Chapter 7 Ambiguity

In English, sentences can be ambiguous , i.e., they can have more than one meaning. There are many sources of ambiguity. One is lexical ambiguity: a sentence can contain words which have more than one meaning. For instance, ‘bank’ can mean the bank of a river, or a financial institution. So I might say that ‘I went to the bank’ when I took a stroll along the river, or when I went to deposit a check. Depending on the situation, a different meaning of ‘bank’ is intended, and so the sentence, when uttered in these different contexts, expresses different meanings.

A different kind of ambiguity is structural ambiguity. This arises when a sentence can be interpreted in different ways, and depending on the interpretation, a different meaning is selected. A famous example due to Noam Chomsky is the following:

1. ​

   Flying planes can be dangerous.

There is one reading in which ‘flying’ is used as an adjective which modifies ‘planes’. In this sense, what’s claimed to be dangerous are airplanes which are in the process of flying. In another reading, ‘flying’ is a gerund: what’s claimed to be dangerous is the act of flying a plane. In the first case, you might use the sentence to warn someone who’s about to launch a hot air baloon. In the second case, you might use it to counsel someone against becoming a pilot.

When the sentence is uttered, usually only one meaning is intended. Which of the possible meanings an utterance of a sentence intends is determined by context, or sometimes by how it is uttered (which parts of the sentence are stressed, for instance). Often one interpretation is much more likely to be intended, and in that case it will even be difficult to “see” the unintended reading. This is often the reason why a joke works, as in this example from Groucho Marx:

1. ​

   One morning I shot an elephant in my pajamas.

2. ​

   How he got in my pajamas, I don’t know.

Ambiguity is related to, but not the same as, vagueness. An adjective, for instance ‘rich’ or ‘tall,’ is vague when it is not always possible to determine if it applies or not. For instance, a person who’s 6 ft 4 in (1.9 m) tall is pretty clearly tall, but a building that size is tiny. Here, context has a role to play in determining what the clear cases and clear non-cases are (‘tall for a person,’ ‘tall for a basketball player,’ ‘tall for a building’). Even when the context is clear, however, there will still be cases that fall in a middle range.

In TFL, we generally aim to avoid ambiguity. We will try to give our symbolization keys in such a way that they do not use ambiguous words or disambiguate them if a word has different meanings. So, e.g., your symbolization key will need two different sentence letters for ‘Rebecca went to the (money) bank’ and ‘Rebecca went to the (river) bank.’ Vagueness is harder to avoid. Since we have stipulated that every case (and later, every valuation) must make every basic sentence (or sentence letter) either true or false and nothing in between, we cannot accommodate borderline cases in TFL.

It is an important feature of sentences of TFL that they cannot be structurally ambiguous. Every sentence of TFL can be read in one, and only one, way. This feature of TFL is also a strength. If an English sentence is ambiguous, TFL can help us make clear what the different meanings are. Although we are pretty good at dealing with ambiguity in everyday conversation, avoiding it can sometimes be terribly important. Logic can then be usefully applied: it helps philosophers express their thoughts clearly, mathematicians to state their theorems rigorously, and software engineers to specify loop conditions, database queries, or verification criteria unambiguously.

Stating things without ambiguity is of crucial importance in the law as well. Here, ambiguity can, without exaggeration, be a matter of life and death. Here is a famous example of where a death sentence hinged on the interpretation of an ambiguity in the law. Roger Casement (1864–1916) was a British diplomat who was famous in his time for publicizing human-rights violations in the Congo and Peru (for which he was knighted in 1911). He was also an Irish nationalist. In 1914–16, Casement secretly travelled to Germany, with which Britain was at war at the time, and tried to recruit Irish prisoners of war to fight against Britain and for Irish independence. Upon his return to Ireland, he was captured by the British and tried for high treason.

The law under which Casement was tried is the Treason Act of 1351. That act specifies what counts as treason, and so the prosecution had to establish at trial that Casement’s actions met the criteria set forth in the Treason Act. The relevant passage stipulated that someone is guilty of treason

if a man is adherent to the King’s enemies in his realm, giving to them aid and comfort in the realm, or elsewhere.

Casement’s defense hinged on the last comma in this sentence, which is not present in the original French text of the law from 1351. It was not under dispute that Casement had been ‘adherent to the King’s enemies’, but the question was whether being adherent to the King’s enemies constituted treason only when it was done in the realm, or also when it was done abroad. The defense argued that the law was ambiguous. The claimed ambiguity hinged on whether ‘or elsewhere’ attaches only to ‘giving aid and comfort to the King’s enemies’ (the natural reading without the comma), or to both ‘being adherent to the King’s enemies’ and ‘giving aid and comfort to the King’s enemies’ (the natural reading with the comma). Although the former interpretation might seem far fetched, the argument in its favor was actually not unpersuasive. Nevertheless, the court decided that the passage should be read with the comma, so Casement’s antics in Germany were treasonous, and he was sentenced to death. Casement himself wrote that he was ‘hanged by a comma’.

We can use TFL to symbolize both readings of the passage, and thus to provide a disambiguiation. First, we need a symbolization key:

$A$:

Casement was adherent to the King’s enemies in the realm

$G$:

Casement gave aid and comfort to the King’s enemies in the realm

$B$:

Casement was adherent to the King’s enemies abroad

$H$:

Casement gave aid and comfort to the King’s enemies abroad

The interpretation according to which Casement’s behavior was not treasonous is this:

The interpretation which got him executed, on the other hand, can be symbolized by:

Remember that in the case we’re dealing with Casement, was adherent to the King’s enemies abroad ($B$ is true), but not in the realm, and he did not give the King’s enemies aid or comfort in or outside the realm ($A$, $G$, and $H$ are false).

One common source of structural ambiguity in English arises from its lack of parentheses. For instance, if I say ‘I like movies that are not long and boring’, you will most likely think that what I dislike are movies that are long and boring. A less likely, but possible, interpretation is that I like movies that are both (a) not long and (b) boring. The first reading is more likely because who likes boring movies? But what about ‘I like dishes that are not sweet and flavorful’? Here, the more likely interpretation is that I like savory, flavorful dishes. (Of course, I could have said that better, e.g., ‘I like dishes that are not sweet, yet flavorful’.) Similar ambiguities result from the interaction of ‘and’ with ‘or’. For instance, suppose I ask you to send me a picture of a small and dangerous or stealthy animal. Would a leopard count? It’s stealthy, but not small. So it depends whether I’m looking for small animals that are dangerous or stealthy (leopard doesn’t count), or whether I’m after either a small, dangerous animal or a stealthy animal (of any size).

These kinds of ambiguities are called scope ambiguities, since they depend on whether or not a connective is in the scope of another. For instance, the sentence, ‘Avengers: Endgame is not long and boring’ is ambiguous between:

1. 1.
   ​

   Avengers: Endgame is not: both long and boring.

2. 2.
   ​

   Avengers: Endgame is both: not long and boring.

Sentence 2 is certainly false, since Avengers: Endgame is over three hours long. Whether you think sentence 1 is true depends on if you think it is boring or not. We can use the symbolization key:

$B$:

Avengers: Endgame is boring

$L$:

Avengers: Endgame is long

Sentence 1 can now be symbolized as ‘$\neg (L\wedge B)$’, whereas sentence 2 would be ‘$\neg L\wedge B$’. In the first case, the ‘$\wedge$’ is in the scope of ‘$\neg$’, in the second case ‘$\neg$’ is in the scope of ‘$\wedge$’.

The sentence ‘Tai Lung is small and dangerous or stealthy’ is ambiguous between:

1. 3.
   ​

   Tai Lung is either both small and dangerous or stealthy.

2. 4.
   ​

   Tai Lung is both small and either dangerous or stealthy.

We can use the following symbolization key:

$D$:

Tai Lung is dangerous

$S$:

Tai Lung is small

$T$:

Tai Lung is stealthy

The symbolization of sentence 3 is ‘$(S\wedge D)\vee T$’ and that of sentence 4 is ‘$S\wedge (D\vee T)$’. In the first, ‘$\wedge$’ is in the scope of ‘$\vee$’, and in the second ‘$\vee$’ is in the scope of ‘$\wedge$’.