Argument with a friend over correct prepositional logic for a sentence

Question

It is possible that Clinton or Sanders and possibly Bush may run for President.

we both agreed on the assignment of prepisitional variables:

P = It is possible that Clinton may run for President

q = It is possible that Sanders may run for President

r = It is possible Bush may run for President.

I argue that the prepositional logic statement would be :

p V q V r

My friend argues that the logic statement is in fact:

(p V q) ^ r

my argument was that "and possibly" is just another term for meaning or.

He argues that you should just ignore the "possibly" and just interpret it as meaning "and".

I argued that the whole point is to capture the original meaning of the sentence.

What do you think is the correct prepositinal logic statement for the sentence?

Answer 1

Your disagreement stems from the ambiguity inherent in an inclusive disjunction.

Here's an easy way to note the semantic difference between an exclusive and inclusive disjunction:

An exclusive disjunction will allow for one or the other, but not both. Imagine you are at a catered event and a waiter asks you, "Would you like chicken or steak?" Since he is using the exclusive disjunction, you are free to choose one or the other. However, he'll likely spit in your food if you insist on both, as it isn't compatible with the semantics of the disjunction.

On the other hand, an inclusive disjunction will allow for one or the other, or both. Imagine after you eat your entree the waiter returns to ask, "Would you like cream or sugar in your coffee?" Since he is now using the inclusive or, you won't run the risk of Tyler Durden having his way with your beverage. You're free to choose one or the other, or both!

Now let's formalize the inclusive disjunction with our cup of coffee:

Let

• p: it is possible to have cream
• q: it is possible to have sugar

Since we're using the inclusive disjunction:

$p \wedge q$ accounts for the semantic meaning of the sentence,

and $p \vee q$ accounts for the semantic meaning of the sentence

Returning to your example, since no candidate is precluded from running in the election should another candidate choose to run we have another use of the inclusive disjunction. In turn, the sentence has been correctly symbolized by both you and your friend.

Answer 2

What you think is correct in my opinion. If you analyze the sentence, you will see that it is logically equivalent. You could try it out with some boolean values and you'll see it's the same. However, I could see why his argument could be valid as well. Your sentence's meaning is a bit ambiguous, so I don't see it having just one answer.

Answer 3

As the sentence stands, the "and" is meant to separate two groups (i.e. political parties). It could be interpreted as, "(I expect that) either Clinton or Sanders AND Bush will be in the general election" - which is akin to what your friend wrote - and it is not unreasonable to assume this. More generally writing

(democrat1 will run or democrat2 will run or ... ) AND (republican1 will run or republican2 will run or ...)

would be entirely reasonable - someone from each party will run.

You interpret it, probably more correctly, as talking merely about the possibility that they will run, and are in that case correct. Your interpretation is more akin to "one of the potential candidates will run" and theirs is "one from each party will run". The fact is that this sentence communicates more than any logical expression would, so there's some ambiguity about reducing it.