** 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**

http://web.gc.cuny.edu/philosophy/events/colloquium/12_fall.htm

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

here.

– – – – 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

assumptions.

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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