Cervelli, Persone e Società ***ABSTRACTS
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Brains, Persons, and Society *** ABSTRACTS Cervelli, Persone e Società ***ABSTRACTS |
Luca
Incurvati
Does a Kripke Semantics Vindicate the Adoption of
Intuitionistic Logic in Set Theory?
Jonathan Lear (1977)
presents an argument for the adoption of intuitionistic logic in our
settheoretic
reasoning.
The argument
consists of three steps: one from the open-endedness of the
concept
of set to
cognitivism, the thesis that we never quantify over all sets, but only
over the
extension
of ‘set’ at a
certain time; one from cognitivism to Kripke semantics; one from Kripke
semantics
to
intuitionistic logic. In two recent articles (Paseau 2001; 2003),
Alexander
Paseau has
criticised
both the step
from cognitivism to Kripke semantics and that from Kripke semantics to
intuitionistic
logic. In
this paper I show that Paseau's criticism of the step from Kripke
semantics to
intuitionistic
logic is
flawed and that the adoption of a Kripke semantics in set theory does
sanction
the
use of
intuitionistic logic.
Paseau
challenges the
step from Kripke semantics to intuitionistic logic on the basis of a
model-theoretic
result
first noticed by Hazen (1982). Hazen observes that if a Kripke model is
directed,
it validates φ v ¬¬φ,
which is generally not
intuitionistically valid. Paseau argues that the
nodes
of the Kripke
model, interpreted as indexing either actual or possible knowledge
states, are
linearly
ordered and,
since linearity is a property stronger than directedness, that the
Kripke model
validates
φ v ¬¬φ on both interpretations. Paseau
argues that possible knowledge states are linearly
ordered
by combining the
Comparability of cardinals with Zermelo’s Categoricity Theorem. He
then
tries to justify
the use of the Comparability of Cardinals, given its equivalence to the
Axiom
of
Choice.
As I show,
however, the argument does not need the Axiom of Choice, since the
levels of
the
hierarchy, rather
than being indexed by cardinals, are indexed by ordinals, and the
ordinals can
be
shown to be
comparable without the Axiom of Choice.
Nevertheless,
I claim,
Paseau’s argument fails because of its use of Zermelo's Categoricity
Theorem,
a theorem of
second-order set theory, which we are not entitled to use in this
context. For
to
argue that the
open-endedness of the concept of set does not lead to the adoption of
intuitionistic
logic
by assuming that
we can grasp the concept well-defined property in such a comprehensive
way
that we can quantify
over all properties would not be a neutral assumption if this concept
were
to
exhibit the same kind
of open-endedness as the concept set. But several considerations, I
argue,
point
in just this
direction. First, the concept of well-defined property is indefinitely
extensible:
once
we have specified a
language and recognised its well-defined properties, it is always
possible
to
specify, by reference
to the expressions of that language, a further well-defined property
which is
not
expressible in it.
Second, second-order set theory decides the Continuum Hypothesis: there
is a
sentence
in the language
of second-order logic which is true if and only if the Continuum
Hypothesis
is true.
This, however, has not led anyone to try to settle the Continuum
Hypothesis by
reflecting
on the acceptability
of this sentence as a logical truth.
I
conclude by providing
an example of a possible evolution of knowledge states that
branches
as different
conceptions of the width and height of the cumulative hierarchy are
combined
and
by offering some final
considerations on the significance of the failure of Paseau’s argument.
References:
Hazen,
A., 1982, On a
possible misinterpretation of Kripke’s semantics for intuitionistic
logic,
Analysis,
42, 128-133.
Lear,
J., 1977, Sets and
Semantics, Journal of Philosophy, 74(2), 86-102.
Paseau,
A., 2001, Should
the logic of set theory be intuitionistic?, Proceedings of the
Aristotelian
Society,
101, 369-378.
Paseau,
A., 2003, The
open-endedness of the set concept and the semantics of set theory,
Synthese,
135, 381-401.