University of Wroclaw

On some variants of discussive logic

Marek Nasieniewski,
in collaboration with Krystyna Mruczek-Nasieniewska
Department of Logic, Institute of Philosophy, Nicolaus Copernicus University in Toruń,
Toruń, Poland
{mnasien, mruczek}@umk.pl
Jaśkowski’s discussive logic was designed to express a logic and, in fact, a conse-
quence relation that would not be explosive. It is achieved through the so-called discussive
connectives of implication, conjunction, and equivalence. Intuitively, these connectives
are used to simulate simple interactions between debaters in a discussion. Discussive
implication is interpreted as meaning ‘if anyone states that p, then q’ (see [1, p. 150,
1969]) or, in modal language, [diamond]p → q. The later-introduced discussive conjunction (see
[2]) can be interpreted as a connection or summary of a statement p made by the given
participant and a statement q made by some participant. This resume is expressed by the
debater formulating the sentence p. So, in the modal language, discussive conjunction
p ∧d q is read as p ∧ [diamond]q. Discussive formulas are translated into the modal language, and
these translations can be called internal ones. Jaśkowski’s model is completed by taking
into consideration the point of view of an external observer. From this external perspec-
tive, all statements made during the discussion are only possible, which corresponds in
the discussive approach to the use of the possibility operator. Thus, a formula including
discussive connectives but also classically understood disjunction and negation is a thesis
of Jaśkowski’s discussive logic D2 iff its ‘internal’ translation to the modal language,
additionally preceded by the possibility operator, is a thesis of the normal modal logic S5.
We discuss selected variants of discussive logic determined by specific changes of the
initial intuitive model. It can be obtained either by changing the initial modal logic (see
e.g., [3]) that is used to interpret discussive connectives or by changing the object language
(see e.g., [4]), for example, by enriching it with modal connectives that are not allowed in
the original formulation of discussive logic. The letter approach can be connected with a
slight change of the intuitive model of discussion.
Reference
[1] Jaśkowski, S., “Rachunek zdań dla systemów dedukcyjnych sprzecznych”, Studia Soci-
etatis Scientiarum Torunensis 1948, Sect. A, I (5): 57–77. In English: “Propositional cal-
culus for contradictory deductive systems”, Studia Logica 1969, 24: 143–157. DOI: https:
//doi.org/10.1007/BF02134311 and “A propositional Calculus for inconsistent deduc-
tive systems”, Logic and Logical Philosophy 1999, 7: 35–56. DOI: http://dx.doi.org/
10.12775/LLP.1999.003
[2] Jaśkowski, S., “O koniunkcji dyskusyjnej w rachunku zda´n dla systemów dedukcyjnych
sprzecznych”, Studia Societatis Scientiarum Torunensis 1949, Sect. A, I (8): 171–172. In
English: “On the discussive conjunction in the propositional calculus for inconsistent deduc-
tive systems”, Logic and Logical Philosophy 1999, 7: 57–59. DOI: http://dx.doi.org/
10.12775/LLP.1999.004
[3] Mruczek-Nasieniewska, K., Nasieniewski M., “A Kotas-Style Characterisation of Minimal
Discussive Logic”, Axioms 2019, 8 (4): 108 (1–17). DOI: https://doi.org/10.3390/
axioms8040108
[4] Mruczek-Nasieniewska, K., Nasieniewski, M., Pietruszczak, A., “A modal extension of
Jaśkowski’s discussive logic D2”, Logic Journal of the IGPL 2019, 27 (4): 451–477. DOI:
https://doi.org/10.1093/jigpal/jzz014