FOM: Truth for Sentence Tokens

[Sandy Hodges]

Dear Moderator: I don't know if this is of interest to the FOM list.

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The Liar Paradox has ben characterized in many ways; recently a popular
idea is to say it lacks a truth value, with the provision that the
status of lacking a truth value must not be treated as a truth value.
This characterization of the Liar, like most others, gives rise to a
revenge form of the Liar.   This version is due in its essential
features to Buridan:

     Plato says: What Aristotle is about to say will be true.
     Aristotle says: What Plato said is either not true, or lacks a
     truth value.
     Robert says: What Plato said is either not true, or lacks a
     truth value.

Plato's and Aristotle's statements together make up a version of the
strong Liar, so presumably if the Liar lacks a truth value both Plato's
and Aristotle's utterances lack a truth value.   In which case Plato's
utterance lacks a truth value, and therefore either lacks a truth value
or is not true.    So we agree with Robert's analysis, and think what
Robert says is true.   But Aristotle and Robert say the same thing;
nonetheless we think Robert's utterance is true, while Aristotle's
utterance lacks a truth value.

One proposal which can confront the Revenge Liar is Gupta and Martin,
1984.   This requires that the semantic predicate "true" be treated as
weak, and also that weak Kleene truth tables be used.     When "true" is
treated as weak, when a sentence designated by "x" lacks a truth value,
so do "x is true" and "x is not true."    So in this case "What Plato
said is not true" lacks a truth value.    With weak Kleene tables, "A or
B" only has a truth value if both A and B do.     So Robert's statement,
like Aristotle's, lacks a truth value.   On the other hand "What Plato
said lacks a truth value" has a truth value (true), and so does "'What
Plato said lacks a truth value" has a truth value."   So the use of a
weak predicate and weak Kleene tables does allow us to make a statement
about the Liar paradox, and have that statement be evaluated as true.

But the use of weak predicates and tables imposes a cost.    Suppose
sentence A is "The sentence on the back of this card is true."    And
suppose we know that either "Twice two is seven." or "This sentence is
not true." is on the back of the card.    If the former, A is not true,
while if the latter, A lacks a truth value.  But we can't say "Either A
is not true or A lacks a truth value." because that claim may lack a
truth value (due to weak Kleene).   In fact my own sentence two before
this one, (the one beginning "If the former, ...") may have no truth
value.   To see this, suppose that "This sentence is not true." is on
the back of the card.    Then "A is not true" lacks a truth value.   So
using weak Kleene tables, "If the former, A is not true." lacks a truth
value.   So therefore "If the former, A is not true, while if the
latter, A lacks a truth value." lacks a truth value.

An alternative to weak predicates and weak Kleene, is to allow that
Aristotle's utterance and Robert's may have different truth values, even
though they say the same thing.    From this point of view, when
Aristotle spoke he faced a dilemma.    He said something which,
ultimately, is the case: what Plato said is indeed either not true or
lacks a truth value.   Nevertheless, if Aristotle's sentence were true,
then Plato's would be true, and thus what Aristotle said would not be
the case after all.    This is the situation that leads to Aristotle's
sentence having no truth value.   But Robert is not in this situation -
Robert can say what is the case, without worrying that his speaking
changes the thing he speaks about.   So Robert's sentence, like any
other ordinary sentence that says what is the case, is true.

When tokens have values, all the semantic predicates can be strong; so
"x is true" always has a truth value, regardless of the status of x.
"x is true" is always either true or not true.    Since the semantic
atoms are always T or F, there is no need for either sort of Kleene
table; ordinary 2-value ones will do. Fixed point systems for tokens are
easily found.  On the other hand, the logic of sentence tokens (as
opposed to formulas) is unfamiliar.

I believe the idea that it is individual utterances (sentence tokens)
which get truth values, rather than propositions or formulas, deserves
more attention.    One treatment that allows sentence tokens to have
different truth values is the Austinian proposal in Barwise and
Etchemendy 1987, but that proposal contains many things besides merely
allowing tokens to have different values.

References

1. Barwise, John, and Jon Etchemendy, 1987.  The Liar: An Essay on Truth
and Circularity, Oxford University Press, Oxford.
2. Buridanus, Johannes, c.1350. Sophismata.  Chapter 8 as translated by
G. E. Hughes, John Buridan on Self-Reference, 1982.    The use of the
word "revenge" is in Martin 1984.
3. Gupta, Anil, and Robert L. Martin, 1984.   A fixed point theorem for
the weak Kleene valuation scheme, Journal of Philosophical Logic
13:131-135.
4. Martin, R. L., 1984.   Forward to Recent essays on truth and the liar
paradox, Clarendon Press, Oxford.

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