2,072 research outputs found
Buying Logical Principles with Ontological Coin: The Metaphysical Lessons of Adding epsilon to Intuitionistic Logic
We discuss the philosophical implications of formal results showing the con-
sequences of adding the epsilon operator to intuitionistic predicate logic. These
results are related to Diaconescuâs theorem, a result originating in topos theory
that, translated to constructive set theory, says that the axiom of choice (an
âexistence principleâ) implies the law of excluded middle (which purports to be
a logical principle). As a logical choice principle, epsilon allows us to translate
that result to a logical setting, where one can get an analogue of Diaconescuâs
result, but also can disentangle the roles of certain other assumptions that are
hidden in mathematical presentations. It is our view that these results have not
received the attention they deserve: logicians are unlikely to read a discussion
because the results considered are âalready well known,â while the results are
simultaneously unknown to philosophers who do not specialize in what most
philosophers will regard as esoteric logics. This is a problem, since these results
have important implications for and promise signif i cant illumination of contem-
porary debates in metaphysics. The point of this paper is to make the nature
of the results clear in a way accessible to philosophers who do not specialize in
logic, and in a way that makes clear their implications for contemporary philo-
sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism.
Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett,
realism, antirealis
CZF does not have the Existence Property
Constructive theories usually have interesting metamathematical properties
where explicit witnesses can be extracted from proofs of existential sentences.
For relational theories, probably the most natural of these is the existence
property, EP, sometimes referred to as the set existence property. This states
that whenever (\exists x)\phi(x) is provable, there is a formula \chi(x) such
that (\exists ! x)\phi(x) \wedge \chi(x) is provable. It has been known since
the 80's that EP holds for some intuitionistic set theories and yet fails for
IZF. Despite this, it has remained open until now whether EP holds for the most
well known constructive set theory, CZF. In this paper we show that EP fails
for CZF
Constructive Mathematics in Theory and Programming Practice
The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishopâs constructive mathematics(BISH). It gives a sketch of both Myhillâs axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focuses on the relation between constructive mathematics and programming, with emphasis on Martin-Lofâs theory of types as a formal system for BISH
On the existence of Stone-Cech compactification
In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann.
Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained
of the locales of which the Stone-Cech compactification can be defined in
constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural
extension of Aczel's system for constructive set theory CZF by a strengthening
of the Regular Extension Axiom REA and the principle of dependent choice. In
this paper I show that this characterization continues to hold over the
standard system CZF plus REA, thus removing in particular any dependency from a
choice principle. This will follow by a result of independent interest, namely
the proof that the class of continuous mappings from a compact regular locale X
to a regular a set-presented locale Y is a set in CZF, even without REA. It is
then shown that the existence of Stone-Cech compactification of a
non-degenerate Boolean locale is independent of the axioms of CZF (+REA), so
that the obtained characterization characterizes a proper subcollection of the
collection of all locales. The same also holds for several, even impredicative,
extensions of CZF+REA, as well as for CTT. This is in contrast with what
happens in the context of Higher-order Heyting arithmetic HHA - and thus in any
topos-theoretic universe: by constructions of Johnstone, Banaschewski and
Mulvey, within HHA Stone-Cech compactification can be defined for every locale
A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
This paper is the first in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper we discuss two different types of language that can be
attached to a system, S. The first is a propositional language, PL(S); the
second is a higher-order, typed language L(S). Both languages provide deductive
systems with an intuitionistic logic. The reason for introducing PL(S) is that,
as shown in paper II of the series, it is the easiest way of understanding, and
expanding on, the earlier work on topos theory and quantum physics. However,
the main thrust of our programme utilises the more powerful language L(S) and
its representation in an appropriate topos.Comment: 36 pages, no figure
- âŠ