# are these propositions equivalent, case xor and then?

I have the following sentence that should be converted into logical connectors:

Either you eat healthy or you will get sick.

One transformation that came into my mind was:

p ---- you do not eat healthy
q------ you will get sick

so if I convert into this form:

¬p V q

meaning: you eat healthy or you will get sick, I can convert that into:

p-->q ------- If you do not eat healthy then you will get sick  (Form 1)

Also I can do the following:

p---you eat healthy
q---you will get sick

¬p--q ------ If you don´t eat healthy then you will get sick (Form 2)

Or by using the XOR and the same structure than Form 2:

p xor q

you eat healthy or you will get sick, and I know that both situations cannot be true or false at the same time (Form 3).

The question that I have is which of these three forms is the correct one for the sentence given. The three truth tables are different and I have specially doubts with the XOR, because when I use the conditional in Forms 2 and 3 and I have two T or two F, then the truth table returns me a value of true; situation that does not happen with the XOR.

So which one is more accurate and if its correct the xor form?

• The first problem is that "Or you eat healthy or you will get sick" is not a grammatical English sentence. I presume you have translated it from some other language where this construction does mean something (for example "aut" in Latin can be used that way, but "or" in English cannot). However, the mostly English-speaking users of this site will not be able to help you with figuring out what the "correct" understanding of the original non-English sentence is. That is not a question of mathematics, but a question of how the original language works. – hmakholm left over Monica Mar 25 '15 at 15:31
• The "meaningful" sentence must be : "if you do not eat healthy, then you will get sick", whose "logical form" is : $p \to q$. In classical logic this is equivalent to $\lnot p \lor q$ that we have to read as : "either you eat healthy, or you will get sick", with the "big" caveat, by Henning's comment, that in propositional calculus the $\lor$ connective is "inclusive" (i.e. $TRUE \lor TRUE$ is $TRUE$) while many natural languages make a difefrence bewtween "inclusive or" (latin : vel) and "exclusive or" (latin : aut). – Mauro ALLEGRANZA Mar 25 '15 at 16:31
• @thanks I will change the question then, "either" was the work I was looking for – Lila Mar 25 '15 at 18:45