Explicating structural ambiguity
Now we can go back to our first example of structural ambiguity: ‘It’s not the case that it’s raining and it’s warm.’ We’ll start by defining basic sentence letters.
- R: It’s raining
- W: It’s warm
The two readings we identified with English sentences were:
- (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
These correspond to the following sentences in our formal language:
- (1a) (~R & W)
- (2a) ~(R & W)
Notice that 1a is the conjunction of ‘~R’ and ‘W’. Its main connective is its ampersand. Sentence 2a is the negation of ‘(R&W)’. Its main connective is its tilde.
Let’s fill a truth-table which lays out the truth-conditions of these two sentences. (You could try writing out this truth-table before reading further, or working along with the steps described below.)
Here’s the unfilled table:
Next, we copy across the truth-values of the basic sentences:
Then we work out the truth-values of the clauses one level up:
Then the truth-values of the clauses at the next level. In the case of these two sentences, these are the whole sentences.
Here, for clarity, is the table showing just the truth-values for the whole sentences (the truth-values under the main connectives).
This table lays out in full the differences between the two readings. The first is only true where ‘It’s raining’ is false and ‘It’s warm’ is true; and it’s false otherwise. The second is only false where ‘It’s raining’ and ‘It’s warm’ are both true. Note that these sentences of our formal language are entirely unambiguous: each one can only express one set of truth-conditions.
Now we’re going to introduce a third connective into our formal language.
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