Propositional formula

Question

how can I transform this statement into a propositional formula?

Theseus must die on Skyros, or else Skyros will be devastated

What symbol is "or else", I can't find anything on the internet.

My suggestion is:

p = theseus must die on Skyros

q = Skyros will be devastated

formula: (¬p->q)

And you can read it as: If theseus doesn't die on Skyros, Skyros will be devastated. But I don't know if my tought is correct.

Thank you

Answer 1

English is often ambiguous, so often you’ll have to use background knowledge and common sense to try and figure out what most likely is being meant ( of course, ideally you can just ask the person making the claim to resolve any ambiguity … in this case maybe your professor?).

Anyway, the ambiguity here is whether the ‘or else’ is used as an inclusive or exclusive or. Now, the expression ‘or else’ is often meant to express an exclusive disjunction … but you can’t use that as a hard rule … just like ‘either … or’ is often used to ecpress an dxvlusive or … but not always.

Now, if we use an inclusive or, we’d get $p \lor q$ … which is equivalent to your $\neg p \to q$. But, in this case I would think a biconditional would probably be more sensible… given what we know about Theseus and Skyros, presumably if Theseus does die on Skyros, then Skyros will not be devastated (at least not in the immediate term, which is probably what they mean), meaning that you also have $p \to \neg q$, which is equivalent to $q \to \neg p$, and together with $\neg p \to q$ that gives you the biconditional $\neg p \leftrightarrow q$ … which is one way to express an exclusive or between $p$ and $q$.

Answer 2

Over-View :

It is "OR" , where "ELSE" is unnecessary & can be eliminated.

Details :

Let $ T=\text{Theseus must die on Skyros} $
Let $ S=\text{Skyros will be devastated} $

Proposition is $ T \text{ OR ELSE } S $
Naturally , "ELSE" means "NEGATION"

Proposition is then $ T \text{ OR } [ \lnot T \text{ AND } S ] $

We can then make it like this :
$ T \lor [ \lnot T \land S ] $

When we "expand" it , we get :
$ (T \lor \lnot T) \land (T \lor S) $
$ (\text{true}) \land (T \lor S) $
$ (T \lor S) $

We have eliminated "ELSE" ...

ADDENDUM 1 :

I have given the Mathematical logic way to think about it.
In the human way to think about it , it means the $T$ is the "Positive/Wanted/favourable" outcome , while $S$ is the "negative/unwanted/unfavourable" outcome.

ADDENDUM 2 :

Whether it is Exclusive OR or regular OR , it will matter on the context , which user "Bram28" alluded to.

It might mean that when $\lnot T$ , we will also have $\lnot S$ :

King : Theseus must die on Skyros, or else Skyros will be devastated.
War Minister : I agree ! We will have to sacrifice Theseus to save Skyros !

Alternately , the Story Dialogue can continue like this :

King : Theseus must die on Skyros, or else Skyros will be devastated.
War Minister : Skyros is DOOMED ! Skyros will be devastated either way ! Let us save the Patriot Theseus !

ADDENDUM 3 :

Simpler Examples to highlight ...

You must hand over your assignment by Monday Morning or else by Monday Evening.
It is Exclusive OR , because you can not hand over the assignment twice.

You must buy the costly textbook or else my cheaper handwritten copy.
It is regular OR here , because it is okay when you buy ( & refer to ) both material.

Summary :

It is "OR" ,
We can eliminate "else" ,
Context will decide whether it is "Exclusive or" or "regular or" . . .