I will be presenting a series of articles introducing various forms of situation theory and situation semantics. I open up with an edited collection notes on an important paper by David Israel and John Perry, entitled "What is Information".
David Israel and John Perry introduce a series of working principles of a theory of semantic information, outline a situation theoretical framework, and apply this framework to an analysis of agent action with respect to the usefulness of having information, attempting to resolve an apparent conflict between a notion of information as relative to external background constraints and facts, and the notion of an agent having information that X and acting appropriately to that information. Israel and Perry wrote a sequel to the article presently being discussed, called Information and Architecture which is highly relevant and which I have blogged about several times before, here, here, here, and here.
By way of introducing their principles of information, Israel and Perry clarify several points about the semantics of the way we talk about information. Some of the terminology they employ is covered in my notes on the article Information and Architecture. They designate sentences like (1) below as an information report.
(1) The clock indicates that it is six o'clock.
In an information report, the initial noun phrase is called the informational context. The proposition indicated by the information context they call the informational content. In our example, the clock is the carrier of the information. The informational context given may be regarded as an abbreviated form of something like:
(2) The fact that the clock's hour hand is pointed at the six and the minute hand is pointed at the zero indicates that it is six o'clock.
The form of information report found in (2) is regarded by the authors as being canonical. The information context expresses a purported fact which they call the indicating fact. It is not entirely clear if the authors' terminology is consistent between the present article and its sequel.
The authors then briefly give a few reasons that information reports are not truth-functional (p.2). These include:
- Logically equivalent statements cannot be substituted in without possibly changing the meaning or truth-value of the information report
- Informational reports do not distribute across logical OR.
We demonstrate this with two examples.
(3) The clock indicates that it is six o'clock and that 2 + 2 = 4. (p. 2)
To be truth-functional means that the truth of a compound sentence is a function of the truth of its components. In (3) the proposition p that it is six o'clock is attached to the necessarily true proposition q that 2 + 2 = 4. In formal logic, the conjunction of a proposition with a logical truth is logically equivalent to just that proposition. For any proposition \(p \land TRUE \equiv p\) (has the same truth values in all cases). But it does not make much sense to say that the clock indicates that 2 + 2 = 4.
(4a) The test indicates that George has disease A and disease B.
If the test indicates that George has both diseases A and B then it indicates that he has disease A and it indicates that he has disease B. But the sentence:
(4b) The test indicates that George has either disease A or disease B.
is different. It needn't be the case that the either the test indicates that George has disease A or the test indicates that George has disease B because the test may test for a symptom common to both, for example. Compare this to the case where George is given either a test for disease A or a test for disease B, but it is unknown which test George was given. In that case, distribution works normally.
Principles of Information
The next part of their paper is concerned with laying out a number of principles of information. Their principles (all quoted):
- Facts carry information (p. 3)
- The information content of a fact is a true proposition. (p. 3)
- The information a fact carries is relative to a constraint. (p. 3)
- The information a fact carries is not an intrinsic property of it. (p. 4)
- The informational content of a fact can concern remote things and situations. (p.4)
- Informational content can be specific; the propositions that are information contents can be about objects that are not part of the indicating fact. (p. 4)
- Indicating facts contain such information only relative to connecting facts; the information is incremental, given those facts. (p. 4)
- Many different facts, involving variations in objects, properties, relations and spatiotemporal locations, can indicate one and the same informational content-- relative to the same or different constraints. (p. 5)
- Information can be stored and transmitted in a variety of forms. (p. 5)
- Having information is good; creatures whose behavior is guided or controlled by information (by their information carrying states) are more likely to succeed than those which are not so guided. (p. 5)
Situation Theoretic Framework
The following is a terse introduction to the version of situation theory used in the article. A state of affairs is a n-ary relation together with its polarity. For example, the relation R of being an author of some text may be written as \(<isAuthorOf, x, y>\) where x is an argument whose appropriate role is that of the person who is an author, and y is that authored by x. For any particular anchoring of x and y to actual objects, x and y either stand in this relation or they do not. The corresponding state of affairs may be written like:
or its dual
Situations make a state of affairs or its dual factual (but not both). It might be the case however that a given situation does not decide the issue. Situations are partial. When a situation s makes a state of affairs σ factual, this is written: \(s \vDash \sigma\). In our example, the situation s of Tolkien having written The Hobbit makes the state of affairs indicated by
The states of affairs used so far have all had individuals as constituents. We can parameterize these states of affairs by abstracting the individuals away. Parametric states of affairs are called infons. A partial function f called an anchor from parameters to objects transforms an infon into a state of affairs. The anchoring used plays an important role in determining whether a situation supports the infon or not.
Situations have types. Their types are delimited by the state of affairs or infon supported: \([ s \mid s \vDash \sigma ]\) where σ is a state of affairs, or \([ s \mid s \vDash i ]\), where i is an infon.. A situation has type T=\([ s \mid s \vDash \sigma ]\) if \(s \vDash \sigma\). A situation has type \(T = [ s \mid s \vDash i ]\) when \(s \vDash i[f]\), that is where the anchor f supplies the objects for the infon i.
A constraint is a relation between types indicating involvement. The authors distinguish two kinds: simple involvment, and relative involvment. Simple involvment 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 \(T′\).
Simple constraints are associated with pure information, while relative constraints are associated with incremental information. The authors use as an example the pure and incremental information involved in an examination of an X-ray indicating a dog's leg having been broken. An examination of the X-ray indicates that there is some dog who is the subject of this X-ray, and that that dog has a broken leg. This is called the pure informational content of the X-ray. The simple constraint is that situations in which an X-ray exhibits such and such pattern then there exists 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 the ternary relative constraint. Notably, the relative constraint involves the same relations as does the simple constraint.
I develop my own example as a variation on the basic pattern of the one outlined above. It's presentation closely follows that of the authors' X-ray example.
Example of Pure and Incremental Information and Simple and Relative Constraints
The Photinus pyralis is a species of firefly common to the eastern United States. The pyralis male advertises his availability for mating by flashing for a short period of time while flying in a particular pattern. A female pyralis responds by waiting two seconds and then giving a half-second flash in return. This process is repeated until mating occurs (Lloyd 1986 113-114).
We might model the female pyralis response in situation theoretic terms, first as an example of pure information, and then as an example of incremental information.
Pure Informational Content with Simple Involvement
We first define our situation types and the constraint between them:
And our simple constraint is:
In plain English this means that if x is a signal having some appropriate pattern F, then x is signaled by some y and that y is an eligible pyralis female.
Given an indicating situation σ:
we anchor our infons by substituting a for x (or more formally using a function from a to x:
The function \(cond(T)\) is a function from the type to its conditioning infon. Applying the anchor f to that infon results in the parameterless state of affairs σ.
Thus P is the proposition that there exists a situation in which a signal a is flashed by some individual y and y is an eligible female pyralis.
Incremental Information with Relative Involvement
We shift the focus of the indicated proposition onto the firefly itself by using a relative constraint and connecting fact.
And the relative constraint between these types of situations is given by the relation:
In natural language this means that in situations where a signal x exhibits pattern F, then given the fact that this signal x is signaled by some particular y means that y is an eligible pyralis female.
Our indicating fact
and connecting connecting fact
together yield us the following incremental proposition: there exists a situation \(s″\) such that b (a specific individual) is an eligible female phyralis.
Agents, Goals, and The Usefulness of 'Having Information'
The authors wish to distinguish between carrying and having information. In their view an agent has some piece of information if its state carries that information and carrying that information has some "effect...that is appropriate given the information."(p.13)
The authors are concerned to dispel worries that the appropriate condition entails action at a distance, since the information carried by an information carrier changes relative to particular constraints and connecting facts, but the local effect of carrying that information is not. They resolve this by noting that appropriateness of an effect or action is relative to an agent's goals, and that the success of the action depends on the agents attunment to environmental constraints.
The pencil sharpener...has been designed so that the state carries the information that a pencil has been inserted sets in motion a chain of events that promote the goal for which it was designed, sharpening pencils. The design will be successful only in an environment in which the depressing of the lever will carry the information that a pencil has been inserted, and the motion of a blade against a pencil will leave it sharpened. We say that the system is attuned to these constraints, relative to the goal of sharpening pencils. (p. 14)
It is here worth while to visit what the authors write in "Information and Architecture":
Informational content is only information when the constraints and connecting facts are actual. If a signal carries the information that P, then P is true. A signal can have the informational content that P relative to a constraint and connecting fact, even though P is not true. This happens when the constraint and connecting "fact" or both are not factual. (Information and Architecture, first page).
Thus, the depressing of the lever has the informational content that a pencil has been inserted, but only carries the information that a pencil has been inserted when the appropriate constraints are factual, and the sharpener has the information when it carries it and its actions allow it to achieve the goal of sharpening pencils.
My firefly example is a little joke. It turns out that this species of firefly is frequently a victim of a predator species of firefly that mimics the pyralis mating ritual. In this context, my example of firefly mating signals might be expanded to include the fact of predatory mimics deceptively using signals with what is apparently the same informational content, but in which the victim neither carries nor has the information that an available mate has been located. In this case, 'environment' may not be the best term to employ, since mimics are a normal part of the environment; a problem to be dealt with. The problem of how to incorporate error in an account of information is a long-standing one.
Israel, David, and John Perry. What is Information. In Information, Language and Cognition, ed. Philip Hanson, 1-19. Vancouver: University of British Columbia Press, 1990. http://www.albany.edu/acc/courses/inf723spring2008/whatisinfo.pdf
Lloyd, James E. Firefly Communication and Deception: Oh, What a Tangled Web. In Deception, perpsectives on human and nonhuman deceit, ed. Robert W. Mitchell and Nicholas S. Thompson, 113-128. SUNY series on animal behavior. Albany, NY: State University of New York Press, 1986.