T2.2: 3 of 5

Disjunction

Now we can think about the wedge, 'v' and "or". This one's not quite as easy as it would seem. Here's the truth table with one missing value:

  P Q PvQ
row one: T T ???
row two: T F T
row three: F T T
row four: F F F

Let's start at the bottom, row four, to begin to see what's going on. If P and Q are both false, then their disjunction is false. This makes good sense of the English. For example, if one says "Either Billy Graham or Pat Robertson is the current US president" then one has spoken falsely.

On the other hand if one utters, "Either Bill Clinton or Billy Graham is the US president in 2000", then one speaks truly. Just as row two would have it.

But what about row one? There are examples and there are examples! Most people think that the question marks highlighted above should be filled in with an 'F'. Here's a standard example of this cited by many logic texts:

With dinner you will receive either soup or a salad.

The idea is that you will get one or the other but not both. So maybe the "???" should be 'F'?

But think of another case.

Chris will get promoted or, at least, will receive a raise.

This statement seems clearly to allow for 'T' in case both disjuncts are true. That is to say, it could easily be true when Chris is to receive both a promotion and a raise.

Many logicians and linguists claim that "or" in English can go either way: There is an exclusive sense of "or" meaning "not both" for which the first row of the truth table should get an 'F' as suggested by the soup/salad example. But also, as suggested by the example of Chris, an inclusive sense allowing (or "including") the possibility that both disjuncts are true while the whole disjunction is true.

So, what's the bottom line? In this course, we will pick inclusive "or" as our model of the wedge:

  P Q PvQ
row one: T T T
row two: T F T
row three: F T T
row four: F F F

This is just a definition. So one doesn't have to wonder if it's correct. We just stipulate that we will use the wedge in this way.

Still, we will have to wonder how well our newly defined 'v' of SL represents English "or". For now, let me just try to show that the proponents of exclusive "or" have not proven their case from the soup/salad example. Just think about your response to a waiter who promises that you will receive soup or salad with your dinner. If he then adds, "if you really want both, I'll see what I can do", is he contradicting himself? Would you think that he had not made good on his promise? Wouldn't it be better to describe the situation in which he made his promise come true and more?

These questions are supposed to make you think that the definition given to the wedge is perhaps not so far from English "or". Later we will see more reasons for this. But for now, keep in mind that our truth table is a definition of the wedge. So, as far as SL is concerned, we have no controversy.

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