Here are some sentences, and I want to know whether I'm thinking about them correctly:
(1) The tooth fairy is not real. Symbolization: (-R)t Truth Value: False
(2) It is not the case that the tooth fairy is real. Symbolization: -(Rt) Truth Value: True
I accept that the statement "There does not exist a tooth fairy" is true. Because of this, in (1) and (2) above, "t" does not refer to a unique object. So, is my symbolization correct, or is there a different way to symbolize the sentences?
Thank you for any input you may be able to provide!
6/6/21 Update - Thank you for very much your input so far which I reviewed in detail. I have some follow-up questions below.
Did I at least get the truth values of (1) and (2) above right? Since the predicate “is real” is contentious, I will change it to “is green”. I believe that the statement “The tooth fairy is not green” is a false statement because there is no tooth fairy and the “not” only negates “is green”. However, the statement “It is not the case that the tooth fairy is green” has the “not” negating the whole statement “The tooth fairy is green”. Thus, “It is not the case that the tooth fairy is green” would be true.
However, my research seems to indicate that classical predicate logic regards both statements to be false. This is unfortunate. I understand that all objects are real, but there is something peculiar about singular terms.
I have two solutions. The first solution is that it is possible to use a term incorrectly. For instance, if one refers to “the bird” as “the cat”, one can arrive at the conclusion “The cat has wings.” This is absurd. This statement can be symbolized as “Wc,” but it is really “Cb &Wb”. Getting back to the statements in question, “t” might represent an object, but just not a tooth fairy as there are no tooth fairies. Thus: “The tooth fairy is not green” = Tt & (-Gt) = false since Tt is false. However, “It is not the case that the tooth fairy is green” = -(Tt & Gt) = (-Tt) v (-Gt) = true a -Tt is true.
The second solution involves getting rid of singular terms altogether. Instead of “t”, we would write (∃1x)(Tx & Ix). “∃1x” is notation I invented that means “There exists exactly one x such that”. Tx means “x is a tooth fairy” and Ix means that “x has these implicit specifying characteristics”. Thus, the statement “The tooth fairy is not green” would be symbolized as: ∃x((∃1y)(x=y & Ty & Iy) & -Gx). Since ∃yTy is false, this entire statement is false.
If both of these solutions are wrong, I can’t avoid a standard contradiction. “The tooth fairy is green” and “It is not the case that the tooth fairy is green” would both be false. Therefore, their negations would both be true. Hence, we would have (-P) & (P).
If you do or do not agree with my reasoning above, please let me know. I welcome all feedback and ideas and appreciate your input very much. Thank you!