"Present king of France" as a typical case of material implication

So why can't I consider "The present king of France is bald" as material implication? P = 'There's a present king of France', Q = 'A present king of France is bald'. Therefore "P -> Q" is true, because it's the case with false antecedent.

Where am I wrong? If you reject this proposition as an example of material implication then please show me what really is such a case. And after that show me the difference between yours example and the proposition about King of France.

• See Descriptions for a clear overview of Russell's analysis of definite and indefinite descriptions. Commented Feb 1 at 13:32
• Because "the present king of France is bald" says something different from "if there is the present king of France then he is bald", which is what your material conditional symbolizes. Just as "wishes are horses and beggars ride them" is different from "if wishes were horses, beggars would ride". Your proposition is a material conditional, "apples are red" is not, and neither is "the present king of France is bald". Commented Feb 1 at 14:47
• @MauroALLEGRANZA "Having said that, what do you mean? (i) ∃x (PresentKingFrance(x) → Bald(x)), or (ii) ∃x PresentKingFrance(x) → ∃x (PresentKingFrance(x) ∧ Bald(x))". Can you explain the difference in natural language? Commented Feb 1 at 16:08
• Ok, if you are not familair with the predicate logic translation, forget it :-) The simple answer is taht of Conifold above: in "The present king of France is bald" there is no "if..., then..." Commented Feb 1 at 16:13

Material implication is the logical rule

(P =-> Q) is equivalent to (not-P or Q)

You give an example for a correct implication P => Q, but the antecendent P is false. Hence you cannot derive the truth value of Q from the implication.

Propositions with false antecedent are useless to derive the truth value of their consequent.

"The present king of France is bald" is one English sentence; "If there is a present king of France, then a present king of France is bald" is another English sentence. Although these two sentences have similarities, they are not the same. Your question amounts to "Why can't I consider that the first sentence is the second sentence?" Well, you can, but it's going to be hard to communicate with other English-speakers unless you warn them beforehand of how you consider some sentences to be other sentences.

Consider these three sentences:

• A "The present king of France is bald";
• B "If there is a present king of France, then a present king of France is bald";
• C "There is a present king of France, and a present king of France is bald".

Sentences B and C both express well-formed propositions; B is true and C is false. I would argue that sentence A does not express a well-formed proposition, because it relies on a false premise: if there is no king of France, then A is meaningless.

If you are writing an essay on logic or teaching a class on logic, and you want to establish a convention that will be used throughout your essay, you probably want to make it explicit how you deal with sentences like A. So at the beginning of your essay, in a paragraph titled "Definitions and conventions", you'll write one of these threes:

• In this work, sentences like A must be interpreted as shorthand for B;
• In this work, sentences like A must be interpreted as shorthand for C;
• In this work, sentences like A are considered badly-formed and cannot be interpreted as propositions.

In other words: you can do whatever you want, but if you want to communicate and be understood, then you must make it clear what it is that you are doing.

If you interpret A as B, and Mauro interprets A as C, and I interpret A as badly-formed, but none of us wrote out interpretation explicitly and we refuse to agree on an interpretation, then we'll be unable to communicate; you'll say that A is true, Mauro will say that A is false, and I'll say that A is meaningless, and our discussion won't go very far.

Note that further variations on B and C are possible, for instance by replacing "There is a present king of France" with "There is one and only one present king of France", or replacing "and/then a king of France is bald" with "and/then this king of France is bald" or "and/then all kings of France are bald".

• Is there some reasons to consider C more valid than B? Or they exist on equal grounds? I know about Bertrand Russell and his C-interpretation. And I also superficially heard about some A-interpretations. But never about B. Why is that? I just didn't explore the topic enough? Commented Feb 2 at 15:52
• @ЕгорГалыкин Because "The king of France is bald" is literally "[Implicit assumption that there is a king of France; and] the king of France is bald." Compare with the question "Did you stop beating your wife?" No matter whether you answer yes or no to this question, by answering yes or no you'll be agreeing that you did beat your wife. That is not the same question as "Is it logically true that if you used to beat your wife then you have now stopped?".
– Stef
Commented Feb 2 at 17:08
• @ЕгорГалыкин Also, because if there is a king and he's bald, then ABC are all true; if there is a king and he's not bald, then ABC are all false; and if there is no king, then A is meaningless/malformed/wrong, C is false, but B is true; and we more naturally equate meaningless/malformed/wrong with false than with true. Since the only difference between B and C is whether the proposition is true or false when there is no king, it is more natural to equate A with C than with B.
– Stef
Commented Feb 2 at 17:16