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Explicating structural ambiguity

In this step we see how truth-tables can be used to spell out the truth-conditions of readings of structurally ambiguous sentences.
Coloured lights through rain
© Max Bender @ Unsplash

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.

© University of York
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Logic: The Language of Truth

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