Intuitively, "The present Kingof France is bald." is false. Yet Bertrand Russell said it would average that "The present Kingof France is not bald.", which appears to it is in false. This supposedly leads come a contradiction.

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Could assertions about things i beg your pardon don"t exist not be false in benidormclubdeportivo.orgematics (or even true)?

For example, walk $\frac10=3$ median anything, due to the fact that $\frac10$ doesn"t exist?


(1) The OP writes:

Intuitively, "The existing King that France is bald." is false. However Bertrand Russell said it would median that "The present King the France is not bald.", which seems to be false. This supposedly leads to a contradiction.

No Bertrand Russell didn"t say fairly that. Rather he differentiated two readings that "The present King that France is not bald." This deserve to be parsed as either "It is no the instance that the-present-King-of-France-is-bald" or "The present King of France is not-bald". (There"s a limit ambiguity -- does the negation take vast scope, the whole sentence, or small scope, the predicate?)

Russell regiments "The existing King that France is bald" as

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

where "$KF$..." expresses "... Is a current King of France" and "$B$..." expresses is outright (there is one and also only one King of France and he is bald). Then the 2 readings the "The current King of France is no bald" space 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 an initial is true, the second false -- no paradox or contradiction. Trouble just arises if girlfriend muddle the two.

(2) The OP additionally writes

Does $\frac10=3$ mean anything, since $\frac10$ doesn"t exist?

Compare: "The (present) King the France" is a coherent expression -- you understand perfectly fine what condition someone would have to meet to be its denotation. In fact, the is because you understand the expression (grasp that meaning) the -- putting that together with your expertise of France"s existing constitutional species -- you understand it lacks a referent. The expression is linguistically meaningful yet happens to denote nothing (with the human being as it is). An in similar way there"s a good sense in which execute you recognize "$\frac10$" perfectly well: it means "the an outcome of splitting one through zero". That is because you know the notation, and because you know that department is a partial duty and return no value when the 2nd argument is zero, the you understand that "$\frac10=3$" isn"t true. The icons aren"t mere rubbish -- you know what duty you are supposed to be using to i m sorry arguments. So, in a great sense, the symbols "$\frac10$" are coherent even despite they failure to signify a value. In Frege"s terms, the expression has actually sense however lacks a reference.

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(3) Marc van Leeuwen writes

Using the definite short article "the" in "the present King the France" implicitly claims there is precisely one person presently King the France; since that is no the case, any phrase that describes this is meaningless."

Not so. Because that example, the sentence "No one is the present King of France" is not only meaningful however true -- so the can"t it is in that just containing the non-referring "the existing King of France" makes for meaninglessness.