This chapter combines the theme of propositional logic from Chapter 6 with that of principal types from Chapter 3. We saw in the Curry-Howard theorem (6B7) that the types of the closed terms are exactly the theorems of the intuitionist logic of implication: hence the principal types of these terms must form a subset of these theorems, and the very natural question arises of just how large this subset is. Do its members form an aristocracy distinguished in some structural way from the general rabble of theorems, or can every theorem be a principal type?

The main result of the chapter will show that there is in fact no aristocracy: if a type τ is assignable to a closed term M but is not the principal type of M, then it is the principal type of another closed term M*.

The proof will include an algorithm to construct M* when τ and M are given. To build M* an occurrence of M will be combined with some extra terms chosen from a certain carefully defined stock of “building blocks”, closed terms with known principal types; and the main aim of this chapter will be to build M* from as restricted a set of building blocks as possible.

The algorithm in the earliest known proof needed full λ-calculus (Hindley 1969), but two later ones used only λI-terms as building blocks (Mints and Tammet 1991, Hirokawa 1992a §3) and another used an even more restricted class (Meyer and Bunder 1988 §9). The algorithm below will be based on the latter very economical one.

To avoid interrupting the main lines of thought in the earlier chapters some concepts were defined there only in outline and their main properties were stated without proof. This chapter gives the full definitions and proofs. It should be read only as required to follow the arguments in the other chapters. Its sections are largely independent of each other.

The structure of a term

From the viewpoint of logical order this section is best read between 1A3 and 1A4.

As remarked in 1A4, a subterm of a term may have more than one occurrence. The present section introduces a precise notation to distinguish such different occurrences; it is rather clumsy and the reader should avoid using it whenever possible, but in some proofs its precision will be vital. The first step is to define a set of expressions called positions that can be assigned to different occurrences of a subterm to show where they occur.

Definition (Positions) A positionp = i1 … im is any finite (perhaps empty) string of symbols such that i1, …, im are integers and im is either an integer or an asterisk, *. Its length is m, and if m = 0 we say p = Ø.

1. If m ≥ 1 and im = 1 we call p a function position;

2. If m ≥ 1 and im = 2 we call p an argument position;

3. if m ≥ 1 and im = 0 we call p a body position; and

4. if m ≥ 1 and im = * we call p an abstractor position.

Given a type τ, how many closed terms can receive type τ in TAλ? As stated this question is trivial, since if the answer is not zero it is always infinite; for example the type a→a can be assigned to all members of the sequence I, II, III, etc. But if we change the question to ask only for terms in normal form, the answer is often finite and interesting patterns show up which are still not completely understood.

The aim of this chapter is to describe an algorithm from Ben-Yelles 1979 that answers the “how many” question for normal forms. For each τ it will decide in a finite number of steps whether the number of closed β-normal forms that receive type τ is finite or infinite, will compute this number in the finite case, and will list all the relevant terms in both cases.

Ben-Yelles' algorithm can be used in particular to test whether the number of terms with type τ is zero or not, and as mentioned in 6B7.3 this gives a test for provability in Intuitionist implicational logic.

The first section below will describe the sets to be counted. The next will show some examples of the algorithm's strategy in action. Then in 8C–D the algorithm will be stated formally, and the rest of the chapter will be occupied by a proof that the algorithm does what it claims to do.

Inhabitants

This section gives precise definitions and notations for the sets to be counted.

Quantum theory is well known for having a nonclassical and somewhat peculiar logic. One approach to trying to make this logic comprehensible is by means of what is called a “manual of experiments.” This is the approach taken, for example, in the book An Introduction to Hilbert Space and Quantum Logic (Cohen, 1989), which we use as our main reference. A manual is thought of as describing a variety of different tests of a system, but where the making of one test may, for some reason, preclude the making of some other test of the same system at the same time. Still, it can happen that from performing one kind of experiment we may get information about what the outcome would have been had we performed another kind of experiment.

Example 21.1. Here is an example taken from 17). Imagine a firefly trapped inside a box, and two kinds of experiments, front and side, one can perform when observing this box at a given instant. In a front experiment one looks in the box from the front, whereas in side one looks from the right-hand side. In front there are three possible outcomes: one may see the firefly light lit up on the right (r), lit up on the left (l), or not lit up at all (n). Similarly, in side there are three possible outcomes: one may see the firefly light at the front (f), at the back (b), or not at all (n).

In this lecture we want to give a simple application of classifications and infomorphisms to analyze J. L. Austin's four-way distinction in “How to talk: Some simple ways” (Austin, 1961). The material in this lecture follows Lecture 4 and is not needed elsewhere in the book.

Truth-Conditional Semantics and Speech Acts

The theory of speech acts owes its origins to Austin's (1961) work. This theory is challenging in a couple of ways. First, Austin's paper is one of his more difficult. It is just hard to figure out what he is saying. Second, the theory of speech acts poses a challenge to certain kinds of semantic theories, and the types of speech acts discussed by Austin in his paper illustrate the challenge very clearly. Austin is saying that there are at least four distinct things a person can be doing with a true utterance as simple as “Figure 4 is a triangle.” One might say that he is arguing that such an utterance can have at least four distinct types of content. The difference is not reflected in the truth conditions of the utterance, but in something else entirely. If this is right, it seems to pose a special problem for semantic theories that try to explicate sentence meaning in terms of truth conditions.

Austin's paper has a special relevance to our project, as well. This work grew out of attempting to flesh out ideas about constraints presented in Situations and Attitudes (Barwise and Perry, 1983), a book that owes much to Austin's general approach to language.

Among the problems that have beset the field of artificial intelligence, or AI, two have a particularly logical flavor. One is the problem of nonmonotonicity referred to in Part I. The other is the so-called frame problem. In this lecture, we suggest that ideas from the theory presented here, combined with ideas and techniques routinely used in state-space modeling in the sciences, suggest a new approach to these problems.

Nonmonotonicity

The rule of Weakening implies what is often called monotonicity:

If Г ⊢ Δ then Г, α ⊢ Δ.

The problem of nonmonotonicity, we recall, has to do with cases where one is disinclined to accept a constraint of the form Г, α ⊢ Δ even though one accepts as a constraint Г ⊢ Δ. The following is an example we will discuss in this lecture.

Example 19.1. Judith has a certain commonsense understanding of her home's heating system – the furnace, thermostat, vents, and the way they function to keep her house warm. Her understanding gives rise to inferences like the following.

(α1) The thermostat is set between sixty-five and seventy degrees.

(α2) The room temperature is fifty-eight degrees.

⊢ (β) Hot air is coming out of the vents.

It seems that α1, α2 ⊢ β is a constraint that Judith uses quite regularly and unproblematically in reasoning about her heating system. However, during a recent blizzard she was forced to add the premise

(α3) The power is off.

In this lecture we investigate a distributed, physical system that is simple enough to explore in some detail but complex enough to illustrate some of the main points of the theory. Readers may work through this lecture as a way of getting a feeling for the motivations behind the mathematics that follow or skip ahead to the theory and come back to this example later. Readers who decide to work through this lecture should accept various assertions on faith, because they are justified by the work that follows.

Informal Description of the Circuit

The example consists of a light circuit LC. The circuit we have in mind is drawn from the home of one of the authors. It consists of a light bulb B connected to two switches, call them SW1 and SW2, one upstairs, the other downstairs. The downstairs switch is a simple toggle switch. If the bulb is on, then flipping switch SW2 turns it off; if it is off, then flipping SW2 turns it on. The upstairs switch SW1 is like SW2 except that it has a slider SL controlling the brightness of the bulb. Full up and the bulb is at maximum brightness (if lit at all); full down and the bulb is at twenty-five percent brightness, if lit, with the change in brightness being linear in between.

The three lectures in the first part of the book present an informal overview of a theory whose technical details will be developed and applied later in the book. In this lecture we draw the reader's attention to the problems that motivated the development of the theory. In Lecture 2 we outline our proposed solution. A detailed example is worked out in Lecture 3.

In the course of the first two lectures we draw attention to four principles of information flow. These are the cornerstones of our theory. We do not attempt to present a philosophical argument for them. Rather, we illustrate and provide circumstantial evidence for the principles and then proceed to erect a theory on them. When the theory is erected, we shall be in a better position to judge them. Until that job is done, we present the principles as a means of understanding the mathematical model to be presented – not as an analysis of all and sundry present day intuitions about information and information flow.

The Worldly Commerce of Information

In recent years, information has become all the rage. The Utopian vision of an information society has moved from the pages of science fiction novels to political manifestos. As the millennium approaches fast on the information highway, the ever-increasing speed and scope of communication networks are predicted to bring sweeping changes in the structure of the global economy.

In Lecture 8 we saw how to construct state spaces from classifications and vice versa. In Lecture 12 we saw how to associate a canonical logic Log(S) with any state space S. In this lecture we study the relation between logics and state spaces in more detail. Our aim is to try to understand how the phenomena of incompleteness and unsoundness get reflected in the state-space framework. We will put our analysis to work by exploring the problem of nonmonotonicity in Lecture 19.

Subspaces of State Spaces

Our first goal is to show that there is a natural correspondence between the subspaces of a state space S and logics on the event classification of S. We develop this correspondence in the next few results.

Definition 16.1. Let S be a state space. An S-logic is a logic ℒ on the event classification Evt(S) such that Log(S) ⊑ ℒ.

The basic intuition here is that an S-logic should build in at least the theory implicit in the state-space structure of S. We call a state σ of S ℒ-consistent if {σ}⊬ℒ and let Ωℒ the set of ℒ-inconsistent states.

Proposition 16.2.If S is a state space and ℒ is an S-logic, then ⊬ℒΩℒ. Indeed, Ωℒ is the smallest set of states such that ⊢ℒΩℒ.

Proof. To prove the first claim, let (Г, Δ) be any partition of the types of Evt(S) with Ωℒ∈Δ. We need to see that Г⊢ℒΔ.