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:
| R | W | (~ | R | & | W) | ~ | (R | & | W) |
|---|---|---|---|---|---|---|---|---|---|
| T | T | ||||||||
| T | F | ||||||||
| F | T | ||||||||
| F | F |
Next, we copy across the truth-values of the basic sentences:
| R | W | (~ | R | & | W) | ~ | (R | & | W) |
|---|---|---|---|---|---|---|---|---|---|
| T | T | t | t | t | t | ||||
| T | F | t | f | t | f | ||||
| F | T | f | t | f | t | ||||
| F | F | f | f | f | f |
Then we work out the truth-values of the clauses one level up:
| R | W | (~ | R | & | W) | ~ | (R | & | W) |
|---|---|---|---|---|---|---|---|---|---|
| T | T | F | t | t | t | T | t | ||
| T | F | F | t | f | t | F | f | ||
| F | T | T | f | t | f | F | t | ||
| F | F | T | f | f | f | F | f |
Then the truth-values of the clauses at the next level. In the case of these two sentences, these are the whole sentences.
| R | W | (~ | R | & | W) | ~ | (R | & | W) |
|---|---|---|---|---|---|---|---|---|---|
| T | T | F | t | F | t | F | t | T | t |
| T | F | F | t | F | f | T | t | F | f |
| F | T | T | f | T | t | T | f | F | t |
| F | F | T | f | F | f | T | f | F | f |
Here, for clarity, is the table showing just the truth-values for the whole sentences (the truth-values under the main connectives).
| R | W | (~ | R | & | W) | ~ | (R | & | W) |
|---|---|---|---|---|---|---|---|---|---|
| T | T | F | F | ||||||
| T | F | F | T | ||||||
| F | T | T | T | ||||||
| F | F | F | T |
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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