Many of the foundational concepts in truth-conditional semantics come from the work of Gottlob Frege, whose distinction between Sense and Denotation we discussed in Chapter 2. The Principle of Compositionality in (2) is often referred to as “Frege’s principle”. Frege himself never expressed the principle in these words, and there is some disagreement as to whether he actually believed it.2 But there are passages in several of his works that seem to imply or assume that sentence meanings are compositional in this sense, including the following:
It is astonishing what language accomplishes. With a few syllables it expresses a countless number of thoughts [=propositions], and even for a thought grasped for the first time by a human it provides a clothing in which it can be recognized by another to whom it is entirely new. This would not be possible if we could not distinguish parts in the thought that correspond to parts of the sentence, so that the construction of the sentence can be taken to mirror the construction of the thought… The question now arises how the construction of the thought proceeds, and by what means the parts are put together so that the whole is something more than the isolated parts.3
In this passage Frege argues for the compositionality of “thoughts”, i.e. propositions; but the same kind of reasoning requires that the meaning of smaller expressions (e.g. noun phrases) be compositional as well. And in many cases, not only senses but also denotations are compositional. One way of seeing this involves substituting one expression for another which is co-referential, i.e., has the same denotation in that particular context.
In our world, the expressions Abraham Lincoln and the 16th president of the United States refer to the same individual. For this reason, if we replace one of these expressions with the other as illustrated in (3–4), the denotation of the larger phrase is not affected.
(3) a. the wife of Abraham Lincoln
b. the wife of the 16th president of the United States
(4) a. the man who killed Abraham Lincoln
b. the man who killed the 16th president of the United States
Both of the NPs in (3) refer to Mary Todd Lincoln; both of the NPs in (4) refer to John Wilkes Booth. This is what we expect if the denotation of the larger phrase is compositional, i.e., predictable from the denotations of its constituent parts: replacing one of those parts with another part having the same denotation does not affect the denotation of the whole. (This principle is referred to as the principle of substitutivity.)
A second way of observing the compositionality of denotations arises when non-referring expressions occur as constituents of a larger expression. In a world where there is no such person as Superman, i.e., a world in which this name lacks a denotation, phrases which contain the name Superman (like those in (5)) will also lack a denotation, i.e. will fail to refer.
(5) a. the mother of Superman
b. the man who Superman rescued
These observations support the claim that the denotation of a complex expression is (often) predictable from the denotations of its constituent parts. Since sentences are formed from constituent parts (words and phrases) which have denotations, this suggests that the denotations of sentences might also be compositional. In his classic paper Über Sinn und Bedeutung ‘On sense and denotation’, Frege (1892) argued that this is true; but he recognized that it may seem odd (at least at first) to suggest that sentences have denotations as well as senses. Sentences are not “referring expressions” in the normal sense of that term, so what could their denotation be?
Frege considered the possibility that the denotation of a sentence is the proposition that it expresses. But this hypothesis leads to unexpected results when we substitute one co-referential expression for another. Samuel Clemens was an American author who wrote under the pen name Mark Twain; so these two names both refer to the same individual. Since the two names have the same denotation, we expect that replacing one name with the other, as illustrated in (6), will not affect the denotation of the sentence as a whole.
(6) a. The Prince and the Pauper was written by Mark Twain.
b. The Prince and the Pauper was written by Samuel Clemens.
Of course, the resulting sentences must have the same truth value; it happens that both are true. However, a person who speaks English but does not know very much about American literature could, without inconsistency, believe (6a) without believing (6b). For Frege, if a rational speaker can simultaneously believe one sentence to be true while believing another to be false, the two sentences cannot express the same proposition.
Examples like (7) lead to the same conclusion. Abraham Lincoln was the 16th president of the United States, so replacing the phrase Abraham Lincoln with the phrase the 16th president of the United States should not change the denotation of the sentence as a whole. But the facts of history could have been different: Abraham Lincoln might have died in infancy, or lost the election in 1860, etc. Under those conditions, sentence (7b) might well be true while sentence (7a) is false. This again is evidence that the two sentences do not express the same proposition, since a single proposition cannot be simultaneously true and false in any single situation.
(7) a. Abraham Lincoln ended slavery in America.
b. The 16th president of the United States ended slavery in America.
Frege concludes that the denotation of a (declarative) sentence is not the proposition which it expresses, but rather its truth value. Frege identifies the proposition expressed by a sentence as its sense.
There are clear parallels between the truth value of a sentence and the denotation of a noun phrase. First, neither can be determined in isolation, but only in relation to a specific situation or universe of discourse. Second, both may have different values in different situations. Third, both are preserved under substitution of co-referring expressions. This was illustrated for noun phrases in (3–4), and for sentences in (6–7). Finally, we noted that NPs which contain non-referring expressions as constituents, like those in (5), will also fail to refer, i.e., will lack a denotation. In the same way, Frege argued that sentences which contain non-referring expressions will lack a truth value. He states that sentences like those in (8) are neither true nor false; they cannot be evaluated, because their subject NPs fail to refer. These parallels provide strong motivation for considering the denotation of a sentence to be its truth value.
(8) a. Superman rescued the Governor’s daughter.
b. The largest even number is divisible by 7.
However, certain types of sentences, such as those in (9), contain a non-referring expression but nevertheless do seem to have a truth value. Even in a world where there is no Santa Claus and no fountain of youth, it would be possible to determine whether these sentences are true or false. Sentences of this type are said to be referentially opaqe, meaning that their denotation is not predictable from the denotations of their constituent parts. In these specific examples, the opacity is due to special properties of verbs like believe and hope. (We will discuss other types of opacity in Chapter 15.)
(9) a. The Governor still believes in Santa Claus.
b. Ponce de León hoped to find the fountain of youth.
2 Specifically, there is debate as to whether Frege believed that compositionality holds for senses, as well as denotations (Gamut 1991b: 12). Pelletier (2001), for example, argues that he did not. A number of modern scholars have argued against the Principle of Compositionality; see Goldberg (2015) for a summary.
3 Frege (1923–1926), “Logische Untersuchungen. Dritter Teil: Gedankengefüge”, quoted in Heim & Kratzer (1998: 2).
