Sten Lindström

(1945 - 2022)

This is a short version of Lindström & Rabinowicz 1991.In earlier papers, we proposed a generalization of the AGM approach to belief revision. The proposal was to view belief revision as a relation rather than as a function on theories (or belief sets). Going relational means that one allows for several equally reasonable revisions of a theory with a given proposition. In the present paper, we show that the relational approach is the natural result of generalizing in a certain way an approach to belief revision due to Adam Grove. In his (1988) paper, Grove presented two closely related modelings of functional belief revision, one in terms of a family of "spheres" around the agent's theory G and the other in terms of an epistemic entrenchment ordering of propositions. The "sphere"-terminology is natural when one looks upon theories and propositions as sets of possible worlds. Grove's spheres may be thought of as possible "fallback" theories relative to the agent's original theory: theories that he may reach by deleting propositions that are not "sufficiently" entrenched (according to standards of sufficient entrenchment of varying stringency). The entrenchment ordering can be recovered from the family of fallbacks by the definition: A is at least as entrenched as B iff A belongs to every fallback to which B belongs. To revise a theory T with a proposition A, we go to the smallest sphere that contain A-worlds and intersect it with A. The relational notion of belief revision that we are interested in, results if one allows that some propositions may be incomparable with respect to epistemic entrenchment. As a result, the family of fallbacks around a given theory will no longer have to be nested. This change opens up the possibility for several different ways of revising a theory with a given proposition.

In this paper, I shall consider the challenge that Quine posed in 1947 to the advocates of quantified modal logic to provide an explanation, or interpretation, of modal notions that is intuitively clear, allows “quantifying in”, and does not presuppose, mysterious, intensional entities. The modal concepts that Quine and his contemporaries, e.g. Carnap and Ruth Barcan Marcus, were primarily concerned with in the 1940’s were the notions of (broadly) logical, or analytical, necessity and possibility, rather than the metaphysical modalities that have since become popular, largely due to the influence of Kripke. In the 1950’s modal logicians responded to Quine’s challenge by providing quantified modal logic with model-theoretic semantics of various types. In doing so they also, explicitly or implicitly addressed Quine’s interpretation problem. Here I shall consider the approaches developed by Carnap in the late 1940’s, and by Kanger, Hintikka, Montague, and Kripke in the 1950’s, and discuss to what extent these approaches were successful in meeting Quine’s doubts about the intelligibility of quantified modal logic.

Årets Hägerströmföreläsningar i Uppsala gavs i februari av den norske filosofen Dagfinn Føllesdal. Ämnet var "Mening og Erfaring". Dagfinn Føllesdal doktorerade 1961 vid Harvard med Willard Van Quine som handledare på en avhandling om kvantifierad modallogik. Han blev internationellt känd främst för studier om Husserls fenomenologi och dess förhållande till Frege samt för sina arbeten om Quines språkfilosofi. Allt sedan 60-talet har Føllesdal delat sin tid mellan Oslouniversitetet och Stanforduniversitetet i Kalifornien. I sina föreläsningar diskuterade Føllesdal meningsbegreppet med anknytning till teorier formulerade av sådana tidigare Hägerströmföreläsare som Quine, Davidson och Kripke. En utgångspunkt för framställningen var Quines empiristiska syn på språket: Allt vi kan lära om omvärlden och varandra, inklusive språket, är grundat på socialt tillgänglig evidens. Enligt Quines uppfattning är språket väsentligen ett socialt fenomen. Detta innebär att: (i) språket har blivit till genom en social process; (ii) individens inlärning av språket sker på basen av socialt tillgänglig information; och (iii) användningen av språket är en social process. Den centrala frågeställningen som diskuterades i föreläsningarna var: Hur skall vi utifrån en empiristisk syn på språket förstå mening och kommunikation? Quine var den förste att på ett helt explicit sätt formulera denna problemställning.

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in the famous Hilbert-Brouwer controversy in the 1920s. The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics.

The present essay is a critical study of Barwise and Perry’s book, emphasizing the logical and model-theoretical aspects of their work. I begin by presenting the authors’ criticism of the classical view of logic and semantics within the tradition of Frege, Russell and Tarski. In this connection, I discuss the so-called Frege argument (“the slingshot”). I try to show that the argument appears inconclusive, not only from a situation-theoretic perspective, but also from such alternative perspectives as orthodox Fregean semantics or Russellian semantics. I then discuss the ontology of situation semantics and the way it is modeled within set theory. In particular, I compare the notion of an abstract situation with that of a possible world. The last two sections concern the model-theoretic aspects of the authors’ theory. In Section 7, I discuss how the “partial” perspective of situation semantics differs from that of classical model theory. Finally, in Section 8, different model-theoretic accounts of attitude reports within situation semantics are discussed, in particular the “relations to situations”-approach presented by the authors in Chapter 9 of S & A. The usual problems of “logical omniscience” that appear in standard Hintikka-style epistemic logic are avoided in situation semantics. I argue, however, that situation semantics is faced with analogous counter-intuitive results, unless the expressive power of the language under study is suitably restricted.

The so called Ramsey test is a semantic recipe for determining whether a conditional proposition is acceptable in a given state of belief. Informally, it can be formulated as follows: (RT) Accept a proposition of the form "if A, then C" in a state of belief K, if and only if the minimal change of K needed to accept A also requires accepting C. In Gärdenfors (1986) it was shown that the Ramsey test is, in the context of some other weak conditions, on pain of triviality incompatible with the following principle, which was there called the preservation criterion: (P) If a proposition B is accepted in a given state of belief K and the proposition A is consistent with the beliefs in K, then B is still accepted in the minimal change of K needed to accept A. (RT) provides a necessary and sufficient criterion for when a 'positive' conditional should be included in a belief state, but it does not say anything about when the negation of a conditional sentence should be accepted. A very natural candidate for this purpose is the following negative Ramsey test: (NRT) Accept the negation of a proposition of the form "if A, then C" in a consistent state of belief K, if and only if the minimal change of K needed to accept A does not require accepting C. This note shows that (NRT) leads to triviality results even in the absence of additional conditions like (P).

This is a thesis in formal semantics. It consists of two parts corresponding to the distinction, due to Richard Montague, between universal grammar and specific semantic theories. The first part concerns universal grammar and is intended to provide a precise and unified conceptual framework within which different theories of formal semantics can be represented and compared. ;The second part of the thesis is concerned with intensional logic, i.e., with the logical analysis of discourse involving so called oblique contexts. These contexts are characterized by the apparent failure of certain classically valid principles of substitutivity. Two kinds of reactions among logicians to the appearance of oblique contexts in natural languages are distinguished: (1) the intensional point of view, according to which oblique contexts are seen as genuine counterexamples to classical logic and to the related semantical principle of extensionality: the principle that the denotation of an expression is a function of the denotations of its parts; and (2) the extensional point of view, according to which the conflict between oblique contexts and classical logic is only apparent. The extensional point of view therefore calls for analyses of oblique contexts that preserve the laws of classical logic and the principle of extensionality. In this thesis, we are concerned with the precise formulation of and comparison between different programs for the formal semantic representation of oblique contexts. In Chapter 5, we are concerned with Church's program of representing oblique contexts within a formal language whose semantics is fully extensional and based on Frege's distinction between sense and denotation. The ambiguity between the "direct" and "indirect" denotation of an expression A is avoided by letting A be replaced in oblique contexts by an expression which denotes A's ordinary sense. We provide a possible-worlds semantics for a version of Church's formal language with the following features: (i) Objectual quantification into oblique constructions is accounted for. The treatment of quantification requires for each entity x an essence of x, by which we mean a sense which determines x in those worlds in which x exists and is undefined in all other worlds. (ii) The variables range over actually existing entities only. It is allowed that entities of certain types are contingent, i.e., the same entity may exist in one world and fail to exist in other worlds. (iii) In order to combine a proper treatment of the cross-world identification of functions with varying domains, partial functions are allowed. (iv) Senses are allowed to be partially defined. For example, the sense of a proper name is undefined in a world in which the denotation of the name does not exist. (v) Senses of functions from objects of type a to objects of type b are identified with functions from senses of objects of type a to senses of objects of type b. In Chapters 6 and 7, we pursue the Russellian Alternative in intensional semantics, where a sense/denotation distinction is avoided. Apparent failures of substitutivity principles for singular terms are explained along the lines of Russell's theory of definite descriptions. Failures of substitutivity of coextensional sentences and predicates are explained by Russell's doctrine that sentences and predicate denote propositions and propositional functions , respectively. Russellian semantics is fully extensional in the sense that the Russellian denotation of a designator is a function of the Russellian denotations of its parts. In Chapter 6, a theory of Russellian propositions is developed within set theory. This theory is then applied in Chapter 7 to the task of providing a set-theoretic semantics for Russellian type theory.

We discuss various possibilities for developing a dynamic doxastic logic (DDL) for introspective agents: agents who have the ability to form higher-order beliefs. Such agents can reflect upon and change their minds about their own beliefs. The project of constructing such a logic, full DDL or DDL unlimited, is ridden with difficulties due to the fact that the agent's own doxastic state now becomes a part of the reality he is trying to explore. When an introspective agent learns more about the world (and himself), the reality he holds beliefs about undergoes a change in its doxastic part, which means that his introspective beliefs have to be adjusted accordingly.

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on in our field—a book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory.

The theories of belief change developed within the AGM-tradition are not logics in the proper sense, but rather informal axiomatic theories of belief change. Instead of characterizing the models of belief and belief change in a formalized object language, the AGM-approach uses a natural language — ordinary mathematical English — to characterize the mathematical structures that are under study. Recently, however, various authors such as Johan van Benthem and Maarten de Rijke have suggested representing doxastic change within a formal logical language: a dynamic modal logic. Inspired by these suggestions Krister Segerberg has developed a very general logical framework for reasoning about doxastic change: dynamic doxastic logic (DDL). This framework may be seen as an extension of standard Hintikka-style doxastic logic with dynamic operators representing various kinds of transformations of the agent's doxastic state. Basic DDL describes an agent that has opinions about the external world and an ability to change these opinions in the light of new information. Such an agent is non-introspective in the sense that he lacks opinions about his own belief states. Here we are going to discuss various possibilities for developing a dynamic doxastic logic for introspective agents: full DDL or DDL unlimited. The project of constructing such a logic is faced with difficulties due to the fact that the agent’s own doxastic state now becomes a part of the reality that he is trying to explore: when an introspective agent learns more about the world, then the reality he holds beliefs about undergoes a change. But then his introspective (higher-order) beliefs have to be adjusted accordingly. In the paper we shall consider various ways of solving this problem.

Is it possible to give a justification of our own practice of deductive inference? The purpose of this paper is to explain what such a justification might consist in and what its purpose could be. On the conception that we are going to pursue, to give a justification for a deductive practice means to explain in terms of an intuitively satisfactory notion of validity why the inferences that conform to the practice coincide with the valid ones. That is, a justification should provide an analysis of the notion of validity and show that the inferences that conform to the practice are just the ones that are valid. Moreover, a complete justification should also explain the purpose, or point, of our inferential practice. We are first going to discuss the objection that any justification of our deductive practice must use deduction and therefore be circular. Then we will consider a particular model of justificatory explanation, building on Georg Kreisel’s concept of informal rigour. Finally, in the main part of the paper, we will discuss three ideas for defining the notion of validity: (i) the classical conception according to which the notion of (bivalent) truth is taken as basic and validity is defined in terms of the preservation of truth; (ii) the constructivist idea of starting instead with the notion of (a canonical) proof (or verification) and define validity in terms of this notion; (iii) the idea of taking the notions of rational acceptance and rejection as given and define an argument to be valid just in case it is irrational to simultaneously accept its premises and reject its conclusion (or conclusions, if we allow for multiple conclusions). Building on work by Dana Scott, we show that the last conception may be viewed as being, in a certain sense, equivalent to the first one. Finally, we discuss the so-called paradox of inference and the informativeness of deductive arguments.