338,565 research outputs found
Injective power objects and the axiom of choice
AbstractâThe axiom of choice states that any set X of non-empty sets has a choice functionâi.e. a function Xâ¶fâX satisfying f(x)âx for all xâX. When we want to generalise this to a topos, we have to choose what we mean by non-empty, since in Set, the three concepts non-empty, inhabited, and injective are equivalent, so the axiom of choice can be thought of as any of the three statements made by replacing ânon-emptyâ by one of these notions.It seems unnatural to use non-empty in an intuitionistic context, so the first interpretation to be used in topos theory was the notion based on inhabited objects. However, Diaconescu (1975) [1] showed that this interpretation implied the law of the excluded middle, and that without the law of the excluded middle, even the finite version of the axiom of choice does not hold! Nevertheless some people still view this as the most appropriate formulation of the axiom of choice in a topos.In this paper, we study the formulation based upon injective objects. We argue that it can be considered a more natural formulation of the axiom of choice in a topos, and that it does not have the undesirable consequences of the inhabited formulation. We show that if it holds for Set, then it holds in a wide variety of topoi, including all localic topoi. It also has some of the classical consequences of the axiom of choice, although a lot of classical results rely on both the axiom of choice and the law of the excluded middle. An additional advantage of this formulation is that it can be defined for a slightly more general class of categories than just topoi.We also examine the corresponding injective formulations of Zornâs lemma and the well-order principle. The injective form of Zornâs lemma is equivalent to the axiom of injective choice, and the injective well-order principle implies the axiom of injective choice
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
Free choice and homogeneity
This paper develops a semantic solution to the puzzle of Free Choice
permission. The paper begins with a battery of impossibility results showing that
Free Choice is in tension with a variety of classical principles, including Disjunction
Introduction and the Law of Excluded Middle. Most interestingly, Free Choice
appears incompatible with a principle concerning the behavior of Free Choice under
negation, Double Prohibition, which says that Mary canât have soup or salad implies
Mary canât have soup and Mary canât have salad. Alonso-Ovalle 2006 and others
have appealed to Double Prohibition to motivate pragmatic accounts of Free Choice.
Aher 2012, Aloni 2018, and others have developed semantic accounts of Free Choice
that also explain Double Prohibition.
This paper offers a new semantic analysis of Free Choice designed to handle the
full range of impossibility results involved in Free Choice. The paper develops the
hypothesis that Free Choice is a homogeneity effect. The claim possibly A or B
is defined only when A and B are homogenous with respect to their modal status,
either both possible or both impossible. Paired with a notion of entailment that is
sensitive to definedness conditions, this theory validates Free Choice while retaining
a wide variety of classical principles except for the transitivity of entailment. The
homogeneity hypothesis is implemented in two different ways, homogeneous alternative semantics and homogeneous dynamic semantics, with interestingly different
consequences
A LEXICO-STYLISTIC ANALYSIS OF KAINE AGARYâS YELLOW-YELLOW
Creativity, the expression of a writerâs imagination, is drawn from the totality of the writerâs experiences. Writers use language, the common middle-ground and the writerâs communication tool, to convey or express their experiences (as in the case of Yellow-Yellow) as well as their cultures and backgrounds which they may or may not share with their readers. A peopleâs culture certainly includes their language (Bodley 2008). This implies that language in use cannot be excluded from the society in which it is used. Moreover, context is a major determinant of any act of language behaviour while choice of words and meaning derivations are a factor of context or environment (Bright 2006). Firthâs (1957:173) view of language as âoccurring in a culturally determined context of situationâ is, therefore, apposite. Eggins (2004:8) states that âour ability to deduce context from text is one way in which language and context are interrelatedâ. In the light of this, this chapter illustrates the relationship between language use and style, the manner in which the writerâs experience is conveyed, at the lexical level to showcase the distinctly Nigerian or Niger-Deltan flavour of Yellow-Yellow
Every metric space is separable in function realizability
We first show that in the function realizability topos every metric space is
separable, and every object with decidable equality is countable. More
generally, working with synthetic topology, every -space is separable and
every discrete space is countable. It follows that intuitionistic logic does
not show the existence of a non-separable metric space, or an uncountable set
with decidable equality, even if we assume principles that are validated by
function realizability, such as Dependent and Function choice, Markov's
principle, and Brouwer's continuity and fan principles
A new foundational crisis in mathematics, is it really happening?
The article reconsiders the position of the foundations of mathematics after
the discovery of HoTT. Discussion that this discovery has generated in the
community of mathematicians, philosophers and computer scientists might
indicate a new crisis in the foundation of mathematics. By examining the
mathematical facts behind HoTT and their relation with the existing
foundations, we conclude that the present crisis is not one. We reiterate a
pluralist vision of the foundations of mathematics. The article contains a
short survey of the mathematical and historical background needed to understand
the main tenets of the foundational issues.Comment: Final versio
Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic
We present a new syntactical proof that first-order Peano Arithmetic with
Skolem axioms is conservative over Peano Arithmetic alone for arithmetical
formulas. This result - which shows that the Excluded Middle principle can be
used to eliminate Skolem functions - has been previously proved by other
techniques, among them the epsilon substitution method and forcing. In our
proof, we employ Interactive Realizability, a computational semantics for Peano
Arithmetic which extends Kreisel's modified realizability to the classical
case.Comment: In Proceedings CL&C 2012, arXiv:1210.289
Strong Normalization for HA + EM1 by Non-Deterministic Choice
We study the strong normalization of a new Curry-Howard correspondence for HA
+ EM1, constructive Heyting Arithmetic with the excluded middle on
Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda
calculus plus an operator ||_a which represents, from the viewpoint of
programming, an exception operator with a delimited scope, and from the
viewpoint of logic, a restricted version of the excluded middle. We give a
strong normalization proof for the system based on a technique of
"non-deterministic immersion".Comment: In Proceedings COS 2013, arXiv:1309.092
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