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

[Sandy Hodges]

Professor Davis:  Sorry about the previous version of the tables, I should have
seen the problem.    I have reformed the data into three very simple and narrow
tables, and I'm pretty sure that these will be readable, if not elegant, in any
computer display.
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Although my main pursuit is "Denotes", I give here an example involving "True",
for the sake of comparing three accounts: dialetheism, weak truth, and
contextualism.   Here's the example:

A.   2+2=4
B.   Sentence A is true
C.   Every sentence in this FOM posting with acronym "SCehtssatLpoif" is true.
D.   Sentence C has the same status as the Liar paradox.
E.   Sentence C is false.
F.   Sentence C is either false or has the same status as the Liar paradox.
G.   Sentence C either has the same status as the Liar paradox or is false.

Of these sentences, A, which refers to no sentence, and B which refers only to
another grounded sentence, are called "grounded."

C and G, which refer to one another, can be called a self-reference loop.

Sentences D, E, and F, each refer to a sentence in a loop, but are not themselves
in a loop.   I will call these "observer" sentences.

Note that while G and  F are logically equivalent, sentence C refers to G but not
to F, so that G and C form a loop, while F is an observer of that loop.

A comparison of the three accounts shows that each departs from intuitions based
on classical logic (logic without a truth predicate).   In contextualism, the
self-referential or "loop" sentences behave in a way very different from classical
logic, but the observer sentences behave in a purely classical way.   The other
accounts are less radical concerning the loop sentences, but the observer
sentences are not entirely classical.    I claim (although I will not try to show
it here) that classical behavior of observer sentences is a requirement of any
formal language, if that language is to be used to make actual assertions, convey
information, and conduct business.   I think highly non-classical behavior of
self-referential sentences can be tolerated, as long as that non-classical
behavior is confined to the self-referential sentences, and does not infect the
observer sentences.
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The dialetheic theory, proposed by Graham Priest, considers that a Liar paradox
sentence is both true and false.    So I will copy the example, replacing "the
status of the Liar" with "both true and false", and adjusting references.

H.   2+2=4
I.   Sentence H is true
J.   Every sentence in this FOM posting with acronym "SJeitafoif" is true.
K.   Sentence J is true and false.
L.   Sentence J is false.
M.   Sentence J either is false or is true and false.
N.   Sentence J either is true and false or is false.

According to dialetheism, the loop sentences (which are J and N) are true and
false.     The observer sentences K, L, and M are also both true and false.
Indeed all observer sentences (if they comment on the truth of loop sentences) are
true and false.    One might think that since, according to dialetheism, the loop
sentences are true and false, then an observer sentence ought to say that the loop
sentences are true and false.   An observer that says this ought to be "right" (in
some sense) and a observer that says otherwise ought to be "wrong."    But in
dialetheism:

   An observer sentence saying J is true and false, is true and false.
   An observer sentence saying J is true, is true and false.
   An observer sentence saying J is false, is true and false.
   An observer sentence saying J is true and is not false, is true and false.

This is a way in which dialetheism does not conform to intuitions molded by
classical ("True"-less) logic.   Every account will violate these intuitions in
some way.
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The weak truth account (due to Martin and Gupta) says that truth is "weak", and it
also uses weak Kleene truth tables.  If X is a statement with the status of the
Liar paradox, then a "strong" account of truth considers "X is true" to be
false.    But a "weak" account of truth considers that "X is true" has the same
status as X does.   If A is true, and B has the status of the Liar, then strong
Kleene tables consider "A v B" to be true (since when A is true it doesn't matter
what B is).    But weak Kleene tables consider "A v B" to have the status of the
Liar.  If any part of a compound has the Liar status, that poisons the whole.

The account using weak truth and weak Kleene considers the Liar to be neither true
nor false.   There is a special predicate, "Neither" which must be used.    If L
is a Liar sentence, then "Neither(L)" is true, but "~True(L) & ~False(L)" is not
(it has the status of the Liar).

I copy the example again, putting in "Neither" for "status of the Liar":

O.   2+2=4
P.   Sentence O is true
Q.   Every sentence in this FOM posting with acronym "SQeiNoif" is true.
R.   Sentence Q is Neither.
S.   Sentence Q is false.
T.   Sentence Q is either is false, or is Neither.
U.   Sentence Q either is Neither, or is false.

By the weak truth account, the loop sentences Q and U are Neither.  Observer
sentence R, which says this, is true.  Observer sentence S, which calls a Neither
sentence false, is therefore itself Neither.  Observer sentence T, which is the
disjunct of a true with an Neither sentence, is therefore Neither.

This is the point where this account departs from standard logic intuitions.    It
seems that if one knows that one of two conditions obtain, then one ought to be
able to say "Either this condition obtains, or that condition does" and have one's
statement be true.   But this does not hold with weak logic.   One might know, of
a certain sentence, X, that in one case it is a false sentence, but in the other
case it is a Neither sentence.    But if one then says "X is either false or
Neither," then this may not be true, because if X is indeed Neither, then the "X
is false" disjunct is Neither, and therefore so is "X is false or X is Neither".
----------
The contextualist account, (I am drawing on Keith Simmons' version), calls the
Liar various things, such as "singularity".    But "neither true nor false" will
do.  Unlike the weak truth account, "~True(X) & ~False(X)" can be used for "X is
neither true nor false", so an undefined "Neither" is not needed.   Here is the
example as modified:

V.   2+2=4
W.   Sentence V is true
X.   Every sentence in this FOM posting with acronym "SXeintnfoif" is true.
Y.   Sentence X is neither true nor false.
Z.   Sentence X is false.
AA.  Sentence X is either is false, or is neither true nor false.
BB.  Sentence X either is neither true nor false, or is false.

The loop sentences are neither true nor false.  Observer sentences Y and AA are
true, and observer sentence Z is false.

All three accounts make use of three statuses: true, false, and a third status
(variously named, but in each case the status that the Liar gets).   All three
accounts have predicates "True(X)" and "False(X)"; the weak truth account also
uses "Neither(X)".  The tables for the predicates are:

---- | ----- dialetheist ---- |
-X- | True(X) | False(X) |
-T- | ----T---- | ----F----- |
-F- | ----F---- | ----T----- |
3rd | ---3rd--- | ---3rd---- |

---- | ------------ weak truth ----------- |
-X- | True(X) | False(X) | Neither(X) |
-T- | ----T---- | ----F----- | ------F----- |
-F- | ----F----- | ----T---- | ------F----- |
3rd | ---3rd--- | ---3rd---- | -----T----- |

---- | ----contextualist---- |
-X- | True(X) | False(X) |
-T- | ----T---- | ----F---- |
-F- | ----F---- | ----T---- |
3rd | ----F---- | ----F---- |

In the contextualist truth table, all the entries are either T or F.  Thus
whatever status X has, anything said about X will have a status of T or F.   Since
the atoms "True(X)" and "False(X)" have status T or F, the status of any compound
can be found by ordinary 2-value truth tables, and will be T or F.

In the contextualist account, observer sentences behave exactly according to
classical, 2-value logic.   The non-classical quality of the contextualist account
is extreme, but this weirdness is entirely confined to the self-referential loop
sentences.

In all three accounts, the loop sentences (at least the ones in these examples)
have the 3rd status.   In the weak truth example, sentence Q calls sentence U
true.   If you look in the table for weak truth, you can see that since U has 3rd
status, True(U) should have 3rd status.  Thus Q should have 3rd status, as it
does.   So in the weak truth account, an assignment of 3rd to the two loop
sentences "solves the equations".   The same applies to the dialetheist account.
In the contextualist example, sentence X calls sentence BB true.   Sentence BB has
3rd status.   If you look in the contextualist table, since BB has 3rd status,
True(BB) should be false.   But X is not false, but has the 3rd status.    Thus
the assignment of 3rd status to the two loop sentences does not "solve the
equations".    From the contextualist point of view, loop sentences are places
where the normal equations do not apply.    The contextualist account has truth
tables, indeed it uses purely classical ones.   But when the equations set up by
those tables have no solution for a loop, the 3rd status is applied to the loop
sentences.  Whatever status those sentences would have from the tables, is just
over-ridden.

Sentences X and BB are a loop.   We try all possible assignments of T and F to
them, and we find that no such assignment works.  There is no solution to these
equations.   So we say both X and BB have 3rd status.  Sentence AA is an observer
sentence; we derive its value using the truth tables.  By those tables, it is
true.   Thus AA is true, while BB is 3rd status.   But BB follows from AA by the
rule "A v B |- B v A".   So this rule is not truth-preserving, and a similar
argument shows many other rules not to be.  We can't give up these rules.    So
instead of saying that a valid rule is always truth-preserving, we say that the
consequence of true premises by a valid rule is never false.   This is a radical
change, but the change effects only sentences in loops, since only sentences in
loops can ever be of 3rd status.   Outside of a loop, the valid consequence of a
true sentence is not false, and can't be 3rd, so it is always true.

Here's an example sentence:

CC.   Every sentence in this FOM posting with acronym "SXeintnfoif" is true.

Sentence CC is the same as sentence X.   Both CC and X refer to sentence BB.   BB
refers to X, but not to CC.   Thus CC is an observer sentence.   CC is false
according to the truth table.    X is 3rd status.   So CC and X are two instances
or utterances of the same formula, but one is false and the other is 3rd
status.    In the contextualist account, a truth value is calculated for a formula
(not a token), just as in a classical situation.   But when 3rd status is assigned
to some loops, over-riding the calculated value, it is sometimes assigned to
tokens, not formulas.   Since it is possible to refer to tokens, it can be tokens,
rather than formulas, that are in loops.   For example X and BB are a loop, and CC
is not in that loop, even though CC is the same formula as X.    Since only things
in a loop can ever be 3rd status, when there is a loop of tokens, it is the tokens
in the loop that get 3rd status - tokens outside the loop don't get 3rd status,
even if they are the same formula as 3rd-status tokens in the loop.

So the contextualist account is non-classical in two profound ways; the meaning of
"valid" is changed, and it is tokens rather than formulas that get a semantic
status.
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So in comparing the three accounts, we see that only contextualism has purely
classical behavior of observer sentences, although it pays a heavy price in terms
of non-classical behavior of the loop sentences.

Next post: Can we use the tables of the contextualist account, without accepting
the contextualist's justification for them?

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