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The feathered end of an archer's arrow.

Structural ambiguity

But now we hit another problematic feature of natural languages.

Suppose someone said:

  • It’s not the case that it’s raining and it’s warm

What would they be saying? Under what sorts of circumstances would what they were saying be true?

It’s actually not clear what they would be saying. The reason is that the English sentence is ambiguous: it can be read or heard with two different meanings, and those different meanings are logically significant, because they involve different truth-conditions. Can you see what the two meanings are? Can you give paraphrases in English which bring them out?

We can bring out the two meanings (as logicians often say, the two readings), as follows.

  1. It’s warm and it’s not the case that it’s raining
  2. It’s not the case that it’s both raining and warm

The first reading (1) is only true where ‘it’s warm’ is true and ‘it’s raining’ is false. The second (2) will be true where at least one of the claims ‘it’s raining’ and ‘it’s warm’ is false. These readings have different truth-conditions. There are kinds of situation in which one would be true and the other would be false. For instance, (2) will be true where it’s raining but not warm; but (1) will be false in those circumstances.

So, ‘It’s not the case that it’s raining and it’s warm’ is a case of what’s called structural ambiguity.

Structural ambiguity is fairly common in natural languages, and doesn’t just involve sentence connectives. Consider:

  • Time flies like an arrow; fruit flies like a banana.

The fact that ‘flies’ is a verb in the first clause leads us to an unusual reading of the second in which ‘flies’ is also taken as a verb in the second — a reading on which the clause says that the manner of flying of fruit is banana-style. On the more natural reading of the second clause ‘fruit flies’ is a noun and the verb in the clause is ‘like’. There’s a clear sense in which readings relate to different structures in the sentence.

Formal languages are designed to make structural ambiguity impossible. When we use a sentence of our formal language, we want to be completely clear about what it means. We also want to be able to use our language to explicate (spell out) ambiguities found in natural language, but it won’t be best suited to that task if it simply mirrors natural language in being structurally ambiguous.

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Logic: The Language of Truth

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