Reduction and Elimination in Philosophy and the Sciences

Context-Based Approaches to the Strengthened Liar Problem
Christine Schurz, Salzburg, Austria
Kripke´s paper (Kripke 1975) was the starting point for
numerous formal languages that are able to express a
theory of their own concept of truth without producing a liar
paradox. However these theories are not regarded as
modeling the concept of truth of natural language. The
main reason for this is the strengthened liar paradox, or
rather a formalized version of this problem. Whether it is
possible to solve this problem without being committed to
something (more or less) like Tarski´s hierarchical
proposal (Tarski 1935), is still a highly disputed matter.
The majority of those who seek for alternatives to Tarski´s
proposal attempt to analyze the strengthened liar problem
by uncovering certain context-dependent elements in a
strengthened liar sentence.
This paper consists of two parts. First, I shall outline
a formal version of the strengthened liar. Thereafter I take
a look at the context-based approach to this problem.
From now on, I shall assume a formal language L of
first order logic that is interpreted by a structure M. M is
assumed to contain names that are assigned to the
sentences of L, and functions that represent a certain
amount of the syntax of L. The symbol Tr is a one-place
predicate of L that is intended to represent “truth in L”. By
the fixed-point-lemma a strengthened liar sentence, i.e., a
sentence φ of the form ¬Tr(l) such that M╞ l = ‘¬Tr(l)’ can
be formed (the term ‘¬Tr(l)’ is the object-language-name of
the formula ¬Tr(l)). I shall assume a theory T of “truth in L”
such that neither T├ φ nor T├ ¬φ. Such a theory results
e.g. from the axioms of (Friedman and Sheard 1987) which
define Tr together with some axioms of a theory of M (if,
for instance, M is a model of arithmetic, then we can take
the axioms of Robinson´s arithmetic).
The theory T matches with the intuition that the
strengthened liar sentence is neither true nor false in as
much as it gives us no information concerning the truthvalue
of φ. But in context of natural language we are also
capable of expressing this intuition by a meaningful
sentence and furthermore infer other statements from such
a sentence (which together lead to the natural language
strengthened liar argument). Thus, if we seek T to be as
close as possible to the concept “is true” of natural
language, we have to investigate whether it is possible for
T to contain any formula of L which represents a
semantical diagnosis as “φ is neither true nor false” of φ. Is
it possible to add such a formula to the axioms of T without
causing T to be inconsistent? This is indeed an intricate
problem.
The first problem: Usually the formula ¬Tr(‘φ’) is
taken to be a most appropriate candidate to represent a
diagnostic statement about φ. Since φ fails to be derivable
from T, it also fails to be true according to T (otherwise it
would be derivable from T), which can in L be represented
by the formula ¬Tr(‘φ’). So ¬Tr(‘φ’) should be derivable
from T. But, on the other hand, we have a “falsity-intuition”
concerning ¬Tr(‘φ’), taking ¬Tr(‘φ’) to represent the
sentence “φ is false”. Since T gives us no information
whether φ is true or false, also no formula representing “φ
is false” will be derivable from T. Therefore ¬Tr(‘φ’) should
not be derivable from T. So in the end we have a “failureintuition”
and a “falsity-intuition” concerning the sentence
¬Tr(‘φ’). Both intuitions display features of how we
310
informally reason about the strengthened liar in natural
language. So they do not have to be considered as rival
intuitions of which we have to select only one that
represents our “actual” reasoning or the way we should
rationally reason. Maybe, if L is supposed to account for
natural language, one has to find a way to model both of
these intuitions and explain how T├ ¬Tr(‘φ’) as well as not
T├ ¬Tr(‘φ’) can be the case. But obviously this can only be
consistently realized if we assume some context-sensitive
element in the sentence ¬Tr(‘φ’).
Before I turn to the second problem, let me note that
one could object against the analysis I just gave that it
shows that ¬Tr(‘φ’) is after all no appropriate choice for a
formula representing a semantical diagnosis of φ. But then
the only alternative is to introduce a new “untrueness”predicate
U to express a formula belonging to T that
represents a statement about φ. But of course this gives
rise to a new strengthened liar sentence φ' of the form U(l')
such that M╞ l' = ‘U(l')’ that leads to just the same
problems. So in the end we would have just the same
conflict of failure- against falsity-intuition.
The second problem: This problem results from our
attempt to model the ‘failure-intuition’ for ¬Tr(‘φ’).
According to this attempt, T contains the sentence
¬Tr(‘φ’), but, on the other hand, it should not contain φ.
These conditions contradict to the rule of substitutivity of
identity (I shall from now on write “(SoI)” to refer to this
rule):
(SoI) Substitutivity of identity:
Let t1 and t2 be any two terms, let P be any oneplace
predicate. Then the following rule is valid:
t1= t2, P(t1)
P(t2)
We have to make a decision whether to give up (SoI) or
again put forward an explanation in terms of a contextshift.
We might even again employ the context-shift we
have already posited in “¬Tr(‘φ’)” occurring in the argument
based on the failure-intuition and the same formula
occurring in the argument based on the falsity-intuition.
Then we not only suppose two different contextual interpretations
of “¬Tr(‘φ’)”, but also two different contextual
interpretations of φ. Indeed some proponents of contextual
approaches (e.g. (Burge 1979) and (Simmons 1993)) argue
for another tension between two semantical views of
the strengthened liar sentence φ. But this point is controversial.
After all, φ is in the first place assumed to be an
entirely pathological sentence to which no classical truth
value can be assigned. ¬Tr(‘φ’) is intended to reflect this
assumption. So if we finally refute that φ is entirely pathological
but consider φ to be also true according to another
reading, then we refute the basic observation of our whole
reasoning.
To conclude, in context of our formal language L,
the strengthened liar paradox is constituted by (at least)
two tensions in our metalinguistic reasoning about φ.
Firstly, the tension between evaluating ¬Tr(‘φ’) as true
according the failure-intuition, and evaluating it as neither
true nor false according to the falsity-intuition. Secondly,