[FOM] procedures to describe lists

[Sandy Hodges]

In order to compare truth theories in a more formal way, consider
procedures which when given a numbered list of statements that refer to
one another by list number, return another list of the same length, with
each statement on the returned list referring to the corresponding
statement on the input list; for example given input list:

(1.1)  0=0
(1.2)  False(1.2)
(1.3)  ~ True(1.2)

a procedure might return:

(2.1) False(1.1)
(2.2) NeitherTrueNorFalse(1.2)
(2.3) True(1.3)

With such procedures, one can combine an input list, with the list the
procedure returned, and submit the combined list to the procedure.
This allows us to check if the procedure calls those sentences True,
which were its own output.    However this quality, of calling its own
conclusions true (which I call self-validation), is only of interest for
those procedures which output something that is not vacuously true, in
the first place.    A test (which I call the test for weak specificity)
for not being vacuous is as follows:    Take any input list, such that
there are two output sentences that differ by more than just their
references.   Swap the references between these two sentences.  Then
submit both the original input combined with the original output, and
the original input combined with the swapped output.    The test is
passed only if both swapped sentences are described differently than
their unswapped versions.    For example, we can swap the references
between (2.1) and (2.3), producing a list
(3.1) False(1.3)
(3.2) NeitherTrueNorFalse(1.2)
(3.3) True(1.1)

Combining original input and output gives:
(4.1)  0=0
(4.2)  False(4.2)
(4.3)  ~ True(4.2)
(4.4) False(4.1)
(4.5) NeitherTrueNorFalse(4.2)
(4.6) True(4.3)
while combining the original input with the swapped output gives:
(5.1)  0=0
(5.2)  False(5.2)
(5.3)  ~ True(5.2)
(5.4) False(5.3)
(5.5) NeitherTrueNorFalse(5.2)
(5.6) True(5.1)

The output for list 4 might include
(7.4) True(4.4)
(7.6)  True(4.6)
while the output for list 5 might include
(8.4)  False(5.4)
(8.6)  False(5.6)
Since (8.4) and (8.6) are both different from (7.4) and (7.6), the weak
specificity test is passed.

Why might this test for weak specificity be something a reasonable truth
theory should pass?   How a theory classifies sentences is up to the
theory itself.   But if a theory has categories A and B, then it should
be possible, in devising a procedure based on the theory, to have it
output for an input sentence it classes as A, a sentence that would
describe the sentences specifically as A.  For a sentence in category A,
the output sentence should not be so broad, that it is also true of a
sentence in category B.  If two input sentences were described
differently, then the procedure considers them different: say they are
in categories A and B.   By swapping references, what was said about the
sentence in category A, is now said about a sentence in category B.   If
what was said was specific, then it should not be true about the
sentence in category B.
----
Looking at procedures that meet these two tests of self-validation and
weak specificity, I think I can reach some results.   If the procedure
also uses truth tables, and always finds a fixed point, I think I can
prove the only possible tables involve weak Kleene treatment of \/ and
-->.   Such a proposal was made by Gupta and Martin in 1983, but it is
not very attractive.   Thus, if you don't want the Gupta-Martin system,
but do want self-validation and weak specificity, the only alternative
is a system that does not find a fixed point of its truth tables (or
does not use tables at all).

As always, I appreciate comments and especially references.   The papers
I've found so far most useful in this connection have been those by
Sheard.



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