678 research outputs found
On the relative strengths of fragments of collection
Let be the basic set theory that consists of the axioms of
extensionality, emptyset, pair, union, powerset, infinity, transitive
containment, -separation and set foundation. This paper studies the
relative strength of set theories obtained by adding fragments of the
set-theoretic collection scheme to . We focus on two common
parameterisations of collection: -collection, which is the usual
collection scheme restricted to -formulae, and strong
-collection, which is equivalent to -collection plus
-separation. The main result of this paper shows that for all ,
(1) proves the consistency of Zermelo Set Theory plus
-collection,
(2) the theory is
-conservative over the theory .
It is also shown that (2) holds for when the Axiom of Choice is
included in the base theory. The final section indicates how the proofs of (1)
and (2) can be modified to obtain analogues of these results for theories
obtained by adding fragments of collection to a base theory (Kripke-Platek Set
Theory with Infinity and ) that does not include the powerset axiom.Comment: 22 page
Dependence Logic with Generalized Quantifiers: Axiomatizations
We prove two completeness results, one for the extension of dependence logic
by a monotone generalized quantifier Q with weak interpretation, weak in the
meaning that the interpretation of Q varies with the structures. The second
result considers the extension of dependence logic where Q is interpreted as
"there exists uncountable many." Both of the axiomatizations are shown to be
sound and complete for FO(Q) consequences.Comment: 17 page
Rich Situated Attitudes
We outline a novel theory of natural language meaning, Rich
Situated Semantics [RSS], on which the content of sentential utterances
is semantically rich and informationally situated. In virtue of its situatedness,
an utteranceâs rich situated content varies with the informational
situation of the cognitive agent interpreting the utterance. In virtue of its
richness, this content contains information beyond the utteranceâs lexically
encoded information. The agent-dependence of rich situated content
solves a number of problems in semantics and the philosophy of language
(cf. [14, 20, 25]). In particular, since RSS varies the granularity of utterance
contents with the interpreting agentâs informational situation, it
solves the problem of finding suitably fine- or coarse-grained objects for
the content of propositional attitudes. In virtue of this variation, a layman
will reason with more propositions than an expert
Eine linguistische Wende in der Logik?
Reporting on the 7th International Congress of Logic, Methodology, and Philosophy of Science, first the main topics and some organisational aspects of the congress are presented; the main part of the report focuses on recent developments in Philosophical Logic (Section 5), in particular the theory of so-called generalized quantifiers as presented at the congress. In addition, some background information on logical language analysis, its possible applications and consequences is provided.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43017/1/10838_2005_Article_BF01800939.pd
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
- âŠ