Bulletin of Symbolic Logic Articles (Project Euclid)
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The latest articles from Bulletin of Symbolic Logic on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 19 May 2011 09:16 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Strong logics of first and second order
http://projecteuclid.org/euclid.bsl/1264433796
<strong>Peter Koellner</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 16, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
In this paper we investigate strong logics of first and
second order that have certain absoluteness properties. We begin
with an investigation of first order logic and the strong logics
ω-logic and β-logic, isolating two facets of
absoluteness, namely, generic invariance and faithfulness. It turns
out that absoluteness is relative in the sense that stronger
background assumptions secure greater degrees of absoluteness. Our
aim is to investigate the hierarchies of strong logics of first and
second order that are generically invariant and faithful against the
backdrop of the strongest large cardinal hypotheses. We show that
there is a close correspondence between the two hierarchies and we
characterize the strongest logic in each hierarchy. On the
first-order side, this leads to a new presentation of Woodin's
Ω-logic. On the second-order side, we compare the strongest
logic with full second-order logic and argue that the comparison
lends support to Quine's claim that second-order logic is really set
theory in sheep's clothing.
</p>projecteuclid.org/euclid.bsl/1264433796_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
Notices, Bull. Symbolic Logic 17, iss. 2 (2011)
http://projecteuclid.org/euclid.bsl/1318855635<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 17, Number 4, 599--603.</p>projecteuclid.org/euclid.bsl/1318855635_Mon, 17 Oct 2011 08:48 EDTMon, 17 Oct 2011 08:48 EDT
The absolute arithmetic continuum and the unification of all numbers great and small
http://projecteuclid.org/euclid.bsl/1327328438<strong>Philip Ehrlich</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 1--45.</p><p><strong>Abstract:</strong><br/>
In his monograph On Numbers and Games , J. H. Conway introduced a real-closed
field containing the reals and the ordinals as well as a great many less
familiar numbers including -ω, ω/2, 1/ω, \sqrt{ω}
and ω-π to name only a few.
Indeed, this particular real-closed field, which Conway calls No , is so remarkably
inclusive that, subject to the proviso that numbers—construed here as members of
ordered fields—be individually definable in terms of sets of NBG
(von Neumann—Bernays—Gödel set theory with global choice), it may be said to contain
“All Numbers Great and Small.” In this respect, No bears much the same relation
to ordered fields that the system ℝ of real numbers bears to Archimedean ordered
fields.
In Part I of the present paper, we suggest that whereas ℝ should merely be
regarded as constituting an arithmetic continuum (modulo the Archimedean
axiom), No may be regarded
as a sort of absolute arithmetic continuum (modulo NBG), and in Part
II we draw attention to the unifying framework No provides
not only for the reals and the ordinals but also for an array of non-Archimedean
ordered number systems that have arisen in connection with the theories of
non-Archimedean ordered algebraic and geometric systems, the theory of the rate of
growth of real functions and nonstandard analysis.
In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich
algebraico-tree-theoretic structure—a simplicity hierarchical structure—that
emerges from the recursive clauses in terms of which it is defined. In the
development of No outlined in the present paper, in which the surreals emerge
vis-à-vis a generalization of the von Neumann ordinal
construction, the simplicity
hierarchical features of No are brought to the fore and play central roles in the
aforementioned unification of systems of numbers great and small and in some of
the more revealing characterizations of No as an absolute continuum.
</p>projecteuclid.org/euclid.bsl/1327328438_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
In praise of replacement
http://projecteuclid.org/euclid.bsl/1327328439<strong>Akihiro Kanamori</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 46--90.</p><p><strong>Abstract:</strong><br/>
This article serves to present a large mathematical perspective
and historical basis for the Axiom of Replacement as well as to
affirm its importance as a central axiom of modern set theory.
</p>projecteuclid.org/euclid.bsl/1327328439_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
Second order logic or set theory?
http://projecteuclid.org/euclid.bsl/1327328440<strong>Jouko Väänänen</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 91--121.</p><p><strong>Abstract:</strong><br/>
We try to answer the question which is the “right” foundation of mathematics,
second order logic or set theory. Since the former is usually thought of as
a formal language and the latter as a first order theory, we have to rephrase
the question. We formulate what we call the second order view and a
competing set theory view , and then discuss the merits of both views.
On the surface these two views seem to be in manifest conflict with each other.
However, our conclusion is that it is very difficult to see any real difference
between the two. We analyze a phenomenon we call internal categoricity
which extends the familiar categoricity results of second order logic to
Henkin models and show that set theory enjoys the same kind of internal
categoricity. Thus the existence of non-standard models, which is usually
taken as a property of first order set theory, and categoricity, which is
usually taken as a property of second order axiomatizations, can coherently
coexist when put into their proper context. We also take a fresh look at
complete second order axiomatizations and give a hierarchy result for second
order characterizable structures. Finally we consider the problem of existence
in mathematics from both points of view and find that second order logic
depends on what we call large domain assumptions , which come quite
close to the meaning of the axioms of set theory.
</p>projecteuclid.org/euclid.bsl/1327328440_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
Reviews, Bull. Symbolic Logic 18, iss. 1 (2012)
http://projecteuclid.org/euclid.bsl/1327328441<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 122--134.</p>projecteuclid.org/euclid.bsl/1327328441_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
2011 Spring Meeting of the Association for Symbolic Logic, Hilton San Diego Bayfront Hotel, San Diego, California, USA, April 21—22, 2011
http://projecteuclid.org/euclid.bsl/1327328442<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 135--141.</p>projecteuclid.org/euclid.bsl/1327328442_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
2010—2011 Winter Meeting of The Association for Symbolic Logic, New Orleans Marriott and Sheraton New Orleans Hotels, New Orleans, LA, January 8—9, 2011
http://projecteuclid.org/euclid.bsl/1327328443<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 142--149.</p>projecteuclid.org/euclid.bsl/1327328443_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
XVI Brazilian Logic Conference (EBL 2011), Petrópolis, Rio de Janeiro, Brazil, May 9—13, 2011
http://projecteuclid.org/euclid.bsl/1327328444<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 150--151.</p>projecteuclid.org/euclid.bsl/1327328444_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
18th Workshop on Logic, Language, Information and Computation (WoLLIC 2011), Philadelphia, PA, USA, May 18—20, 2011
http://projecteuclid.org/euclid.bsl/1327328445<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 152--153.</p>projecteuclid.org/euclid.bsl/1327328445_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
Notices, Bull. Symbolic Logic 18, iss. 1 (2012)
http://projecteuclid.org/euclid.bsl/1327328446<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 1, 154--160.</p>projecteuclid.org/euclid.bsl/1327328446_Mon, 23 Jan 2012 09:21 ESTMon, 23 Jan 2012 09:21 EST
A survey of Mučnik and Medvedev degrees
http://projecteuclid.org/euclid.bsl/1333560805<strong>Peter G. Hinman</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 2, 161--229.</p><p><strong>Abstract:</strong><br/>
We survey the theory of Mučnik (weak) and Medvedev
(strong) degrees of subsets of ω ω with particular attention
to the degrees of Π 0 1 subsets of ω 2. Sections 1—6 present
the major definitions and results in a uniform notation. Sections 7—16 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.
</p>projecteuclid.org/euclid.bsl/1333560805_Wed, 04 Apr 2012 13:34 EDTWed, 04 Apr 2012 13:34 EDT
On Tarski's foundations of the geometry of solids
http://projecteuclid.org/euclid.bsl/1333560806<strong>Arianna Betti</strong>, <strong>Iris Loeb</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 2, 230--260.</p><p><strong>Abstract:</strong><br/>
The paper [Tarski: Les fondements de la géométrie des corps ,
Annales de la Société Polonaise de Mathématiques ,
pp. 29—34, 1929] is in many ways remarkable.
We address three historico-philosophical issues that force themselves upon
the reader. First we argue that in this paper Tarski did not live up to his
own methodological ideals, but displayed instead a much more pragmatic approach.
Second we show that Leśniewski's philosophy and systems do not play the
significant role that one may be tempted to assign to them at first glance.
Especially the role of background logic must be at least partially allocated
to Russell's systems of Principia mathematica .
This analysis leads us, third, to a threefold distinction of the technical
ways in which the domain of discourse comes to be embodied in a theory. Having
all of this in place, we discuss why we have to reject the argument in
[Gruszczyński and Pietruszczak: Full development of Tarski's
Geometry of Solids , The Bulletin of Symbolic Logic ,
vol. 4 (2008), no. 4, pp. 481—540] according to which Tarski has made
a certain mistake.
</p>projecteuclid.org/euclid.bsl/1333560806_Wed, 04 Apr 2012 13:34 EDTWed, 04 Apr 2012 13:34 EDT
The stable core
http://projecteuclid.org/euclid.bsl/1333560807<strong>Sy-David Friedman</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 2, 261--267.</p><p><strong>Abstract:</strong><br/>
Vopěnka [2] proved long ago that every set of ordinals is set-generic over
HOD, Gödel's inner model of hereditarily ordinal-definable
sets. Here we show that the entire universe V is class-generic over
(HOD,S), and indeed over the even smaller inner model
𝕊=(L[S],S), where S is
the Stability predicate . We refer to the inner model 𝕊 as the
Stable Core of V .
The predicate S has a simple
definition which is more absolute than any definition of HOD;
in particular, it is possible to add reals which are not set-generic
but preserve the Stable Core (this is not possible for HOD by
Vopěnka's theorem).
</p>projecteuclid.org/euclid.bsl/1333560807_Wed, 04 Apr 2012 13:34 EDTWed, 04 Apr 2012 13:34 EDT
Reviews, Bull. Symbolic Logic 18, iss. 2 (2012)
http://projecteuclid.org/euclid.bsl/1333560808<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 2, 268--274.</p>projecteuclid.org/euclid.bsl/1333560808_Wed, 04 Apr 2012 13:34 EDTWed, 04 Apr 2012 13:34 EDT
2011 North American Annual Meeting of the Association for Symbolic Logic, University of California at Berkeley, Berkeley, CA, USA, March 24—27, 2011
http://projecteuclid.org/euclid.bsl/1333560809<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 2, 275--305.</p>projecteuclid.org/euclid.bsl/1333560809_Wed, 04 Apr 2012 13:34 EDTWed, 04 Apr 2012 13:34 EDT
Notices, Bull. Symbolic Logic 18, iss. 2 (2012)
http://projecteuclid.org/euclid.bsl/1333560810<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 2, 306--310.</p>projecteuclid.org/euclid.bsl/1333560810_Wed, 04 Apr 2012 13:34 EDTWed, 04 Apr 2012 13:34 EDT
Erratum
http://projecteuclid.org/euclid.bsl/1333560811<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 2, 311--311.</p>projecteuclid.org/euclid.bsl/1333560811_Wed, 04 Apr 2012 13:34 EDTWed, 04 Apr 2012 13:34 EDT
Gentzen's proof systems: byproducts in a work of genius
http://projecteuclid.org/euclid.bsl/1344861886<strong>Jan von Plato</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 3, 313--367.</p><p><strong>Abstract:</strong><br/>
Gentzen's systems of
natural deduction and sequent calculus were byproducts in his program
of proving the consistency of arithmetic and analysis. It is suggested
that the central component in his results on logical calculi was the use
of a tree form for derivations. It allows the composition of derivations
and the permutation of the order of application of rules, with a full
control over the structure of derivations as a result.
Recently found documents shed new light on the discovery of these calculi.
In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934—35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG . After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.
</p>projecteuclid.org/euclid.bsl/1344861886_Mon, 13 Aug 2012 08:45 EDTMon, 13 Aug 2012 08:45 EDT
Model theory of analytic functions: some historical comments
http://projecteuclid.org/euclid.bsl/1344861887<strong>Deirdre Haskell</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 3, 368--381.</p><p><strong>Abstract:</strong><br/>
Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal
work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's
proof of o-minimality of the theory of the reals with the exponential function, and the formulation of
Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments
and to reflect on today's open problems, in particular for theories of valued fields.
</p>projecteuclid.org/euclid.bsl/1344861887_Mon, 13 Aug 2012 08:45 EDTMon, 13 Aug 2012 08:45 EDT
Vaught's theorem on axiomatizability by a scheme
http://projecteuclid.org/euclid.bsl/1344861888<strong>Albert Visser</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 3, 382--402.</p><p><strong>Abstract:</strong><br/>
In his 1967 paper Vaught used an ingenious argument to show
that every recursively enumerable first order theory that directly
interprets the weak system VS of set theory is axiomatizable by
a scheme. In this paper we establish a strengthening of Vaught's theorem
by weakening the hypothesis of direct interpretability of
VS to direct interpretability of the finitely axiomatized fragment
VS 2 of VS. This improvement significantly increases the scope of
the original result, since VS is essentially
undecidable, but VS 2 has decidable extensions. We also
explore the ramifications of our work on finite axiomatizability
of schemes in the presence of suitable comprehension principles.
</p>projecteuclid.org/euclid.bsl/1344861888_Mon, 13 Aug 2012 08:45 EDTMon, 13 Aug 2012 08:45 EDT
Reviews, Bull. Symbolic Logic 18, iss. 3 (2012)
http://projecteuclid.org/euclid.bsl/1344861889<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 3, 403--412.</p>projecteuclid.org/euclid.bsl/1344861889_Mon, 13 Aug 2012 08:45 EDTMon, 13 Aug 2012 08:45 EDT
In Memoriam: Ernst Specker, 1920—2011
http://projecteuclid.org/euclid.bsl/1344861890<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 3, 413--417.</p>projecteuclid.org/euclid.bsl/1344861890_Mon, 13 Aug 2012 08:45 EDTMon, 13 Aug 2012 08:45 EDT
2011 European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium '11, Barcelona, Catalonia, Spain, July 11—16, 2011
http://projecteuclid.org/euclid.bsl/1344861891<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 3, 418--476.</p>projecteuclid.org/euclid.bsl/1344861891_Mon, 13 Aug 2012 08:45 EDTMon, 13 Aug 2012 08:45 EDT
Notices
http://projecteuclid.org/euclid.bsl/1344861892<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 3, 477--480.</p>projecteuclid.org/euclid.bsl/1344861892_Mon, 13 Aug 2012 08:45 EDTMon, 13 Aug 2012 08:45 EDT
The philosophy of logic
http://projecteuclid.org/euclid.bsl/1352802979<strong>Penelope Maddy</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 4, 481--504.</p><p><strong>Abstract:</strong><br/>
This talk surveys a range of positions on the fundamental metaphysical and
epistemological questions about elementary logic, for example, as a starting
point: what is the subject matter of logic—what makes its truths true?
how do we come to know the truths of logic? A taxonomy is approached by
beginning from well-known schools of thought in the philosophy of
mathematics—Logicism, Intuitionism, Formalism, Realism—and sketching
roughly corresponding views in the philosophy of logic. Kant, Mill, Frege,
Wittgenstein, Carnap, Ayer, Quine, and Putnam are among the
philosophers considered along the way.
</p>projecteuclid.org/euclid.bsl/1352802979_Tue, 13 Nov 2012 05:37 ESTTue, 13 Nov 2012 05:37 EST
Fifty years of the spectrum problem: survey and new results
http://projecteuclid.org/euclid.bsl/1352802980<strong>Arnaud Durand</strong>, <strong>Neil D. Jones</strong>, <strong>Johann A. Makowsky</strong>, <strong>Malika More</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 4, 505--553.</p><p><strong>Abstract:</strong><br/>
In 1952, Heinrich Scholz published a
question in The Journal of Symbolic Logic
asking for a characterization of spectra, i.e., sets of natural numbers
that are the cardinalities of finite models of first order sentences.
Günter Asser in turn asked whether the complement of a spectrum is always a spectrum.
These innocent questions turned out to be seminal for the development
of finite model theory and descriptive complexity.
In this paper we survey developments over the last 50-odd years
pertaining to the spectrum problem.
Our presentation follows conceptual developments rather than
the chronological order.
Originally a number theoretic problem, it has been approached by means
of recursion theory, resource bounded complexity theory,
classification by complexity of the defining sentences,
and finally by means of structural graph theory.
Although Scholz' question was answered in various ways, Asser's question remains open.
</p>projecteuclid.org/euclid.bsl/1352802980_Tue, 13 Nov 2012 05:37 ESTTue, 13 Nov 2012 05:37 EST
The graph-theoretic approach to descriptive set theory
http://projecteuclid.org/euclid.bsl/1352802981<strong>Benjamin D. Miller</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 4, 554--575.</p><p><strong>Abstract:</strong><br/>
We sketch the ideas behind the use of chromatic numbers in establishing descriptive
set-theoretic dichotomy theorems.
</p>projecteuclid.org/euclid.bsl/1352802981_Tue, 13 Nov 2012 05:37 ESTTue, 13 Nov 2012 05:37 EST
Reviews, Bull. Symbolic Logic 18, iss. 4 (2012)
http://projecteuclid.org/euclid.bsl/1352802982<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 4, 576--580.</p>projecteuclid.org/euclid.bsl/1352802982_Tue, 13 Nov 2012 05:37 ESTTue, 13 Nov 2012 05:37 EST
Officers and Committees of the Association for Symbolic Logic
http://projecteuclid.org/euclid.bsl/1352802983<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 4, 581--585.</p>projecteuclid.org/euclid.bsl/1352802983_Tue, 13 Nov 2012 05:37 ESTTue, 13 Nov 2012 05:37 EST
Members of the Association
http://projecteuclid.org/euclid.bsl/1352802984<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 4, 586--639.</p>projecteuclid.org/euclid.bsl/1352802984_Tue, 13 Nov 2012 05:37 ESTTue, 13 Nov 2012 05:37 EST
Notices, Bull. Symbolic Logic 18, iss. 4 (2012)
http://projecteuclid.org/euclid.bsl/1352802985<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 18, Number 4, 640--644.</p>projecteuclid.org/euclid.bsl/1352802985_Tue, 13 Nov 2012 05:37 ESTTue, 13 Nov 2012 05:37 EST
Descriptive inner model theory
http://projecteuclid.org/euclid.bsl/1368716862<strong>Grigor Sargsyan</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 1--55.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large.
</p>projecteuclid.org/euclid.bsl/1368716862_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Non-standard lattices and o-minimal groups
http://projecteuclid.org/euclid.bsl/1368716863<strong>Pantelis E. Eleftheriou</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 56--76.</p><p><strong>Abstract:</strong><br/>
We describe a recent program from the study of definable groups in
certain o-minimal structures. A central notion of this program is that of a
(geometric) lattice . We propose a definition of a lattice in an
arbitrary first-order structure. We then use it to describe, uniformly,
various structure theorems for o-minimal groups, each time recovering
a lattice that captures some significant invariant of the group at hand.
The analysis first goes through a local level, where a pertinent notion
of pregeometry and generic elements is each time introduced.
</p>projecteuclid.org/euclid.bsl/1368716863_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
The hyperuniverse program
http://projecteuclid.org/euclid.bsl/1368716864<strong>Tatiana Arrigoni</strong>, <strong>Sy-David Friedman</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 77--96.</p><p><strong>Abstract:</strong><br/>
The Hyperuniverse Program is a new approach to set-theoretic
truth which is based on justifiable principles and leads to the resolution
of many questions independent from ZFC. The purpose of this paper is
to present this program, to illustrate its mathematical content and
implications, and to discuss its philosophical assumptions.
</p>projecteuclid.org/euclid.bsl/1368716864_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
The Horn theory of Boole's partial algebras
http://projecteuclid.org/euclid.bsl/1368716865<strong>Stanley N. Burris</strong>, <strong>H. P. Sankappanavar</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 97--105.</p><p><strong>Abstract:</strong><br/>
This paper augments Hailperin's substantial efforts (1976/1986) to place Boole's
algebra of logic on a solid footing. Namely Horn sentences are used to give a modern
formulation of the principle that Boole adopted in 1854 as the foundation
for his algebra of logic—we call this principle The Rule of 0 and 1 .
</p>projecteuclid.org/euclid.bsl/1368716865_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Reviews, Bull. Symbolic Logic 19, iss. 1 (2013)
http://projecteuclid.org/euclid.bsl/1368716866<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 106--118.</p>projecteuclid.org/euclid.bsl/1368716866_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
In Memoriam: Michael Dummett, 1925—2011
http://projecteuclid.org/euclid.bsl/1368716867<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 119--122.</p>projecteuclid.org/euclid.bsl/1368716867_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
In Memoriam: Ruth Barcan Marcus, 1921—2012
http://projecteuclid.org/euclid.bsl/1368716868<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 123--126.</p>projecteuclid.org/euclid.bsl/1368716868_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
2011 Winter Meeting of the Association for Symbolic Logic, Washington Marriott Wardman Park Hotel, Washington, DC, December 27—29, 2011
http://projecteuclid.org/euclid.bsl/1368716869<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 127--134.</p>projecteuclid.org/euclid.bsl/1368716869_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Conference on Computability, Complexity and Randomness, Isaac Newton Institute, Cambridge, UK, July 2—6, 2012
http://projecteuclid.org/euclid.bsl/1368716870<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 135--136.</p>projecteuclid.org/euclid.bsl/1368716870_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Notices, Bull. Symbolic Logic 19, iss. 1 (2013)
http://projecteuclid.org/euclid.bsl/1368716871<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 1, 137--143.</p>projecteuclid.org/euclid.bsl/1368716871_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
The foundational problem of logic
http://projecteuclid.org/euclid.bsl/1368716899<strong>Gila Sher</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 2, 145--198.</p><p><strong>Abstract:</strong><br/>
The construction of a systematic philosophical foundation for logic is
a notoriously difficult problem. In Part One I suggest that the problem
is in large part methodological, having to do with the common philosophical
conception of “providing a foundation”. I offer an alternative to
the common methodology which combines a strong foundational
requirement (veridical justification) with the use of
non-traditional, holistic tools to achieve this result. In Part Two I
delineate an outline of a foundation for logic, employing the new
methodology. The outline is based on an investigation of why logic
requires a veridical justification, i.e., a justification which
involves the world and not just the mind, and what features or aspect
of the world logic is grounded in. Logic, the investigation suggests,
is grounded in the formal aspect of reality, and the outline proposes
an account of this aspect, the way it both constrains and enables
logic (gives rise to logical truths and consequences), logic's role
in our overall system of knowledge, the relation between logic and
mathematics, the normativity of logic, the characteristic traits of
logic, and error and revision in logic.
</p>projecteuclid.org/euclid.bsl/1368716899_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Shift-complex sequences
http://projecteuclid.org/euclid.bsl/1368716900<strong>Mushfeq Khan</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 2, 199--215.</p><p><strong>Abstract:</strong><br/>
A Martin-Löf random sequence is an infinite binary sequence with the
property that every initial segment $\sigma$ has prefix-free Kolmogorov
complexity $K(\sigma)$ at least $|\sigma| - c$,
for some constant $c \in \omega$. Informally,
initial segments of Martin-Löf randoms are highly
complex in the sense that they are not compressible by more
than a constant number of bits. However, all Martin-Löf randoms
necessarily have contiguous substrings of arbitrarily low complexity.
If we demand that all substrings of a sequence be uniformly complex,
then we arrive at the notion of shift-complex sequences. In this paper,
we collect some of the existing results on these sequences and contribute
two new ones. Rumyantsev showed that the measure of oracles that
compute shift-complex sequences is zero. We strengthen this result by
proving that the Martin-Löf random sequences that do not
compute shift-complex sequences are exactly the incomplete ones,
in other words, the ones that do not compute the halting problem.
In order to do so, we make use of the characterization by Franklin
and Ng of the class of incomplete Martin-Löf randoms via a
notion of randomness called difference randomness . Turning
to the power of shift-complex sequences as oracles, we show that
there are shift-complex sequences that do not compute
Martin-Löf random (or even Kurtz random) sequences.
</p>projecteuclid.org/euclid.bsl/1368716900_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Reviews, Bull. Symbolic Logic 19, iss. 2 (2013)
http://projecteuclid.org/euclid.bsl/1368716901<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 2, 216--222.</p>projecteuclid.org/euclid.bsl/1368716901_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
2011—2012 Winter Meeting of the Association for Symbolic Logic, John B. Hynes Veterans Memorial Convention Center Boston Marriott Hotel, and Boston Sheraton Hotel, Boston, MA, January 6—7, 2012
http://projecteuclid.org/euclid.bsl/1368716902<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 2, 223--235.</p>projecteuclid.org/euclid.bsl/1368716902_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
2012 North American Annual Meeting of the Association for Symbolic Logic, University of Wisconsin, Madison, WI, USA, March 31—April 3, 2012
http://projecteuclid.org/euclid.bsl/1368716903<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 2, 236--256.</p>projecteuclid.org/euclid.bsl/1368716903_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Twelfth Asian Logic Conference, Victoria University of Wellington, Wellington, New Zealand, December 15—20, 2011, with Student Day on December 14, 2011
http://projecteuclid.org/euclid.bsl/1368716904<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 2, 257--283.</p>projecteuclid.org/euclid.bsl/1368716904_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Notices, Bull. Symbolic Logic 19, iss. 2 (2013)
http://projecteuclid.org/euclid.bsl/1368716905<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 2, 284--288.</p>projecteuclid.org/euclid.bsl/1368716905_Thu, 16 May 2013 11:08 EDTThu, 16 May 2013 11:08 EDT
Logic in the 1930s: type theory and model theory
http://projecteuclid.org/euclid.bsl/1388953941<strong>Georg Schiemer</strong>, <strong>Erich H. Reck</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 433--472.</p><p><strong>Abstract:</strong><br/>
In historical discussions of twentieth-century logic, it is typically
assumed that model theory emerged within the tradition that adopted
first-order logic as the standard framework. Work within the type-theoretic
tradition, in the style of Principia Mathematica , tends to be
downplayed or ignored in this connection. Indeed, the shift from type theory
to first-order logic is sometimes seen as involving a radical break that
first made possible the rise of modern model theory. While comparing several
early attempts to develop the semantics of axiomatic theories in the 1930s,
by two proponents of the type-theoretic tradition (Carnap and Tarski)
and two proponents of the first-order tradition (Gödel and Hilbert),
we argue that, instead, the move from type theory to first-order logic is
better understood as a gradual transformation, and further, that the
contributions to semantics made in the type-theoretic tradition should
be seen as central to the evolution of model theory.
</p>projecteuclid.org/euclid.bsl/1388953941_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
Analytic equivalence relations and the forcing method
http://projecteuclid.org/euclid.bsl/1388953942<strong>Jindřich Zapletal</strong><p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 473--490.</p><p><strong>Abstract:</strong><br/>
I describe several ways in which forcing arguments can be used to yield
clean and conceptual proofs of nonreducibility, ergodicity and other results
in the theory of analytic equivalence relations. In particular, I present
simple Borel equivalence relations $E, F$ such that a natural proof of
nonreducibility of $E$ to $F$ uses the independence of the Singular
Cardinal Hypothesis at $\aleph_\omega$.
</p>projecteuclid.org/euclid.bsl/1388953942_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
Reviews, Bull. Symbolic Logic 19, iss. 4 (2013)
http://projecteuclid.org/euclid.bsl/1388953943<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 491--496.</p>projecteuclid.org/euclid.bsl/1388953943_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
2012—2013 Winter Meeting of the Association for Symbolic Logic, San Diego Convention Center, San Diego Marriott Marquis and Marina, San Diego, CA, January 11—12, 2013
http://projecteuclid.org/euclid.bsl/1388953944<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 497--511.</p>projecteuclid.org/euclid.bsl/1388953944_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
2013 Winter Meeting of the Association for Symbolic Logic, The Hilton New Orleans Riverside Hotel, New Orleans, Louisiana, February 20—21, 2013
http://projecteuclid.org/euclid.bsl/1388953945<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 512--518.</p>projecteuclid.org/euclid.bsl/1388953945_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
24th European Summer School on Logic, Language and Information (ESSLLI 2012), Opole University, Opole, Poland, August 6—17, 2012
http://projecteuclid.org/euclid.bsl/1388953946<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 519--522.</p>projecteuclid.org/euclid.bsl/1388953946_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
Officers and Committees of the Association for Symbolic Logic
http://projecteuclid.org/euclid.bsl/1388953947<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 523--527.</p>projecteuclid.org/euclid.bsl/1388953947_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
Members of the Association
http://projecteuclid.org/euclid.bsl/1388953948<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 528--583.</p>projecteuclid.org/euclid.bsl/1388953948_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST
Notices, Bull. Symbolic Logic 19, iss. 4 (2013)
http://projecteuclid.org/euclid.bsl/1388953949<p><strong>Source: </strong>Bull. Symbolic Logic, Volume 19, Number 4, 584--588.</p>projecteuclid.org/euclid.bsl/1388953949_Sun, 05 Jan 2014 15:32 ESTSun, 05 Jan 2014 15:32 EST