OCIE Seminar Double Session: Brice Halimi and Walter Carnielli

[Walter Carnielli]

A Double Session of the
OCIE Seminar in the History and Philosophy
of Mathematics and Logic



Please note the early start time (3 p.m.) for this double session.


Iterated modalities

Speaker:
Brice Halimi

(Université Paris Cité)

Friday, March 13, 2026
3 - 4:30 p.m. (PST)
Keck Center 153
Chapman University

Also in hybrid format on Zoom:
https://chapman.zoom.us/j/99548174703
Meeting ID: 995 4817 4703
Passcode: 000000

Abstract:

Modal iteration is the superposition of modal clauses, as when some
proposition is said to be, for instance, necessarily necessarily true.
It has a very strong meaning: Saying that a proposition is necessarily
necessarily true amounts, intuitively, to saying that that proposition
is necessarily true “whatever the range of the possible itself may
be.” In terms of possible worlds, the latter phrase implies that the
collection of all possible worlds has itself many possible
configurations, corresponding to “second-order” possible worlds. The
admissibility of iterated modalities is not self-evident at all and is
a fundamental issue throughout the philosophical tradition, which
deserves to be highlighted more than it has been so far. That will be
my starting point.

Then, assuming the meaningfulness of iterated modalities, how to
specify their semantics? In terms of possible worlds, the semantic
counterpart of modal iteration is a “change of scale,” i.e., the shift
to ranges of possible worlds in the form of second-order possible
worlds, then to ranges of second-order possible worlds in the form of
third-order possible worlds, and so on. Such a progression thus refers
to higher-order possible worlds, based on an open-ended collection of
ranges of possible worlds lying at higher and higher levels, in sharp
contrast to the metaphysical single-levelness that Leibniz bestowed on
his possible worlds.

I will argue that Kripke semantics for propositional modal logic
remains too Leibnizian in that respect. I will thus put forward
another semantic framework, geared to better formalizing modal change
of scale and the concept of higher-order possible world. The ensuing
modal semantics, developed with tools coming from differential
geometry, aims to generalize Kripke semantics and to endow modal logic
with a deepened geometric meaning.


Gödel's Incompleteness Theorems Beyond the Classical: LFIs and Provability Logic

Speaker:
Walter Carnielli

(Centre for Logic, Epistemology and the History of Science, University
of Campinas, Brazil)

Friday, March 13, 2026
4:30 - 6 p.m. (PST)
Keck Center 153
Chapman University

Also in hybrid format on Zoom:
https://chapman.zoom.us/j/99548174703
Meeting ID: 995 4817 4703
Passcode: 000000

Abstract:

Gödel's Incompleteness Theorems rank among the deepest results in the
foundations of mathematics. Their classical proofs, however, rely on
the Principle of Explosion — a classical principle long regarded as
logically heavy-handed, lacking both constructive force and logical
relevance: it derives anything whatsoever from a contradiction, with
no constructive justification. This raises a natural question: are
Gödel's results truly universal, or do they depend on the particular
logic in which they are formulated?

This talk investigates whether Gödel's theorems resist the pressure
when classical logic is replaced by a more flexible framework: the
Logics of Formal Inconsistency (LFIs). Unlike classical logic, LFIs
allow contradictions to occur in a controlled, local way without
trivializing the whole system — by means of an explicit consistency
operator that governs when Explosion may be applied.

We show that both Incompleteness Theorems can be reconstructed within
this paraconsistent setting, combined with tools from provability
logic and modal logic. The price to pay is explicit: classical global
assumptions must be replaced by careful local consistency conditions.
Once these are made transparent, Gödel's arguments go through.

The conclusion is philosophically significant: Gödel survives.
Incompleteness is not an artifact of classical logic, but a deep
structural boundary of formal reasoning — one that persists even when
consistency and contradiction are carefully pulled apart.

This is joint work with D, Fuenmayor, Bamberg, Germany.


This seminar session will also count as a session of the MPP Seminar /
Graduate Colloquium. The MPP Seminar will also meet on Thursday, March
12, at 11:30am.

For more information, please contact Marco Panza (pa...@chapman.edu)
or Lisa Beesley (bee...@chapman.edu. See the complete OCIE Seminar
Series calendar here.




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Walter Carnielli
CLE and Department of Philosophy
University of Campinas –UNICAMP, Brazil

AI2- Advanced Institute for Artificial Intelligence
Blog https://waltercarnielli.com/