Chapter Four, Tutorial Three
Finally, we need to think a bit more about how well SL can be used to symbolize English statements.
In some cases, it is uncontroversial that SL can be used to closely translate the meaning of English. (For example, many uses of "and" are nicely handled by the ampersand.) For other types of expressions, it is uncontroversial that SL cannot well symbolize the English. (Any non-truth functional context provides an example. The counterfactual conditionals, often of the form "if P were true, then Q would be true" provide a case in point.)
However, in between there are a number of unclear, even controversial cases. Here we look to the cases of English "or", "unless", and "if...then...".
In chapter two, we distinguished between an exclusive sense of "or" and an inclusive sense. Briefly, the difference concerns the "first row" of a table:
|P||Q||P or Q|
Is row one, the possible case in which P and Q are both true, to be included amongst the rows making "P or Q" true? If a disjunction is taken to be true in this case, then it is inclusive. Otherwise, it is exclusive.
Back in chapter two, we saw examples of English "or" which seemed best interpreted as inclusive and others which seemed to be exclusive. Our example of the latter was:
With dinner you will receive either soup or a salad.
At least superficially, this example suggests an exclusive reading: one or the other but not both. From such examples, some would argue that English "or", unlike SL's wedge, is (usually) exclusive.
But chapter two provides some reason to believe that this conclusion is too hasty. (The idea there is that a waiter who first promises you one or the other and then adds "if you really want both, I'll see what I can do" is not contradicting himself.)
The following thinking provides another reason to think that English "or" is inclusive.
Let us suppose for a moment that English "or" is exclusive and consider the results of this assumption. We will find that these consequences are not good and will then blame the assumption. (This kind of reasoning is sometimes called indirect or reductio ad absurdum thinking: one makes an assumption, notices that the assumption has silly or absurd consequences, and concludes that the assumption is faulty. Remember this from 1.4?)
So, assume for a moment that "or" in English is typically exclusive: i.e., suppose that an English disjunction is usually false when both disjuncts are true. Then, consider (a typical instance) "Either Bill or Hillary is human" and use 'B' and 'H' to symbolize "Bill is human" and "Hillary is human" respectively.
If English "or" were exclusive then (in the real world) we would be in the first row of a truth table as such:
B H B or H T T T F T
"B or H" is false, given the assumption that "or" is exclusive. But, and now we note the simple consequence, if "B or H" is false, then "neither B nor H" is true:
B H ~ (B or H) T T T T F T
But this means that someone committed to saying that "or" is exclusive for the Bill and Hillary example, is also committed to the conclusion that neither of the two former denizens of the US White House are human! This is an absurd result showing that the original assumption is faulty. That is, "or" is not exclusive, at least not for this case.
Now, notice that you can run this same example for the soup or salad case. Just imagine a time when the waiter (feeling generous?) happens to bring both. The proponent of exclusive "or" is committed to saying that he brings neither! (This for the same reasoning as above.) Obviously the exclusive "or" assumption has led us astray.
All this is meant to show that English "or" cannot be thought of as exclusive. But if you feel queasy about translating it with the wedge, you are not alone. We need to investigate a bit further...