[FOM] Some logics of "True(x)" compared

[Sandy Hodges]

In the previous post, I showed that only the contextualist account, of the three
accounts considered, provided for classical behavior of the observer sentences,
but it did so at the cost of very unclassical behavior of the loop sentences.
While the other two accounts are so structured that there is always a "solution to
the equations," the contextualist account allows that sometimes there is no
solution (or more than one).    When this happens the tokens involved are
considered to have Liar status.    This status assignment over-rides the account's
own table.

---- | ----contextualist---- |
-X- | True(X) | False(X) |
-T- | ----T---- | ----F---- |
-F- | ----F---- | ----T---- |
Liar | ---F---- | ----F---- |

For example, of these three sentence tokens:

A.   Token B is true.
B.   Token A is false or is neither true nor false.
C.   Token A is false or is neither true nor false.

We can try any guess of T and F for tokens A and B, and calculate values using the
table, but no guess is confirmed.
Since no solution exists, we declare A and B to be of Liar status.   Having
assigned them Liar status, we can calculate values using the table.   Tokens A and
B are:

A.   True(B)
B.   False(A) v (~True(A) & ~False(A) )

B is considered Liar, so True(B) should be false, according to the table.
A is also considered Liar, so "False(A) v (~True(A) & ~False(A) )" should be true,
according to the table.

Thus the account calls A and B both Liar, although the consequence of that,
according to the account's own table, should be that they are F and T,
respectively.    Thus the contextualist account does not even have
compositionality (for loop sentences).

The contextualist account attempts to make the logical behavior of loop sentences
more normal, by claiming that the words in loop sentences change in meaning, as a
result of the context (namely being in a loop).   Simmons writes: "The predicates
'true' and 'false' are context-sensitive terms, shifting their extensions
according to context."   Tokens B and C, although the same formula, are called
Liar and true, respectively, by the account.     According to the idea of words
shifting in extension, B in C have different status because words, such as "true",
have a different extension in the loop context of B, than in the observer context
of C.    Thus it is no more surprising that B and C have different status, than it
is surprising that:

John smokes grass.

can have a different truth status according to whether "grass" extends over
marijuana or over plants of the family Gramineae.   According to the
extension-shift account, the "true" in B has a different extension than the "true"
in C, just as "grass" can have different extension in tokens of "John smokes
grass."

One would not however have a formal language in which "Smokes(John,grass)" and
"~Smokes(John,grass)" both hold.   One would use terms whose extension was fixed:
perhaps "marijuana" and "gamineae".   Suppose in a formal language the two (or
three) supposed interpretations of "true" were represented by different symbols:
"true-L", "true-P", "true-R."    "True-R" is supposed to be the meaning that
"true" takes on in observer contexts.    In such a system, we could create this
example of three tokens:

D.   True-R(E)
E.   False-R(D) v (~True-R(D) & ~False-R(D) )
F.   False-R(D) v (~True-R(D) & ~False-R(D) )

That is, since we can make the meaning of "true" explicit, we force the uses of
"true" inside the loop, to have the meaning that "true" is only  supposed to have
in observer contexts.    There are still no value of T and F for D and E that will
be confirmed.    So we still have no choice but to say the loop tokens, D and E,
have Liar status.    The observer token F works out to be true.    So we have not
overcome the problem of having different tokens of the same formula have different
status.

I conclude from this that what happens when a token in a loop has the status of
Liar, is not the same thing as happens when the words in a sentence shift in
extension according to context.   Thus I think this attempt to explain away the
weirdness of some loop sentences, fails.  For my part, I seek only to describe the
properties of loop sentences.   I do not expect loop sentences to behave like
grounded sentences.

------ -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.