Collaborative articulation of how abstraction and language is employed in the computational manifestation of numbers -- including analysis of the role of syntax, semantics, and meaning in the specification and use of software interfaces.

[update 2008-06-17T16:08Z: I added t0' because I couldn't stand not to show the better refactoring of t0. I added mention of equivalence classes in an interpretation and also came up with conditions other than invalidity for finding an interpretation unusable.]

In The Logic of Ot, I said that I would use informal expressions of Ot, the logical theory that applies to Miser Obs. Now that there has been some use of the special characters and notations of First-Order Logic with equality, I want to take advantage of that to talk about interpretations of identity in models of Ot. The ability to identify and distinguish has great bearing on computational systems, and identity as an interpretation is particularly useful to explore.

= as Equivalence Relation

With FOL=, identity and the relational operator, "=", are taken as given, and the following hold:

=1: ∀x(x = x)

=2: ∀x∀y(x = y → y = x)

=3: ∀x∀y∀z(x = y ∧ y = z → x = z)

≠0: ∀x∀y(x ≠ y ↔ ¬(x = y))

The first three are the common properties of equivalence relationships: "=" is reflexive (=1), symmetrical (=2), and transitive (=3). The final condition is essentially the definition of "≠" in terms of "=".

What Are We Talking About?

In a first-order logical theory, the variables (x, y, and z as seen in ∀x∀y∀z) are understood to refer to objects in the domain of discourse. We only know what there is to know about that domain from the introduction of constants and expression of conditions that are theoretically required to be satisfied over that domain. For FOL=, we are given an equivalence relation (expressed with the symbol "="), which tells us very few things about conditions under which variables can be taken as referring to the same object of the domain of discourse.

It should be apparent that having "=" doesn't tell us much about the theoretical objects, although it is more than not having "=" (and its partner, "≠").

An intended interpretation could well be that objects in the domain of discourse be identifiable (we can tell when we are referring to the same one) and discernable (we can tell when we aren't). Let's see how that might work.

A Tiny Domain

To provide some practice with ideas of practical interpretation, consider the logical theory obtained by adding the following conditions:

t0: ∃x∃y∃z∀u( x ≠ y ∧ y ≠ z ∧ x ≠ z ∧ (u = x ∨ u = y ∨ u = z) )

or the equivalent,

t0': ∃x∃y∃z( x ≠ y ∧ y ≠ z ∧ x ≠ z ∧ ∀u(u = x ∨ u = y ∨ u = z) )

The intended reading of this is as

There are at least three different objects in the domain of discourse, and

Any object in the domain of discourse is one of those

which is to say, there are exactly three objects in the domain of discourse.

If we only have that one additional condition (t0) in our logical theory, we know nothing beyond that.

Notice that we have not labeled the three theoretical objects in any way. All we have provided for is that there be exactly three.

It will be useful to appear to be more specific by naming them:

t1: A ≠ B ∧ B ≠ C ∧ A ≠ C

t2: ∀u(u = A ∨ u = B ∨ u = C)

Here, A, B, and C are constants for objects in the domain of discourse. We haven't provided much more than what (t0) assures us of, although if there is more to say about the ways that the three objects differ, having the constants to refer to may be useful.

Illustrative Interpretations

The first interpretation will be in terms of numbers. Assume the system of numbers and arithmetic. Take that system as separate from the logical theory consisting of FOL= plus t0 and t1. (t2 is actually a consequence of that much.)

One interpretation of our tiny theory in number theory would be by saying that the interpretation of A is any n < 0, the interpretation of B is 0, and the interpretation of C is any n > 0. We could be specific, say, with interpretation of A as -1, B as 0, and C as +1. We could also say that A is all n < 0, B is 0 only, and C is all n > 0. That is, A, B, and C correspond to distinct classes. Since they have no members in common, these are known as equivalence classes. We'll explore that further when we return to exploration of Ot.

It doesn't matter, here, how the interpretation is chosen, so long as, having made it, we stick to it. The system in which we make the interpretation is a model (in the loose sense of Reality is the Model) provided that all deductions in our logical theory hold in the interpretation.

Because the logical theory says nothing about aspects of the model that are not accounted for in the logical theory, those matters are irrelevant to the conditions of the logical theory. It does not matter how many different ways the interpretation could be made in the model, so long as when one is made, the logical theory is seen to hold for the interpretation. There is a common fallacy involving reasoning about extra-theoretical characteristics of the model to argue that the theory is incorrect or inapplicable, when the disagreement is more-appropriately viewed as one over choice of interpretation. It helps to carefully separate the theory from its interpretations and models to avoid that pitfall. The abstract theory and the logical formalism is helpful in that regard, even if it feels quite unnatural.

"="/"≠" Are Interpreted Too

It is easy to overlook one important feature of an interpretation of our tiny theory: There must be an interpretation for "="/"≠" in the model as well. That comes along too easily in our choice of interpretations in the system of numbers and arithmetic and it is easy to overlook. When we dig into computational systems and the details of Miser, the ability to discriminate "="/"≠" in particular interpretations becomes very important.

Finally, as an interpretation in reality: let A be earth, B be wind, and C be fire. This is a questionable interpretation quite apart from the omission of water. The difficulty is assuring that these are cleanly distinguishable concepts. What do we do with flaming molten lava and the sucking wind of a forest fire?. We will stumble here at least in an effort to have well-determined "="/"≠" and understandable communication of the conditions that others can accept and apply. The simple, practical conclusion may be that the interpretation is invalid (or simply meaningless/useless) and the theory is inapplicable in that case.

In other cases, a certain conceptual sloppiness, if carefully circumscribed, may be tolerable in having useful interpretations in reality. It remains to be seen whether that is ever very workable.