, a radically different approach, according to which no belief is contingently true, is entirely defensible. The key point in this alternative approach is that each belief concerns the world in which the believer is (or would be) situated, which makes it the case that, say, the actual belief that Kofi Annan is not bald is different from the belief, in any other world, that Kofi Annan is not bald. This difference is further backed up by considerations related to disagreement between believers, and to knowledge. The most important objection to this alternative approach is that it cannot be right since it makes all true beliefs necessarily true. It will be shown, as a reply to this objection, that under this alternative approach it can still be said truly, for instance, that Annan is not bald but could have been so.

We examine carefully the rationale underlying the approaches to belief change taken in the literature, and highlight what we view as methodological problems. We argue that to study belief change carefully, we must be quite explicit about the ontology or scenario underlying the belief change process. This is something that has been missing in previous work, with its focus on postulates. Our analysis shows that we must pay particular attention to two issues that have often been taken for granted: the first is how we model the agent's epistemic state. (Do we use a set of beliefs, or a richer structure, such as an ordering on worlds? And if we use a set of beliefs, in what language are these beliefs are expressed?) We show that even postulates that have been called beyond controversy are unreasonable when the agent's beliefs include beliefs about her own epistemic state as well as the external world. The second is the status of observations. (Are observations known to be true, or just believed? In the latter case, how firm is the belief?) Issues regarding the status of observations arise particularly when we consider iterated belief revision, and we must confront the possibility of revising by and then by ¬.

A semantics is presented for belief revision in the face of common announcements to a group of agents that have beliefs about each other’s beliefs. The semantics is based on the idea that possible worlds can be viewed as having an internal-structure, representing the belief independent features of the world, and the respective belief states of the agents in a modular fashion. Modularity guarantees that changing one aspect of the world (a belief independent feature or a belief state) has no effect on any other aspect of the world. This allows us to employ an AGM-style selection function to represent revision. The semantics is given a complete axiomatisation (identical to the axiomatisation found by Gerbrandy and Groeneveld for a semantics based on non-wellfounded set theory) for the special case of expansion.

A semantics is presented for belief-revision in the face of common announcements to a group of agents that have beliefs about each other's beliefs. The semantics is based on the idea that possible worlds can be viewed as having an internal structure, representing the belief independent features of the world, and the respective belief states of the agents in a modular fashion. Modularity guarantees that changing one aspect of the world (a belief independent feature or a belief state) has no effect on any other aspect of the world. This allows us to employ an AGM-style selection function to represent revision. The semantics is given a complete axiomatisation (identical to the axiomatisation found by Gerbrandy and Groeneveld for a semantics based on non-wellfounded set theory) for the special case of expansion.

The principle of belief persistence, or conservativity principle, states that ’\Nhen changing beliefs in response to new evidence, you should continue to believe as many of the old beliefs as possible' (Harman, 1986, p. 46). In particular, this means that if an individual gets new information, she has to accommodate it in her new belief set (the set of propositions she believes), and, if the new information is not inconsistent with the old belief set, then (1) the individual has to maintain all the beliefs she previously had and (2) the change should be minimal in the sense that every proposition in the new belief set must be deducible from the union of the old belief set and the new information (see, e.g., Gardenfors, 1988; Stalnaker, 1984). We focus on this minimal notion of belief persistence and characterize it both semantically and syntactically. A ’possible world' semantic formalization of the principle easily comes to mind. The set of all the propositions that the individual believes corresponds to the set of states of the world that she considers possible and is a subset of the set of states that are not ruled out by the individual's information (or knowledge). It is required that, if the individual considers a state possible and her new information does not exclude this state, then she continue to consider it possible. Furthermore, if the individual regards a particular state as impossible, then she should continue to regard it as impossible unless her new information excludes all the states that she previously regarded as possible. This is closely related to the..

0. Relativistic Content In standard semantics, propositional content, whether it be the content of utterances or mental states, has a truth-value relative only to a possible world. For example, the content of my utterance of ‘Jim is sitting now’ is true just in case Jim is sitting at the time of utterance in the actual world, and the content of my belief that Alice will give a talk tomorrow is true just in case Alice will give a talk on the day following the occurrence of my belief state in the actual world. Let us call propositional content which has a truth-value relative only to a possible world ‘non-relativistic content’. Non-relativistic content can be treated as either structured or unstructured. On the unstructured-content view, non-relativistic content is a set of possible worlds and bears the truth-value true just in case the actual world is a member of that set. For example, the content of my utterance of ‘Jim is working now’ at time t is the set of worlds in which Jim is working at t, and this content is true just in case the actual world is among those worlds. On the structured-content view, non-relativistic content is a set or conglomeration of properties and/or objects, where properties are features which objects possess regardless of who considers or observes them and regardless of when they are being considered or observed. Such properties are said to be (or represent) functions from possible worlds to extensions. Relative to a possible world they determine a set of objects instantiating the property. For example, relative to the actual world the property of being human determines the set of actual humans. Not all content is non-relativistic. Let us say that propositional content is relativistic just in case it possesses a truth-value only relative to a centered world. A centered world is a possible world in which an individual and a time are marked, where the marked individual..