Negation and logical paraphrase

The result of applying tilde to a sentence is the negation of that sentence. The negation of a sentence is true when that sentence is false, and false when that sentence is true. So exactly one of the pair made up from a sentence and its negation will be true.

In ordinary English, ‘it is not the case that’ is not particularly common. Instead, we find that people often express negations by inserting ‘not’ into a ‘positive’ sentence. (We saw this in our example arguments when we started looking at formal validity.)

For example, we’d say ‘It’s not raining’, rather than ‘It is not the case that it’s raining’, and ‘It’s not cold’, rather than ‘It is not the case that it’s cold’.

As we saw when we started looking at formal validity, it can help in uncovering significant structure in arguments to rephrase (or paraphrase / logical paraphrase) a sentence into a form which contains a sentential clause and a sentence operator. In the argument

• Mattie is tidying or Mattie is gaming online
• Mattie is not tidying
• So, Mattie is gaming online

We paraphrased ‘Mattie is not tidying’ as ‘It is not the case that Mattie is tidying’ to reveal the fact that the same sentential clause occurs in the first and second premises.