4

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?

giusti
  • 145
  • 2
    I will say this is a case where propositional logic can't quite capture the nuances of language. As I understand it, the second clause means that it is possible to change clothes and not go swimming. This notion of possibility doesn't exist in propositional logic. Another way to look at it is that this clause is consistent with any single observation. So your translation does feel closer (but still not perfect). – Tipping Octopus Aug 24 '18 at 00:00
  • I fully agree with your analysis. The first sentence states that there are at least two reasons for changing clothes, namely going to bed and going swimming. The second sentence seems to draw a Sherlock Holmes type of conclusion from the first. Therefore, Dr Watson, from the fact that clothes were changed we can not conclude that the victim intended to go swimming. There might have been a different reason for doing so. – M. Wind Aug 24 '18 at 01:42
  • @PoonLevi the possibility of changing clothes and not going swimming does exist in propositional logic and is represented by $S \rightarrow C$, or, in this case, $S \vee B \rightarrow C$. Both truth tables show that in line (3), where $C$ is true, and $S$ is false. – giusti Aug 24 '18 at 04:34

3 Answers3

4

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.

Ross Millikan
  • 374,822
  • I don't understand why $\lnot(C \rightarrow S)$ can be deleted. – quasi Aug 24 '18 at 00:37
  • 1
    No, as I said, I don't think $\lnot (C \rightarrow S)$ is a correct translation of "changing clothes does not mean going swimming". I can't find a way to translate it without the "it is possible" quantifier. That is what made me think it is meaningless fluff. – Ross Millikan Aug 24 '18 at 00:42
  • I think the translation is correct: "changing clothes does not mean going swimming" I would interpret as "changing clothes does not imply going swimming", which is the same as "it's not true that (changing clothes implies going swimming). Thus, to me it means $\lnot (C \rightarrow S)$. – quasi Aug 24 '18 at 00:45
  • 1
    But your "it's not true that (changing clothes implies going swimming)" guarantees that we change clothes and do not swim. That is the only way to make the implication false. I don't read that in the original statement. If I delete the however and everything else, I think I know as much about the world as if it is there. – Ross Millikan Aug 24 '18 at 00:48
  • It guarantees that there is an instance where we change clothes while not swimming. Without the however statement, we don't know that. – quasi Aug 24 '18 at 00:50
  • 1
    That was my thought behind suggesting the it is possible quantifier. When we are not quantifying, I don't see how to write it. I have explained why I think $\lnot (C \rightarrow S)$ is incorrect. Do you have a better idea? – Ross Millikan Aug 24 '18 at 00:53
  • Ok, now I think I understand your point. There's no way to get the existential quantifier into the truth table. Is that it? – quasi Aug 24 '18 at 00:56
  • That is correct. In propositional logic each basic sentence is either true or false. There is not a universe of things that a predicate can apply to and be true sometimes and false sometimes. – Ross Millikan Aug 24 '18 at 01:53
  • @quasi it actually guarantees that it is impossible to change clothes and go swimming. And I think this is the answer I was looking for. Here's why: $¬(C\rightarrow S) \equiv ¬(¬C \vee S) \equiv C \wedge ¬S$. – giusti Aug 24 '18 at 04:38
  • Yes, I get it now, thanks, and a belated thanks to Ross Millikan who put some effort into resolving my doubts. And (+1) to you both. – quasi Aug 24 '18 at 04:45
  • @giusti: I said in my original answer that it required changing clothes and not swimming, which is what your symbols say – Ross Millikan Aug 24 '18 at 04:46
  • Does "might cloths" mean anything? – William Elliot Aug 24 '18 at 22:25
1

Consider using $\neg(C \leftrightarrow S)$
instead of $\neg(C \rightarrow S).$

William Elliot
  • 17,598
0

Writing a "however" sentence into propositional form

Either going to bed or going swimming is a sufficient condition for changing clothes; however, changing clothes does not mean going swimming.

$B$: I $\color{brown}{\text{am going}}$ to bed
$S$: I $\color{brown}{\text{am going}}$ swimming
$C$: I $\color{brown}{\text{am changing}}$ clothes $$(B \vee S \rightarrow C) \wedge ¬(C \rightarrow S) \tag1$$

The word "however" above is a red herring; the issue seems to be that the boldfaced phrases indicate that analytic implications are being asserted, which suggests perhaps $$\forall x\:(Bx\lor Sx\to Cx)\;\land\;\exists x\:¬(Cx → Sx)\tag2$$ or $$(B \vee S \models C) \quad\text{and}\quad (C \not\models S).\tag3$$ In comparison, the suggested answer above feels lacking in "logical force"; since it is synthetically false whenever I am not changing clothes, it is true only in very restricted contexts, so feels unsatisfyingly unuseful.

Your suggestion to discard the right conjunct of (1) is related to the fact that the right conjunct of (3) (S is not a logical consequence of C) is indeed a redundant tautology.

The left conjunct of (3) is incorrect, so (2) is the only remaining reasonable translation among the three.

ryang
  • 38,879
  • 14
  • 81
  • 179