I'm working through a problem now that asks whether it is possible to logically formalize two sentences:
"Say of each of the following pairs of English sentences whether there is a sentence φ of L1 such that φ and ¬φ provide correct formalizations. Justify your answer, specifying any ambiguity and explaining how it affects things.
Let me give you two worked examples; consider the pairs (a) and (b) of sentences:
(a) Someone is smoking. It's not the case that someone is smoking.
(b) Someone is smoking. Someone is not smoking.
The answer to (a) is "yes", since we can take the dictionary "P: someone is smoking" and correctly formalize the pair of sentences as P and ¬P, respectively. But the answer to (b) is "no", since neither the dictionary above nor any other dictionary will do - nor is there any more complex formal sentence φ, with whatever dictionary, that gives you a pair of correct formalizations φ and ¬φ - since neither of the English sentences is the negation of the other."
I understand the basic premise and have gotten through many of the problems. But I've hit a road block on the following:
"(vi) It’s moving. It’s not moving.
(vii) It’s moving. It’s unmoving.
(viii) It’s moving. It’s motionless.
(ix) It’s moving. It’s still."
While in many cases (as in (b) above) it is possible to disprove the possibility of formalizing the two sentences because they can either both be true (for example, 'somebody is smoking' = T and 'somebody is not smoking' = T) or both be false, it appears as if in problems xi-ix, there is no way for both premises to be true or both to be false. Does that mean, then, that xi-ix can all be formalized as φ and ¬φ? It just seems like this is a trick question, but I can't think of any other answer. Thanks!