Paraconsistency and the need for inﬁnite semantics
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
We show that most of the paraconsistent logics which have been investigated in the literature have no ﬁnite characteristic
matrices, and in the most important cases not even ﬁnite characteristic non-deterministic matrices (Nmatrices).
There are many paraconsistent logics that have been devel-
oped over the years. The simplest of them, like Asenjo–
Priest’s logic LP (Asenjo 1966;Priest1979), D’Ottaviano’s
(D’Ottaviano 1985; Epstein 2012), and Sette’s logic
(Sette 1973), were based from the start on ﬁnite matrices
(actually: three-valued matrices). However, it is known that
most of the paraconsistent logics that were designed on the
basis of other ideas, like relevance or formal inconsistency,
do not have ﬁnite characteristic matrices. Our goal in this
paper is to show that the no-ﬁnite-semantics phenomenon in
paraconsistent logics goes much further than has been known
so far. Many more such logics have no ﬁnite characteristic
matrices, and in the most important cases they do not have
even ﬁnite characteristic non-deterministic matrices (Nma-
2 Basic concepts
In what follows we denote by L a propositional language with
,...} of propositional variables, and
use p, q, r to vary over this set. The set of the well-formed
formulas of L is denoted by W(L), and ϕ, ψ,σ will vary
It should be emphasized that these results do not imply that the logics
investigated in this paper are undecidable. In fact, with one exception,
all of them are decidable. More information on this topic is given below
in Notes 3.3, 4.4, 4.10, 4.13,and4.18.
Communicated by C. Noguera.
School of Computer Science, Tel Aviv University, Tel Aviv,
over its elements. Var(ϕ) denotes the set of variables which
occur in ϕ.
Deﬁnition 2.1 A (propositional) logic is a pair L =L,
such that L is a propositional language, and is a structural
and non-trivial Tarskian consequence relation for L.
Now we deﬁne the notion of “paraconsistent logic”. A
very useful general deﬁnition can, e.g., be found in Arieli
and Avron (2015). However, for the purposes of this paper
the following much weaker notion used in Avron and Béziau
(2017) (following Marcos 2005) would do.
Deﬁnition 2.2 Let L =L,
be a propositional logic
whose language L includes the unary connective ¬.
1. ¬ is called a weak negation for L if the following condi-
tions are satisﬁed:
¬ p if p is atomic.
p if p is atomic.
2. A logic L =L,
is ¬-paraconsistent if ¬ is a weak
negation for L, and there are propositional variables p, q
such that p, ¬ p
From now on we assume that L includes ¬, and usually
write just “paraconsistent” instead of “¬-paraconsistent”.
Next we describe the semantics of non-deterministic
matrices, which generalizes the semantics of ordinary (deter-
ministic) matrices, but preserves most of its important
is structural, this implies that p, ¬ p
q whenever p and
q are distinct propositional variables.
Nmatrices were introduced in Avron and Lev (2001), Avron and
Lev (2005). Independently, non-deterministic truth tables were used
in Crawford and Etherington (1998)andIvlev(2000). Special cases