Re: All panduks are green

Newberry says...

1. We need a sensible definition of truth. "A sentence is true iff it
corresponds to an existing state of affairs" is a sensible definition.

Well, whether a sentence corresponds to an existing state of affairs
depends on the *interpretation* of the sentence. We can't avoid that.

Similarly "a sentence is meaningful iff it is a picture of a possible
state of affairs" is a sensible definition. Accordingly we have "all
round squares are green" is meaningless because it is not a picture of
a possible state of affairs.

I don't see any problem in coming up with a picture for that
sentence. You have a set R of all round objects. You have a set S
of all square objects. You have a set G of all green objects.

The claim "all round squares are green" means that the intersection
of R and S is a subset of the set G. That's not hard to picture.
Use Venn diagrams.

2. I could turn the question around and ask why we need to assign a
truth value if we do not have to? What is the tradeoff?

It enormously complicates reasoning to say that "All Xs are Ys"
is meaningless when there are no Xs. As I pointed out, a typical
way that someone *proves* that there are no Xs, is to prove that

All Xs are Ys.
All Xs are Zs.
No Y is a Z.

These three statements allow us to conclude that there are no
Xs. If you want to say that the first sentence is meaningless if
there are no Xs, that means that it is possible to prove meaningless
statements, and that meaningless statements are sometimes necessary
in order to prove meaningful statements. So what is the point of
calling them meaningless?

There is one advantage in not assigning a truth value to the
so called "vacuously true" sentences - most logic puzzles will
disappear.

If we decide to treat the "vacuously true" propositions as
neither true nor false then

(x)(x # x --> x > x)

does not have a truth value because ~(Ex)(x # x). Analogically, if

~(Ex)Pxm

then

~(Ex)(Ey)(Pxy & Qy)

does not have a truth value, if Q is satisfied only by y = m. Do I
need to go any further in describing what P and m might be?

I can't imagine why you consider this an advantage. Why do you
want such statements not to have truth values?

--
Daryl McCullough
Ithaca, NY

.

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