This Week in Logic at CUNY


November 27, Time 2:00 – 4:00 PM, Room 3309
Speaker: Melvin Fitting, CUNY
Title: Possible world semantics for first order LP

Abstract: Propositional Justification Logics are modal-like logics
in which the usual necessity operator is split into a family of more
complex terms called justifications. Instead of KA one finds t : A,
which can be read “t is a justification for A.” The structure of t
embodies, in a straightforward way, how we come to know A or verify A.
Many standard propositional modal logics have justification logic
counterparts, where the notion of counterpart has a precise definition
via what are called Realization Theorems. One can think of
justification logics as explicit versions of modal logics, with
conventional modal operators embodying justifications in an implicit
way. The first propositional justification logic was LP, the Logic of
Proofs, an explicit version of propositional S4. It was introduced by
Artemov as part of a project to provide an arithmetic semantics for
propositional intuitionistic logic. Since this initial work there has
been much study of propositional justification logics. But to
reiterate, all this was at the propositional level.

Recently Artemov and Yavorskaya defined a first-order extension of the
logic of proofs, FOLP. The original results on propositional LP were
shown to extend to the first-order case as well. This completed the
arithmetic semantics project for intuitionistic logic, but it also
introduced a new family of interesting explicit logics to study.

In an earlier article we introduced a possible world semantics for LP,
and for a few other propositional justification logics. On the one
hand this semantics elaborates the familiar Kripke semantics for modal
logics by adding machinery to model the behavior of explicit reasons,
and on the other hand it extends, in a direct way, an earlier LP
semantics of Mkrtychev. The purpose of the present work is to extend
this propositional semantics to a first-order version. The resulting
possible world semantics obeys a monotonicity condition, familiar from
propositional modal logics. This is natural because of the intended
application to intuitionistic logic. We postpone to future work the
study of constant domain versions. The current work is specifically
for the first-order version of LP. Simple modifications adapt the
results to several other first-order logics, and we discuss this
briefly too. The work appeared in a Tech Report. The current
presentation is much simpler, and hopefully more intuitive.

– – – – Wednesday, Nov 28, 2012 – – – –

GC Philosophy Colloquium
November 28, 2012, 4:15 pm GC Room 9204
Speaker: Joel David Hamkins, CUNY
Title: “Pluralism in set theory: does every mathematical statement have a
definite truth value?”

Abstract: I shall give a summary account of some current issues in the
philosophy of set theory, specifically, the debate on pluralism and the
question of the determinateness of set-theoretical and mathematical truth.
The traditional Platonist view in set theory, what I call the universe view,
holds that there is an absolute background concept of set and a
corresponding absolute background set-theoretic universe in which every
set-theoretic assertion has a final, definitive truth value. What I would
like to do is to tease apart two often-blurred aspects of this perspective,
namely, to separate the claim that the set-theoretic universe has a real
mathematical existence from the claim that it is unique. A competing view,
which I call the multiverse view, accepts the former claim and rejects the
latter, by holding that there are many distinct concepts of set, each
instantiated in a corresponding set-theoretic universe, and a corresponding
pluralism of set-theoretic truths. After framing the dispute, I shall argue
that the multiverse position explains our experience with the enormous
diversity of set-theoretic possibility, a phenomenon that is one of the
central set-theoretic discoveries of the past fifty years and one which
challenges the universe view. In particular, I shall argue that the
continuum hypothesis is settled on the multiverse view by our extensive
knowledge about how it behaves in the multiverse, and as a result it can no
longer be settled in the manner formerly hoped for. Commentary can be made

– – – – Thursday, Nov 29, 2012 – – – –

NY Philosophical Logic Group
Time: Thurs November 29, 7:15 p.m. to 9:15 p.m.
Place: 2nd floor seminar room, NYU Philosophy Dept (5, Washington Place)
Speaker: Branden Fitelson, Rutgers University
Title: A Framework for Grounding (Formal) Coherence Requirements

Abstract. In this talk, I explain how one can (vastly) generalize the
arguments of de Finetti and Joyce (for probabilism as a coherence
requirement for degrees of belief) to ground (formal) coherence
requirements for many different types of judgments. I will begin by
rehearsing the arguments of de Finetti and Joyce. Then, I will
present a general framework which was inspired by those arguments.
Finally, I will show how our framework can be used to yield (new)
coherence requirements for full belief and comparative confidence.
Applications to various “paradoxes of consistency” will also be
discussed. This talk draws on joint work with Rachael Briggs,
Fabrizio Cariani, Kenny Easwaran, and David McCarthy.

– – – – Friday, Nov 30, 2012 – – – –

Set Theory Seminar
Friday, November 30, 2012, 10:00am GC 6417
Speaker: Kaethe Minden
Title: The strength of the failure of the Kurepa Hypothesis

Abstract: I will show that Diamond Plus holds in inner models of the
form L[A], for subsets A of aleph one in the sense of L[A]. Putting
this together with the result from last meeting, that Diamond Plus
implies the Kurepa Hypothesis, I will show that if the Kurepa
Hypothesis fails, then aleph two is an inaccessible cardinal in L.
Again, putting this together with another result from the previous
seminar meeting, that one can force the failure of Kurepa’s Hypothesis
over a model with an inaccessible cardinal, this shows the
equiconsistency of the failure of Kurepa’s Hypothesis with the
existence of an inaccessible cardinal, over ZFC. These results are
mainly due to Silver and Solovay.

Model Theory Seminar
Friday, November 30, 2012, 12:30pm-1:45pm, GC 6417
Whanki Lee (CUNY Grad Center)
Thesis Defense

Logic Workshop
Friday, November 30, 2012 2:00 pm GC 6417
Prof. Alan D. Taylor (Union College)
The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems
(based on joint work with Christopher S. Hardin)

ABSTRACT: We consider problems in infinitary combinatorics related to
guessing the values of
a function at various points based on its values at certain other
points, often presented by way
of a hat-problem metaphor: there are a number of players who will have
colored hats placed on
their heads, and they wish to guess the colors of their own hats. A
directed graph often specifies
who can see which hats. The starting point here is an easy but
remarkable observation of Yuval
Gabay and Michael O’Connor: If each player sees all the others, then
— regardless of the size of
the set of players or the number of colors — there is a strategy
ensuring that only finitely many
agents guess incorrectly.

In an article entitled “Limit-like predictability for discontinuous
functions”, Chris Hardin and
I asked the following more general question: To what extent is a
function’s value at a point x
of a topological space determined by its values in an arbitrarily
small (deleted) neighborhood
of x? We showed that the best one can ever hope to do is to predict
correctly except on a
scattered set (a notion going back to Cantor). Moreover, we gave a
predictor whose error set,
in T_0 spaces, is always scattered. One consequence of this is a
result we published earlier in the Monthly with the deliberately
provocative title: “A peculiar connection between the axiom of
choice and predicting the future.” That article, in turn, led the
philosopher Alexander George
to publish a paper entitled “A proof of induction?” that revisited
some of David Hume’s original

In this seminar I’ll survey these results and present some new results
from two recent papers.
The first discusses some hat problems whose solution turns out to be
independent of ZFC + 2^omega = omega_2. The second connects hat
problems with Ramsey ultrafilters and p-point ultrafilters on omega.

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