# Usage of adjective and imperative in statement logic

I think I know how to form sentences in statement logic if it's an "if statement" like (A) and (B) below, but how do I express adjective like "not so easy" or imperative like "Choose X or Y", as shown in (C) and (D) below?

(A) If Lionel Messi makes another goal, then he will make a hat trick and Barcelona will win the match"
“Lionel Messi makes another goal” → “he will make a hat trick” Ʌ “Barcelona will win the match”

(B) If Musikhjälpen 2019 gains over 11 million Euro, then it becomes a record, otherwise Musikhjälpen 2018 holds the record.
“Musikhjälpen 2019 gains over 11 million Euro” → “2019 becomes a record”
“Musikhjälpen 2019 doesn't gain over 11 million Euro” → “2018 holds the record”

(C) It is not so easy to get both a job that is interesting and a job with a high salary even if you search many companies and it is easy to just find a job itself.
"??????"

(D) Choose a ferry trip with a cabin or a flight and hotel, but not both of them
(“a ferry trip with a cabin” V “a flight and hotel”) Ʌ ¬(“ferry with a cabin” Ʌ “flight and hotel”)


It's a little bit unclear what the person who has written these exercises is expecting of you, but here is one way that these might be reasonably interpreted:

It seems for (C) that you need to have a partial ordering over the set of potential jobs that tells you whether one jobs is easier to obtain than another.

If we call $$J$$ the complete set of "jobs", then we can define a partial ordering relation over those jobs that orders them by how difficult they are to obtain by constructing $$R \subseteq J\times J$$ and saying that for $$a,b\in J$$, "$$a$$ is at least as easy to obtain as $$b$$ iff $$(a,b)\in R$$."

You may then construct some predicates on $$J$$, such as $$\mathrm{isInteresting} : J \to \mathrm{Boolean}$$ and $$\mathrm{hasHighSalary} : J \to \mathrm{Boolean}$$. (Predicates are in general the usual way one might handle something like "adjectives" logically.)

Then the statement of (C) is just that $$\forall a,b \in J, \mathrm{isInteresting}(b) \land \mathrm{hasHighSalary}(b) \implies (b,a) \not\in R$$

For question (D), I think the purpose of the imperative is to have you thinking about the solution as involving applying the constraints to a particular element. If you think about a type (or set) of all possible travel options, which we might call $$T$$, then you could perhaps phrase this by asserting the existence of a particular choice of travel option and given certain constraints, perhaps as $$\exists c\in T, c=\textrm{ferry trip with cabin''} \lor c=\textrm{flight and hotel''}\land \neg (c=\textrm{ferry trip with cabin''} \land c=\textrm{flight and hotel''})$$

Although without knowing the context that you're studying this in, I can't predict for certain that this is what is expected of you.

• Note that instead of using the more common set-theoretical definition of a partial ordering as I described above, you could also define it as a predicate on two arguments with something like $\mathrm{isAtLeastAsEasyAs}: J\times J \to \mathrm{Boolean}$ and then replace $(b,a)\not\in R$ with $\neg \mathrm{isAtLeastAsEasyAs}(a,b)$. There are many ways you could do something like this. – Jack Crawford May 19 at 12:50
• Constructing predicates like hasHighSalary, isInteresting in "job statement" and adding one more variable like c in "trip statement" totally make sense. This is exactly what I was looking for. You interpreted really excellent and that was quite helpful. Thank you very much! – Shinichi Takagi May 19 at 17:53