Reduction and Elimination in Philosophy and the Sciences

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Structure of the Paradoxes, Structure of the Theories: A Logical Comparison of Set Theory and Semantics — Giulia Terzian
(b) A logical analysis of the two conceptions favours
IT, insofar as it involves only intuitive notions, while
LIM presupposes an understanding of more sophisticated
concepts of set theory (cf. Boolos 1989).
(c) Boolos 1989 shows that IT on its own does not
entail the axioms of replacement and extensionality;
these follow from LIM, but other axioms of ZF do
not. The upshot is that it is possible to bridge the
gap between the historical and the logical reconstructions
of ZF set theory by understanding LIM
and IT as jointly necessary, individually not sufficient
premises to a full explication of the notion of set: in
Boolos' words, “there are at least two thoughts ‘behind’
set theory” (1989, p.103).
4 Too simple?
In order to assess the soundness of the analogy account,
the key question to be answered is the following: Are there
any semantic principles that could be identified as counterparts
to IT and LIM?
The first part of this section addresses this question;
the second part raises two more issues of methodology.
A semantic limitation of size principle LIMT should
e.g. guarantee the assertability of some sentences of the
form ∀x (P(x) → T(x)) for some predicate P. To this end,
LIMT should presumably place a cardinality constraint on
the extension of P, so as to ensure that problematic (Liar)
sentences do not end up in the extension of T.
Suppose the extension of P is ℒT, i.e., the set of all
sentences including those which contain the truth
predicate. Then among these will be many (Liar-like)
sentences we would not want in the extension of T: so it
might seem reasonable to invoke a principle LIMT requiring
the cardinality of the extension of T to be smaller than that
of ℒT.
Now suppose the extension of P is ℒarith (the
language of arithmetic): applying T to purely arithmetical
sentences does not lead to any inconsistency, so this
choice should be allowed by LIMT. However ℒT and ℒarith
are both countably infinite: hence LIMT gives contradictory
verdicts for languages with the same cardinality.
Finally suppose the extension of P is simply λ . In
principle LIMT should obviously apply in this case: insofar
as λ should definitely not be in the extension of T, the
minimal language which it forms is already ‘too big’. This is
an undesirable result which makes it starkly clear that the
quantitative constraint in LIMT is inadequate for the
semantic context, where instead the key question is about
which sentences are put inside the extension of T.
It is fairly natural to understand IT as embodying “a
fundamental relation of [...] dependence between
collections." (Potter 2004, p.36) In semantics, too, one can
talk about a fundamental relation of dependence between
some sentence φ and a set of sentences of the language.
Semantic dependence subtends the notion of semantic
groundedness 1 : a sentence containing T is grounded if its
truth value ultimately depends on non-semantic states of
affairs, so that working back along the dependence
relation leads to a sentence which does not contain T. Liar
1 First formally discussed in Kripke 1975; a more thorough analysis is found in
Yablo 1982 and Leitgeb 2005.
sentences are ungrounded, because their dependence
path is not linear but circular; in standard truth-gap
accounts, this is equivalent to saying that they lack a truth
value in the least fixed point of the (Kripkean) jump
operator 2 .
Dependence seems to provide a more promising
case for the analogy account. To follow up this conjecture,
we look at the features of a typical construction in both
contexts. Take Ø as the starting point. At each level of the
cumulative hierarchy, all individuals and sets appearing at
all previous levels are collected into a new set: the iterated
power-set operation produces a strictly increasing
progression of sets, from which no element of the universe
is left out.
On the other hand, the iterated jump operation
produces a ‘semi’-hierarchy of extensions (anti-extensions)
of the truth predicate, but in this case not all sentences of
the language are guaranteed a place. Specifically, only the
grounded sentences will make a regular appearance at
each level (and will also be consistently part of the same
semi-hierarchy); so the iterated collecting operation is here
constrained to filter out the ungrounded sentences.
Thus on closer inspection dependence actually
appears to be a fairly weak link between set theory and
semantics, and moreover reveals further differences
between them. Set-theoretic dependence is central to the
construction of the hierarchy, in the sense that a set exists
in virtue of being made up of lower-level elements: it is so
to speak a by-product of IT. But in semantics there is no
question about the ‘existence’ of a sentence: what matters
instead is whether the correct ones are put inside the
extension of T. The key relation here is groundedness,
which determines whether a sentence can be evaluated
for truth; moreover, this can only be established once we
reach the least fixed point – while the existence of a set is
‘determined’ at every level at which it appears.
Finally, groundedness not only presupposes
dependence, but on some accounts (e.g. Yablo 1982) also
an understanding of partial predicates, non-classical
logics, etc.: so there is the additional worry that the
relevant set-theoretic and semantic relation are
mismatching in another respect, namely that the former but
not the latter underlies a natural conception (cf. Section 3).
In giving a formal theory of truth, the central aim is
to explicate this fundamental semantic notion so as to
account for its non-problematic use in everyday speech.
For any such theory, a ‘sample basis’ of sentences
containing the truth predicate should thus naturally be
expected to be as broad as possible. Paradoxical
sentences, which constitute an extremely restricted
sample, could then be used to test the formalism, as a
measure of its efficacy.
One could also choose a different approach: start
from some pre-existing formalism and subject it to
successive revisions, constrained primarily by the rule:
“avoid contradictions”.
These diametrically opposite strategies might be
called respectively ‘constructive’ and ‘regressive’; the
choice of these terms is intended to parallel the distinction
made by Potter 2004 between “intuitive” and “regressive”
methodologies in set theory. As in the case of set theory
2 Let (A+;A-) be a partial interpretation of T; then the jump is a monotone
inductive operator defined by j (A+;A-) = (j + (A+;A-); j - (A+;A-)), where j
+(A+;A-) = {φ: (A+;A-) ⊨ φ} and j - (A+;A-) = {φ: (A+;A-) ⊨ ¬φ}.