# Making comparisons

Making comparisons

Bernard D. Katz

Much of the philosophical discussion concerning attributive adjectives, like “tall”, “large”, “heavy”, and “old”, has focused on the correct representation of their positive forms. It is clear, for a number of reasons, that we cannot represent the adjective “tall” in the sentence

(1) Brendan is a tall man

as an ordinary, unstructured one-place predicate, in the manner, say, of

(2) Tall (Brendan) & Man (Brendan).

For one thing, such a predicate would both be true of Brendan and not true of him: since Brendan is a man, he is also a mammal, though doubtless not a tall one. A further difficulty with taking “tall” as a simple one-place predicate is that doing so would obscure its logical connection with the comparative form of the adjective. The sentence “Any man who is taller than a tall man is a tall man” seems to be logically true and to depend for its truth on its logical form. “Taller than”, of course, is standardly taken as a two-place predicate. Thus, there would be no formal connection between “taller than” and “tall” if the latter were construed as an unstructured one-place predicate.

Things are quite different when it comes to the superlative form. Taking the superlative form of “tall” in the sentence

(3) Brendan is the tallest applicant

as a simple one-place predicate would, of course, also be problematic. Again, for example, it would be difficult to provide a satisfactory account of the formal connection between the comparative and the superlative forms and, thus, of the logical truth “If Brendan is the tallest applicant, he is taller than any other applicant”. But in fact there is not much temptation to represent “tallest” as a simple one-place predicate. Given the resources of first-order logic with identity, we may represent (3) in the manner of

(3[prime]) Applicant (Brendan) & [for every]x (Applicant (x) & x [not equal to] Brendan [right arrow] Taller (Brendan, x)).

An adequate representation of the superlative form in terms of the comparative is, as John Wallace notes (1972, p. 773), one of first-order logic’s success stories. The model is undoubtedly impressive, and it suggests a pattern that might somehow be extended to the positive form. Accordingly, Wallace, and others, have attempted to show that the positive form may be construed in terms of the comparative.

In this paper, however, I want to consider a prior question, namely, whether the comparative form is itself basic. Wallace remarks that “it is hard to imagine any improvement on the standard representation in quantification theory of the comparative and superlative forms of adjectives” (1972, p. 773). It seems to me, however, that there is room for improvement. I do not dispute the adequacy of the standard representation of the superlative in terms of the comparative, but I think that there is more to be said about the comparative. Construing the comparative form of the likes of “tall” as an unstructured two-place predicate does not get to the bottom of things, or so I shall argue.

I

In treating the comparative form of “tall” in the sentence

(4) Alfred is taller than Brendan

as a simple two-place predicate, we are apt to overlook the fact that “than” in (4) is a grammatical conjunction and that “Brendan” is a grammatical subject. One sign of this is that “Brendan” gives way to a pronoun in the nominative case: for example, “Alfred is taller than he”. But if “than” is a grammatical conjunction and “Brendan” a grammatical subject, it would seem that (4) should contain a sentential clause, of which “Brendan” is the grammatical subject, following “than”. And, in fact, it does, for (4) is elliptical for “Alfred is taller than Brendan is”, which itself is short for

(5) Alfred is taller than Brendan is tall.

The use of a construction like (5) in place of one like (4) is no doubt rare; (4) by itself is sufficient to indicate what comparison between Alfred and Brendan is at stake. But when the point of comparison is not obvious, we do use a longer form; for example, “The window is taller than it is wide” or “The cabinet is longer than the table is high”.

Since the sentence “Brendan is tall” appears in (5), it is natural to suppose that the sentence “Alfred is tall” does so as well. We can make this more explicit if we rewrite (5) in the manner of

(6) Alfred is tall er-than Brendan is tall.

Here “er-than” is a grammatical conjunction connecting two sentential clauses. What is the logical form of (6)? In particular, what is the logical role of the expression “er-than” in (6)?

Someone who takes the surface grammar of (6) as displaying its logical form would conclude that “er-than” forms a sentential connective of some sort and that “Alfred is tall” and “Brendan is tall” are not only grammatical constituents of (6) but are also logical ones. Aside from the grammar of (6), however, there is little to recommend such an interpretation. And there is a lot to be said against it. For one thing, as we saw earlier, we cannot construe the adjective “tall” as a simple one-place predicate; consequently, sentences such as “Alfred is tall” and “Brendan is tall” are at best incomplete: we are unable to assign a truth-value to such a sentence unless the context in which it occurs enables us to connect “tall” to an appropriate noun or noun phrase. It is hard to see that in the present case the context does provide such information.

Even if we assume that “Alfred is tall” and “Brendan is tall”, as they occur in (6), are elliptical for sentences that combine “tall” with an appropriate noun or noun phrase, there is a further difficulty with the idea that “er-than” in (6) is a sentential connective, which is that it conflicts with a well-known piece of reasoning, derived from Frege. The problem is this. On the one hand, the context created by “taller than” permits substitution of singular terms: substituting another name, or a definite description, having the same reference for “Alfred”, or for “Brendan”, will not alter the truth-value of (4). If Brendan is the tallest applicant, then it follows that Alfred is taller than the tallest applicant. It is also natural to suppose that the context created by “taller than” tolerates the replacement of constituent expressions with ones that are logically equivalent. So, “taller than” creates a context that allows substitution both of co-extensional names and definite descriptions and of logically equivalent components within its scope salva veritate, in which case “er-than” in (6) does as well. But, on the other hand, if “er-than” is a sentential connective, it is not a truth-functional one. Suppose that the biconditional

(7) Brendan is tall [equivalence] Charles is tall

is true. It may be that while both Brendan and Charles are tall, Alfred is taller than the former but not the latter, which is to say that it is possible that (6) and (7) are true but that

(8) Alfred is tall er-than Charles is tall

is false. (8) comes from (6), however, by substitution of “Charles is tall” for “Brendan is tall”, that is, of one constituent of (7) for the other. Since substituting another sentence having the same truth-value for “Brendan is tall” may alter the truth-value of (6), “er-than” must be a non-truth-functional sentential connective (if it is a sentential connective at all). Hence, the supposition that “er-than” is a sentential connective commits us to the consequence that “er-than” creates a sentential context that allows both substitution of co-extensional singular terms and interchange of logical equivalents but is not truth-functional. This outcome, however, contradicts the conclusion of the Fregean argument mentioned earlier, which asserts that: if a sentential context permits both substitution of co-extensional names and definite descriptions and of logically equivalent components within its scope salva veritate, then it must be truth-functional.(1)

Anyone who accepts this principle as well as our earlier claims about “taller than” must discard the idea that “er-than” creates a sentential context, and together with it the idea that it constitutes a sentential connective.

This principle connecting referential transparency and truth-functionality has attracted considerable comment and criticism.(2) So it is worth pointing out that for our present purposes, we can do without it. Using reasoning similar to that underlying the Fregean argument, one can easily see that even if non-truth-functionality is compatible with referential transparency, the conjecture that “er-than” creates a sentential context is incompatible with the idea that substitution of co-extensional singular terms and interchange of logical equivalents are both permissible operations in the likes of (6). Suppose that both constituents of (7) are true. Since “Brendan” denotes the same individual as the expression” ??x(x = Brendan & Charles is tall)”, we may substitute the latter for the former in (6), thereby obtaining

(9) Alfred is tall er-than ??x(x = Brendan & Charles is tall) is tall.

But” ??x(x = Brendan & Charles is tall) is tall” is logically equivalent to “??x(x = Charles & Brendan is tall) is tall”; so we may infer

(10) Alfred is tall er-than ??x(x = Charles & Brendan is tall) is tall

from (9). Finally, substituting “Charles” for” ??x(x = Charles & Brendan is tall)” in (10), we obtain (8), which by hypothesis is false. This by itself does not show truth-functionality; but it is sufficient to show that if the context created by “taller than” tolerates substitution of co-extensional singular terms and interchange of logical equivalents, there is something amiss in the idea that “er-than” creates a sentential context.

Since it is clear that the context created by “taller than” permits substitution of singular terms within its scope salva veritate, someone might contend that the foregoing reasoning shows only that it is a mistake to suppose that “er-than” creates a sentential context that allows substitution of logical equivalents within its scope salva veritate. It is, no doubt, true that if “er-than” creates a sentential context, it is not one that allows the requisite substitutions. This, I think, casts doubt on the idea that we have hit on the correct analysis of “taller than”. But it also points to a further difficulty with the hypothesis under consideration: replacing “Brendan is tall” in (6) with a logically equivalent sentence may take us not only from truth to falsehood but also from sense to nonsense.(3) For example, the result of replacing “Brendan is tall” in (6) with the logically equivalent “Brendan = ??x(x = Brendan & x is tall)” is clearly incoherent. Such a consequence is, however, inconsistent with the idea that “Brendan is tall” occurs as a sentential constituent of (6). A sentential context will permit the interchange of logically equivalent sentences within its scope salva congruitate (even if not salva veritate). On the other hand, the replacement of an expression that occurs merely as a grammatical segment of some sentence with a logically equivalent expression may well fail to preserve the grammatical character of that sentence. For example, though the sentences “[there exists]x(x walked & x = Alfred)” and “Alfred walked” are logically equivalent, the result of substituting “[there exists]x(x walked & x = Alfred)” for “Alfred walked” in the sentence “Alfred walked a mile” is nonsensical. This shows (what we already knew) that “Alfred walked” does not figure as a sentential constituent of “Alfred walked a mile”. Since a sentential context will permit interchange of logically equivalent sentences within its scope salva congruitate, if a context fails to tolerate such substitutions, it cannot be a sentential context. As we have seen, supplanting “Brendan is tall” in (6) with a logically equivalent sentence may yield nonsense; it follows that “Brendan is tall” is not a sentential constituent of (6).

I conclude that, appearances to the contrary, “er-than” does not create a pair of sentential contexts within (6), and the sentences “Alfred is tall” and “Brendan is tall” are not sentential constituents of (6). Considering the very limited range of sentences that may be substituted salva congruitate for “Alfred is tall” and “Brendan is tall”, one may find this conclusion plausible independently of the arguments that I have given for it. But our earlier question concerning the logical form of (6) and the logical role of “er-than” remains.

Essentially the same problem arises for other forms of the adjective. According to traditional grammar, adjectives are said to have three degrees of comparison, positive (“tall”), comparative (“taller”), and superlative (“tallest”). The so-called comparative degree, however, is in fact just one of many constructions of the adjective available for making comparisons: two individuals may be as tall as each other; one may be taller, or less tall, than another; one may be twice as tall as another; and so on. The difficulty raised earlier in connection with the logical form of (6) also arises in connection with these other constructions. Consider, for example,

(11) Alfred is as tall as Charles.

As with (4), we may expand (11) to reveal a pair of sentential clauses joined by a grammatical conjunction. Recasting (11) in the manner of (6) gives us something along the lines of

(12) Alfred is tall as Charles is tall.

It is evident from our discussion of (6) that (12) does not accurately display the logical form of (11), that is, that “as” does not create a pair of sentential contexts within (12) and that the sentences “Alfred is tall” and “Charles is tall” are not logical constituents of (12). Hence, we have a puzzle similar to the one that arises in connection with (6): What is the logical form of (12)? In particular, what is the logical role of the expression “as”?

Our questions concerning the logical form of (4) and (11) are closely connected. It is evident that (4) and (11), taken together, imply

(13) Charles is taller than Brendan.

This inference holds, moreover, as a matter of logical form, so an acceptable account of the likes of (4), (11), and (13) should enable us to show that the premises of this inference are logically related to the conclusion. Of course, if we think of “taller than” and “as tall as” in terms of unrelated two-place predicates, as in fact we do on the usual way of formalizing these sentences, we stand little chance of showing any formal connection between these various sentences. On the other hand, if we think of (13) in the manner of

(14) Charles is tall er-than Brendan is tall,

we find ingredients that these sentences have in common, namely, “Alfred is tall”, “Brendan is tall”, and “Charles is tall”. But our earlier discussion shows that it is a mistake to suppose that these grammatical ingredients are genuine logical constituents of (6), (12), and (14). So, how can we explain the validity of the foregoing inference? What formal connection holds between the premises and conclusion?

II

One possible answer to our question concerning the logical form of the likes of (6) and (12) is suggested by Frege’s treatment of temporal terms such as “before” and “after”. Certainly, “after” in

(15) Alfred arrived after Brendan departed

is a grammatical conjunction joining two sentential clauses. The argument sketched earlier, however, shows that “after” in (15) cannot be regarded as a sentential connective and that “Alfred arrived” and “Brendan departed” cannot be regarded as sentential components of (15). Of course, we are not ordinarily inclined to construe “after” as a sentential connective, nor are we apt to construe “Alfred arrived” and “Brendan departed” as logical constituents of (15). Our standard procedure, essentially Frege’s, is to regard such a sentence as saying something like: Alfred arrived at sometime, Brendan departed at sometime, and the former time was after the latter; in other words,

(15[prime]) [there exists]t[there exists]t[prime] (Arrived (Alfred, t) & Departed (Brendan, t[prime]) & After (t, t[prime]))

It turns out, as a consequence, that (15) is really an existentially quantified conjunction, with variables ranging over times. Contrary to grammatical appearance, “arrived” and “departed” are two-place predicates rather than one-place predicates, and “after” is a two-place predicate, true of pairs of times, rather than a sentential connective. This explains why (15) permits substitution of co-referential expressions (the positions occupied by “Alfred” and “Brendan” in (15[prime]) are purely referentially) and also why we may not, in general, substitute other sentences having the same truth-value for “Alfred arrived” and “Brendan departed”: “Alfred arrived” and “Brendan departed” occur in (15) not as sentential components but as parts of the predicate expressions “Alfred arrived at t” and “Brendan departed at t[prime]”, respectively.

Donald Davidson employs a similar device in his analysis of action sentences and singular causal sentences (Davidson 1980 and 1980a). For example, the causal sentence

(16) Jack fell down, which caused it to be the case that Jack broke his crown

appears to contain a sentential connective linking the constituent clauses “Jack fell down” and “Jack broke his crown”. But according to Davidson’s well-know analysis, that sentence is to be construed along the lines of

(16[prime]) [there exists]e[there exists]e[prime] (Fell (Jack, e) & Break (Jack’s crown, e[prime]) & Caused (e, e[prime])).

The variables here are taken as ranging over events, so that the sentence may be understood as saying: some fall of Jack’s caused some breaking of his crown.(4)

Using this pattern as a model, we might take (6) and (12) in the manner of

(17) [there exists]x[there exists]x[prime] (Tall (Alfred, x) & Tall (Brendan, x[prime]) & Er-than (x, x[prime]))

and

(18) [there exists]x[there exists]x[prime] (Tall (Alfred, x) & Tall (Charles, x[prime]) & As (x, x[prime])),

respectively. The basic idea here is that we interpret “tall”, in the likes of (6) and (12), as a two-place predicate, that is, as containing one more place than it appears to, and take “er-than” and “as” as two-place predicates, rather than sentential connectives. Accordingly, the logical form of (6), and hence of (4), would be that of an existentially quantified conjunction; likewise with (12) and (11). Again, this would explain why we may substitute other terms having the same reference for “Alfred” and “Brendan” in (4) or (6), as well as in (11) or (12), and yet may not, in general, substitute other sentences having the same truth-value for “Alfred is tall” and “Brendan is tall”. Of course, if we do proceed in this way, we must take up two crucial questions. First, what are we supposed to make of this new two-place predicate introduced by “tall”? How are we supposed to read, or understand, an expression like “Tall (Alfred, x)”? Secondly, what are the values of the variables in (17) and (18)? What are the items that the relations expressed by “er-than” and “as” order?

The answer to the first question is, I think, surprisingly straightforward. At first blush, it may seem odd, if not bizarre, to take “tall” as a relative term. It may, in fact, seem an artificial construction, having little connection with the way in which we ordinarily use the adjective “tail”. Such a conclusion would be hasty. Concentration on the positive and comparative forms may obscure a commonplace fact, which is that we standardly use “tail” as a relational expression, as in

(19) Alfred is 6 feet tall.

It is natural to take “tall” in a sentence such as this as representing a two-place relation, relating Alfred to something else.

I propose that we take this last use of “tall” as basic to our understanding of the comparatives “taller” and “as tall as”, as well as of the positive “tall” and the superlative “tallest”.(5) We saw earlier that it is standard to represent the superlative, in quantificational theory, in terms of the comparative and noted that various philosophers have tried to extend this pattern to the positive. My suggestion is that we take “tall” in its relational use as the basic component in our representation of all of these forms, and perhaps others as well.

This approach, if sound, supplies us with a natural and intuitive way of connecting apparently disparate uses of the adjective. It shows us, of course, a syntactic ingredient common to the likes of (4), (11), and (13). It also shows us how these sentences are related to the likes of (19). Sentences such as these stand in various logical relations to each other and to other sentences containing occurrences of “tall”. We already noted that (4) and (11), taken together, imply (13). (4) and (19), taken together, imply

(20) Brendan is under 6 feet tall,

while (11) and (19), taken together, imply

(21) Charles is 6 feet tall.

Similarly, from (19) and (20) we may infer (4); and from (19) and (21) we may infer (11). Since these inferences are formally valid, an account of the relevant premises and conclusions should provide the resources for explaining this fact. As I mentioned earlier, we stand little chance of showing any formal connection between these various sentences if we construe the various forms of “tall” occurring in those sentences as syntactically unconnected, if we construe them in terms of distinct, simple two-place predicates. We have, moreover, the prospect of a great many such predicates, for in addition to “taller” and “just as tall as”, there are “half as tall as”, “twice as tall as”, and so forth. On the present proposal, however, we can define the various expressions in terms of a single, common underlying component, in the manner suggested by (17) and (18). Accordingly, we have the promise of an explanation of the validity of the various inferences involving “tall”, “taller than”, “as tall as”, and “tallest”.

III

An important characteristic of “taller than” is that it stands for a relation that is transitive, asymmetric, and irreflexive. Accordingly, we assume that the relation represented by “er-than” does as well. It is plausible to suppose that “er-than” is short for “more than”, in which case we can take it simply as representing the relation of being greater than. Thus, (17) becomes:

(22) [there exists]x[there exists]x[prime] (Tall (Alfred, x) & Tall (Brendan, x[prime]) & x [greater than] x[prime]).

Similarly, “as tall as” stands for a relation that is transitive, symmetric, and reflexive; thus, we assume that “as” does as well. It is plausible to think of “as” here as short for “as much as”, in which case we can take it simply as representing the relation of identity. Hence, we may put (18) as:

(23) [there exists]x[there exists]x[prime] (Tall (Alfred, x) & Tall (Charles, x[prime]) & x = x[prime]).

Consider, once again, the inference from (4) and (11) to (13). Using the current format, we may put (13) as:

(24) [there exists]x[there exists]x[prime](Tall (Charles x) & Tall (Brendan, x[prime]) & x [greater than] x[prime]).

Hence, if (22), (23), and (24) give us the correct form of the comparatives mentioned earlier, we should be able to derive (24) from (22) and (23). Reflection reveals that we cannot: the argument from (22) and (23) to (24) is not valid. Apparently something has gone wrong.

It is not difficult, I think, to locate the defect. The truth of (22), our formulation of the claim that Alfred is taller than Brendan, as well as that of (23), our formulation of the claim that Alfred is as tall as Charles, ensures that the open formula “Tall (Alfred, x)” is true for at least one value of “x”; but that formula may not be true for the same values of “x” in both (22) and (23). In other words, our formulation allows that “tall” may relate Alfred to two distinct values. This, however, makes no clear sense, for it implies, in effect, that something could be taller than itself. Accordingly, we must add the requirement that a formula such as “Tall (Alfred, x)” is true for just one value of “x”.(6) Thus, we may reformulate (22), (23), and (24) as:

(22[prime]) [there exists]x[there exists]x[prime] ([for every]y(Tall (Alfred, y) [equivalence] y = x) & [for every]y(Tall (Brendan, y) [equivalence] y=x[prime]) & x [greater than] x[prime])

(23[prime]) [there exists]x[there exists]x[prime] ([for every]y(Tall (Alfred, y) [equivalence] y = x) & [for every]y(Tall (Charles, y) [equivalence] y = x[prime]) & x = x[prime])

(24[prime]) [there exists]x[there exists]x[prime] ([for every]y(Tall (Charles, y) [equivalence] y = x) & [for every]y(Tall (Brendan, y) [equivalence] y = x[prime]) & x [greater than] x[prime]).

This gives us a straightforward account of the inference from (4) and (11) to (13). The first premise says, in effect, that the extent to which Alfred is tall exceeds that to which Brendan is tall; the second premise says that Alfred and Charles are tall to the same extent; and the conclusion says that the extent to which Alfred is tall exceeds that to which Charles is tall.

Given the assumption that “tall” relates a thing to exactly one value (at least if “tall” relates that thing to any value), we may construe this two-place relation as a function. Accordingly, our analysis allows us to abbreviate (22[prime]), (23[prime]), and (24[prime]) as:

(22[double prime]) tall (Alfred) [greater than] tall (Brendan),

(23[double prime]) tall (Alfred) = tall (Charles),

and

(24[double prime]) tall (Charles) [greater than] tall (Brendan)

respectively, where “tall (x)” represents the relevant function. Taken in this manner, we see that the conclusion follows from the premises by substitutivity of identity.

It is easy to see that our analytic sentences now capture the desired characteristics of “taller than” (for example, transitivity, asymmetry, and irreflexivity) and “as tall as” (for example, transitivity, symmetry, and reflexivity). Given, for example, both that the spire is taller than the chimney and that the chimney is taller than the house, it follows that the spire is taller than the house. Using the present format, we have:

(25) tall (the spire) [greater than] tall (chimney) (26) tall (chimney) [greater than] tall (house) (27) tall (spire) [greater than] tall (house).

It is evident that in the presence of the assumption that “[greater than]” is transitive, we may infer (27) from (25) and (26).

We can also keep track of various inferences connecting “tall” and other adjectives, such as “wide” and “long”. For example, the sentences “The cabinet is taller than the table is wide” and “The table is as wide as the desk is long”, taken together, imply “The cabinet is taller than the desk is long”. Applying to “wide” and “long” an account similar to the one proposed for “tall”, we may represent the premises and conclusion of this inference as

(28) tall (cabinet) [greater than] wide (table) (29) wide (table) = long (desk)

and

(30) tall (cabinet) [greater than] long (desk)

respectively, where “wide (x)” represents the function associated with the predicate “Wide (x, y)” and “long (x)” the function associated with “Long (x, y)”.

We can, of course, extend this analysis to forms of the adjective that we already understood in terms of the comparative, such as the superlative. Thus, we may put “Molly is the tallest student in the school” in the manner of

(31) Student (Molly) & [for every]x(Student (x) & x [not equal to] Molly [right arrow] tall (Molly) [greater than] tall (x)).

Can we also extend this analysis to the positive form of “tall”? I think that we can at least show how the logical role of the positive form of “tall” is related to that of the other forms of the adjective, in particular, to the comparative. The positive form of “tall”, as we have seen, requires a reference class: things are tall (or not) only relative to some comparison class. Thus, the sentence

(32) Laura is a tall ballerina

tells us that Laura is a ballerina and that compared to other ballerinas she is tall. The superlative, of course, also requires a reference class: the tallest student in the school may not be the tallest person. Unlike the superlative, however, the positive form of “tall” is vague: in particular, there are, or may be, ballerinas of whom the adjective “x is a tall ballerina” is neither determinately true nor determinately false. So the condition that Laura must satisfy, if she is tall compared to the class of ballerinas, will be vague as well.

Laura is tall, or not, compared to the class of ballerinas depending on how tall Laura is in comparison with how tall ballerinas in general are; depending, in other words, on where the value of “tall (Laura)” falls in the distribution of the range of values of “tall (x)”, as applied to the class of ballerinas. Presumably, if she is tall compared to this class, her height must be above the average height of ballerinas; but how far above? It seems unlikely that any precise, determinable specification will support the application of a vague notion such as being tall compared to ballerinas. The best we can say, I think, is something like this: Laura is tall in the class of ballerinas just in case Laura’s height is comparatively great; in other words, just in case tall (Laura) has a relatively high place in the ordering of the elements of the set {tall (x)[where](Ballerina (x)} with respect to the “greater than” relation.(7) Suppose that “g(s)” represents a standard that separates those elements of the set s that have a relatively high location in the relevant ordering from those that do not. Then we can represent (32) in the following manner:

(33) Ballerina (Laura) & tall (Laura) [greater than] g({tall (x)[where](Ballerina (x)}).

So construed, we see that (32) entails “Laura is a ballerina”, as it should. We also see why if Laura is a tall ballerina, then any ballerina who is as tall as, or taller than, Laura is also a tall ballerina, and why Laura is taller than any ballerina who is not a tall ballerina. And we see that if all and only members of the dance troupe are dancers in the corps de ballet, then a tall member of the dance troupe must be a tall dancer in the corps de ballet. On the other hand, we can explain why a tall ballerina may not be a tall dancer.

Since the extension of the predicate “x is a tall ballerina” is indeterminate, so is the position of g({tall (x)[where](Ballerina (x)}). Does it nevertheless partition that class precisely, dividing it, in the manner of a sharp boundary, into two exhaustive groups? In other words, does the predicate “x is a tall ballerina” have a precise, though indeterminate, extension? This issue, though crucial to an understanding of vagueness, goes beyond our present concern. In fact, I am inclined to the view that there is an exact, though indeterminate, boundary between those ballerinas that are tall and those that are not.(8) But I do not see that the foregoing account of the likes of (33), by itself, commits us to the view that there is: for all that we have said, “g({tall (x)[where](Ballerina (x)})” may indicate a penumbra rather than a definite border.

IV

The proposal we have been sketching takes as basic the idea that “tall” in certain of its uses represents a two-place relation and, using that relative term, provides a procedure for representing various other forms of the adjective “tall”. I now turn to the question: what is the range of this relation? What exactly are the values of the bound variables introduced in the likes of (22[prime])-(24[prime]) or the function represented in (22[double prime])-(24[double prime])?

The idea that the relevant values are simply numbers may be appealing. Take the set of humans (or spires) and factor out by the equivalence relation of being as tall as: we thus obtain a structure which is an ordered set, in particular, a strict linear ordering (that is, one that is transitive, asymmetric, and connected). It may be tempting to think of this simply as a subset of the real numbers ordered by the “greater than” relation. There are good reasons, however, for resisting this strategy.

One problem with the idea that the values of the variables are simply numbers is that it interferes with a straightforward account of the validity of various natural inferences. For example, given that Brendan is 5 1/2 feet tall and that Alfred is taller than Brendan, we may infer that Alfred is taller than 5 1/2 feet. Or given that given that Brendan is 5 1/2 feet tall and that Alfred is taller than 5 1/2 feet, we may infer that Alfred is taller than Brendan. What makes such inferences valid? The obvious explanation takes “Brendan is 5 1/2 feet tall” and “Alfred is taller than 5 1/2 feet” as involving reference to a particular height (among other things). If we take “Brendan is 5 1/2 feet tall” and “Alfred is taller than Brendan” in the manner of

(34) tall (Brendan) = 5 1/2 feet and

(35) tall (Alfred) [greater than] tall (Brendan),

respectively, we can explain why their truth ensures that of “Alfred is taller than 5 1/2 feet”, that is,

(36) tall (Alfred) [greater than] 5 1/2 feet.

The first premise of the argument is true just in case “tall (Brendan)” denotes a certain measure of height, namely, one of 5 1/2 feet; and the second premise is true just in case the denotation of “tall (Alfred)” is greater than that of “tall (Brendan)”. So it follows that “tall (Alfred)” denotes a measure of height greater than 5 1/2 feet, in which case the conclusion is true. But if we interpret these sentences in this way, we are construing them as involving reference to, or quantification over, quantities, in particular, heights.

Some philosophers have urged, however, that it is not necessary to import items such as 5 1/2 feet to make sense of talk of height. Quine, for example, urges that we can accommodate terms that purport to denote units of measure as parts of relative terms, for example, “length in miles”. According to Quine,

… “length in miles” is to be understood as true of this and that number relative to this and that body or region. Thus instead of “length of Manhattan = 11 miles” we would now say “length-in-miles of Manhattan = 11”. (Quine 1960, p. 245)

Hence, we have no reason to take expressions like “11 miles” as singular terms or units of measure as occupying positions accessible to variables of quantification.

Davidson as well thinks that we can do without quantities. He argues that while talk about weight, for example, requires an object to be weighed, it does not require that there be weights for objects to have. He points out that when we assign weights to things we are reporting on relations among objects such as weighing the same as, weighing more than, and weighing twice as much as. But according to Davidson, this does not require that we countenance such items as weights. He says:

Thus we say that the Koh-i-noor diamond weighs 109 carats or 345 grams. But talk of this sort does not require us to include carats or grams in our ontology. The only objects we need are numbers and the things that have weight. To say the weight in carats of the Koh-i-noor diamond is 109 does not commit us to weights as objects: it is just to assign the number 109 to the diamond as a way of relating it to other objects on the carat scale. (Davidson 1989, p. 10)

I do not see that these proposals are consistent with an adequate semantics for the sentences in question. I presume that we are to understand “feet” in “Brendan is 5 1/2 feet tall” and “Alfred is taller than 5 1/2 feet” as part of the relative term “tall”. Thus, instead of “tall (Brendan) = 5 1/2 feet” and “tall (Alfred) [greater than] 5 1/2 feet”, we have

(34[prime]) tall-in-feet (Brendan) = 5 1/2

and

(36[prime]) tall-in-feet (Alfred) [greater than] 5 1/2,

respectively. But then, how are we to understand the second premise of the earlier inference, that is, “Alfred is taller than Brendan”? The validity of this syllogism requires that we find a syntactic ingredient common to the two premises and the conclusion, and the obvious candidate is “tall-in-feet (x)”. The problem, however, is that the second premise does not make reference to any particular unit of measure and, so should be analysed in the same way regardless of whether the first premise and conclusion speak of inches, feet, yards, rods, or meters.

Sentences such as “Alfred is taller than Brendan” and “Alfred is as tall as Charles” compare objects in respect of their height and, as I argued earlier, compare heights as well. But they do this without mentioning specific units of measure. Nor do their meanings vary according to the units of measure we might use to size up the individuals whose heights we are comparing: it is not as though “Alfred is taller than Brendan” means one thing when we infer its truth from the information that Alfred is 6 feet tall and Brendan is 5 1/2 feet and means something else when we infer its truth from the information that Alfred is 1.8288 meters and Brendan is 1.6764 meters. Perhaps one cannot know that such comparisons hold without knowing something about some unit of measure or other; but it is clear that two individuals can both know that these comparisons hold, they can both reach the conclusion that Charles is just as tall as Alfred, without using the same scale of measure. If we can understand sentences such as “Alfred is taller than Brendan” and “Alfred is just as tall as Charles” without knowing anything about some particular scale of measure – without knowing, for example, anything about feet as a unit of measure – then it is hard to see that a correct analysis of them can incorporate predicates whose application presupposes that we know how to use that scale.

There is further problem with the idea that the relevant values are numbers, which is that it would allow for comparisons that make, no clear sense. We may represent the comparatives “The cabinet is taller than the table is wide” and “The cabinet is heavier than the table” as

(37) tall (cabinet) [greater than] wide (table)

and

(38) heavy (cabinet) [greater than] heavy (table),

respectively, where “heavy (x)” represents the function associated with the predicate “Heavy (x, y)”. I suggested earlier that we understand such sentences as speaking of, and comparing, heights and widths, in the first case, and weights, in the second. If the range of the relevant relations, or functions, is simply a subset of the real numbers, then we are speaking of, and comparing, numbers, in both cases. Though we can, and do, compare heights and widths, it makes no sense to compare a height and a weight, to say that a height of 5 1/2 feet is greater than, or less than, a weight of 150 pounds. On the other hand, any two numbers are comparable: if “tall (the cabinet)” and “heavy (table)” denote numbers, then unless they denote the same number, the value of one is greater than the value of the other. Therefore, on the view that the range of such functions is simply a subset of the real numbers, it follows that either the cabinet is taller than the table is heavy, the table is heavier than the cabinet is tall, or the cabinet is as tall as the table is heavy. I take this consequence to show that the proposal under consideration is radically defective.

It is plausible to suppose that the “greater than” relation on a domain of heights forms a strict linear ordering, and similarly for the “greater than” relation on a domain of weights; but it is not at all plausible to suppose that it also does on a domain of heights and weights taken together: for, as we have just seen, it makes no sense to say of a specific height and a specific weight that they are the same, or that one is greater than the other. If so, we cannot take the range of the relations, or functions, under discussion simply as a subset of the real numbers, for the “greater than” relation on a subset of the real numbers forms a linear ordering. Evidently, we must distinguish those numbers used to measure height from those used to measure weight, and of course we normally do so by attaching units of measure to them, feet to the former and pounds to the latter, for example.

This encourages the conclusion that “5 1/2 feet” in “Brendan is 5 1/2 feet tall” should be taken as a term that stands for a particular height. More generally, it supports the conclusion that the values of the bound variables introduced in our analysis of comparatives are quantities. Of course, if we treat certain expressions as referring to heights and take certain sentences as involving quantification over heights, then we are including them in our ontology.

V

According to the preceding account, the adjective “tall” has a relative form, corresponding to its use in a metrical sentence such as “Alfred is 6 feet tall”. This form of “tall” may be rendered as a two-place predicate “Tall (x, y)” and is true of ordered pairs consisting of things and their heights. Using that predicate, we may represent various other forms of the adjective, in particular, “taller”, “as tall as”, and “tallest”. It is evident that if this account of “tall” is right, then we can, without too much difficulty, extend it to many other attributive adjectives, certainly to “large”, “dense”, “fast”, “warm”, and “old”, as well as “heavy”, “wide”, and “long”.

Though we have well-defined procedures for measuring the quantities associated with the adjectives mentioned earlier, we also speak of degrees and make comparisons using adjectives where we have neither access to precise numerical measures nor well-developed methods for producing them: thus, for example, with “graceful”, “creative”, and “useful”. While we sometimes infer the truth of a comparison from that of metrical sentences, it is obvious that the ability to make comparisons does not presuppose either knowledge of numerical measures or the ability to produce them: if it did, we would not be entitled to make many of the comparisons that we do make. But it is no part of the proposed analysis that we have knowledge of, or the ability to produce, the requisite numerical measures. On the current view, a comparative such as “The coffee is warmer than the cream” is true if and only if the coffee has a unique temperature, the cream has a unique temperature, and the former temperature exceeds the latter. Since we can know that an existential generalization is true without knowing that any particular instance of it is, we can have reasons for believing that the comparative is true without producing a precise measure of the quantities involved. Our account does, of course, presuppose that when we compare things using some adjective, there is something to be measured, that the characteristic on the basis of which we apply the adjective comes in degrees. Though we may not have great confidence in any procedure for assigning numbers to measure, say, intelligence or beauty, I do not see that we could make sense of the claim that one individual is smarter, or more beautiful, than a second unless we also were able to attach a sense to talk of degrees of intelligence, or of beauty.

Though an ability to compare things in the manner under discussion does not presuppose an ability to assign numerical measures, there is one way in which they are closely connected, which is that we employ numerical measures as a device for keeping track of comparisons.(10) If we are able to compare things in terms of the degree to which they have some characteristic in such a way as to produce a ranking of those things, then we have taken an important step in the direction of a adopting a scale of measure and assigning numerical measures. And if we are able to compare things in terms of a fixed standard and determine numerical ratios, then we have a system of numerical measurement.

The present analysis reads the construction “taller than” as “more tall than”, construing “more than” in terms of the relation of being greater than. Similarly, it interprets “as tall as” in terms of the relation of identity. A variation required for an analysis of some adjectives, however, is that the relation of being greater than gives way to that of being less than. Given the meaning of “tall” and “short”, for example, we know that “shorter” determines the same relation as “less tall than”, which is, of course, the converse of that represented by “taller”. Accordingly, we may represent the truth that Alfred is taller than Brendan if and only if Brendan is shorter than Alfred as:

tall (Alfred) [greater than] tall (Brendan) [equivalence] short (Brendan) [less than] short (Alfred)

Both “taller” and “shorter” order things in terms of their height, but the former does so from the greater to the lesser, while the latter does so from the lesser to the greater.

It is easy to see why on the present analysis the comparative “taller” represents a relation that is transitive, asymmetric, and irreflexive. Underlying the adjective “tall” is a function that assigns to a given object a unique measure or degree of tallness. The domain of these values – that is, of these measures of tallness – may be ordered according to their size. Thus, the function underlying “tall” assigns a unique height to each thing having a height, and this height gives that item a unique location in a ranking. Finally, the comparative “taller” is true of a pair of objects, a and b, just in case the location of a in that ranking is greater than the location of b in the ranking. Given the nature of this ranking – namely, that it has the order of the real numbers – transitivity, asymmetry, and irreflexivity follow.(11)

Although “taller” stands for a transitive, asymmetric, and irreflexive relation, I do not take this to be part of its logical form. I think that something like this is generally true of comparatives of this sort: that is, where F is an adjective that “admits of degrees” and has an underlying function that assigns to a given object a unique degree of F-ness, the comparative form F-er stands for a transitive, asymmetric, and irreflexive relation, though it does not do so as a matter of logical form. It makes sense to speak of degrees of F-ness or measures of F-ness, I think, only where it makes sense to suppose that such items may be ordered by the relation of being greater than. Assuming that degrees of F-ness or measures of F-ness may be so ordered, we can generalize the reasoning of the preceding paragraph: If the function underlying the adjective F assigns a unique location in a ranking of things with respect to degrees of F-ness to each item in its domain, and if “more” with respect to F-ness is understood in terms of relative location in such a ranking, then a is F-er than b will stand for a relation that is transitive, asymmetric, and irreflexive. But this feature of comparatives is a consequence, at least in part, of our interpretation of “more”, which, it seems to me, is not a logical particle; hence, I do not take this to be an aspect of the logical form of such constructions. Thus, while not formally true, it is logically true in a broad sense that such a construction is transitive, asymmetric, and irreflexive.(12) This assumes, of course, that we may speak of degrees of F-ness or measures of F-ness and that we may rank things with respect to these characteristics. But without these assumptions, or similar ones, I do not see that we can make sense of the likes of a is F-er than b or a is as F as b.

VI

There is a further point beating on the logical grammar of comparatives which confirms the foregoing analysis, a point that may be of historical interest. A sentence reporting a comparison may, of course, occur in an oblique context: for example, “Mary thinks that the wall is greener than the ceiling” or “John says that the first contestant is just as strong as the last”. In “On Denoting”, Russell uses such a construction, one in which a comparative is embedded in an ascription of a propositional attitude, to illustrate the distinction between primary and secondary occurrences of denoting phrases. He provides the following vignette:

I have heard of a touchy owner of a yacht to whom a guest, on first seeing it, remarked, “I thought your yacht was larger than it is”; and the owner replied, “No, my yacht is not larger than it is”. (Russell 1905, p, 489)

Russell’s point, of course, is that the sentence

(39) I thought that your yacht was larger than it is

is ambiguous. He goes on to say that it is equivocal between

(40) The size that I thought your yacht was is greater than the size your yacht is

and

(41) I thought the size of your yacht was greater than the size of your yacht.

Thus, the ambiguity rests on the relative scope of the intensional operator “I thought that” and of an occurrence of the definite description “the size of your yacht”: interpreted in the manner of (40), the description has wide scope (or, as Russell calls it, a primary occurrence), and interpreted in the manner of (41), it has narrow scope (or a secondary occurrence).

While Russell’s account of the ambiguity in (39) has become part of philosophical lore, we may fail to notice, perhaps because of our familiarity with it, that it is inconsistent with the common assumption that “a is larger than b” is an unstructured predicate. Russell’s contention that the ambiguity in (39) is one of structure, involving permutation of scope, requires that there be some ingredient in the embedded sentence whose scope might conflict with that of the psychological verb in the requisite manner. Consider, for example, the sentence, “I thought that some ship in the harbour was larger than your yacht”. This sentence has two readings, which we may distinguish in terms of how we fix the scope of the phrase “some ship in the harbour” relative to that of the operator “I thought that”. Similarly, if there is an ambiguity in (39) resulting from an ambiguity of scope, there will be an item in the embedded sentence whose scope may vary relative to that of “I thought that”. But if the predicate “a is larger than b” is a simple, two-place predicate, the embedded sentence lacks any structural feature that could generate the relevant ambiguity of scope. The embedded sentence does contain occurrences of the definite description “your yacht”; but it is easy to see that the scope of these descriptions plays no significant role in the ambiguity under consideration, for we can produce the same ambiguity using a name in the place of “your yacht”: for example, “I thought that Paris was larger than it is”.(13) So, if the sentence is ambiguous in the manner that Russell contends, that is, if the ambiguity is one of scope, the predicate “a is larger than b” must be logically complex.

This conclusion is reinforced by another example. A more politic guest might have said:

(42) Your yacht is larger than I thought that it was.

Though not another reading of (39), it shows another way of arranging the constituents of (39), a way that would not be possible unless the predicate “a is larger than b” has a structure that is amenable to such rearrangement.

As we have seen, Russell thinks that the source of the ambiguity in (39) is a conflict between the scope of the psychological verb and that of a definite description referring to the size of the yacht. Of course, when we turn to the embedded sentence “Your yacht is larger than your yacht”, we do not find explicit reference to, or mention of, sizes. Thus, Russell’s explanation presupposes not only that the predicate “a is larger than b” is logically complex but more specifically that it somehow introduces a pair of descriptive phrases that denote the size of whatever “a” and “b” denote, respectively. Given Russell’s views about definite descriptions, this amounts to the claim that “a is larger than b” has the form of an open sentence containing a pair of existential quantifiers as well as predicates with variables that range over sizes. In fact, in the presence of Russell’s theory of descriptions, (40) and (41) come to much the same as

(40[prime]) [there exists]x[there exists]x[prime](I thought that: [for every]y(Large (your yacht, y) [equivalence] y = x) & [for every]y(Large (your yacht, y) [equivalence] y = x[prime]) & x [greater than] x[prime])

and

(41[prime]) I thought that: [there exists]x([for every]y(Large (your yacht, y) [equivalence] y = x) & x [greater than] x),

respectively. In other words, Russell’s explanation of the ambiguity presupposes an analysis of comparatives with the same elements as the one I have urged.

Russell does not discuss the likes of (42), but we can adapt the pattern of (40[prime]) and (41[prime]) to accommodate that sentence as well:

(42[prime]) [there exists]x[there exists]x[prime]([for every]y(Large (your yacht, y) [equivalence] y = x) & I thought that: [for every]y(Large (your yacht, y) [equivalence] y = x[prime]) & x [greater than] x[prime]).

Thus, we see that (39) and (42) contain the same elements structured differently.

Ascriptions of propositional attitudes, especially those involving quantification into the scope of the attitude, raise a number of well-known problems that I shall not discuss here. It is difficult, however, to think of a plausible alternative to Russell’s claim that the ambiguity in the likes of (39) is structural, consisting of an ambiguity of scope. Accordingly, an adequate account of such a sentence will enable us to provide for its different readings, showing that they are due to structural variation. Such an account will also enable us to distinguish between either of the readings of (39) and that of (42), again in terms of the form of those sentences. Such an account seems impossible if we understand “a is larger than b” as an unstructured, two-place predicate. Assuming that we have eliminated the hypothesis (discussed in [section]I above) that this predicate might be analysed in terms of a sentential connective such as “er-than”, we have no plausible alternative to the idea that it should be understood as a conjunction quantified by a pair of existential quantifiers. Thus, at least some of the scope permutations that emerge when comparatives interact with ascriptions of propositional attitudes are intelligible only on the assumption that the logical form of comparatives may be given in a manner like that proposed here.(14)

BERNARD D. KATZ Department of Philosophy University of Toronto Toronto, Ontario M5S 1A1 Canada

1 The thesis that referential transparency and non-truth-functionality are irreconcilable is the conclusion of an argument that has as its source an argument of Frege’s (1952) to show that sentences alike in truth-value have the same reference. Versions of Frege’s argument may be found in: Church 1943; Church 1956, p. 25; and Godel 1944, p. 129. Versions of the argument showing that inferential transparency entails truth-functionality may be found in: Quine 1953, p. 159; Quine 1966, pp. 161-2; Davidson 1980, pp. 117-8; and Davidson 1980a, pp. 152-3. My use of the principle connecting referential transparency and truth-functionality is derived from Davidson’s (1980a) application of the argument.

2 Objections to the truth-functionality argument, or to various of its applications, may be found in: Sharvey 1970; Cummins and Gottlieb 1972; Horgan 1978; Barwise and Perry 1981; and Widerker 1983. It goes beyond the scope of the present paper to discuss the criticisms these authors advance against the argument (some of which are, in any case, not really germane to the present application).

3 I am indebted to Michael Kremer for drawing to my attention the significance of this point.

4 Davidson remarks that “once events are on hand, an obvious economy suggests itself: [a sentence such as (12[prime])] may as well be construed as about events rather than times” (1980a, p. 154).

5 Wheeler construes the positive “tall” and the comparative “taller” in terms of a distinct common element, a primitive two-place relation; but, it is one quite different from the relation that I am suggesting. On Wheeler’s account, the common element is a primitive two-place relation between an individual and a class. He says: “The idea is that ‘Tall (x, yFy)’ is a primitive relation behind both ‘tall’ and ‘taller than’. It is the relation which I, for instance, bear to the class of dwarfs and to the unit class of my sister” (1972, p. 316).

6 At least, it only relates Alfred to one value at a given time. No doubt, Alfred will be taller at one time than he is at another. To avoid complexity not germane to the present discussion, I will suppress temporal references.

7 Similarly, Wallace (1972, p. 776) takes the sentence “Jumbo is a large elephant” as expressing “a relationship between Jumbo, the linear ordering larger-than, and the property of being an elephant”; a large elephant, on his account, is distinguished from other elephants by lying relatively high in that ordering.

8 I am drawn to this view by considerations of classical logic and predication. Given that classical logic – in particular, the classical law of excluded middle – holds for vague predicates, then in the presence of certain plausible assumptions about predication, it seems to follow that even a vague predicate divides a domain of objects into exactly two partitions. Suppose that [Alpha] is a ballerina. The requisite assumptions about predication, applied to “x is a tall ballerina”, say this: [Alpha] belongs to the positive extension of “x is a tall ballerina” if [Alpha] is a tall ballerina; and [Alpha] belongs to the negative extension of “x is a tall ballerina” if it is not the case that [Alpha] is a tall ballerina”. If we suppose, in addition, that “Either [Alpha] is a tall ballerina or it is not the case that [Alpha] is a tall ballerina” is true, then we may infer that either [Alpha] belongs to the positive extension of “x is a tall ballerina” or that it belongs to the negative extension of that predicate. Since [Alpha] might be any ballerina, this seems to show that every ballerina belongs either to the positive, or to the negative, extension of “x is a tall ballerina”. Horwich (1990, pp. 81-8) urges a similar view about vagueness for related reasons.

9 Wheeler (1972, p. 331) connects comparatives with predicates that “admit of degrees”.

10 Davidson (1989, p. 10) makes this point in support of his claim, mentioned above, that we need not suppose that in addition to numbers and things that weigh something, there are weights. I think that the premise is true, but I do not see that the conclusion follows.

11 Since the “greater than” relation on a domain of heights forms a strict linear ordering, our account of “taller” and “as tall as” may be misleading in an important respect. The “greater than” relation on a domain of heights not only is transitive, symmetric, and irreflexive, but also is strictly connected; that is, if two individuals are not as tall as each other, then one must be taller than the other. While an analogous point holds for many comparatives, it would be a mistake to presume that it must hold for all, and it is certainly not a presumption of our present analysis that it does. Suppose, for example, that we use “large” to order people in terms of both their height and weight. In other words, suppose that the predicate “Large (x,y)” is true of a pair whose first element is a person and whose second element is another pair consisting of that person’s height and weight. A natural ranking for pairs of heights and weights would be a product ordering: one person is larger than a second just in case (i) the first is either taller or heavier than the second and (ii) the second is neither taller nor heavier than the first. It will be a consequence of such an arrangement that two individuals are not comparable if one is the taller of the two and the other the heavier of the two. Thus, the “greater than” relation will form only a strict partial ordering on the domain of elements in terms of which we are assessing and comparing the largeness of people. The relevant ranking has the order, not of the real numbers, but of a product ordering of n-tuples of real numbers. I should emphasize that I am not claiming that we must rank sizes this way; rather I am claiming we might do so. (Of course, we might also figure out a procedure for producing a weighted average of the height and weight, in which case we would be back to a linear order.) Something like this, I believe, is true of comparisons of artistic performances, literary works, and athletic achievement, where we tend to compare things on the basis of a number of factors. For example, we might compare musical performances by rating them, on some scale, according to such factors as technique, expressiveness, and interpretation; in the case of a competition, we are apt to devise procedures for producing weighted averages of some sort.

12 I do not mean to deny that one might, perhaps, fabricate an expression that resembles an “-er” construction and that represents a relation lacking one or other of the characteristics of transitivity, asymmetry, and irreflexivity. What I find doubtful is that one could do so using “more” in a conventional manner and using it to order things simply in terms of the degree to which they have the characteristic on the basis of which we apply the adjective in question.

13 In fact, we can produce essentially the same ambiguity simply using a pronoun: for example, “Kurtz thinks that he is smarter than he is”.

14 I am grateful to Jim Brown, Jack Canfield, Avrum Fenson, Elmar Kremer, Michael Kremer, and the Editor of Mind for helpful comments and suggestions.

REFERENCES

Barwise, Jon and John Perry 1981: “Semantic Innocence and Uncompromising Situations”, in Peter A. French, Theodore E. Uehling, Jr., and Howard K. Wettstein (eds.), Midwest Studies in Philosophy, vol. 6, The Foundations of Analytic Philosophy. Minneapolis: University of Minnesota Press, pp. 387-404.

Church, Alonzo 1943: Review of Carnap’s Introduction to Semantics. Philosophical Review, 52, pp. 298-304.

—– 1956: Introduction to Mathematical Logic. Princeton: Princeton University Press.

Cummins, Robert and Dale Gottlieb 1972: “On an Argument for Truth-Functionality”. American Philosophical Quarterly, 3, pp. 265-9.

Davidson, Donald 1980: “The Logical Form of Action Sentences”, in his Essays on Actions and Events. Oxford: Clarendon Press, pp. 105-122. Originally published in 1967 in N. Rescher (ed.), The Logic of Decision and Action. Pittsburgh: University of Pittsburgh Press, pp. 81-95.

—– 1980a: “Causal Relations”, in his Essays on Actions and Events. Oxford: Clarendon Press, 1980, pp. 149-62. Originally published in 1967 in The Journal of Philosophy, 64, pp. 691-703.

—– 1989: “What is Present to the Mind?” Grazer Philosophische Studien, 36, pp. 3-18.

Frege, Gottlob 1952: “On Sense and Reference”, in Peter Geach and Max Black (eds.), Translations from the Philosophical Writings of Gottlob Frege. Oxford: Basil Blackwell, pp. 56-78. Originally published in 1892 as “Uber Sinn and Bedeutung” in Zeitschrift fur Philosophie und philosophische Kritik, 100, pp. 25-50.

Godel, Kurt 1944: “Russell’s Mathematical Logic”, in P. Schilpp (ed.), The Philosophy of Bertrand Russell. La Salle, Ill.: Open Court, pp. 125-53.

Horgan, Terence 1978: “The Case Against Events”. The Philosophical Review, 87, pp. 28-47.

Horwich, Paul 1990: Truth. Oxford: Basil Blackwell.

Quine, W.V. 1953: “Reference and Modality”, in his From a Logical Point of View. Cambridge, Mass.: Harvard University Press, pp. 139-59.

—–1960: Word and Object. Cambridge, Mass.: MIT Press.

—–1966: “Three Grades of Modal Involvement”, in his The Ways of Paradox and Other Essays. New York: Random House, pp. 156-74. Originally published in 1954 in Proceedings, XIth International Congress of Philosophy, Brussels 1953, vol. 14. Amsterdam: North-Holland, pp. 65-81.

Russell, Bertrand 1905: “On Denoting”. Mind, 14, pp. 479-93. Reprinted in R. C. Marsh (ed.), Logic and Knowledge. London: George Allen and Unwin, pp. 41-56.

Sharvey, R. 1970: “Truth-Functionality and Referential Opacity”. Philosophical Studies, 21, pp. 5-9.

Wallace, John 1972: “Positive, Comparative, Superlative”. The Journal of Philosophy, 69, pp. 773-82.

Wheeler, Samuel C. 1972: “Attributives and their Modifiers”. Nous, 6, pp. 310-34.

Widerker, David 1983: “The Extensionality Argument”. Nous, 17, pp. 457-68.