Introduction to Situation Theory Part I

What Is Situation Theory?

Situation theory is an information theoretic mathematical ontology developed to support situation semantics, an alternative semantics to the better known possible world semantics originally introduced in the 1950s. Rather than a semantics based on total possible worlds, situation semantics is a relational semantics of partial worlds called situations. Situations support (or fail to support) items of information, variously called states of affairs or infons. The partial nature of situations gives situation theory and situation semantics a flexible framework in which to model information and the context-dependent meaning of sentences in natural language.

Beginning with the work and collaboration of Jon Barwise and John Perry in (1981, 1983, 1985), situation semantics and situation theory developed rapidly, and at times changed radically, in a more than decade long regime of vigorous collaboration between a diverse group of researchers. Much of the work in situation theory and situation semantics occurred under the auspices of the Situation Theory and Situation Semantics research group (STASS) at Stanford University's Center for the Study of Language and Information (CSLI) These researchers include Jon Barwise, John Perry, Robin Cooper, Keith Devlin, John Etchemendy, David Israel, Jerry Seligman and many others. An excellent introduction and overview to Jon Barwise's many publications on situation semantics is given in (Devlin 2004). A brief and non-technical introductions to situation theory and situation semantics can be found in (Devlin 2006), as well as the ecyclopedia entries (Perry 1998) and (Kratzer 2009). A more thorough and highly technical overview of situation theory may be found in (Seligman and Moss 1997).

Situation theory and situation semantics have been applied to many problem domains, particularly in philosophical linguistics. These include famously a solution to the so-called liar paradox (Barwise and Etchemendy 1989) and various puzzles associated with propositional attitudes, indexicals, pronouns, conditionals, reference, metaphor, and anaphora (Barwise 1989; Barwise and Perry 1981, 1983, 1985; Devlin 1991). Barwise (1989) applies situation theory to a novel analysis of common knowledge. Other topics studied in relation to situation semantics and situation theory include information architecture and information flow (Israel and Perry 1990, 1991; Barwise 1989), diagrammatic reasoning (Shin 1991; Stenning and Oberlander 1991), relevant logic and other paraconsistent logics (Restall 1996), episodic logics and natural language processing (Hwang and Schubert 1993), event and processe logic (Georgeff et al. 1993), cooperative action and information systems design (Devlin and Rosenberg 1993, 1996, 2008), legal reasoning systems (Tojo and Wong 1996), computational linguistics (Rieger 1995), linguistic disambiguation (Prashant 1990), and Japanese honorifics (Sugimura 1986).

Development of a graphical notation for situation theory may be found in (Barwise and Cooper 1991, 1993). In addition, there have been several attempted (if partial) implementations of situation theory as Prolog-esque programming languages, most notably PRO-SIT (Nakashima et al. 1988) and BABY-SIT (Tin and Akman 1995, 1996).

Caveats

First, situation theory and situation semantics has always been a moving target. The theory changed rapidly throughout its development, and apart from general agreement on the ideas motivating situation theory and situation semantics and on some of mechanics of the theory, at least later in its development, the theory never has completely settled into a single universally accepted version. On some of the thornier points, there remains substantive disagreement, and there exists some diversity in the notation employed. We will try to make it clear when an item of the theory is not yet settled. However, one major consequence of this is that an account of situation theory cannot be both comprehensive and totally coherent (see for example Braisby and Cooper 1996). Since our objective is to give a more or less representative example of situation theory we have chosen to draw from a few sources, mainly(Devlin 1991), (Barwise and Perry 1983), and from (Barwise 1989). While some details are particular to each account, it should be understood that the gist of the theory is the product of a broad and sustained effort by multiple researchers; the description given here represents a broad if not exceptionless concensus on the main points.

Overview of Situation Theory

We begin with a simple overview of the main components of situation theory, followed by a brief overview of situation semantics. We will then proceed to flesh out the basic theory in the next sections.

States of Affairs, Infons, and Anchors

A state of affairs or concrete infon defines an issue which may or may not be decided by some part of the world. A state of affairs is represented by an n-ary relation whose arguments are filled with appropriate objects*1*, together with a polarity indicating whether the relation holds or does not hold. Very often special arguments for spatial and temporal locations are included in the relation.

We might represent the schema of the relation of someone being the author of some book by writing that \(R=<isAuthorOf,\dot{x},\dot{y}>\), where \(\dot{x}\) is an argument whose appropriate role is that of a person who is an author, and \(\dot{y}\) is that of a book. If the author is J.R.R. Tolkien and the book is The Hobbit, then the corresponding state of affairs maybe be written as:

This indicates a state of affairs in which J.R.R. Tolkien is the author of The Hobbit. The dual of this state of affairs is the state of affairs in which Tolkien is not the author of The Hobbit, written*2*:

Note that an infon is not usually taken to be something that in of itself can be true or false. Infons are not the kinds of entities that bear truth values; they are not propositions.*3* The polarity of the infon serves a function similar, but not exactly the same, to that of negation in propositional logic.

A parametric infon is an infon in which some of the argument places are filled with parameters of the an appropriate type*4*. For example,

is a parametric infon. A parameter abstracts away from a concrete object of the theory, serving as a place-holder in an infon for some object to fill. Although not strictly necessary (Crimmins 1993), parameters are useful. For example, the parameter \(\dot{x}\) links the following two parametric infons together in a useful way:

and

If parameters are abstractions of objects, then we need a way to move away from the abstract to the concrete. A partial function f from a domain of parameters to objects is called an anchor. The anchor replaces the parameters of the parametric infon with the objects in the range of the anchor. If f is an anchor mapping J.R.R. Tolkien onto \(\dot{x}\) and The Hobbit onto \(\dot{y}\), then

The existence of a parameter in an infon should not be interpreted as having an existential import. The infon \(\sigma_{3} = <<livingOn,\dot{y},Venus;1>>\) is perfectly legitimate, but we have no reason to believe that there is any situation for which there exists some constituent object to anchor the parameter \(\dot{y}\).

Situations and the Support Relation

A situation is a limited part of the world that supports various states of affairs. When a situation supports a state of affairs, we say that that situation makes that concrete infon factual, written \(s\vDash\sigma\) . A situation may make a parametric infon or its dual factual relative to an anchor if all the parameters in the infon are replaced with appropriate objects by that anchor.

Given a basic state of affairs \(\sigma\) *5* and a situation s, three possibilities arise:

1. s makes \(\sigma\) factual, i.e. \(s\vDash\sigma\)
2. s makes its dual \(\bar{\sigma }\) factual, i.e. \(s\vDash\bar{\sigma}\) in which case s does not support \(\sigma\) .
3. s neither makes \(\sigma\) factual nor its dual \(\bar{\sigma }\) factual.

The first and second possibilities are quite ordinary, but the third possibility deserves comment. The partial nature of situations means that a situation will not in general be able to resolve every issue. A logic of propositions in situation theory will not include, unmodified, all of the usual logical laws of propositional logic, like the law of the excluded middle*6*.

Consider a non-parametric infon about the famous parrot Alex, who was the subject of a 30 year Avian Language EXperiment (ALEX) (Pepperbert 2008, 84).

, where t and l are temporal and spatial locations. \(\mu\) is factual if and only if there is some situation s such that \(s\vDash\mu\) . In other words, the infon \(\mu\) is made factual by some situation s if and only if in s, the parrot Alex is flying at time t and location l. A situation may fail to support an infon without supporting that infon's dual. For example, Alex may not be part of a given situation, and so that situation will fail to support the infon regardless of whatever Alex may in fact be doing.

In the next part of this series, we will look in more depth at the ontology of objects that make situation theory so rich.

Footnotes and References