Here's a condensed version of a solution to the Liar that I posted on
this board last century... =A0The full version can be found at:

http://www.andrewboucher.com/papers/paradoxes.htm

First, what is a solution to the Liar? =A0First, it says exactly *which*
step in the Liar reasoning, a reasoning which leads to a
contradiction, is invalid. =A0Secondly, it explains *why* it is invalid.

So, what is this normal reasoning? =A0It goes like this:

Consider:
[L] "Not L is true."

Then:
Step 1. =A0Suppose L is true.
Step 2. =A0"Not L is true" and L are the same thing, so by substitution
(of equals for equals), "Not L is true" is true.
Step 3. =A0Using the rule "If "S" is true, then S", one concludes that
not L is true.
Step 4. =A0Statements 1 and 3 are contradictory, so by reductio ad
absurdam, the supposition is not so, i.e. not L is true.
Step 5. =A0From "If S, then "S" is true", it can be inferred that "Not L
is true" is true.
Step 6. =A0"Not L is true" and L are the same thing, so by substitution
(of equals for equals), L is true.
Step 7. But this is a contradiction.

I assert that the flaw in the reasoning is Step 5, because the rule
"Is S, then "S" is true" is invalid.

Now I have to explain *why* it is flawed. =A0There may be other meanings
of "truth", but I will consider the one where a proposition is true if
the fact which it asserts corresponds with reality. =A0E.g. "George
Washington was the first American President" is true because it
asserts the fact that George Washington was the first American
President, and this corresponds with reality. =A0On the other hand,
"(Classic) Coca-cola is white" is not true, because it asserts a fact
which does not correspond with reality.

First, let's note that certain referring expressions depend on the
reference of other expressions, e.g.

[A] "the referent of [B]"
[B] "George Washington"

So [A] refers to George Washington.

This allows for the possibility of viciously referring expressions:
expressions which depend on what their own referents would be in order
to refer, e.g.

[C] "the referent of [C]"

and

[D] "the referent of [E]"
[E] "the referent of [D]"

None of [C], [D], and [E] refer successfully.

As an aside, note that the problem is not self-reference, since

[F] "the referring expression [F]"

refers successfully, to [F]. =A0Viciousness comes when a supposition is
made about one's own referent, in order for a referring expression to
refer, not from self-reference.

Okay, so now let's return to propositions and truths. =A0A proposition
is true if and only if the fact which the proposition asserts
corresponds with the world (i.e. the fact "holds").

So a proposition is not true in two cases: =A0(a) the fact which it
asserts does not correspond with the world; or (b) "the fact which the
proposition asserts" does not refer successfully. =A0In case (a) we say
that the proposition is false; in (b) that it is empty. =A0Note that one
could use "meaningless" or "paradoxical" or "X" for case (b); the name
isn't at all important. =A0The important point is coming.

Consider:
[G] "The fact which [H] asserts holds" i.e. "[H] is true"
[H] "Snow is white"

The fact which [H] asserts is that snow is white. =A0Since this holds,
[G] is true.

But now we can create vicious cases of reference with "the fact", e.g.

[I] "The fact which [I] asserts holds" i.e. "[I] is true".

The referring expression in this proposition, "the fact which [I]
asserts," is vicious, because its reference depends on what it itself
would refer to. =A0So [I] does not assert any fact; it is empty.

Similarly,

[J] "The fact which [J] asserts does not hold," i.e. "[J] is false,"

is empty.

Now "not true" is "false or empty," so the Liar is equivalent to:

[K] "The fact which [K] asserts does not hold, or [K] does not assert
a fact."

Now "the fact which [K] asserts" is vicious, because its reference
depends on what fact it itself would assert. =A0So, [K] does not assert
a fact. =A0To repeat:

[K] does not assert a fact. =A0(Call this [M]).

Thus:

The fact which [K] asserts does not hold, or [K] does not assert a
fact. (Call this [N]).

But [N] is just the assertion [K]. =A0Now normally this would mean that
[K] would be true (using the rule "If S, then "S is true""), but
hopefully it is clear why this rule does not apply. =A0It applies in the
case when S asserts a fact, but here there is none.

Indeed, the reason why the Liar is special is that it is a
disjunction, where one disjunct is empty, and the other disjunct is
true. =A0The over-all disjunction is still empty, because emptiness is
contagious. =A0And this means that the over-all disjunction is not true,
even if one disjunct is true. =A0(Emptiness takes precedence over truth
in a disjunction.)

Finally, I'd like to emphasize that there is no change of logic here.
Logic stays at it is. =A0If A, then A or B. =A0What you *don't" have is
the truth table rule, "If A is true, then (A or B) is true." =A0This
holds when statements are normal, where the rule "If S, then "S is
true"", but not necessarily in the special cases where it does not.