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Consider the following sentence:

"Either Santa Claus is hungry or Santa Claus is not hungry."

This seems to be a straightforward application of the law of excluded middle. However, it also seems like both disjuncts could be seen as problematic. Santa Claus has to exist to be hungry, but Santa Claus doesn't exist by stipulation, so that disjunct can't be correct. And if similar reasoning applies to his being not hungry, then we would have a situation where the LEM sanctioned a false disjunction. So what's going on here? Do we need to modify our commitment to the LEM? Or is there some reason that it actually isn't violated in this case?

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    Options: convert into a hypothetical: "If Santa Claus exists, he is either hungry or not hungry." (Then say, "If Santa Claus does not exist, then he is not hungry, as he satisfies no descriptions, as such, at all, in the first place.") Or: "According to the story of Christmas, Santa Claus is either hungry or not hungry."
    – Kristian Berry
    Commented Nov 21, 2021 at 5:38
  • In first-order logic you would have to write "Santa Claus is hungry" as something like "there exists an x such that SantaClaus(x) and hungry(x)", and the negation of that would be "there does not exist an x such that SantaClaus(x) and hungry(x)", which is in fact true if there is no x satisfying SantaClaus(x). So if you stick the the formalism of first-order logic, this would not be an exception to the rule that anytime you pair a proposition with its own negation, one must be true and the other false.
    – Hypnosifl
    Commented Nov 21, 2021 at 5:48
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    If Santa Claus is not in the domain of discourse then it is a non-referring term, and in standard logic expressions with non-referring terms are not well-formed expressions. So LEM does not sanction anything involving them. But existing "in reality" is not a precondition for being in the domain of discourse, we routinely include fictional entities into such domains. If Santa Claus is there then LEM applies and the disjunction is true. Alternatively, you can paraphrase "Santa Clause" into a definite description, as Hypnosifl suggested, and LEM will still apply, but the negation is different.
    – Conifold
    Commented Nov 21, 2021 at 7:54
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    On dealing with non-referring terms systematically without fictionalizing the domain of discourse, see SEP, Free logic, which extends the standard logic to them.
    – Conifold
    Commented Nov 21, 2021 at 7:56

6 Answers 6

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Regarding the existence of Santa Claus and the excluded middle, Bertrand Russell dealt with puzzles like these in his paper 'On Denoting'.

The classic example is 'The King of France is bald' and 'The King of France is not bald'. Russell distinguishes two scopes in which it can be read, narrow and wide*.

For example, Russell identifies a sentence 'I thought your yacht was larger than it is' as to have:

A narrow scope reading:

  • I thought the size of your yacht is bigger than the size of your yacht.

A wide scope reading:

  • There is a size x, such that the size of your yacht is x, and I thought that the size of your yacht is bigger than x.

So, the proposition:

  • Either Santa Claus is hungry, or Santa Claus is not hungry.

Expresses truth only if two disjuncts contradict each other. The left disjunct is obviously false, but the right one has two possible readings:

1. A wide scope reading:

  • ∃x (Sx ∧ ∀y (Sy → y = x) ∧ ¬Hx)
  • There is a unique x who is Santa Claus, and x is not hungry.

2. A narrow scope reading:

  • ¬∃x (Sx ∧ ∀y (Sy → y = x) ∧ Hx)
  • It is not the case that there is a unique x who is Santa Claus, and x is hungry.

If the scope is wide (1.), it is simply false. If the scope is narrow (2.), it is true. Only the narrow reading (2.) is the negation of the left disjunct, so the law of excluded middle is preserved.

(For further reading, I encourage engaging with the paper itself.)

*- A narrow scope is also called a secondary occurrence, and the wide scope is called a primary occurrence.

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In everyday language, “Santa Claus is hungry” means “Santa Claus exists, and Santa Claus is hungry”. “Santa Claus is not hungry” means “Santa Claus exists, and Santa Claus is not hungry”. The correct negation of the first sentence is “Either Santa Claus doesn’t exist, or Santa Claus exists and is not hungry”.

So with the correct negation, there is no “excluded middle”. With the given, incorrect negation, there is the supply excluded “there is no Santa Claus”.

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"Either Santa Claus is hungry or Santa Claus is not hungry."

The following would be subject to the Law of the Excluded Middle: "Either it is the case that Santa Claus is hungry or it is not the case that Santa Claus is hungry." That is true because either Santa Claus exists and is hungry, or else either Santa Claus exists but is not hungry or Santa Claus does not exist and therefore is not hungry.

It's the meaning, not the structure of the sentence, that matters when applying the Law of the Excluded Middle. The English construction "Santa Claus is not hungry" is ambiguous: It can mean either "Santa Claus exists and is not hungry" or "Either Santa Claus exists and is not hungry or Santa Claus does not exist and therefore is not hungry." "Santa Claus is not hungry" must be construed the second way in order for the Law of the Excluded Middle to apply to "Either Santa Claus is hungry or Santa Claus is not hungry." Construing it the first way is what gives an apparent (but not real) violation of the Law of the Excluded Middle. It's unreal because on that first construal, the entire sentence does not have the form . Only on the second construal does the entire sentence have the form .

As I'm sure someone has noted, the classic example is "Either the king of France is bald or the king of France is not bald," at a time when there is no king of France. Here, too, the ambiguous phrasing of the second disjunct is the problem; so construing "the king of France is not bald" as not to allow for the possibility of the king's nonexistence is to construe the whole thing in a way that is not of the form . "Either the king of France is bald or it is not the case that the king of France is bald," which allows for the possibility that the king of France does not exist, has the form and obeys the Law of the Excluded Middle perfectly well.

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The model theory for classical FOL satisfies the formula ~H(s) if s is not in the extension of the predicate H, where H means “is hungry” and “s” refers to Santa Claus.

Further, we can distinguish between the truth of a formula in a model and what that formula is supposed to mean in real life. That is, if pushed to choose H(s) or ~H(s), I’d choose ~H(s) since it’s easy to create a model in which whatever “Santa Claus” actually refers to is not hungry. This is a bit of a stretch, but clearly we can speak of Santa meaningfully, else there’d be no worry in spoiling the truth that there is no Santa. The term ‘Santa Claus’ does mean a fat old man in a Red Suit, etc. but when we refer to ‘Santa Claus’ we refer to the idea/mythos behind him, and the way that those fictional things mingle with our real-world dealings with ‘Santa Claus’ in order to make sense of an object ‘Santa Claus’.

You can also do free logic, which does not allow for existential quantification unless it has been established that for a term t ∃x x=t. That is, the formula ∀x(H(x)∨~H(x)) is valid, but universal elimination on it is only valid for terms t that meet the requirement. So, we wouldn’t be able to derive a formula representing the proposition “Santa Claus is either hungry or not” without deriving that there is someone called Santa Claus.

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Logic has the concept of well-formed sentences. Before being correct or incorrect, your sentence must be well formed, and yours wouldn't be.

Most valid logical formulas are based on existential quantification i.e. in order to say something about an object that don't exist by axiom, you have to say "There exists an object X such that object X is Santa Claus and object X is hungry."

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The common sense solution adopted by Gottlob Frege was that a sentence whose subject does not exist does not have a truth value, with the consequence that it is not a proposition and is therefore outside the scope of the law of excluded middle. Problem solved.

The solution proposed by Bertrand Russell in his paper On denoting published in 1905 requires to assume that the sentence "Santa Claus is hungry" says that Santa Claus exists, but this is clearly false, on the face of the sentence itself.

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  • Do you at least agree with Russell that "Santa Claus is hungry" would be false if Santa Claus doesn't exist (and never did and never will), even if you disagree with his rationale?
    – Hypnosifl
    Commented Nov 21, 2021 at 20:03
  • @Hypnosifl I'm sure you don't believe in Santa Claus so it should be easy for you just to think about "Santa Claus is hungry". Still, If it was false, then its negation would have to be true, but, linguistic fact, its negation is "Santa Claus is not hungry", but according to Russell, it is also false. It just doesn't work, unless you want to believe in Russell's idea that "Santa Claus is hungry" says "Santa Claus exists and Santa Claus is hungry", which is as fantastic as the idea that Santa Claus exists. You don't believe in Santa Claus, but you still believe in Russell's Santa Claus theory.
    – Speakpigeon
    Commented Nov 22, 2021 at 11:10
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    There's no single notion of "linguistic negation", "It's not the case that Santa Claus is hungry" could also be seen as a linguistic negation of "Santa Claus is hungry". If you translate "Santa Claus is hungry" into 1st order logic and take its negation, the former ordinary-language statement would be a better translation. Are you arguing that "X is not hungry" is the true negation of "X is hungry" in terms of your belief that there's a unique "correct" version of logic rooted in biology? Can you point to animal behavior where they implicitly use this type of negation for nonexistent things?
    – Hypnosifl
    Commented Nov 22, 2021 at 16:58
  • @Hypnosifl 1. "no single notion" I didn't say there is only one form possible, but it is a linguistic fact that 99% of the time we use "X is not F" to negate "X is F". Russell produced a logical model of a non-existent language. 2. "this type of negation for nonexistent things" There is nothing special about the grammar of talking about non-existence things: "Your friend is an imaginary being" and "My friend is not an imaginary being". This is standard English. How come Russell doesn't speak standard English?
    – Speakpigeon
    Commented Nov 22, 2021 at 17:19
  • Why should logic reflect "standard English", which is highly context-dependent? For example, "Santa Claus is jolly" could be judged either true or false in English, depending on whether the speaker wanted to emphasize the defining characteristics of a fictional character, or to emphasize that Santa Claus doesn't exist. Logic is supposed to be about some abstract structure of assertions that allows us to judge certain inferences true or false without needing to know the meaning of adjectives or speaker's intentions. Do you judge "Santa Claus is jolly" to be definitely true or false logically?
    – Hypnosifl
    Commented Nov 22, 2021 at 18:08

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