Exclusive Or vs Inclusive Or

Question

I have a doubt on constructing English statements from propositions. I went through three exercises and every time I faced the same situation I can not solve the proposition properly

1. Let p and q be the propositions

p : I bought a lottery ticket this week.

q : I won the million dollar jackpot.

I have to find the English translation ¬p ∨ (p ∧ q)

My answer is: I did not buy a lottery ticket this week or I bought the lottery ticket this week and I won the million dollar jackpot.

I found on the web that they used either, "Either I didn't buy a lottery ticket this week, or I did and I won the million dollar jackpot."

I did not understand what is the need for using exclusive or instead of inclusive or?

Then I tried another exercise

2. Let p and q be the propositions “Swimming at the New Jersey shore is allowed” and “Sharks have been spotted near the shore,” respectively.

compound proposition is ¬p ∧ (p∨ ¬q)

My answer is : Swimming at the New Jersey shore is not allowed and swimming at the New Jersey shore is allowed or sharks have not been spotted near the shore.

I found on the web: Swimming at the New Jersey shore is not allowed and either swimming at the New Jersey shore is allowed or sharks have not been spotted near the shore.

Here again, people used exclusive or instead of inclusive or.

Then I went though the third example,

3. Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively.

The compound proposition is ¬q ∨ (¬p ∧ q)

Now, I used exclusive or after seeing so many examples.

My answer is: Either the votes have not been counted or the election is not decided and the votes have been counted.

This time I found on the Web two variations

1. The votes have not been counted, or they have been counted but the election is not(yet) decided.

2. The votes have not been counted, or the votes have been counted but the election is not decided.

My confusion is when to use exclusive or and when to use inclusive or?

Thank you.

Answer 1

I have to find the English translation ¬p ∨ (p ∧ q)

My answer is: I did not buy a lottery ticket this week or I bought the lottery ticket this week and I won the million dollar jackpot.

The problem with this phrasing is that it is unclear how you are supposed to associate it.

If I say "P or Q and R", it is not clear if I mean "P, or else Q and R", or if I mean "P or Q, and in addition to that also R". That is, whether I mean to say $(P\vee Q)\wedge R$ or $P\vee(Q\wedge R)$.

The answer you found on the web eliminates the ambiguity with the use of "either" and the comma. By writing

"Either I didn't buy a lottery ticket this week , or I did and I won the million dollar jackpot." the "Either" and the comma signal that the stiff in italics is the first clause of the "or" statement, and the stuff after the comma is the second clause (which happens to be a conjunction).

This is not a matter of exclusive vs. inclusive or, but rather of signaling which ones are the clauses of the "or" and which ones are the clauses of the "are". In this particular case, because the two clauses are mutually incompatible, you can express it in natural language with an exclusive or, making it easier to signal the start and end of the clauses in natural language. If instead we had $P\vee (Q\wedge R)$, we would want to say something like "P, or we have both Q and R" with the comma and "both" signaling the clauses.

The examples you have are using exclusive or in natural language because the exclusivity is implied by the specifics (one clause has $\neg p$, the other one has $p$). They take advantage of that in order to clarify how to signal the clauses. But you don't have to use an "exclusive or" phrasing to distinguish them. You could have said

I did not buy a lottery ticket this week, or I bought a ticket and won a million dollars.

Here the comma does the work of letting you know what are the clauses.

Again, compare the first two below, which are clear and mean different things, and the third, which is ambiguous and unclear:

$P$, or $Q$ and $R$.

$P$ or $Q$, and $R$.

$P$ or $Q$ and $R$.

Answer 2

Contrary to popular belief, the use of 'either' does not necessarily mean that you are dealing with an exclusive or ... and the lack of 'either' does not mean you are dealing with an inclusive or.

Consider the following two perfectly fine uses of English:

A mathematician says: "Every whole number is even or odd"

We all understand that the mathematician means the or to be exclusive here, despite not using the word 'either': the mathematician would be upset if we ever found a number that is even and odd!

I say: "When I grow old, I want to be either rich or happy".

Here, I do use the word 'either', but do you think I will be upset if I turn out to be happy and rich? Of course not: I mean this or to be inclusive.

The moral is: the English language is slippery and there are typically no hard and set rules for how something should be interpreted by the expressions alone: often background knowledge and common sense will have to be brought into play to understand what is really meant.