[FOM] Another easy solution does not work

[Andrew Boucher]

Dean Buckner wrote:

>
>6.  Now consider
>
>What this sentence says is true.
>
>But what does the sentence say?  It's referring to the content of some
>sentence, which happens to be the sentence itself.  Thus, if I'm right, and
>every sentence consists of a content-part, analogous to a noun clause, and
>an assertion part, analogous to a verb, then the content-part is what is
>signified by "What this sentence says".  But we can't tell what it
>signifies.  It's exactly the same as the expression "what this expression
>signifies" - which signifies nothing.
>
>If we suppose that there is an implicit assertion, in sentences like "what
>Richard said is / is not true", that Richard said anything in the first
>place, then the Liar sentence comes out false.  Thus there is no paradox.
>
I think you are right to consider "What this sentences says" and relate 
it to failed reference.  But I think you were a bit brief and you need 
to flesh out your discussion to dissipate paradox more completely.  

First, in questions of failed reference, you have to be careful about 
the scope of negation.  Is "The King of France is not bald" true or not 
true?   To clarify, say it is *not* true, i.e. stipulate it is making an 
assertion of the King of France, which cannot be true because he does 
exist.  And use "Not the King of France is bald" for the full negation 
(which is true, again because there is no King of France).  So 
essentially (by stipulation)

    (*) "Not the King of France is bald"
    = "The King of France is not bald or there is no King of France."

Secondly, instead of overloading "true" to attribute both of statements 
and content, use "hold" to make assertions about content (and keep 
"true" for assertions about statements).  Assume "Statement x is true" = 
"The content of statement x holds."  (I prefer to talk of "facts" rather 
than "content," but let's split the difference and keep your term on 
that one.)

OK, now consider:

(A) "Not the content of statement A holds."

The full negation splits, by (*), into

(B) "The content of statement B does not hold or there is no content to B."

Now "the content of statement B" is vicious, in the way that "the 
referent of this referring term" is vicious.  "The referent of this 
referring term" would refer to whatever it refers to:  but that doesn't 
tell you what it refers to, so it doesn't refer at all.  Its reference 
fails.  Similarly, whatever content the statement B has, depends on the 
reference of "the content of statement B", which depends on the content 
of statement B.  Because it is vicious, it does not refer, and so there 
is no content to B.  

Remark that (B) is a disjunction.  The first disjunct has no content, 
because "the content of the first disjunct" would depend on the 
reference of "the content of statement B," which in turn depends on the 
content of the first disjunct.  So "the content of the first disjunct" 
is vicious, and so the first disjunct has no content. The second 
disjunct does *not* have any vicious reference, and so it has content 
and indeed content which holds.  Thus, we can really assert that

(C) There is no content to B.

Thus, logically, we can infer that

(D) The content of statement B holds or there is no content to B.

That is, we can assert that (D) even though (D) has no content.  We can 
do this because it is a disjunction, with one disjunct (the second one) 
full of content and indeed content which holds.  

Return now to talking about statements, and call a statement "false" if 
it has content which does not hold, and "empty" if it does not have 
content.  Consider (analogous to (B)):

(E) "Statement E is false or statement E is empty."

"E is false" is empty (it is contentless).  "E is empty" has content and 
is true.  But because E is empty, it is contentless, so cannot be true. 
  That is, E is not true, but

(F) Statement E is empty,

and indeed

(G) Statement E is false or statement E is empty.  

So we can assert that (G) even though E is not true.   Evidently, the 
T-Schema falls down - so those who want to "save" the T-schema, even at 
the expense of altering logic, will not be happy.  But doesn't logic 
fall down as well for us? Well, yes, as far as the truth-table rules go, 
but not "real" logic, so to speak. "Statement E is empty" is True, 
"Statement E is false" is Not-true, and their disjunct is Not-True, 
contra the truth-table rule for disjunction.  But logic *should* only 
say:  from "p" or "q", infer "p or q", and this still is so, since we 
*are* asserting that G.  That is, the truth-table rules of logic have 
implicitly assumed the T-schema; when the T-schema goes, so will the 
truth-table rules.  *But the rules of logical inference all stay.*  

Remark that E is not true, and "E is not true" is not true.  So ""E is 
not true" is not true" is not true, and so on.

Although our examples so far employ vicious reference (reference that 
loops back), the real culprit is failed reference.  Endless reference 
also fails.  That is,

"the referent of the next referring term"
"the referent of the next referring term"
etc.

all fail to refer.  And analogously

"The next statement is not true"
"The next statement is not true"
etc.

are all not true, because they are all empty.  Similarly, for every 
statement in Yablo's Paradox:  they are *all* empty, so not true.

Anyway, my apologies for the long and non-standard post. For those who 
have made it this far and have the stamina and will to continue (an 
empty set?), this is written up in somewhat more detail on my web site, see:

www.andrewboucher.com/papers/paradoxes.htm

Andrew Boucher
Unaffiliated