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Assume following statements are true

  1. Bob's favorite color is blue.
  2. Sally's favorite color is red.
  3. Bob and Sally always tell the truth.

I ask Bob the following question: "If you are Sally, is your favorite color blue?"

According to the definition of logical implication, since "you [Bob] are Sally" is false, the above compound proposition must be true ($P\implies Q$ is true when $P$ is false). However, the statement seems to be false as interpreted in standard written English. Is this just an unfortunate quirk of the English words used to designate mathematical implication or is something deeper going on?

Mithrandir
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    I think you want your example to reach a different truth value in "normal English conversation" than in propositional logic for the purposes of your question, rather than the same one, no? Otherwise, what's with the "However"? – Arturo Magidin Mar 19 '22 at 21:28
  • There is ambiguity in the English version, namely, about the meaning of the question: is it "if you are a person who's not Sally, is your favorite color red?", or "if you're Sally, which you're not, because you're Bob, ...", or "if someone is Sally, is their favorite color red?"... – paul garrett Mar 19 '22 at 21:29
  • @ArturoMagidin I got my wires crossed and edited accordingly. – Mithrandir Mar 19 '22 at 21:35
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    Also, propositions in propositionsl calculus are not supposed to be context-dependent. "You are Bob" has different assigned values depending on context, that is, semantics; that is, meaning. Propositional calculus is syntactic: it does not depend on the assigned meaning, but only on the syntax of the sentence. – Arturo Magidin Mar 19 '22 at 21:36
  • @ArturoMagidin Thanks for these helpful comments! Hopefully, my edited question makes sense outside the context of propositional calculus. – Mithrandir Mar 19 '22 at 21:46
  • I think a more straightforward way of asking Bob is thus: "Is Sally's favorite color blue?". This is the question youre trying to get Bob to answer, not the conditional you have. I think the issue here is that youre not writing your intended question to Bob with the correct logical structure. Why must your question to Bob be written as an implication? Here's a question to you. Why would Bob answer blue or red, yes or no, to logical question thats supposed to have a true or a false value? If you want a yes or a no from Bob, youre already outside the realm of logic. – SquishyRhode Mar 19 '22 at 22:36
  • Questions cannot be written in logic; only statements. And Id even go so far as to argue semantically and grammatically correct statements, else you run risks of paradoxes. And all such statements boil down to a true or a false, and no other value. So yes, no are out of the question. There is no purely propositional way to ask a question with a non truth value, and compel an answer, especially in this form. – SquishyRhode Mar 19 '22 at 22:46
  • See https://en.wikipedia.org/wiki/Counterfactual_conditional You can, as Dan Christensen does in their Answer, use truth-functional/propositional logic, bite the bullet, and say that Bob should answer 'Yes'. ... but that is indeed so counter-intuitive that maybe we shoulkd look for a different approach .. but what? The link provides some suggestions. – Bram28 Mar 20 '22 at 15:42
  • @ryang I found both your and BrianO's answers helpful and was hoping for a third-party upvote to guide my decision. I chose your answer because I found the post you linked particularly enlightening. – Mithrandir Mar 23 '22 at 13:25

3 Answers3

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In English, one would say “if you were Sally“, not “if you are Sally“. We can paraphrase it as “If in some possible world you are Sally, then in that possible world is your favorite color blue?“. It’s what’s called the subjunctive mood, not the present tense. The statement is hypothetical, counterfactual, and like all counterfactual conditionals it’s trivially true because the antecedent is false. Counterfactual statements are not subject to the simple rules of propositional calculus.

BrianO
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  1. If the question was instead phrased as “If you were Sally, is your favorite color blue?” then it's natural to interpret it as “Is Sally's favorite color blue?”, whose answer is No.

  2. “If you are Sally, is your favorite color blue?”

    Technically, it is unclear whether “your” is pointing back at Bob or still at Sally (I discussed such ambiguity of variable recycling in this answer).

  3. Let's convert the given question into a statement:
      “If Bob is Sally, then their favorite color is blue.”

    With the implicit premise that Bob is Sally, regardless of whether “their” refers to Bob or Sally, the above (implication) statement is vacuously true, and the argument valid, since its associated conditional $$P\to\big(\lnot P\to Q\big)$$ is a tautology.

ryang
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(Corrections)

Assume following statements are true

  1. Bob's favorite color is blue.
  2. Sally's favorite color is red.
  3. Bob and Sally always tell the truth.

I ask Bob the following question: "If you are Sally, is your favorite color blue?"

Suppose your name is not Sally. Then it is true that if your name is Sally (assumed to be false) then your favourite colour is blue. We say that it is vacuously true in this case.

$~~~~\neg S \to (S \to B)$

Where:

$~~~~S =$ Your name is Sally

$~~~~B =$ Your favourite colour is blue

Here is the truth table:

enter image description here

Source: https://www.erpelstolz.at/gateway/TruthTable.html

There is no ambiguity or inconsistency here. It's just that we have an implication $(S \to B)$ with an antecedent $(S)$ that is known to be false. Such implications are simply not very useful in daily discourse since you will not be able to use that them to infer anything about the truth value of the consequent $(B)$.

Similarly, it is also vacuously true that your favourite colour is red.

$~~~~\neg S \to (S \to R)$

Where:

$~~~~R =$ Your favourite colour is red

In fact, any proposition whatsoever can be used:

$~~~~\neg S \to (S \to P)$

Where:

$~~~~P = $ Pigs can fly!

Though not very useful in daily discourse, implications that are vacuously true can be useful in very technical arguments. In set theory, for example, we have:

Proposition: For every set $A$, the empty set $\emptyset$ is a subset of $A$.

Proof: By definition, for all $x$, we have $x\notin \emptyset$. Therefore, it is vacuously true that, for all $x\in \emptyset$, we have $x\in A$.

Dan Christensen
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