Rip its 2:37

[criticslovesnatch]

Notes on Compounds, Part Two

1. In our last set of notes, we ended with the question of whether "" has a truth table. In fact, it isn't obvious what the right answer really is. Some things are clear, nonetheless. Think about each of the following sentences:

If Boston is in Massachusetts then Seattle is in Costa Rica.
If George Washington was an American president then Ronald Reagan was once King of England.
If horses are mammals then elephants have wings.
These are all false. The sure clue is that in each case, the antecedent is true and the consequent is false. So whether or not there is a complete truth table for "if... then," we can at least fill in one row:
P Q P Q
1 1
1 0 0
0 1
0 0

Can we get any farther than this?
Perhaps we can. It will take a bit of reasoning, but here's how we might proceed. Consider this statement:

Either Paula or Quincy is in New York.
Suppose you know that this is true. I would suggest that you know something else: if Paula isn't in New York, then Quincy is. If this isn't obvious, here's a way to think about it.
We are given that either Paula or Quincy is in New York. We're trying to draw a hypothetical conclusion: if Paula is not in New York, then Quincy is. So suppose -- make the hypothetical assumption -- that Paula is not in New York. It follows by v- that Quincy is. That seems pretty clearly to establish the hypothetical relationship: if Paula is not in New York, Quincy is in New York.

There seems to be a moral here: when we have a disjunction that we believe to be true, there is a related hypothetical or conditional statement that we should also accept. Now as a special case, consider a disjunction like this one:

Either Paula is not in New York or Quincy is.
Suppose we know this. By exactly the sort of reasoning we just considered, it follows that if Paula is in New York, Quincy is in New York.
Why?

Symbols will help us see. Our original statement is

~P v Q
Suppose, hypothetically, that P. We have a case of v-:
~P v Q
P
|= Q
Thus, when we know that "~P v Q" is true, we also know that if P is true, so is Q. That is, we know that "if P then Q" -- "P Q" -- is true.
For the algebraically inclined, here's another way to think about it. From "P or Q" to "If not-P then Q" seems pretty much obvious:

P v Q
|= ~P Q

Now think about

~P v Q

and use "X" to stand for "~P". We get

X v P

and so it should follow that

~X P

But since "X" is "~P", "~X" is "~~P". And that's just another way of saying "P." So no matter which way you look at it, if we agree that "P v Q" implies
"~P Q", we have to agree that "~P v Q" implies "P Q."

This gives us everything we need. Let's see why.

First, we've just seen that if "~P v Q" is true, so is "P Q." So "~P v Q" implies "P Q". But what if "~P v Q" is false? What would that amount to? An "or" statement is false if both of its parts are false. In this case, that would mean "~P" is false and "Q" is false. But for "~P" to be false is for "P" to be true. So for "~P v Q" to be false is for "P" to be true and "Q" to be false. But we saw above: when that happens, "P Q" is false:

P Q P Q
1 0
1 0 0
0 1
0 0

Put this together: when "~P v Q" is true, so is "P Q". And when "~P v Q" is false, so is "P Q." "~P v Q" and "P Q" are true and false in exaclty the same cases.
What we seem to have discovered is that

~P v Q
and
P Q
mean the same thing. But "~P v Q" has a truth table. It's true if it has at least one true part. That means it's true if "P" is false or if "Q" is true:
P Q ~P v Q
1 1 1
1 0 0
0 1 1
0 0 1

But if "~P v Q" means the same thing as "P Q," then we have a truth-table for "P Q":
P Q P Q
1 1 1
1 0 0
0 1 1
0 0 1

This is the standard table for what is called the material conditional. In spite of the argument we've given, whether English "if... then" really works this way is a matter of some dispute. The results of assuming that it does are odd, to say the least. We would have to agree that the likes of
If Denmark is not a country then there are goblins in the White House
or
If there is life on other planets, then Arizona is an American State
are true. (Exercise: explain why.) But we don't normally admit any such thing.
In fact there's no harm in admitting that these peculiar statements are true. They sound odd because they would typically be pointless or even misleading. Suppose I say that if there is life on other planets, then Arizona is an American State. This is twice bizarre. First, the two clauses seem to have no interesting relation to one another. But we normally only use "if.. then" when there is some reasonably transparent relationship between the two clauses. It's also bizarre because it suggests that I'm not sure whether Arizona is a state; very misleading.

The first worry is smaller than it seems. Just what is related to or relevant to what is often dependent on context. With enough imagination, we could dream up a context where the question of life on other planets was somehow relevant to what I should believe about the political status of Arizona. The second worry has to do with a perfectly sensible convention in conversation: we usually don't make a less informative statement when we could make a relevant more informative one. The only reason I believe

If there is life on other planets, then Arizona is an American State
is that I believe its consequent. If I'm being a good conversational partner, I won't disguise my knowledge in this way. But while that may be so, it doesn't mean that "If there is life on other planets, then Arizona is an American State" isn't true. "Misleading" is one thing; "false" is another.
In any case, we will operate on the assumption that "if.. then," represented by "", has a truth table:

P Q P Q
1 1 1
1 0 0
0 1 1
0 0 1

There. thats logic. thats part of my notes.