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