# The king of France is bald - true or false?

Western logic is by convention "two-valued", ie any statement must be either true or false.

Consider the statement "The king of France is bald". Is this true or false?

France is a republic. There is no king of France at the present time.

Clearly therefore the statement "The king of France is bald" is not true.

But neither is the statement "The king of France is bald" false (this would imply the king of France is not bald, which is also incorrect).

Conventional logicians may resort to saying that the statement "The king of France is bald" is in fact meaningless.

But we can argue the statement (as an English language statement) is certainly NOT meaningless. The fact that there is no king of France at the present time does NOT make the statement "The king of France is bald" a meaningless statement.

What does that leave us with? Perhaps "The king of France is bald" is neither true nor false, but indeterminate. But this implies that 2-valued logic is inadequate - we need to resort to 3-valued logic to solve the question.

MF

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Perhaps. But you might also consider the definition of bald as "having no hair". There is no king of France; so, naturally, the king of France has no hair. So "The king of France is bald." is actually a true statement.

But that doesn't mean your idea won't hold in principle.

honestrosewater
Gold Member
In formal logic, the thing that tells you the truth-value of a statement is defined under some interpretation on some language (you have to know what the statement means before you can try to determine whether it's true or not). Under one interpretation on one first-order language, the king of France is bald may be stated as

$$\forall x (Kx \rightarrow Bx)$$

where Kx means x is the King of France and Bx means x is bald. If Kx is false for all x (i.e. there is no King of France), the truth-value of Bx will not affect the truth-value of $\forall x (Kx \rightarrow Bx)$, because if an implication's antecedent is false, the implication is true regardless of the truth-value of its consequent ('false implies false' and 'false implies true' are both true). I can try to explain what that means if you don't happen to know yet, but I'm a bit rusty, so it may take me a while to get back into the swing of things. Actually, I may have messed up something already, so I'll brush up and get back to you. I just wanted to get things on the right track.

AKG
Homework Helper
Normally statements where the subject is described (and not named) are existential statements, or at least, some like Russell argue that this is the case. Naturally, this makes sense. To say that the King of France is bald seems to imply that the King of France exists, and he happens to be bald, and not that all Kings of France are bald. Well it seems to say both, sort of, which is why this sentence and such sentences are controversial. If we treat it as an existential sentence, it becomes:

$$(\exists x)(Kx \wedge (\forall y)(Ky \to (x = y)) \wedge Bx)$$

The stuff in the middle there says that that which satisfies K is unique. It's not really relevant to this problem, but in general sentences with definite descriptions are (at least according to people like Russell) to be interpreted this way. However, if we get rid of the stuff in the middle because it's irrelevant, we are left with:

$$(\exists x)(Kx \wedge Bx)$$

This statement is false. So what is the appropriate interpretation?
But neither is the statement "The king of France is bald" false (this would imply the king of France is not bald, which is also incorrect).
This is not true. To say that "The King of France is bald" is false is not necessarily incorrect. It seems you're implicitly assuming that saying that that sentence is false implies that the King of France has hair. "The King of France has hair" would be as false as "The King of France is bald." Do you see the distinction between saying, "'The King of France is bald' is false" and "The King of France has hair"?

Now there is a different justification for saying that the sentence is meaningless. How about the sentence "joook is large". Well 'joook' doesn't refer to anything, so the sentence is meaningless. 'joook' has no meaning, since it has no referrant. Similarly, 'the King of France' is a description with no referrant, so it is arguably meaningless when looking at it from the perspective of evaluating the truth of the sentence. Suppose I give you the sentence "_______ is bald." Now it doesn't make sense to call this "sentence" true or false, but this is no reason to reject the principle of bivalence. Some argue that "the King of France" is just like putting in a "_____".

It seems there's two ways of looking at the problem. One is to say that the sentence "The KOF is bald" implies that KOF exists, because to say KOF is bald is to say KOF has a property, which is to say KOF exists, because things exist iff they have properties. But if "The KOF is bald" implies KOF exists, and KOF doesn't exist, then "The KOF is bald" is false. On the other hand, you can look at as the second person in the following conversation.

"I think the KOF is bald."
"What is it that you think is bald?"
"The KOF"
"What KOF?"

The first person can't say what he's talking about. So he's not talking about anything, it's like his sentence is meaningless. And depending on how you want to look at it, you can either say that this "sentence" is not properly a sentence, or that it is a sentence with a different truth-value, this value being something like "meaningless."

So, in other words, the description KOF has intention but no extension. And from one point of view, if you have a sentence with a description that has no extension, it is like a sentence with a gap in it, a meaningless sentence. On the other hand, the description does have an intention, so it seems hard to say that the sentence really is meaningless.

honestrosewater
Gold Member
Well, first, do we have any reason to think that there is a suitable translation from English to FOL? Perhaps there is none, perhaps many?

Yes, $(\exists x)(Kx \wedge Bx)$ was my first reading (uniqueness aside). But, IIRC, categorical logic treats statements about individuals as A statements (Socrates is mortal becomes All S are M), so I went with that.
Are these the choices (is this how you would translate (2)):

1) The KOF is bald
1a) $$\forall x (Kx \rightarrow Bx)$$
1b) $$\exists x (Kx \wedge Bx)$$

2) The KOF is not bald
2a) $$\forall x (Kx \rightarrow \neg Bx)$$
2b) $$\exists x (Kx \wedge \neg Bx)$$

When there is no KOF, (1a) and (2a) are true, and (1b) and (2b) are false.
When there is a (unique) KOF and he is bald, (1a) and (1b) are true, and (2a) and (2b) are false.
When there is a (unique) KOF and he is not bald, (1a) and (1b) are false, and (2a) and (2b) are true.
So are they not both acceptable?
It seems there's two ways of looking at the problem. One is to say that the sentence "The KOF is bald" implies that KOF exists, because to say KOF is bald is to say KOF has a property,
This translation would make KOF an individual, say, k, and B the predicate: Bk. Yes? And Bk implies Ex(Bx), yes?

(:rofl: Oops, had my left and right mixed up. Oy.)

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AKG
Homework Helper
honestrosewater said:
Well, first, do we have any reason to think that there is a suitable translation from English to FOL? Perhaps there is none, perhaps many?

Yes, $(\exists x)(Kx \wedge Bx)$ was my first reading (uniqueness aside). But, IIRC, categorical logic treats statements about individuals as A statements (Socrates is mortal becomes All S are M), so I went with that.
'Socrates' is a name, 'the King of France' is a definite description. I would think that such a sentence would just be Ms, not "all Socrates are mortal." But when you have a description, I've read that they should be treated as existentially quantified statements. Of course, I don't know that it's a rule, it was someone's (Russell's) position. It does seem that one could also make a case for interpreting it as a universally quantified statement as well.
This translation would make KOF an individual, say, k, and B the predicate: Bk. Yes? And Bk implies Ex(Bx), yes?

(:rofl: Oops, had my left and right mixed up. Oy.)
If the correct interpretation is Bk, then yes, Bk implies Ex(Bx), but Ex(Bx) is not the point of contention. We're worried about the sentence "the King of France is bald" not the sentence "there exists something that is bald." The problem, however, is that Bk is not the correct interpretation, at least not at first glance. k is supposed to pick out a member of the universe of discourse. Which member of the universe does k denote? If k is supposed to denote the King of France, then k denotes nothing, contradicting the fact that it is supposed to. This is why definite descriptions ought to be interpreted in some quantified sentence. If Kx iff x has the property of being the King of France, then the sentence would be interpreted:

Ax(Kx -> Bx)

or

Ex(Kx & Bx)

but Bk simply doesn't make sense if k denotes nothing.

honestrosewater
Gold Member
AKG said:
'Socrates' is a name, 'the King of France' is a definite description. I would think that such a sentence would just be Ms, not "all Socrates are mortal."
I did look into how individual (or singular) terms were dealt with in categorical/syllogistic/Aristotelian logic, and I found conflicting accounts of what Aristotle said (how, I don't know). But I don't want to track down his words, and I personally tend to trust SEP, so:
Subjects and predicates of assertions are terms. A term (horos) can be either individual, e.g. Socrates, Plato or universal, e.g. human, horse, animal, white. Subjects may be either individual or universal, but predicates can only be universals: Socrates is human, Plato is not a horse, horses are animals, humans are not horses.
- http://plato.stanford.edu/entries/aristotle-logic/#premises

[Compare with:]

Aristotle excluded what are called individual subjects or singular terms from his logic.
...
The inability to deal with individual subjects is considered to be one of the greatest flaws of Aristotelian logic. Although many have argued that we could universally predicate an individual subject such as Socrates', Aristotle himself disagreed:  Socrates' is not predicable of more than one subject, and therefore we do not say `every Socrates' as we say 'every man'.''([AM] 5:9).
- http://planetmath.org/encyclopedia/AristotelianLogic.html [Broken]
If anyone is interested in other people's opinions:
Medieval Theories of Singular Terms; Singular Terms in Logic
And how existential import was dealt with:
But when you have a description, I've read that they should be treated as existentially quantified statements. Of course, I don't know that it's a rule, it was someone's (Russell's) position.
Yes, I vaguely remember reading the same thing. I don't know of any such rules - not ones that are part of the formal language anyway.

[Thinking aloud]I guess it might be possible to make rules for translating from a natural language to a formal language using something like a generative grammar, but that's a pretty wild guess. I don't think you'd want to let all grammatical, 'natural' sentences, for example, This sentence is false, become 'formal' sentences. I can't imagine how you'd distinguish This sentence is false from That joke is old by syntax alone.[/thinking aloud]
If the correct interpretation is Bk, then yes, Bk implies Ex(Bx), but Ex(Bx) is not the point of contention. We're worried about the sentence "the King of France is bald" not the sentence "there exists something that is bald." The problem, however, is that Bk is not the correct interpretation, at least not at first glance. k is supposed to pick out a member of the universe of discourse. Which member of the universe does k denote? If k is supposed to denote the King of France, then k denotes nothing, contradicting the fact that it is supposed to. This is why definite descriptions ought to be interpreted in some quantified sentence. If Kx iff x has the property of being the King of France, then the sentence would be interpreted:
Ax(Kx -> Bx)
or
Ex(Kx & Bx)
but Bk simply doesn't make sense if k denotes nothing.
Right, I agree. I may have read "to say KOF is bald is to say KOF has a property" too closely; I took "KOF has a property" to mean that you were looking at KOF as an individual in that case.

So does anyone think that (bivalent) FOL cannot handle The King of France is bald?

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AKG
Homework Helper
honestrosewater said:
[Thinking aloud]I guess it might be possible to make rules for translating from a natural language to a formal language using something like a generative grammar, but that's a pretty wild guess. I don't think you'd want to let all grammatical, 'natural' sentences, for example, This sentence is false, become 'formal' sentences. I can't imagine how you'd distinguish This sentence is false from That joke is old by syntax alone.[/thinking aloud]
I think Tarski said something about it being impossible to define a truth predicate in your object language. If j denotes that joke, then you could say Oj to say that that joke is old. If s represents "this sentence", you can't have a predicate T such that Ts means 'this sentence is true' and so you don't have a T such that ~Ts means 'this sentence is false'. So if I'm right about Tarski, and if Tarski himself is right, then your two italicized sentences would be quite different syntactically.

The cheque is in the post

I think that in any reasonable interpretation 'The king of France is bald' is false. 'The cheque is in the post' has exactly the same form, and I don't think that anyone would argue that it wasn't false if there wasn't in fact a cheque.

honestrosewater
Gold Member
AKG said:
I think Tarski said something about it being impossible to define a truth predicate in your object language. If j denotes that joke, then you could say Oj to say that that joke is old. If s represents "this sentence", you can't have a predicate T such that Ts means 'this sentence is true' and so you don't have a T such that ~Ts means 'this sentence is false'. So if I'm right about Tarski, and if Tarski himself is right, then your two italicized sentences would be quite different syntactically.
I don't know what he said. One solution that makes me happy: If all predicates are symbols of the object language and no truth-values are symbols of the object language, then no truth-values are predicates. You get to define them, so do whatever you want to. That's pretty much how I feel about it.
As for the other thing, context-free grammars are the only kind I'm the least bit familiar with, but:
Chomsky has argued that it is impossible to describe the structure of natural languages using context free grammars (at least if these descriptions are to be judged on vaguely Chomskian criteria).
- http://en.wikipedia.org/wiki/Transformational_grammar
I can't really explain the idea anyway.
I meant that the two English sentences were indistinguishable in some way, whatever you want to call it (it's syntax to me). They have the same form:
Determiner - Noun - Verb - Adjective

AKG said:
'Socrates' is a name, 'the King of France' is a definite description. I would think that such a sentence would just be Ms, not "all Socrates are mortal." But when you have a description, I've read that they should be treated as existentially quantified statements. Of course, I don't know that it's a rule, it was someone's (Russell's) position.
Yes. "[URL [Broken] Theory of Descriptions[/url]. According to Russell, the statement "The present king of France is bald" should be analysed as three distinct assertions:

1) there is at least one present King of France;
2) there is at most one present King of France;
3) the individual specified by clauses 1 and 2 is bald.

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