Symbolization and Evaluation of Some Simple Sentences

Question

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!

Answer 1

In some cases, at least, definite descriptions such as "the tooth fairy" do not function like names, but more like predicates that may or may not be satisfied. Russell proposed to treat definite descriptions of the form "the F is G" as meaning, "there is one and only one thing that is F and that thing is also G". Not all definite descriptions work this way, but some do. On Russell's account, "the F is G" can be expressed formally as:

(∃x)(∀y)(Fx ∧ (Fy ⊃ x=y) ∧ Gx)

where ⊃ is material implication.

So, "the tooth fairy is green" would be glossed as, "there is one and only one thing that is the tooth fairy and that thing is green". This comes out false because there is no tooth fairy. The narrow scope negation, "the tooth fairy is not green" would be glossed as, "there is one and only one thing that is the tooth fairy and that thing is not green". This also is false. But the wide scope negation, "it is not the case that the tooth fairy is green" comes out true on this account, since it is indeed not the case that there is a unique thing that is the tooth fairy, green or otherwise.

Russell's account is not universally accepted, by any means, but it appears to do a reasonable job in this instance. Alternative accounts might have it that the above sentences lack truth values because the expression "the tooth fairy" fails to refer to anything, or that it refers to a possible object that does not exist in the actual world.

Answer 2

Your new example "The tooth fairy is not green" with imagined non-existent object is a problematic edge case in classic logic if you treat "tooth fairy" as a proper name singular term (not a predicate), since classical logic’s singular terms must denote existing things which can be really quantified. One way out is to follow Frege-Russell's theory of descriptions to treat "tooth fairy" as a definite description as referenced here:

As France is currently a republic, it has no king. Bertrand Russell pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald." The sentence does not seem to be true: if we consider all the bald things, the present King of France is not among them, since there is no present King of France. But if it is false, then one would expect that the negation of this statement, that is, "It is not the case that the present King of France is bald", or its logical equivalent, "The present King of France is not bald", is true. But this sentence does not seem to be true either: the present King of France is no more among the things that fail to be bald than among the things that are bald. We therefore seem to have a violation of the law of excluded middle.

Russell proposed to resolve this puzzle via his theory of descriptions. A definite description like "the present King of France", he suggested, is not a referring expression, as we might naively suppose, but rather an "incomplete symbol" that introduces quantificational structure into sentences in which it occurs... More briefly put... ∃x(Kx ∧ ∀y(Ky → x=y) ∧ Bx). This is false, since it is not the case that some x is currently King of France.

So via definite description interpretation you proposition "The tooth fairy is not green" is clearly false under classic logic, no more weirdness. Similar conclusions about its negation with 2 different scope interpretations are also discussed in the same reference as totally settled.

Finally as Saul Kripke emphasized most definite descriptions are not proper names, so how can we deal with it when "tooth fairy" or "Pegasus" cannot be treated as definite descriptions? One common approach is to use Free logic where a monadic existence predicate E! is introduced while classic FOL only has existential quantifier ∃. Now your imagined proposition can be expressed as (E!t ∧ ¬Gt), and E!t is again indisputably false under this logic when the variable t=tooth fairy.