Definite descriptions

Definite descriptions, such as "the badly juggling clown," succeed when the intended individual is indeed characterized by the property. But how does the logic work when characterizations fail?

We use a definite description when we make an assertion about an object or individual, referring to that individual by means of a property that uniquely picks them out. When I say, “the badly juggling clown in the subway car has a sad expression” I am referring to the clown by describing a property that uniquely determines the individual to whom I refer, namely, the clown that is badly juggling in the subway car, that clown, the one fitting this description. Definite descriptions in English typically involve the definite article “the” as a signal that one is picking out a unique object or individual.

If there had been no clown in the subway car, then my description wouldn't have succeeded—there would have been no referent, no unique individual falling under the description. My reference would similarly have failed if there had been a clown, but no juggling clown, or if there had been a juggling clown, but juggling very well, and so no badly juggling clown.

My reference would have failed in a different way if the subway car was packed full of badly juggling clowns, for in this case the description would not have succeeded in picking out just one of them. In each of these failing cases, there seems to be something wrong or insensible with my statement, “the badly juggling clown in the subway car has a sad expression.” What would be the meaning of this assertion, after all, if there was no such clown, if for example all the clowns were juggling very well?

Bertrand Russell emphasized that when one makes an assertion involving a definite description like this, then part of what is being asserted is that the definite description has succeeded. According to Russell’s account, when I say, “the book I read last night was fascinating,” then I am asserting first of all that indeed there was a book that I read last night and exactly one such book, and furthermore that this book was fascinating. For Russell, the assertion “the king of France is bald” asserts first, that there is such a person as the king of France and second, that the person fitting that description is bald. Since there is presently no such person as the king of France (and there wasn’t when Russell considered the assertion), Russell takes the statement to be false.

Is he bald or not?

Iota expressions

Let us introduce a certain notational formalism, originating in Russell and Whitehead’s Principia Mathematica (1910-13), to assist with our analysis of definite descriptions. Namely, the inverted iota notation ℩x ψ(x) is a formal term denoting “the x for which ψ(x),” such as “the first prime number after 100” in the integer ring or “the house in which I lived as a child.” Such a reference succeeds in a model M precisely when there is indeed a unique x for which ψ(x) holds,

\( M ⊨ ∃!x\ ψ(x),\)

or in other words, when

\( M ⊨ ∃x\,∀y\ \bigl(x = y ↔ ψ(y)\bigr).\)

The value of the term (℩x ψ(x)) interpreted in M is this unique object fulfilling property ψ. The use of iota expressions is perhaps the most meaningful when this property is indeed fulfilled, that is, when the reference succeeds, and we might naturally take them otherwise to be meaningless or undefined, a failed reference.

Because the iota expressions are not always meaningful in this way, their treatment in formal logic faces many of the same issues faced by a formal treatment of partial functions, functions that are not necessarily defined on the whole domain of discourse. According to the official semantics for first-order logic as we described it in an earlier post, after all, the interpretation of a function symbol f is a function defined on the whole domain of the model—in other words, in first-order logic we always interpret function symbols with total functions.

But partial functions commonly appear throughout mathematics, and we might naturally seek a formal treatment of them in logic. One immediate response to this goal is simply to point out that partial functions are already fruitfully and easily treated in first-order logic by means of their graph relations, the relation holding of a pair (x,y) for a given function g when it lies on the graph, that is, when y = g(x). One can already express everything one would want to express about a partial function g by reference to the graph relation—whether a given point is in the domain of the function and if it is, what the value of the function is at that point and so on. In this sense, first-order logic already has a robust relational treatment of partial functions.

In light of that response, the dispute here is not about the expressive power of the logic, but is rather entirely about the status of terms in the language, about whether we should allow partial functions to appear as terms. To be sure, mathematicians often form partial term expressions, such as the terms √(9 - x²) or 1/x in the context of the real numbers ℝ, which are not defined for all values of x in the domain. Allowing partial functions as terms would therefore seem to align with this aspect of mathematical practice.

But the semantics are a surprisingly subtle matter. The main issue is that when a term is not defined it may not be clear what meaning, if any, to ascribe to assertions formed using that term. To illustrate the point, suppose that e(x) is a term arising from a partial function or from an iota expression that is not universally defined in a model M, and suppose that R is a unary relation that holds of every individual in the model. Do we want to say that M ⊨ ∀x R(e(x))? Is it true that for every person, the elephant they are riding is self-identical? Well, some people are not riding any elephant, and so perhaps we might say, no, that shouldn't be true, since some values of e(x) are not defined, and so this statement should be false.

Perhaps someone else suggests that it should be true, because R(e(x)) will hold whenever e(x) does succeed in its reference—in every case where someone is riding an elephant, this elephant is self-identical. Or perhaps we want to say the whole assertion is meaningless? If we say it is meaningful, but false, however, then it would seem we would want to say M ⊨ ¬∀x R(e(x)) and consequently also M ⊨ ∃x ¬R(e(x)). In other words, in this case we are saying that in M that there is some x such that e(x) makes the always-true predicate R false—there is a person for which the elephant they are riding is not self-identical. That seems weird and probably undesirable, since it only works because e(x) must be undefined for this individual x. Furthermore, this situation seems to violate the Russellian injunction that assertions involving a definite description are committed to the success of that reference, for in this case, the truth of the assertion ∃x ¬R(e(x)) would be based entirely on the failure of the reference e(x).

Ultimately we shall face such decisions in how to define the semantics in the logic of iota expressions and more generally in the first-order logic of partial functions as terms.

The strong semantics for iota expressions

Let me begin by describing what I call the strong semantics for the logic of iota expressions. A bit later we shall also have the weak semantics, as well as what I call the natural semantics.