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I'm taking a discrete maths course and we were tasked with converting the following English sentence into propositional logic:

"You can take a picture of the painting only if your camera's flash is off or you are not close to the painting."

We split the sentence into the propositions
p: You can take a picture of the painting
q: Your camera's flash is off
r: You are close to the painting.

Everyone agreed on the "or" and the "not". However, the class was pretty evenly split regarding the "only if" part of the statement: some were saying that the correct translation is p ↔ (q ∨ ¬r), while others were saying that it should be p → (q ∨ ¬r). Which is correct?

I think the biconditional statement is the correct one, because the original sentence can be rewritten as the following two sentences while keeping the same meaning and also fulfill (p → q) ∧ (q → p) = p ↔ q:

  1. If you can take a picture of the painting, then your camera's flash is off or you are not close to the painting.
  2. If your camera's flash is off or you are not close to the painting, then you can take a picture of the painting.

I believe they both imply each other. If you can take a picture of the painting, then it is implied that you are not close to the painting or your camera's flash is off. If your camera's flash is off or you are not close to the painting, then it is implied that you can take a picture of the painting.

ryang
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mars_plastic
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    When you state that "personally, I think..." #2, you are saying what you believe to be true of the scenario based on your personal knowledge of the subject matter. But the question is only about what the statement in bold is equivalent to. I believe that is often what makes exercises like this confusing when first learning logic. You have to ignore any knowledge you have of the subject matter, and focus only on the logical form of the statement. – Joe Sep 12 '21 at 14:46
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    Why do you think the sentence is "A if and only if B" when all it says is "A only if B"? – ancient mathematician Sep 12 '21 at 14:49
  • @Joe "You have to ignore any knowledge you have of the subject matter, and focus only on the logical form of the statement." Thank you, this helps a lot. – mars_plastic Sep 12 '21 at 15:15
  • You're welcome. – Joe Sep 12 '21 at 15:34

1 Answers1

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  1. Think about it:

    • $$\text{A is true only if B is true}\\\text{B being true is a necessary condition for A to be true}$$
    • $$\text{if B is false, then A must also be false}\\\lnot B\implies\lnot A$$
    • $$\text{there is no way for A to be true yet B false}\\\text{if A is true, then B must be true too}\\\text{B is true if A is true}\\A\implies B\\\text{A being true is a sufficient condition for B to be true}$$

    I've divided the above into three groups, but all nine statements are in fact logically equivalent to one another.

    Thus, the correct translation of “only if” as a logical connective is $\to.$

  2. Echoing the first two comments: while in natural language “[blah blah] only when [bleh bleh]” is generally contextually understood to be bi-implicational, in formal logic, mathematics, and computer science, “only if” $(\to,\implies)$ and “if and only if” $(\leftrightarrow,\iff)$ are well-defined to have distinct meanings.

    To be clear: “only if” $(\to,\implies)$ is not a stronger form of “if” $(\leftarrow,\impliedby).$

ryang
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