Writing a “however” sentence into propositional form

discrete mathematicspropositional-calculus

This is an exercise from Judith L. Gersting's book "Mathematical Structures for Computer Science":

Write the following as propositions: Either going to bed or going swimming is a sufficient condition for changing clothes; however, changing clothes does not mean going swimming.

The solution from the textbook and the solution we achieved in class was

$B$: I will go to bed
$S$: I will go swimming
$C$: I will change clothes

$$(B \vee S \rightarrow C) \wedge ¬(C \rightarrow S) $$

However, the truth-table for this expression does not seem compatible with the statement in English:

    B   C   S   | (((S ∨ B) → C) ∧ ¬(C → S))
    -----------------------------------------
(1) F   F   F   |              F
(2) F   F   T   |              F
(3) F   T   F   |              T
(4) F   T   T   |              F
(5) T   F   F   |              F
(6) T   F   T   |              F
(7) T   T   F   |              T
(8) T   T   T   |              F

The mistakes seem to be:

  • In (1): if all possible antecedents are false, then the entire expression should be true (if no conditions are satisfied, then the outcome could be anything)
  • In (4): going swimming (S) was satisfied, and that's sufficient to change clothes (C). So this should be true
  • In (8): both conditions are satisfied, and clothes have been changed. So this should also be true

It seems to me that the correct answer should be simply

$$B \vee S \rightarrow C$$

This produces the following truth-table:

    B   C   S   | ((S ∨ B) → C)
    ---------------------------
(1) F   F   F   |      T
(2) F   F   T   |      F
(3) F   T   F   |      T
(4) F   T   T   |      T
(5) T   F   F   |      F
(6) T   F   T   |      F
(7) T   T   F   |      T
(8) T   T   T   |      T

Which seems correct to me. Whenever $B \vee S$ are true and $C$ is also true, the entire expression is true (sufficient condition confirmed). The second part of the statement, to me, only explains (3) and (7), where clothes have been changed, but no swimming happened.

It's hard to question the truth table. But we're talking about an exercise of the 7th edition of a highly reputable source. Also, I talked with a colleague about this exercise and we couldn't reach an agreement.

Is the book wrong? Did I get something wrong? How do I convincingly explain that the truth table proves that the book is wrong?

Best Answer

I agree with your $(B \vee S) \rightarrow C$. The however phrase does not seem to add anything except to point out that you can might clothes without swimming. It certainly does not mean $\lnot(C \rightarrow S)$, which would guarantee changing clothes and not swimming. That would be changing clothes means not swimming.

It is true that however should usually be rendered by and. Here I would see the phrase as meaningless fluff and render it as "and True", which can be deleted.

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