# Does the king of France really have hair?

AFAIK, every statement must be either true or false (or a paradox, but that's not what I am talking about now), by the law of the excluded middle.

The statement "The king of France is bald." is false, because there is no king of France. But then, by the same law, shouldn't "The king of France is not bald.", that is, "The king of France has hair.", be true?

The only possibility I can think of to solve this confusion is that the opposite of a statement is not what we usually think it is, and as "The king of France is bald." means "The king of France exists." and "If he does exist, he is bald.", that is "If the king of France exists, then he is bald.".

Maybe as "If the king of France exists, then he is bald." equals "The king of France doesn't exist and/or he is bald.", the opposite should be (by DeMorgan's laws) "The king of France exists and he isn't bald.", but that's even more awkward.

So what should the opposite of this should be? The converse? The inverse? The contrapositive cannot be, it would be false as well. Can you give me a hand here?

• You're making a mistake with "every statement must be either true or false or a paradox". Some statements are meaningless. For example, "my height smells of mangoes". Your example sentence makes an invalid presupposition. Also, is it true that you have stopped beating your wife? Nov 20, 2013 at 20:13
• As I never beat my wife, it is not true that I have stopped beating my wife. It is perfectly allowable that a statement implies another truth. You imply that "At some point I beat my wife" when you make your statement. That statement being false makes yours false too. And your height does not smell of mangoes, so that is false too.
– Ryno
Nov 20, 2013 at 20:28
• @prash, so you reject the law of the excluded middle? Nov 20, 2013 at 20:46
• @prash - the statement "my height smells of mangoes" is actually provably false, as we can determine by hypostatic abstraction. "my height smells of mangoes" can be decomposed into the set of statements: { "i have a height, H" , "H has a smell, S1" , "mangoes have a smell, S2", "S1 = S2" }. The proposition "H has a smell, S1" is a necessary component of the statement "my height smells of mangoes" and is known to be false, thereby making the entire statement false. Nov 20, 2013 at 21:21
• To deny that a meaningless sentence has a truth value isn't to deny the law of excluded middle. Truth values attach to propositions. By claiming the sentence is meaningless, you are denying that it expresses a proposition--- no proposition, no truth value. Nov 21, 2013 at 19:06

The statement "The King of France is bald" would usually mean "The King Of France exists, and has no hair". It could be taken to mean a few other things, but this is the meaning I will assume for this answer, and regardless of what you assume it to mean, you can use the same method I will use below to derive the opposite.

Let X = The King of France
Let A = X exists
Let B = X has no hair

Truth Table:
BT     BF
AT    .
AF

The dot in that truth table represents the situation "The King of France is Bald".

The opposite would be
BT     BF
AT           .
AF    .      .

Which is (A&!B) or (!A&B) or (!A&!B)
which simplifies to (!A or (A&!B))
It can also simplify to (!B or (!A&B))

which in english is "The King of France does not exist, or he exists and has hair"
or "The King of France has hair, or he does not exist and has hair".

As the second translation doesn't really make sense, we take the first. (As, in this case, the second sentence only has a value if the first is true).

If you interpret the original sentence to be a different combination of facts, or some dependency between the two sentences, then your original truth table is different, but to find the opposite, you take the truth table with every combination being opposite, and interpret from there.

Another popular interpretation would be "There exists a King of France such that it is bald." The opposite of that one is "There does not exist a King of France such that it is bald," which is quite clearly true.

• I should have made a truth table before asking indeed... My problem was in (!B or (!A&B)), which as you referred, is a senseless interpretation in this case. Thanks for your time. Nov 20, 2013 at 21:39
• Yep - the difference between logical/syntactic sense, and "English" sense... :)
– Ryno
Nov 20, 2013 at 23:04