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Intuitively, "The present King of France is bald." is false. But Bertrand Russell said it would mean that "The present King of France is not bald.", which seems to be false. This apparently leads to a contradiction.

Could assertions about things which don't exist not be false in mathematics (or even true)?

For example, does $\frac{1}{0}=3$ mean anything, since $\frac{1}{0}$ doesn't exist?

jinawee
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    Both statements are false. They are not mutually exclusive - the present king of France is neither bald nor not bald, as he does not exist. – preferred_anon Oct 03 '13 at 11:40
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    Alternatively, both statements are vacuously true, since the present king of France doesn't exist. This really depends on how you interpret the sentence; you can say "There is a king of France and he is bald," or "Every person who is the king of France is bald." – Miha Habič Oct 03 '13 at 12:03
  • Yes, $1/0=3$ means something, it means $1=0\times3$, so it means $1=0$. Of course, it's false. – Gerry Myerson Oct 03 '13 at 12:36
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    @GerryMyerson: No! By that reasoning $0/0=3$ would mean $0=0\times 3$ which is true, and also $1/0\neq 3$ would be true, but in fact both are meaningless statements (unless one uses the definition that $1/0=\infty$, but in OP that was obviously not meant). – Marc van Leeuwen Oct 03 '13 at 12:44
  • @Marc, yes, very careless of me. – Gerry Myerson Oct 03 '13 at 12:50
  • @Gerry Myerson: I think it's a good answer- you are just illustrating that division is occasionally multivalued, when "$a/b$" is taken to mean "a value $c$ with $bc = a$". – Carl Mummert Oct 03 '13 at 20:32
  • Surely, 'the present King of France is not bald' is true, as there is no present King of France and therefore he cannot possibly be bald. – sean_robbins Mar 03 '16 at 23:53
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    Your title question is a main topic studied in formal linguistic semantics and philosophy of language known as definite description. As the reference mentions Russell's analysis is the most influential (the puzzle about the the negation of your title sentence lies in the scope interpretation of the negation, no LEM violation here). P. F. Strawson and K. Donnellan criticized while S. Kripke defended Russell's analysis... – cinch Oct 14 '21 at 06:23

4 Answers4

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(1) The OP writes:

Intuitively, "The present King of France is bald." is false. But Bertrand Russell said it would mean that "The present King of France is not bald.", which seems to be false. This apparently leads to a contradiction.

No Bertrand Russell didn't say quite that. Rather he distinguished two readings of "The present King of France is not bald." This can be parsed as either "It is not the case that the-present-King-of-France-is-bald" or "The present King of France is not-bald". (There's a scope ambiguity -- does the negation take wide scope, the whole sentence, or narrow scope, the predicate?)

Russell regiments "The present King of France is bald" as

$$\exists x(KFx \land \forall y(KFy \to y = x) \land Bx)$$

where '$KF$...' expresses '... is a present King of France' and '$B$...' expresses is bald (there is one and only one King of France and he is bald). Then the two readings of "The present King of France is not bald" are respectively

$$\neg\exists x(KFx \land \forall y(KFy \to y = x) \land Bx)$$

$$\exists x(KFx \land \forall y(KFy \to y = x) \land \neg Bx)$$

The first is true, the second false -- no paradox or contradiction. Trouble only arises if you muddle the two.

(2) The OP also writes

Does $\frac{1}{0}=3$ mean anything, since $\frac{1}{0}$ doesn't exist?

Compare: "The (present) King of France" is a meaningful expression -- you know perfectly well what condition someone would have to satisfy to be its denotation. In fact, it is because you understand the expression (grasp its meaning) that -- putting that together with your knowledge of France's current constitutional arrangements -- you know it lacks a referent. The expression is linguistically meaningful but happens to denote nothing (with the world as it is). Similarly there's a good sense in which do you understand "$\frac{1}{0}$" perfectly well: it means "the result of dividing one by zero". It is because you understand the notation, and because you know that division is a partial function and returns no value when the second argument is zero, that you know that "$\frac{1}{0}=3$" isn't true. The symbols aren't mere garbage -- you know what procedure you are supposed to be applying to which arguments. So, in a good sense, the symbols "$\frac{1}{0}$" are meaningful even though they fail to denote a value. In Frege's terms, the expression has sense but lacks a reference.

(3) Marc van Leeuwen writes

Using the definite article "the" in "the present King of France" implicitly claims there is exactly one person presently King of France; since that is not the case, any phrase that refers to this is meaningless."

Not so. For example, the sentence "No one is the present King of France" is not only meaningful but true -- so it can't be that just containing the non-referring "the present King of France" makes for meaninglessness.

Peter Smith
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    Natural language is not like formal logic. "Nobody is the present King of France" is not a sentence that refers to "the present King of France", and it does not express a property to the designated person like "the present King of France is nobody" would maybe do (although being nobody is unclear too). And this in spite of the fact that being the same (person) is normally a symmetric relation. In fact that sentence is linguistically not very clear; it probably expresses the same thing as "nobody is presently the King of France" would, which is the negation of a (unique?) existence statement. – Marc van Leeuwen Oct 03 '13 at 20:56
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    @Marc: But the fact remains that while the phrase the present King of France has no referent, it is meaningful. Indeed, the fact that we know that it has no referent shows that it’s meaningful. And it’s not true that use of the definite article always implies uniqueness: consider The man in the street thinks ..., The man on the Clapham omnibus would never ..., The tiger is fearsome beast, etc. The semantics of the definite article are extremely complicated. – Brian M. Scott Oct 03 '13 at 22:25
  • @BrianM.Scott: I'll agree that "the" in natural language can have many meanings. But I maintain that, for instance, say in ring theory that the maximal ideal $m$ of $R$ has some property $P(m)$ implicitly asserts that $R$ is a local ring (has a unique maximal ideal); if not (either there are multiple maximal ideals, or none at all), then neither $P(m)$ nor $\lnot P(m)$ hold, which is why I say that stating $P(m)$ is meaningless in such cases. This does not of course imply that we don't know what a maximal ideal is. – Marc van Leeuwen Oct 03 '13 at 23:14
  • @Marc: I think that you’re confusing pragmatics with semantics. However, I will agree that in mathematical writing we are relatively intolerant of violations of pragmatic expectations. – Brian M. Scott Oct 03 '13 at 23:18
  • I have argued that 1/0=3 is "meaningless" in the sense that, using the usual definition of division on R, no truth value can be assigned to it. – Dan Christensen Mar 10 '24 at 21:09
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This is the sort of thing that free logic can handle in ways classical logic cannot. In logic jargon, the phrase "the present king of France" is a singular term that does not denote any object. Depending on which semantics for free logic you use, the sentence "the present king of France is bald" might be true, or false, or truth-valueless.

Separately, in normal first-order logic, we often translate a "the" sentence of English into a formal sentence with a quantifier. In this case, "the present king of France is bald" might become "for every person $P$, if $P$ is the present king of France then $P$ is bald" (which is a true sentence, classically, in the intended interpretation) or "there is a person $P$ who is the present king of France and is bald" (which is false, classically).

Carl Mummert
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  • Something is amiss with trying to elucidate "the present K of F is bald" using some analysis that still uses "the present K of F". Whatever the failings of Russell's analysis, at least it avoids that pitfall! – Peter Smith Oct 03 '13 at 22:13
  • What else is "the present king of France" but a nondenoting term? Is there something special about it, as compared to any other nondenoting term? It seems to me that I could just as well ask whether unicorns wear roller skates. – Carl Mummert Oct 03 '13 at 23:06
  • Part of the issue, of course, is whether we see "the present K of F" as a closed term, or whether we see "$x$ is the present K of F" as a predicate of $x$, in which case we need a quantifier over $x$. – Carl Mummert Oct 03 '13 at 23:08
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Mathematical expressions or phrases that are not defined, or that make an implicit claim that is not satisfied, are meaningless, and no (truth) value is ascribed to them. Examples are for example expressions containing a division by $0$, limits of divergent sequences, or taking the minimum of a set of numbers that turns out to be empty; there are many more examples. Using the definite article "the" in "the present King of France" implicitly claims there is exactly one person presently King of France; since that is not the case, any phrase that refers to this is meaningless.

Marc van Leeuwen
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Contrary to Russell, given that there is presently no King of France, it is vacuously true that if x is presently The King of France, then x is bald.

PROOF

Let K and B be unary predicates such that:

  • K(x) = "x is presently The King of France"
  • B(x) = "x is bald"
  1. $\forall a: \neg K(a)~~~~$ (Assume)
  2. $K(x)~~~~$ (Assume)
  3. $\neg B(x)~~~~$ (Assume)
  4. $\neg K(x)~~~~$ (U Spec, 1)
  5. $K(x) \land \neg K(x)~~~~$ (Join 2, 4)
  6. $\neg \neg B(x)~~~~$ (Discharge 3, 5)
  7. $B(x)~~~~$ (Elim $\neg\neg$, 6)
  8. $\forall a: [K(a) \implies B(a)]~~~~$ (Discharge 2, 7)
  9. $\forall a: \neg K(a) \implies \forall a: [K(a) \implies B(a)]~~~~$ (Discharge 1, 8)

Note that we could as easily have proven that the nonexistent king of France is NOT bald:

$\forall a: \neg K(a) \implies \forall a: [K(a) \implies \neg B(a)]$

Hint: Change line 3 to "$B(x)$" instead of "$\neg B(x)$".

EDIT

Re: $\frac{1}{0} = 3$

The expression $\frac{1}{0} = 3$ is "meaningless" in the sense that its truth value cannot be determined.

We can define division on the reals as follows:

$\forall x,y,z \in R: [y\neq 0 \implies [\frac{x}{y} =z \iff x=z\cdot y]]$

To determine the truth value of $\frac{1}{0} = 3$, we set $x=1, y=0, z=3$.

Substituting, we have:

$[0\neq 0 \implies [\frac{1}{0} =3 \iff 1=3\cdot 0]]$

Now, $0\neq 0$ false. Therefore, this implication is vacuously true, but we cannot infer that its consequent is true. As such we cannot determine the truth value of the expression $\frac{1}{0} =3$.

Dan Christensen
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    Of course this is vacuously true, but the question is whether ‘The present King of France is bald’ means this in the first place. – Toby Bartels Mar 05 '24 at 06:18
  • @TobyBartels I interpret both predicates to be in the present tense. The statement $\forall a: [K(a) \implies B(a)]$ on its own can be interpreted as: All kings of France are (present tense) bald. This is not inconsistent with there being (present tense) no kings of France. – Dan Christensen Mar 05 '24 at 14:17
  • The dispute is not about present tense; the word ‘present’ is usually included to head off that issue. The problem with this answer is that we're not talking about the statement ‘All kings of France are bald’ at all. We're talking about ‘The king France is bald’, and the meaning of ‘the’ is key. If you want to interpret that as meaning ‘All kings of France are bald’ (which everyone agrees is true), then that's the argument that you have to make, and your answer just skips over that. – Toby Bartels Mar 05 '24 at 20:50
  • @TobyBartels You could easily introduce an axiom that there is at most one king (The King): $\forall a, b: [K(a) \land K(b) \implies a=b]$. You could still, however, obtain the same result as above, ignoring that axiom. – Dan Christensen Mar 05 '24 at 22:06
  • So your advocating that ‘the’ should not include existential content, only uniqueness? Then that should be the point of your answer: the King of France is bald, because vacuously, any two kings of France are identical, and any one king of France is bald. And $ \frac 1 0 = 3 $, because any two reciprocals of $ 0 $ are identical, and any one of them is equal to $ 3 $. – Toby Bartels Mar 06 '24 at 22:51
  • @TobyBartels No. In my proof, uniqueness is not ruled out. It simply turns out to be immaterial in this analysis. It is vacuously true that $x$ is the king of France implies $x$ is bald. It is also vacuously true that $x$ is NOT bald. – Dan Christensen Mar 07 '24 at 00:36
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    You clearly don't understand what this question is about, and you apparently don't want to understand. I explained my downvote, and I'm done. – Toby Bartels Mar 07 '24 at 01:05