Institut Jean Nicod

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Mini Workshop on Logical Consequence

Organized by: Emmanuel Chemla & Paul Egré

With the support of the program "New Ideas in Mathematical Philosophy"

Vendredi 15 décembre 2017

Salle Pasteur (Pavillon Pasteur, 45 rue d'Ulm)



Joao Marcos: Do not be afraid of the Unknown
- Logical consequence explicated in terms of cognitive attitudes


Lorenzo Rossi: Graphs, truth, and conditional(s)


Joao Marcos (UFRN, BR & RUB, DE)

The received notions of logical consequence, either introduced by
semantical means or by way of some convenient proof formalism,
or even studied in their own right as abstract relations/operations
between sentences or collections of sentences, are often explicated
in terms of standard judgments such as assertion and refutation/denial.
As a matter of fact, from the semantical viewpoint such judgments are
often confused with truth-values.  For a fresh view on the matter, we
propose substituting judgments by a richer collection of primitive
cognitive attitudes concerning acceptance or rejection, on the part of
an agent, of a given piece of information, and organize such attitudes
into an opposition structure from which we show how to extract a
generous bidimensional notion of entailment, henceforth called
B-entailment, that generalizes the well-known approaches by Tarski,
by Scott, and by Shoesmith & Smiley ([7]).  We study and prove
a general characterization result about the underlying abstract
consequence relations in terms of a bilattice-based structure of
truth-values ([1]), show that it extends earlier results by G. Malinowski
and S. Frankowski ([6,4]), use it to explain some failures in the
meta-properties of reflexivity and transitivity of consequence,
and discuss how this connects to older and newer research on
the structure of truth-values or of the space of valuations ([2,8,5]).
Finally, we also prove a normal form result that shows how the
B-entailment formalism is expressive enough so as to define
any 4-valued (partial) nondeterministic matrix ([0,3]).

This reports on joint work with Carolina Blasio.


[0] Arnon Avron and Iddo Lev, Non-deterministic multiple-valued
structures, Journal of Logic and Computation, vol. 15 (2005), no. 3,
pp. 241-261.

[1] Carolina Blasio, Joao Marcos, and Heinrich Wansing,
An inferentially many-valued two-dimensional notion of entailment,
to appear in the Bulletin of the Section of Logic (2017).

[2] Alexander Bochman, Biconsequence relations: A four-valued
formalism of reasoning with inconsistency and incompleteness, Notre
Dame Journal of Formal Logic, vol. 39 (1998), no. 1, pp. 47-73.

[3] Matthias Baaz and Ori Lahav and Anna Zamansky, Effective
finite-valued semantics for labelled calculi, IJCAR 2012 (Manchester),
(B. Gräamlich and D. Miller and U. Sattler, editors), LNAI vol. 7364,
Springer, 2012, pp. 52-66.

[4] Szymon Frankowski, Formalization of a plausible inference,
Bulletin of the Section of Logic, vol. 33 (2004), pp. 41-52.

[5] Ole Hjortland, Speech acts, categoricity, and the meanings of
logical connectives, to appear in Notre Dame Journal of Formal Logic.

[6] Grzegorz Malinowski, Q-consequence operation, Reports on
Mathematical Logic, vol. 24 (1990), no. 1, pp. 49-59.

[7] D. J. Shoesmith and Timothy J. Smiley, Multiple-Conclusion Logic,
Cambridge University Press, Cambridge / MA, 1978.

[8] Yaroslav Shramko and Heinrich Wansing, Truth and Falsehood: An
inquiry into generalized logical values, Springer, 2011.


Lorenzo Rossi (Salzburg)
Graphs, truth, and conditional(s)

This paper proposes a novel analysis of semantic paradoxes,
employing some graph-theoretic techniques. I develop a new semantic
theory of truth that validates as many instances of the T-Schema (φ if
and only if ‘φ is true’) as possible, while blocking the ensuing
paradoxes by assigning to the paradoxical sentences special semantic
values that encode and represent their semantic behaviour. The theory
presented here differentiates between four main kinds of paradoxical
sentences – Liar-like paradoxes, Truth-teller-like paradoxes, Revenge
paradoxes, and Non-well-founded paradoxes – and accounts for their
semantics in one single model, thus providing a unified theory of
truth and paradoxes.