The issue is that umbrella(smith)
means "the thing that is an umbrella of Smith".
It has to refer to some specific thing so that a claim can be made about it.
In natural language, "Smith's umbrella" is ambiguous, and that ambiguity can be handy. But the goal of expressing it symbolically is to get rid of that ambiguity.
The second approach is to interpret Found(smith, umbrella(smith))
as saying "Smith has an unique umbrella and Smith found it."
So if Smith has no umbrella or multiple umbrellas, Found(smith, umbrella(smith))
comes out false.
This is intuitively appealing: it's false, after all, that the present king of France is in my living room. So too would it be false that Smith found his nonexistent umbrella (or his nonexistent unique umbrella).
However, one might be tempted to make a more sweeping claim about how this works, like, "Sentences about nonexistent things are false."
The problem is, Found(smith, umbrella(smith))
is false, which makes ¬Found(smith, umbrella(smith))
true.
This is actually a very deep problem, and the major alternative to function symbols has it too. But one part of the appeal of function symbols, in my view, is the illusion that they are simpler than they are.
One can altogether avoid this problem--or, a cynic might say, replace it with other problems. One approach is to use
free logic, which lets atomic sentences whose arguments fail to refer to anything be true. I have not studied free logic. Another approach is to use a logic with more than two truth values. Multivalued logics aren't necessary for the things people often assume they're necessary for, but I do think they have some value.
There are also, um,
wrong "approaches," such as pretending that some of one's sentences do not really have a truth value, but then still relying on parts of standard logic that are incompatible with that, such as the
law of the excluded middle and proof techniques that rely on it (like
proof by contradiction).
I like this much better. It is very powerful. Anything you can do with function symbols, you can do with predicates and definite descriptions. It also does not conceal its existence and uniqueness claims.
So, without function symbols or definite descriptions, we had:
39 mins ago, by
Eliah Kagan ∃x (Umbrella(x, smith) ∧ (∀y (Umbrella(y, smith) → y = x)) ∧ Found(smith, x))
With definite descriptions:
Found(smith, ɿx (Umbrella(x, smith)))
"Smith found the thing that is an umbrella of Smith."
The symbol "ɿ" should be a turned lower-case Greek iota (a turned "ι") in the same way that the symbols "∀" and "∃" are a turned "A" and a turned "E", respectively, but Unicode doesn't quite have anything for that, so I used the closest I could find. It would be excellent if SE chat had MathJax enabled so
LaTeX could be used (though writing the upturned iota is a bit challenging even with LaTeX -- at least it's challenging for me).
With either function symbols or a notation for definite descriptions, the notation for functions in mathematics like sin
and +
can be introduced as alternate forms involving particular function symbols or definite descriptions. I say "alternate forms involving" rather than "alternate forms of" because the ways of doing the translation that avoid diminishing the power such notations are expected to have may be somewhat surprising.
This is what I alluded to when I mentioned that function symbols don't really capture what is going on with mathematical functions. I still hope to get to this.
What either function symbols or definite descriptions buys us, though, is that notation like cos x
or a + b
, can be defined as an alternate form of something, and the thing it is an alternate form of is the same no matter what bigger thing it is part of,
There, by "thing" I mean term or formula/sentence.
In contrast, imagine expressing "addition is associative" with just a AddsTo
predicate (used like: AddsTo(x, y, z)
to express what we'd prefer to say by x + y = z
).
You can do it, of course. It's not even all that bad.
But what you get doesn't really seem like it clearly conveys the meaning of what you are trying to say.