Reduction and Elimination in Philosophy and the Sciences

there is a tension between evaluating ¬Tr(‘φ’) and φ
equally, and evaluating ¬Tr(‘φ’) and φ differently. The first
tension indicates that ¬Tr(‘φ’) is context-dependent.
Concerning the second tension we have two possible lines
of explanation available. We can either again trust into
effects of a context-shift, but we can alternatively argue
against the rule (SoI).
On a first assessment one might for economical
reasons decide to make an effort to solve the problem by
employing an explanation in terms of context-shift only. I
shall simply call this the context-based approach in order
to distinguish it from the other strategies. Generally, any
context-based approach has to clarify what kind of contextdependence
is present in the strengthened liar reasoning.
It has to determine where a context-dependent element is
hidden in ¬Tr(‘φ’).
All context-based approaches agree that a certain
change of context affects our interpretation of the truth
predicate Tr occurring in φ. A very direct realization of this
comes from (Burge 1979). He proposes to add an index i
to each occurrence of Tr, i.e. “Tri”. This reminds very much
of Tarski´s proposal, but Burge`s system has several
advantages. Formulas, such as “Tri(‘Tri(‘P(c)’)’)” where
“P(c)” represents “Snow is white”, that are meaningless or
ill-formed according to Tarski´s account, become
meaningful within Burge´s system. Further developments
of Burge´s approach are (Koons 1992) and (Simmons
1993).
A metaphysically more sophisticated approach is
posed by (Parsons 1974), (Barwise and Etchemendy
1987) and (Glanzberg 2004). According to Parsons and
Glanzberg, a context-shift can lead to a shift from M to
another L-structure M* which is regarded to be an
expansion of M. They take propositions as truth-bearers,
and define an expression relation between sentences and
propositions. The universe of discourse D of their model M
is proposed to contain all propositions that can be
expressed by L M (in Glanzberg´s system, D actually
contains all truth conditions that can constitute a
proposition and that can be expressed by L M ). The
universe can expand as context shifts. In the first place,
therefore, quantifiers represent the context-dependent
element in (Glanzbergs and Parsons versions of) the
strengthened liar sentence. But as Glanzberg and Parsons
note, it is then only a definitional matter to work with a
context-dependent truth predicate, and in fact both end up
with working with a context-dependent truth predicate.
(Barwise and Etchemendy 1987) give a very similar
account.
Context-based approaches employ an analysis that
inevitably gives rise to a certain kind of hierarchy of truth
predicates. This is obvious for proposals which attach an
index belonging to a partially ordered set to all
occurrences of “Tr”. Approaches that appeal to an
expansion M* of M with a more comprehensive domain D*
lead to a hierarchical view as well. The language L M*
interpreted by M* is used to express a true statement
about φ' (the “ ' ” indicates that φ' is actually a translation of
the original φ belonging to L M into the language L M* ): M* ├
Tr*(‘φ'’), and furthermore M* ├ ¬Tr(‘φ'’), where the
Context-Based Approaches to the Strengthened Liar Problem — Christine Schurz
predicate Tr represents “truth in L M ” and Tr* represents
“truth in L M* ”. A new strengthened liar sentence φ* of the
form ¬Tr*(l) such that l = ‘¬Tr*(l)’ can be formed that gives
rise to another context-shift that is associated with another
model M** to reason about φ*. M** in turn has its own
truth-predicate Tr** and so on and so forth.
Of course the hierarchies advanced by contextbased
approaches differ from Tarski´s hierarchies of truthpredicates
in several respects. First, as I have already
mentioned, this approach features some advantages,
since several cases of formulas that correspond to
ordinary and meaningful sentences in English, but that
become pathological in Tarski´s system, are evaluated as
our intuitions would suggest it. Furthermore, these
alternative hierarchies are considered to model natural
language as closely as this can possibly be done, or even
to represent an intrinsic feature of natural language. The
strengthened liar problem does not show that the concept
of truth in natural language is inconsistent, but that the
predicate “is true” is context-dependent.
Still it is prima facie undesired that the concept of
truth is context-dependent. So context-based approaches
have to give a motivation for this context-dependency. A
very essential question is, what kind of context-shift
happens to effect a shift in the extension of Tr and how it
does this. Earlier proponents of context-based approaches
do not employ any rigid formal concept of a context.
Therefore they only give rough and informal explanations
for this problem, if at all. Sometimes the presence of the
strengthened liar problem is taken to provide enough
evidence to prove a context-shift. In contrast, (Glanzberg
2004) gives a rigid proof of how a context-shift leads to a
shift from M to M*, which is based on precise notions of a
context, a proposition and a truth predicate.
An additional problem which I think deserves
attention apart from any profound explanation concerning
the previous question is that in the languages proposed
there is no way to reflect over all contexts. No formula can
be used to represent an ultimate or universal perspective.
Proponents of a contextual analysis have to clarify why this
is so, and why our naive view that we can reflect over
absolutely everything is wrong.
Apart from these two problems a context-based
approach also has to motivate why it fares better than
alternative approaches to explain the strengthened liar
problem. Given all the difficulties related with a hierarchical
account I have just outlined, it could e.g. be that contrary
to our first assessment an alternative approach that relies
in addition on restricting (SoI) in the end provides a better
trade-off than a purely context-based approach. It has
sometimes been argued against restricting (SoI) that this
line of solution was ad-hoc and unmotivated. For the most
part this point applies to approaches put forward so far
(e.g. (Skyrms 1970)), since they are not very elaborated
concerning their applicability to natural language (besides,
they do not invoke any contextual considerations, and
therefore do not explain the first problem from which the
strengthened liar problem arises). Therefore it could be
interesting to search for other kinds of contextual
approaches and turn our attention to (SoI).
311