Entailment

Definition 3.1   A set of formulae semantically entails the formula if and only if there is no structure on which and the members of are defined, and all the members of are true whilst is false. (Hodges, 2001, 128)

Symbolise the sentences in the following arguments, using the suggested interpretation. By means of a tableau determine whether or not the resulting premisses entail the conclusion.

1. P: He claims that no politicians ever tell the truth;
   Q: What he says is true;
   R: He himself is a politician.

   He claims that no politicians ever tell the truth. If what he says is true, then if he is himself a politician, what he says is not true. But he is. Hence, what he says cannot be true.

2. P: There will be a climate change;
   Q: Something we shall do is going to prevent a climate change;
   R: Something we shall do is going to cause a climate change.

   Either there will be a climate change, or there won't. If there will, nothing we shall do is going to prevent it. If there won't, nothing we shall do is going to cause it. Hence, nothing we shall do is going to prevent or cause a climate change.