# Formalizing negations

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!

• Try to express formulas using quantifiers and see whether quantifiers are negated. Apr 21, 2018 at 20:28
• @rus9384 I'm sorry, could you please elaborate? Apr 21, 2018 at 20:35
• I take it that you are studying mathematical logic. You are discovering the difference between contradiction and the other kinds of inconsistency. The negation in math means the contradictory of the original claim. Both claims cannot both be true & both cannot also be false. Other inconsistencies can be contrary. This is where both cannot be true but both can be false. Then there is also the sub contrary. This is where both claims cannot be false. These relationships come from the original square of opposition not the modern square. Apr 21, 2018 at 22:50
• @Logikal, "Both claims cannot both be true & both cannot also be false." - it depends on where the negation operation is applied. E.g. in case (b) it is applied in a way that it is not really a negation of the whole formula. Apr 21, 2018 at 22:59
• @rus9384, I gave concept definitions. That does not change where negation are applied. If you reject the definitions I gave please express what is wrong with them. I distinguished contradictory from contraries and sub contraries. If you are not familiar with these then you study math. These concepts are basic in philosophy. Apr 21, 2018 at 23:08

"Someone" means existential quantifier. In case of (a) you can see that someone is under negation and the first formula is "∃x: x is smoking" where second is "¬(∃y: y is smoking)". In (b) second formula is "∃y: ¬(y is smoking)". Here we take x and y as different variables since "someone" in first and second sentences may refer to different objects.

When speaking about "it's" we take the same variable in both sentences. And we do not even consider quantification. Thus any of them should be "x is moving. ¬(x is moving)".

The authors of forall x: Calgary Remix provide this rule for negations on page 26:

A sentence can be symbolized as ¬A if it can be paraphrased in English as ‘It is not the case that. . . ’.

Consider the four examples in the OP's question:

(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.

Let A be "It's moving." If the corresponding sentences can be paraphrased as "It is not the case that it's moving" then the sentence could be symbolized with ¬A. It appears to me that each of them can be so paraphrased.

Reference

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/