Basic, yet confusing translation within Propositional Logic

Question

This is embarrassing to ask, but

If the wff is '¬(P∧¬S)' and 'P' stands for 'I will buy the pants' and 'S' for 'I will buy the shirt', why does my book in the appendix says that the answer is (a) and not (b) (my translation)?

(a) I won't buy the pants without the shirt. (b) I won't buy both the pants and not the shirt.

I'm guessing both might be correct, but merely paraphrased differently; clearly, (a) is intuitively more understandable than than (b)...

Questions

(1) If there is a negation ('¬') before parentheses, shouldn't 'both' be used in the English translation, i.e. as in (b)? I'm asking this because in one other textbook of Logic I remember myself including the 'both' in my translations quite frequently

(2) Why does '¬S' translates into 'without the shirt' and not 'not the shirt'? Why 'without'? Is it because of my initial, possibly correct guess, i.e. that the book's author merely translated a stiff/rigid translation into a simpler and more comprehensible translation?

(3) Are there any tips you can give me when paraphrasing stiff/rigid translations such as (b), into more comprehensible and fluid translations such as (a)?

Thank you!

Answer 1

It might help to first look at it literally. If P = 'I will buy the pants' and S = 'I will buy the shirt', then ¬(P∧¬S) should translate to 'It is false that [I will buy the pants and it is false that (I will buy the shirt)].

(1) Yes, that does tend to be a nice way to clarify negation scoping over conjuncts in plain English e.g. 'I won't eat both ham and rye today'. However, it seems that the author is attempting to phrase the proposition in more natural language. However, this is just a guideline, not a hard and fast rule. The goal from translating from object language to plain English should include render an accurate yet natural reading of the proposition.

It sounds somewhat stiff to say, 'I won't buy both the pants and not the shirt.' Or consider even two negated propositions in the conjunct, 'I won't buy both not the pants and not the shirt.' A more natural rendition would be to 'I won't buy the pants nor the shirt,' or 'I will buy neither the pants or the shirt.' In this case, we are being indicated that it won't be the case that the pants are bought while the shirt is not - so we might translate this in plain English as simply, 'I won't buy pants while buying no shirt,' or, 'I won't buy pants without buying a shirt,' or as they translated, 'I won't buy pants without a shirt.'

(2) '¬S' should translate to 'It is false that I will buy the shirt,' which we might write more naturally as 'I will not buy the shirt.' The author combined both conjuncts in plain English to make the phrase more natural.

(3) I would start with the most literal rendition of the proposition and then try to work from there to phrase in more natural language.

Answer 2

If the wff is '¬(P∧¬S)' and 'P' stands for 'I will buy the pants' and 'S' for 'I will buy the shirt', why does my book in the appendix says that the answer is (a) and not (b) (my translation)?

(a) I won't buy the pants without the shirt. (b) I won't buy both the pants and not the shirt.

I'm guessing both might be correct, but merely paraphrased differently; clearly, (a) is intuitively more understandable than than (b)...

¬(P∧¬S) becomes ¬PvS, by Godwin's law and the law of double negation.

This says "I will not buy the pants or I will buy the shirt". Note that the or operator, if no specific indication is given, is inclusive, meaning it's actually an and/or operator.

(a) I won't buy the pants without the shirt can be translated as "if I buy the pants, then I buy the shirt (P=>S). If you buy the pants (P), then ¬S (I will not buy the shirt) must be the case (¬PvS, P ⊢ S), thus (¬PvS, P ⊢ P=>S). So (a) is correct.

(b) "I won't buy both the pants and not the shirt", while a strange sentence, can be translated as ¬(P∧¬S), which was our first premise. Obviously, ¬(P∧¬S) ⇔ ¬(P∧¬S), so (b) must be correct.

(1) If there is a negation ('¬') before parentheses, shouldn't 'both' be used in the English translation, i.e. as in (b)? I'm asking this because in one other textbook of Logic I remember myself including the 'both' in my translations quite frequently

I would say, get rid of the parentheses, then translate.

(2) Why does '¬S' translates into 'without the shirt' and not 'not the shirt'? Why 'without'? Is it because of my initial, possibly correct guess, i.e. that the book's author merely translated a stiff/rigid translation into a simpler and more comprehensible translation?

I think your guess is correct.

(3) Are there any tips you can give me when paraphrasing stiff/rigid translations such as (b), into more comprehensible and fluid translations such as (a)?

Yes, try to understand the meaning of the statements rather than translate literally without figuring out the meaning of a statement first.

Answer 3

Assuming "I will buy" is distributive and its negation is "I will not buy", then (a).

I will not (buy the pants [without the shirt]) <=> I will not (buy the pants [and not the shirt]).

Note that "without" and "and not" are logically equivalent.

(b) is really awkward, but logically equivalent. Given the above demonstrated equivalence between "and not" and "without", and the superfluousness of "both", we get:

I won't (buy both the pants [and not the shirt]). <=> I won't (buy the pants [and not the shirt]). <=> I won't (buy the pants [without the shirt]).

Answer 4

I won't repeat the previous answers, but I imagine the quip with (b), i.e. "I won't buy both the pants and not the shirt" is that it's very common for beginning students in logic to simply (and thoughtlessly) replace every wedge with "and" and every negation with "not". But this doesn't always yield the best (or even good) translations from formal logic to English, and most authors want to emphasize you can't just do that.