Convert logical forms into English sentences

Question

P = I will buy a pant

S= I will buy a shirt

a. $ \lnot (P \land \lnot S) $

I won't buy a pant or i will buy a shirt

b.$\lnot P \land \lnot S$

I won't buy pant and i won't buy the shirt

c.$\lnot P \lor \lnot S$

i won't buy pant or i won't buy shirt

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S= Steve is happy

G= George is happy

d. $(S \lor G) \land (\lnot S \lor \lnot G )$

Both Steve and George aren't happy but either Steve or George is happy

e.$[S ∨ (G ∧ ¬S)] ∨ ¬G$

Steve is happy or George is happy and Steve is not or George is not happy

f.$S ∨ [G ∧ (¬S ∨ ¬G)]$

Steve and George are not both happy and George is happy or Steve is happy

Answer 1

Although the balance between maintaining fidelity and avoiding awkardness can be a tricky to maintain, a translation should retain the sense of order and structure of the origial statement; avoid the tempation to simplify phrases while expressing them in natural language.

Punctuation can help avert any ambiguity caused by removing parenthesis.

P = I will buy a pant

S= I will buy a shirt

a. $ \lnot (P \land \lnot S) $

I won't buy a pant or i will buy a shirt

You've simplified the expression.   Remember to retain the original structure without being awkard.

Such would be : "I will not buy a pant without a shirt."

b.$\lnot P \land \lnot S$

I won't buy pant and i won't buy the shirt

That is okay.

c.$\lnot P \lor \lnot S$

i won't buy pant or i won't buy shirt

That is okay, too.

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S= Steve is happy

G= George is happy

d. $(S \lor G) \land (\lnot S \lor \lnot G )$

Both Steve and George aren't happy but either Steve or George is happy

Retain the order of the predicates, and avoid trying to simplify the statement.

"Either Steve or George is happy, and either Steve or George is unhappy."

A more natural phrasing might be: "Of Steve and George, one is happy and the other is unhappy," although that is not strictly a dirrect translation.

e.$[S ∨ (G ∧ ¬S)] ∨ ¬G$

Steve is happy or George is happy and Steve is not or George is not happy

Punctuation retains the association and removes ambiguity.

"Either: Steve is happy, George is happy while Steve is not, or George is not happy."

f.$S ∨ [G ∧ (¬S ∨ ¬G)]$

Steve and George are not both happy and George is happy or Steve is happy

Again, keep it a direct translation, retain the order, and use punctuation for clarrity.

"Either Steve is happy, or both George is happy and at least one of Steve or George is unhappy."

Answer 2

a. ¬(P∧¬S)

I won't buy pants or I will buy a shirt

It is not true that I will buy the pants and not the shirt or alternatively:

I will not buy the pants and not buy the shirt

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Also at d. you applied some simplification that may not be correct.

I would say something like

Either Steve is happy or George is happy, and either Steve is not happy or George is not happy.

(Effectively One is happy and the other is not, but that is only a literal translation after applying simplification rules)

Answer 3

Your translation to d) is confusing: in the first half you say that 'both Steve and George aren't happy', which makes it sound like Steve is not happy, and George is not happy (i.e. that they are both not happy). So, instead say that 'Steve and George aren't both happy' for that first part. (In other words, there is an important difference between 'both not' and 'not both'.

Also, you translations to e) and f) are very confusing/ambiguous. For example, when you say for e) 'Steve is happy or George is happy and Steve is not or George is not happy', that could mean the following as well:

$(S \lor G) \land (\neg S \lor G)$

which is not equivalent to the original. (In general, $P \lor (Q \land R)$ is not the same as $(P \lor Q) \land R$, but the directtranslation of both would be 'P or Q and R'. In other words, you need to be very careful when mixing and's and or's.

To avoid these kind of ambiguities you can 'group' statements together using 'both ... and ..' and 'either ...or ...'

Thus, for example, $P \lor (Q \land R)$ can be translated into 'either P or both Q and R', while $(P \lor Q) \land R$ can be translated to 'Both either P or Q and R' ... Though the latter is still rathr confusing, and can be made bit more clear by saying 'Both R and either P and Q'

Applied to e), you would get something like:

'Either Steve is happy, or both George is happy and Steve is not happy, or George is not happy'

... Which is still rather confusing ...

And so ... if you are allowed to... You could try and simplify the statements rather than trying to transla them directly. For example, d) effectively ends up saying that 'Exactly on of Steve and George is happy'

e). Can be simplified to (using the Reduction equivalences $P \lor (\neg P \land Q) \Leftrightarrow P \lor Q $ and $P \land (\neg P \lor Q) \Leftrightarrow P \land Q):

$(S \lor G) \lor \neg G$ ('Steve is happy or George is Happy or George is not happy')

Which actually is equivalent to a tautology ...