Introduction to Situation Theory Part III

Posted on Sun 17 October 2010 in Situation Theory tutorial

Infon Logic

Devlin develops an infon logic framework to be used either by a theorist or by an agent. The description of this logic is relatively informal. Infon logic combines basic infons to form non-basic compound infons. Conjuction and disjunction of infons may be recursively applied to create larger expressions.

Conjunction

Definition 1.3 Conjunction of Infons (Devlin 1991, 132) For any situation, s, we have $$s\vDash\sigma \wedge \tau$$ if and only if $$s \vDash \sigma$$ and $$s\vDash\tau$$ .

Disjunction

Definition 1.4 Disjunction of Infons (Devlin 1991, 132) For any situation, s, we have $$s\vDash\sigma \vee \tau$$ if and only if $$s\vDash\sigma$$ or $$s\vDash\tau$$ (or both).

Quantification

Devlin's version of situation theory uses a bounded notion of quantification. Unrestricted quantification of infons does not have the desirable property of persistence.

Definition 1.5 Persistence (Devlin 1991, 126) If $$s\vDash<<R,a1..an,i>>$$ for any situation, s, and appropriate objects $$a_{1}..a_{n}$$ in s, then $$s'\vDash<<R,a_{1}..a_{n},i>>$$ whenever s' is a situation that extends s'.

Generally we would wish that some fact that is true for a given situation stays true for a larger situation. There are cases of infons which by their nature seem to violate this principle, such as the infon $$s\vDash<<Alone,a,t,i>>$$. I might be alone in my office, but I may not be alone in the building. But (Devlin 1991, 126-127) argues that this infon, and others like it are malformed infons failing to meet appropriate minimality conditions for the relation. For example, an infon with the Alone relation requires the specification of some location that one is alone in.

How does this relate to quantification in situation theory? Universal quantification threatens the principle of persistence in the following way. Suppose that for some given situation everyone is wearing a watch. Is it true then that everyone in some larger situation s' is wearing a watch? Of course not. Therefore, a universal quantifier must be bound in its scope. Existential quantification does not present the same difficulty, but Devlin bounds its scope in a similar fashion as he does universal quantification.

Definition 1.6 Existential Quantification (Devlin 1991, 134) If $$\sigma$$ is an infon, or compound infon, that involves the parameter $$\dot{x}$$ and u is some set, then ($$\exists \dot{x}\in u)\sigma$$ is a compound infon, and for any situation, s, that contains, as constituents, all members of $$u: s(\exists \dot{x}\in u)\sigma$$ if and only if there is an anchor, f, of $$\dot{x}$$ to an element of u, such that$$s\vDash\sigma [f]$$.

Definition 1.7 Universal Quantification (Devlin 1991, 135) If $$\sigma$$ is an infon, or compound infon, that involves the parameter $$\dot{x}$$ and u is some set, then $$(\forall \dot{x}\in u)\sigma$$ is a compound infon, and for any situation, s, that contains, as constituents, all members of u, $$s\vDash(\forall \dot{x}\in u)\sigma$$ if and only if for all anchors, f, of $$\dot{x}$$ to an element of u, $$s\vDash\sigma [f]$$.

Clearly these definitions limit the scope of quantification to the constituents of a given situation. The semantics of quantification as found natural language is given more detailed treatment in chapter 9.1 of (Devlin 1991).

Negation

Devlin's situation theory does not include negation on infons, relying on the polarity of an infon to supply a means of asserting that something is or is not the case (Devlin 1991, 136). Other versions of situation theory do include some limited forms of negation on infons.

The semantics of negation as found in natural language is represented or handled in a more subtle fashion beyond the scope of the present discussion. Those interested may consult chapter 9.2 in (Devlin 1991).

Now that we have finished our discussion of infon logic, we talk about one of the most important notions in situation theory, that of constraints.

Constraints

Beginning with (Barwise and Perry 1983) the notion of constraint gained ever increasing prominence in the development of situation theory. A constraint is generally conceived as a regularity or uniformity between types. The most commonly cited example is that smokey situations involve fire-y situations. Agents attuned to those constraints may make inferences between types of situations.

Given two types T and T', we write $$C=T\Rightarrow T'$$ to mean indicate that T involves (or means) T'. Some authors, e.g. (Israel and Perry 1990, 1991) use an infon to indicate a constraint:

$$C=<<INVOLVES,T,T';1>>$$

to indicate that T involves (or means) T'.

For example, if an agent is attuned to the constraint C, and the agent individuates a situation s as having type T, then the agent make infer that there exists some situation s' having type T'. The situations s and s' may or may not be the same situation.

David Israel and John Perry (1990) differentiate between simple and relative constraints. A simple constraint involves two types T and T' such that whenever a situation has type T, then there exists some situation (possibly the same one) having type T'. Relative involvement involves three types T, T', and T'' where T involves T' relative to T'' so that when a situation $$s_{1}$$ has type T and a situation $$s_{2}$$ has type T'', then there exists some situation $$s_{3}$$ having type T'. Note: $$s_{1}$$, $$s_{2}$$ and $$s_{3}$$ need not be distinct situations.

Simple constraints are associated with what Perry and Israel call pure information, while relative constraints are associated with what they call incremental information. The authors (Perry and Israel 1990, 9-11) illustrate this distinction using the example of an X-ray of a dog's broken leg.

The pure informational content of the X-ray is that the X-ray is of some dog, and that dog's leg is broken. The simple constraint involved is that situations in which an X-ray exhibits such and such pattern involves a situation in which the dog in the X-ray has a broken leg.

The incremental information shifts the focus away from the X-ray to the dog by way of a third piece of information about the identity of the dog.

For an example of pure and incremental contents, and a more detailed discussion of Perry and Israel's paper, please see my blog post What is Information: some notes.

Next: Situation Semantics

Situation theory was developed to support situation semantics. I will give a basic account of situation semantics as a situated relational theory of meaning, and work out a few illustrative examples in the next article in this introduction to situation theory.