Semantic guilt
When reading technical material outside philosophy, I am often struck by the widespread use of non-rigid names and variables. A typical example goes like this. You introduce 'X' to stand for, say, the velocity of some object under investigation. When you want to say that at time t1, the velocity is 10 units, you put exactly this into symbols: 'at t1, X = 10'. If the velocity changes, we get a violation of the necessity of identity:
At t1, X = 10.
At t2, X = 20.
Or suppose you have a population of n objects with various velocities. Your statistics textbook will tell you that the variance of the velocity in the population is defined as
Var(X) = Exp[X - Exp(X)]^2,
where Exp(Phi) is the average Phi value weighted by its frequency. What does 'X' stand for in this equation? On the right-hand side, it seems to denote a number, since you subtract another number from it. But it clearly doesn't stand for any particular number, such as the velocity of the third object in the population. On the other hand, it also isn't a bound variable: it represents the same magnitude on the right-hand side that it represents on the left-hand side (where, incidentally, it cannot just stand for a number, since it makes no sense to speak of the variance of a number).
As a final example, consider the description of dynamic systems in control theory. The probability that the system moves from a certain state to another at stage k is standardly written
P(x_{k+1} / x_k).
The two terms 'x_{k+1}' and 'x_k' are supposed to denote states. But if we just call these two states s1 and s2, and if the system can't be in two different states at once, then P(s1/s2) would have to be either zero or one, depending on whether s1 = s2. So 'x_{k+1}' doesn't just denote the state s1, but somehow says that the system is in s1 at stage k+1, which is thus compatible with 'x_k'.
Publications in statistics, computer science, engineering, and physics are full of examples like this. Nevertheless, my first reaction is always to blame the authors for sloppy writing. I think to myself that 'P(x_{k+1} / x_k)' is really short-hand for
P(the state at k+1 is x_{k+1} / the state at k is x_{k}),
and 'Var(X) = Exp[X - Exp(X)]^2' is short-hand for something like
Var(X) = Exp_i[Val_i(X) - Exp_i(Val_i(X))]^2
where 'i' ranges over the individuals in the population and 'Val_i(X)' denotes the value of the maginture X (i.e. velocity) for individual i.
However, the original expressions are completely in order if we interpret the problematic terms as intensional variables. An intensional variable denotes different things relevant to different points of evaluation. It has two semantic values: an extension and an intension. Predicates and functors can apply to either of these values. Thus in 'Var(X)', 'Var' applies to the intension of 'X'. The minus sign, on the other hand, applies to the extension of 'X'. 'Exp' is a modal operator that quantifies over points of evaluation. Similarly for 'x_{k+1}'. Here 'x' is an intensional function symbol. Its extension is a function from times, given by the subscripts, to states. Its intension is the functional concept "the system's state at --". 'P' in 'P(x_{k+1} / x_k)' is a binary intensional functor. The formal semantics of such expressions has been worked out in much detail by people like Church, Montague, Cocchiarella, Garsons and Fitting, and there is little doubt about its consistency and coherence.
So maybe I should stop accusing everyone else of sloppiness, and rather start accusing us philosophers of technical narrow-mindedness. It looks like we have been indoctrinated with the idea of semantic innocence -- the idea that individual variables must be "directly referring" -- oblivious to the fact that everyone else happily violates our doctrine. Shouldn't we rather join the party and give up on our innocence?
I think that in a good statistics text they're pretty explicit about something close to this. (At any rate, this is what I remember being taught.) "X" denotes a "random variable", which is a real-valued function on some measure space. You can trivially define operations that combine random variables and constant numbers: e.g., (X - a)(p) = X(p) - a for each point p in the underlying space. (In general, you can take names for real numbers as ambiguously referring to constant functions. This is pretty standard practice in math generally.) Exp and Var are function operators. (No need to posit an implicit variable ranging over the elements of the underlying spaceâwhen the space is infinite, that isn't even really correct.)
Similarly, if X_k is a random variable, then conventionally the lower-case "x_k" is used ambiguously between a value for X_k and an *event*âi.e., the subset of the underlying space on which X_k achieves that value. The probability talk is perfectly literal for the second use. This is also the standard thing to say, I think.
I think this is basically isomorphic to what you said, and it's nice to note that some of these systematic ambiguities track the traditional extension/intension distinction. I wonder how this thought might handle some of Kit Fine's applications of his theory of "arbitrary individuals".