Is a proposition about something which doesn't exist true or false?

Question

Let S = {x | x is not an element of x }

The set S doesn't exist. Then, would a proposition such as "The cardinality of S is 1," be true or false?

Equivalently, I could have made a proposition, "the unicorns are red." Would the proposition be false since unicorns don't exist, or would it be true?

Ideally, I'd like a propositional/predicate logic explanation. Thanks in advance for the help.

Answer 1

In normal first-order logic, you cannot refer to something that does not exist. So, for example, you cannot directly say "The cardinality of $S$ is 1." This is because every term, in first-order logic, always refers to an actual object, and so there is no way to make a term for $S$. This is one reason that not every English expression can be translated directly into first-order logic.

What you can do is to use quantifiers and a definition of $S$ to simulate referring to $S$. For example, you can say $$ (\forall z)[ (z = \{ x : x \not \in x\}) \to ( |z| = 1)] $$ or $$ (\exists z)[ z = \{ x : x \not \in x\} \text{ and } |z| = 1] $$

The first of these, with a $\forall$, will come out to be true, because there is no $z$ to match the hypothesis of the implication. The second, with an $\exists$, will come out false, essentially for the same reason.

For the purposes of formalizing mathematics, this system work perfectly well. After all, in mathematics we are interested in objects that do exist. Experience shows that we don't need more than first-order logic allows when we want to write axiom systems for set theory.

However, for formalizing natural language, first-order logic may leave something to be desired. The field of free logic studies logics in which some terms may not denote actual objects - some terms are "undefined", such as $1/0$. In free logic, statements do not have to be true or false, and in particular statements like "$|S| = 1$" will not be true or false, because they are atomic formulas with undefined terms in them.

Answer 2

The lesson in Russel's paradox isn't that $S$ doesn't exist, the lesson is that if you accept that you can define such an $S$ you get a contradiction.

However, to answer the question in the title, the statement "$\left( \exists x \mbox{ such that } P(x)\right) \implies Q$", or in written language "If (there exists an $x$ such that $P(x)$), then $Q$." is true, regardless of the statement $Q$, if no such $x$ exists. This is in the same way that "false implies true" is a true statement.

So, "The cardinality of $S$ is 1" is either:

1. True, since every statement is true IF you allow the construction of $S$ (the principle of explosion)
2. Undefined, since $S$ is left undefined so as to prevent a contradiction,
3. True, if you give a lot of wiggle room and tell me that you really mean: "If $S=\{ ...\}$ and $S$ is a set, then the cardinality of $S$ is $1$." It's true because the formalization of classes vs. sets makes it so that $S$ defined as such is not a set.

I'd go for number 2, but you can argue for any case.

Answer 3

Your question is essentially a matter of philosophy. Here are some statements about fictional entities that have definite truth values.

• "A unicorn has one horn." This is true. And the proposition "A unicorn has two horns" is false. Why is this? It's because the definition of a unicorn includes the condition that a unicorn has one horn. We may in fact take the definition to be: "A unicorn is a fictional creature that has one horn." So we are entitled to assign definite truth values to the statements I gave.

• "Ahab is captain of the Pequod." This is also a true statement about a fictional entity. Captain Ahab doesn't exist and the novel Moby Dick is fiction. Yet this statement is true; but for a different reason than the unicorn example. In this case, "Ahab is captain of the Pequod" is true because we are implicitly appending the clause "... in the novel Moby Dick." But that's no objection to assigning a truth value; after all, "1 + 1 = 2" also has an implicitly appended clause: "... in ordinary arithmetic."

What kinds of statements about fictional entities don't have truth values? How about: "Captain Ahab likes to eat scrambled eggs." The novel Moby Dick says nothing about Ahab's food preferences so we have no basis to assign a truth value.

And your example "Unicorns are red" is very difficult to analyze one way or another. I could certainly say that some unicorns are red. In particular, a red unicorn is a fictional animal that's red and has one horn." So a red unicorn is red. That is a true statement. Or, we could say that the definition of a unicorn does not provide enough information to assign a truth value one way or another.

There's much more that could be said, but it's a matter of philosophy and not math; and in any event, I'm not qualified to expound on it. But these links may be of interest.

• The Wiki article on propositions. A proposition is the "primary bearer of truth value" in philosophy. The term goes all the way back to Aristotle.

• The Wikipedia article on truth-bearers. A truth-bearer is a thing that is either true or false and nothing else. Again, many learned philosophers have thought about what that means.

• The SEP entry on mathematical fictionalism. This is a reaction to the doctrine of Platonism, which claims that mathematical statements are about something real. Fictionalism considers the oppposite point of view.

To sum up: It is definitely the case that some statements about fictional entities do have definite truth values; and that this can happen for a variety of reasons. And we can even argue the case that mathematical statements are of the same type: statements about fictional entities that have definite truth values.

Answer 4

Using predicate logic with $\in$ as a binary infix relation:

1. $\forall x:[x\in S \iff x\notin x]$ (Premise)

2. $S\in S \iff S\notin S$ (Universal Specification, 1) (a contradiction)

3. $\neg \exists S:\forall x:[x\in S \iff x\notin x]$ (Conclusion, 1, 2)

Since $S$ cannot exist, we cannot apply the definition of cardinality (whatever it might be) to $S$, so we cannot prove anything about its cardinality.

Note that this is not a property peculiar to $\in$ as set membership, or to $S$ being a set. We could have as easily have proven $\neg \exists y:\forall x:[R(x,y) \iff \neg R(x, x)]$ as follows:

1. $\forall x:[R(x,y) \iff \neg R(x,x)]$ (Premise)

2. $R(y,y) \iff \neg R(y,y)$ (Universal Specification, 1) (a contradiction)

3. $\neg \exists y:\forall x:[R(x,y) \iff \neg R(x, x)]$ (Conclusion, 1, 2)

Answer 5

I'd say that if logic were done to more reflect (useful) mathematical usage the answer would be that such propositions, if atomic, are silly. The silly truth value or values can be thought of as something intermediate between the true and false truth values. If logic is done to make a Heyting prealgebra with an order-reversing involutive negation $\neg$ with a few extra simple rules, then one can define $\operatorname{false} A$ as $A \rightarrow \bot$, $\operatorname{true} A$ as $\neg A \rightarrow \bot$ and $\operatorname{silly} A$ as $((A \wedge \neg A) \rightarrow \bot) \rightarrow \bot$. Each predicate and function symbol can implicitly be taken to have its own domain of definition associated with it. In particular, predicate symbols send undefined terms and n-tuples of terms outside their domain of definition to silly propositions, and function symbols send undefined terms and n-tuples of terms outside their domain of definition to undefined terms.

As to whether silly propositions should be assertible, I would say that generally speaking, yes. That way mathematicians can avoid the pedantic bother of forever having to restrict things to where they make sense. E.g., one wants to be able to assert statements like "$x/x = 1$" or "$\operatorname{not} x/x =2$" without forever having to preface the assumption that $x \neq 0$. Or when discussing functions that are continuous at $c$ or that are not continuous at $c$, one doesn't want to have to talk about the case in which $c$ is not in the domain of the functions under consideration. Of course, non-atomic sentences about undefined things can be true, false, or silly. E.g., "it is not true that $1/0 = 1$" is true. Quite frequently, one does need to assert of a statement that it is true. I would say that, ideally, assertion should be acceptable if the statement asserted is true or silly, which corresponds to the most basic use of the subjunctive. Ideally, most math should be done in subjunctive mood and the indicative should be reserved for assertions of truth. But this would make mathematicians sound like they are using so-called (undignified) pirate talk, using "be" for "is" almost everywhere and rarely conjugating the third person singular. But really it wouldn't actually be like (ridiculous) pirate talk because not infrequently the indicative would in fact be used. It would be a discriminating sort of talk, actually. This sort of approach suggests where laughter comes from, as well. Laughing $A$ corresponds to $\operatorname{ha} A$, which one may define as $A \wedge \operatorname{silly} A$. The reason $1/0 = 1$ doesn't make people laugh (even if "$=$" is read as "equal" rather than "equals") is that $1/0 = 1$ is so obviously silly there is not felt to be any communicative purpose in laughing at it, much as farce is so obviously silly that it is not as funny to people with a sense of humor as humor that is less obviously silly. Philosophically, math silliness is a kind of ultrafarce--too farcical to cause laughter. Math is a very serious subject even if it is ideally (in my opinion) not totally serious.

Even in the very basics of logic (e.g., when dealing with parsing) silliness and partially defined function symbols are useful. E.g., it can be useful in parsing to think of a wedge operation corresponding to the operator $\wedge$ which doesn't make sense on expressions that have $\vee$ operators in them outside of parentheses. And parsing properly concerns occurrences of symbols rather than the symbols themselves, and if you don't assume that any occurrence of an expression is undefined if the occurrences of symbols in it don't all differ from one another, while at the same time allowing yourself to substitute occurrences directly from one occurrence of an expression into another occurrence of an expression, you get into sloppiness or pedantic levels of difficulty.

I am inclined to think, were one to ramp things up into even more efficiency and exactness, it could be useful to have for atomic statements an unassertible type of silliness as well an assertible type of silliness (which together correspond to the truth values not affected by negation). E.g., when asserting that $\frac {d y^n}{dx} = {n y}^{n - 1} \frac {dy}{dx}$, one probably also wishes to assert that $\frac {d y^n}{dx}$ is defined if $\frac{dy}{dx}$ is defined. One could add an article, say, "ye" (I'm not really happy about the options, which I have yet to consider exhaustively and carefully), before the $\frac {d y^n}{dx}$ sign and have various logical rules guaranteeing that atomic statements are unassertibly silly if they are silly merely on account of function symbols preceded by "ye" taking values at out-of-domain but defined values. To keep the inference rules of logic coherent, one also would want two truth values corresponding respectively to the meet (not assertible) and to the join (assertible) of an unassertibly silly atomic statement with an assertibly silly atomic statement. So in total there would be four silly truth values. But I haven't yet written down the four-valued silliness approach exactly as I have done (in the draft of the logic book I have been writing for the past decade or so) with the approach that considers silliness as being just one truth value, and so I am not nearly so confident as to its usefulness or whether a modification be preferable.