22.1 Names and Objects
We said last chapter that FOL is bivalent: every sentence has exactly one truth value, true or false.
But what if the name “pia” in FOL could refer to two different people?
The Guilty(pia) might be true for one and false for the other.
If we allowed that sort of confusion, then it would be impossible to analyze arguments properly.
So in order to maintain bivalence, names in FOL must obey a key rule:
RULE: Each name picks out exactly one object, no more, no less.
It is essential to understand, though, that the rule does not mean each name has to pick out a different object.
One object can have multiple names. But one name cannot have multiple objects.
In the same vein: there can be objects with no names. That does not cause a problem with the logic. But there cannot be names with no object; that does break the logic.
It might seem weird to have an object with no name. How could we refer to it?, you might be asking.
Quantifiers apply to all the objects we are talking about, not just the ones with names.
For example, let’s say the animal shelter has two cats and a dog. The cats came into the shelter with names, Socks and Mittens, but the dog didn’t.
Even though the dog doesn’t have a name, the quantifiers apply to him too!
Now let’s review.