Paraconsistent Logic

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Paraconsistent Logic

First published Tue Sep 24, 1996; substantive revision Fri Mar 20, 2009

The contemporary logical orthodoxy has it that, from contradictory
premises, anything can be inferred. To be more precise, let ⊨ be
a relation of logical consequence, defined either semantically or
proof-theoretically. Call ⊨ explosive if it validates
{A , ¬A} ⊨ B for every A
and B (ex contradictione quodlibet (ECQ)). The
contemporary orthodoxy, i.e., classical logic, is explosive, but also
some ‘non-classical’ logics such as intuitionist logic and
most other standard logics are explosive.

The major motivation behind paraconsistent logic is to
challenge this orthodoxy. A logical consequence relation, ⊨, is
said to be paraconsistent if it is not explosive. Thus, if
⊨ is paraconsistent, then even if we are in certain
circumstances where the available information is inconsistent, the
inference relation does not explode into triviality. Thus,
paraconsistent logic accommodates inconsistency in a sensible manner
that treats inconsistent information as informative.

There are several reasons driving such motivation. The development of
the systems of paraconsistent logic has depended on these. The prefix
‘para’ in English has two meanings: ‘quasi’
(or ‘similar to, modelled on’) or
‘beyond’. When the term ‘paraconsistent’ was
coined by Miró Quesada at the Third Latin America Conference on
Mathematical Logic in 1976, he seems to have had the first meaning in
mind. Many paraconsistent logicians,
however, have taken it to mean the second, which provided different
reasons for the development of paraconsistent logic as we will see
below.

This article is not meant to be a complete survey of paraconsistent
logic. The modern history of paraconsistent logic maybe relatively
short. Yet the development of the field has grown to the extent that a
complete survey is impossible. The aim of this article is to provide
some aspects and features of the field that are philosophically
salient. This does not mean that paraconsistent logic has no
mathematical significance or significance in such areas as computer
science and linguistics. Indeed, the development of paraconsistent
logic in the last two decades or so indicates that it has important
applications in those areas. However, we shall tread over them lightly
and focus more on the aspects that are of interest for philosophers
and philosophically trained logicians.


1. Paraconsistency

A logic is said to be paraconsistent iff its logical
consequence relation is not explosive. Paraconsistency is thus a
property of a consequence relation and of a logic. In the literature,
especially in the part of it that contains objections to
paraconsistent logic, there has been some confusion over the
definition of paraconsistency. So before going any further, we make
one clarification.

Paraconsistency, so defined, is to do with the inference relation
{A , ¬A} ⊨ B for every A
and B (ex contradictione quodlibet (ECQ)).
Dialetheism,

on the other hand, is the view that there are true contradictions. If
dialetheism is to be taken as a view that does not entail everything,
then a dialehtiest’s preferred logic must better be
paraconsistent. For dialetheism is the view that some
contradiction is true and it does not amount to trivialism
which is the view that everything, including every
contradiction, is true.

Now, a paraconsistent logician may feel the force pulling them towards
dialetheism. Yet the view that a
consequence relation should be paraconsistent does not entail the view
that there are true contradictions. Paraconsistency is a
property of an inference relation whereas dialetheism is a view about
some sentences (or propositions, statements, utterances or whatever,
that can be thought of as truth-bearers). The fact that one can define
a non-explosive consequence relation does not mean that some sentences
are true. That is, the fact that one can construct a model where a
contradiction holds but not every sentence of the language holds (or,
if the model theory is given intensionally, where this is the case at
some world) does not mean that the contradiction is true per
se
. Hence paraconsistency must be distinguished from dialetheism.

Moreover, as we will see below, many paraconsistent logics validate
the Law of Non-Contradiciton (LNC) (⊨ ¬(A
¬A)) even though they invalidate ECQ. In a discussion of
paraconsistent logic, the primary focus is not the obtainability of
contradictions but the explosive nature of a consequence relation.

2. Motivations

The reasons for paraconsistency that have been put forward seem
specific to the development of the particular formal systems of
paraconsistent logic. However, there are several general reasons for
thinking that logic should be paraconsistent. Before we summarise the
systems of paraconsistent logic and their motivations, we present some
general motivations for paraconsisent logic.

2.1 Inconsistent but Non-Trivial Theories

A most telling reason for paraconsistent logic is the fact that there
are theories which are inconsistent but non-trivial. Once we admit the
existence of such theories, their underlying logics must be
paraconsistent. Examples of inconsistent but non-trivial theories are
easy to produce. An example can be derived from the history of
science. (In fact, many examples can be given from this area.)
Consider Bohr’s theory of the atom. According to this, an
electron orbits the nucleus of the atom without radiating
energy. However, according to Maxwell’s equations, which formed
an integral part of the theory, an electron which is accelerating in
orbit must radiate energy. Hence Bohr’s account of the behaviour
of the atom was inconsistent. Yet, patently, not everything concerning
the behavior of electrons was inferred from it, nor should it have
been. Hence, whatever inference mechanism it was that underlay it,
this must have been paraconsistent.

2.2 Dialetheias (True Contradictions)

Despite the fact that dialetheism and paraconsistency needs to be
distinguished, dialetheism can be a motivation for paraconsistent
logic. If there are true contradictions (dialetheias), i.e., there are
sentences, A, such that both A and ¬A
are true, then some inferences of the form {A ,
¬A} ⊨ B must fail. For only true, and not
arbitrary, conclusions follow validly from true premises. Hence logic
has to be paraconsistent. One candidate for a dialetheia is
the liar paradox. Consider the sentence: ‘This sentence
is not true’. There are two options: either the sentence is true
or it is not. Suppose it is true. Then what it says is the
case. Hence the sentence is not true. Suppose, on the other hand, it
is not true. This is what it says. Hence the sentence is true. In
either case it is both true and not true. (See the entry on
dialetheism

in this encyclopedia for further details.)

2.3 Automated Reasoning

Paraconsistent logic is motivated not only by philosophical
considerations, but also by its applications and implications. One of
the applications is automated reasoning (information
processing
). Consider a computer which stores a large amount of
information. While the computer stores the information, it is also
used to operate on it, and, crucially, to infer from it. Now it is
quite common for the computer to contain inconsistent information,
because of mistakes by the data entry operators or because of multiple
sourcing. This is certainly a problem for database operations with
theorem-provers, and so has drawn much attention from computer
scientists. Techniques for removing inconsistent information have been
investigated. Yet all have limited applicability, and, in any case,
are not guaranteed to produce consistency. (There is no algorithm for
logical falsehood.) Hence, even if steps are taken to get rid of
contradictions when they are found, an underlying paraconsistent logic
is desirable if hidden contradictions are not to generate spurious
answers to queries.

2.4 Belief Revision

As a part of artificial intelligence research,
belief revision

is one of the areas that have been studied widely. Belief revision is
the study of rationally revising bodies of belief in the light of new
evidence. Notoriously, people have inconsistent beliefs. They may even
be rational in doing so. For example, there may be apparently
overwhelming evidence for both something and its negation. There may
even be cases where it is in principle impossible to eliminate such
inconsistency. For example, consider the ‘paradox of the
preface’. A rational person, after thorough research, writes a
book in which they
claim A1,…, An. But
they are also aware that no book of any complexity contains only
truths. So they rationally believe ¬(A1
∧…∧ An) too. Hence,
principles of rational belief revision must work on inconsistent sets
of beliefs. Standard accounts of belief revision, e.g., that of
Gärdenfors et al., all fail to do this, since they are
based on classical logic. A more adequate account is based on a
paraconsistent logic.

2.5 Mathematical Significance

Another area of significance for paraconsistent logic concerns certain
mathematical theories. Examples of such theories are
formal semantics, set theory,
and arithmetic. The latter concerns Gödel’s
Theorem
.

Formal Semantics and Set Theory

Semantics is the study that aims to spell out a theoretical
understanding of meaning. Most accounts of semantics insist that to
spell out the meaning of a sentence is, in some sense, to spell out
its truth-conditions. Now, prima facie at least, truth is a
predicate characterised by the Tarski T-scheme:

T(A) ↔ A

where A is a sentence and A is its
name. But given any standard means of self-reference, e.g.,
arithmetisation, one can construct a sentence, B, which says
that ¬T(B). The T-scheme gives
that T(B) ↔
¬T(B). It then follows
that T(B) ∧
¬T(B). (This is, of course,
just the liar paradox.)

The situation is similar in set theory. The naive, and intuitively
correct, axioms of set theory are the Comprehension Schema
and Extensionality Principle:

yx(xy
A)

x(x
yxz) → y = z

where x does not occur free in A. As was discovered
by Russell, any theory that contains the Comprehension Schema is
inconsistent. For putting ‘yy
for A in the Comprehension Schema and instantiating the
existential quantifier to an arbitrary such object
r’ gives:

y(yryy)

So, instantiating the universal quantifier to ‘r’ gives:

rrrr

It then follows that rrrr.

The standard approaches to these problems of inconsistency are, by and
large, ones of expedience. However, a paraconsistent approach makes it
possible to have theories of truth and sethood in which the
mathematically fundamental intuitions about these notions are
respected. For example, as Brady (1989) has shown, contradictions may
be allowed to arise in a paraconsistent set theory, but these need not
infect the whole theory.

Arithmetic

Unlike formal semantics and set theory, there may not be any obvious
arithmetical principles that give rise to contradiction. Nonetheless,
just like the classical non-standard models of arithmetic, there is a
class of inconsistent models of arithmetic (or more
accurately models of inconsistent arithmetic) which have an
interesting and important mathematical structure.

One interesting implication of the existence of inconsistent models of
arithmetic is that some of them are finite (unlike the classical
non-standard models). This means that there are some significant
applications in the metamathematical theorems. For example, the
classical Löwenheim-Skolem theorem states that Q
(Robinson’s arithmetic which is a fragment of Peano arithmetic) has
models of every infinite cardinality but has no finite
models. But, Q can be shown to have models of finite size too
by referring to the inconsistent models of arithmetic.

Gödel’s Theorem

It is not only the Löwenheim-Skolem theorem but also other
metamathematical theorems can be given a paraconsistent treatment. In
the case of other theorems, however, the negative results that are
often shown by the limitative theorems of metamathematics may no
longer hold. One important such theorem is Gödel’s theorem.

One version of Gödel’s first incompleteness theorem states
that for any consistent axiomatic theory of arithmetic, which can be
recognised to be sound, there will be an arithmetic truth – viz., its
Gödel sentence – not provable in it, but which can be established
as true by intuitively correct reasoning. The heart of
Gödel’s theorem is, in fact, a paradox that concerns the
sentence, G, ‘This sentence is not
provable’. If G is provable, then it is true and so not
provable. Thus G is proved. Hence G is true and so
unprovable. If an underlying paraconsistent logic is used to formalise
the arithmetic, and the theory therefore allowed to be inconsistent,
the Gödel sentence may well be provable in the theory
(essentially by the above reasoning). So a paraconsistent approach to
arithmetic overcomes the limitations of arithmetic that are supposed
(by many) to follow from Gödel’s theorem. (For other
‘limitative’ theorems of metamathematics, see Priest
2002.)

3. A Brief History of ex contradictione quodlibet

It is now standard to view ex contradictione quodlibet as a
valid form of inference. This contemporary view, however, should be
put in a historical perspective. It was towards the end of the 19th
century, when the study of logic achieved mathematical articulation,
that an explosive logical theory became the standard. With the work of
logicians such as Boole, Frege, Russell and Hilbert, classical logic
became the orthodox logical account.

In antiquity, however, no one seems to have endorsed the validity of
ECQ. Aristotle presented what is sometimes called the connexive
principle
: “it is impossible that the same thing should be
necessitated by the being and by the not-being of the same
thing.” (Prior Analytic II 4 57b3). (See the entry on
connexive logic

that has been developed based on this principle.) This principle
became a topic of debates in the Middle Ages or Medieval time. Though
the medieval debates seem to have been carried out in the context of
conditionals, we can also see it as debates about consequences. The
principle was taken up by
Boethius

(480–524 or 525) and
Abelard

(1079–1142), who considered two accounts of consequences. The
first one is a familiar one: it is impossible for the premises to be
true but conclusion false. The first account is thus similar to the
contemporary notion of truth-preservation. The second one is less
accepted recently: the sense of the premises contains that of the
conclusion. This account, as in relevant logics, does not permit an
inference whose conclusion is arbitrary. Abelard held that the first
account fails to meet the connexive principle and that the second
account (the account of containment) captured Aristotle’s principle.

Abelard’s position was shown to face a difficulty by Alberic of Paris
in the 1130s. Most medieval logicians didn’t, however, abandon
the account of validity based on containment or something
similar. (See, for example, Martin 1987.) But one way to handle the
difficulty is to reject the connexive principle. This approach, which
has become most influential, was accepted by the followers of Adam
Balsham or Parvipontanus (or sometimes known as Adam of The Little
Bridge) (12th CE). The Parvipontanians embraced the truth-preservation
account of consequences and the ‘paradoxes’ that are
associated with it. In fact, it was a member of the Parvipontanians,
William of Soissons, who discovered in the 12th century what we now
call the C.I. Lewis (independent) argument for ECQ. (See Martin 1986.)

The containment account, however, did not
disappear.
John Duns Scotus

(1266–1308) and his followers accepted the containment
account (see Martin 1996). The Cologne School of the late 15th
century argued against ECQ by rejecting disjunctive
syllogism
(see Sylvan 2000).

Now, the history of logic in the ‘East’, or more
specifically Asia, is moot. There is a tendency, for example, in Jaina
and Buddhist traditions to consider the possibility of statements
being both true and false. Moreover, the logics developed by the major
Buddhist logicians, Dignāga (5th century) and Dharmakīrti
(7th century) do not embrace ECQ. Their logical account is, in fact,
based on the ‘pervasion’ (Skt: vyāpti,
Tib: khyab pa) relation among the elements of an
argument. Just like the containment account of Abelard, there must be
a tighter connection between the premises and conclusion than the
truth-preservation account allows. (For the logic of Dharmakīrti
and its subsequent development, see for example Dunne 2004 and
Tillemans 1999.)

4. Modern History of Paraconsistent Logic

In the 20th century, the idea of challenging the explosive orthodoxy
occurred to different people at different times and places
independently of each other. They were often motivated by different
considerations. The earliest paraconsistent logics in the contemporary
era seem to have been given by two Russians. Starting about 1910,
Vasil’év proposed a modified Aristotelian syllogistic
including statements of the form: S is both P and
not P. In 1929, Orlov gave the first axiomatisation of the
relevant logic R which is paraconsistent. (On
Vasil’év, see Arruda 1977 and Arruda 1989, pp. 102f. On Orlov, see Anderson, Belnap
and Dunn 1992, p. xvii.)

The work of Vasil’év or Orlov did not make any impact at
the time. The first (formal) logician to have developed paraconsistent
logic was the Polish logician, Jaśkowski, who was a student of
Łukasiewicz, who envisaged paraconsistent logic in his critique
of Aristotle on LNC (Łukasiewicz 1951).

Paraconsistent logics were also developed in South America by Asenjo
(1954) and da Costa (1963) in their doctoral dissertations. Since
then, an active group of logicians has been working on paraconsistent
logic in Brazil, especially in Campinas and in São Paulo.

Paraconsistent logics in the forms of relevant logics were proposed in
England by Smiley in 1959 and also at about the same time, in a much
more developed form, in the USA by Anderson and Belnap. An active
group of relevant logicians grew up in Pittsburgh including Dunn and
Meyer. The development of paraconsistent logics (in the form of
relevant logics) was transported to Australia. R. Routley (later
Sylvan) and V. Routley (later Plumwood) discovered an intentional
semantics for some of Anderson/Belnap relevant logics. A school
developed around them in Canberra which included Brady and Mortensen,
and later Priest who, together with R. Routley, incorporated
dialetheism to the development.

By the mid-1970s, the development of paraconsistent logic became
international. In Belgium, a group of logicians around Batens in Ghent
grew up and remains active. Paraconsistent logic is also actively
investigated in Canada by Jennings, Schotch and their student
Brown. In 1997, the
First World Congress on Paraconsistency
was held at the University of Ghent in Belgium. The
Second World Congress
was held in São Sebastião (São Paulo, Brazil) in
2000, the
Third

in Toulous (France) in 2003 and the
Fourth

in Melbourne (Australia) in 2008. We now see logicians
working on paraconsistent logic in Bulgaria, China, France, Germany,
Italy, Japan, New Zealand to name just a few.

5. Systems of Paraconsistent Logic

A number of formal techniques to invalidate ECQ have been
devised. Most of the techniques have been summarised elsewhere, for
example Brown 2002 and Priest 2002. As
the interest in paraconsistent logic grew, different techniques
developed in different parts of the world. As a result, the
development of the techniques has somewhat a regional flavour (though
there are, of course, exceptions, and the regional differences can be
over-exagerated). (See Tanaka 2003.)

Most paraconsistent logicians do not propose a wholesale rejection of
classical logic. They usually accept the validity of classical
inferences in consistent contexts. It is the need to isolate an
inconsistency without spreading everywhere that motivates the
rejection of ECQ. Depending on how much revision one thinks is needed,
we have a technique for paraconsistency. The taxonomy given here is
based on the degree of revision to classical logic. Since the logical
novelty can be seen at the propositional level, we will concentrate on
the propositional paraconsistent logics.

5.1 Discussive Logic

The first formal paraconsistent logic to have been developed
was discussive (or discursive) logic by the
Polish logician Jaśkowski (1948). The thought behind discussive
logic is that, in a discourse, each participant puts forward some
information, beliefs or opinions. Each assertion is true according to
the participant who puts it forward in a discourse. But what is true
in a discourse on whole is the sum of assertions put forward by
participants. Each participant’s opinions may be self-consistent, yet
may be inconsistent with those of others. Jaśkowski formalised
this idea in the form of discussive logic.

A formalisation of discussive logic is by means of modelling a
discourse in a modal logic. For simplicity, Jaśkowski
chose S5. We think of each participant’s belief set as the
set of sentences true at a world in a S5
model M. Thus, a sentence A asserted by a
participant in a discourse is interpreted as “it is possible
that A” (◊A). That is, a
sentence A of discussive logic can be translated into a
sentence ◊A of S5. Then A holds in a
discourse iff A is true at some world
in M. Since A may hold in one world but not in
another, both A and ¬A may hold in a
discourse. Indeed, one should expect that participants disagree on
some issue in a rational discourse.

To be more precise, let d be a translation function from a
formula of discussive logic into a formula of S5. Then
(p)d = ◊p. For complex
formulas

A)d =
¬(Ad)

(AB)d =
AdBd

(AB)d =
AdBd

(AB)d =
AdBd

(AB)d =
AdBd

It is easy to show that B is a discussive consequence
of A1, …, An iff
the formula ◊A1d
(… ⊃ (◊And
⊃ ◊Bd)…) is a theorem
of S5.

To see that discussive logic is paraconsistent, consider a S5
model, M, such that A holds
at w1, ¬A holds at a different
world w2 but B does not hold at any world
for some B. Then both A and ¬A hold,
yet B does not hold in M. Hence discussive logic
invalidates ECQ.

However, there is no S5 model where A
¬A holds at some world. So an inference of the form
{A ∧ ¬A} ⊨ B is valid in
discussive logic. This means that, in discussive
logic, adjunction ({A, ¬A}
A ∧ ¬A) fails. But one can define a discussive conjunction, ∧d,
as A ∧ ◊B (or ◊A
B). Then adjunction holds for
d (Jaśkowski 1949).

One difficulty is a formulation of a conditional. In S5, the
inference from ◊p and ◊(pq)
to ◊q fails. Jaśkowski chose to introduce a
connective which he called discussive implication,
d, defined as ◊AB. This
connective can be understood to mean that “if some participant
states that A, then B”. As the inference from
AB and ◊A to
B is valid in S5, modus ponens for
d holds in discussive logic. A discussive
bi-implication, ≡d, can also be defined as
(◊AB) ∧ ◊(◊A
B) (or ◊(◊AB) ∧
(◊AB)).

5.2 Non-Adjunctive Systems

A non-adjunctive system is a system that does not validate adjunction
(i.e., {A, B} ⊭ A
B). As we saw above, discussive logic without a
discussive conjunction is non-adjunctive. Another non-adjunctive
strategy was suggested by Rescher and Manor 1970-71. In effect, we
can conjoin premises, but only up to maximal
consistency. Specifically, if Σ is a set of premises, a
maximally consistent subset is any consistent subset Σ′
such that if A ∈ Σ − Σ′ then
Σ′ ∪ {A} is inconsistent. Then we say
that A is a consequence of Σ iff A is a
classical consequence of Σ′ for some maximally consistent
subset Σ′. Then {p, q}
pq but {p, ¬p}
p ∧ ¬p.

5.3 Preservationism

In the non-adjunctive system of Rescher and Manor, a consequence
relation is defined over some maximally consistent subset of the
premises. This can be seen as a way to ‘measure’
the level of consistency in the premise set. The level of
{p, q} is 1 since the maximally consistent subset is
the set itself. The level of {p, ¬p}, however,
is 2: {p} and {¬p}.

If we define a consequence relation over some maximally consistent
subset, then the relation can be thought of as preserving the level of
consistent fragments. This is the approach which has come to be
called preservationism. It was first developed by the
Canadian logicians Ray Jennings and Peter Scotch.

To be more precise, a (finite) set of formulas, Σ, can be
partitioned into classically consistent fragments whose union is
Σ. Let ⊢ be the classical consequence
relation. A covering of Σ is a set
i : iI}, where
each member is consistent, and Σ
= iI
Σi. The level of
Σ, l(Σ), is the least n such that
Σ can be partitioned into n sets if there is
such n, or ∞ if there is no such n. A
consequence relation, called forcing, [⊢, is defined as
follows. Σ [⊢ A iff l(Σ) =
∞, or l(Σ) = n and for every covering of%0

antics.

The semantics for relevant logics were developed by Fine (1974), Routley and Routley (1972), Routley and Meyer (1993) and Urquhart (1972). (There are also algebraic semantics. See for example Dunn and Restall 2002, pp. 48ff.) In the Routleys-Meyer semantics, based on possible-world semantics (which is the most studied semantics for relevant logics, especially in Australia), conjunction and disjunction behave in the usual way. But each world, w, has an associate world, w*, and negation is evaluated in terms of w*: ¬A is true at w iff A is false, not at w, but at w*. Thus, if A is true at w, but false at w*, then A ∧ ¬A is true at w. To obtain the standard relevant logics, one needs to add the constraint that w** = w. As is clear, negation in these semantics is an intensional operator.

The primary concern with relevant logics is not so much with negation as with a conditional connective → (satisfying modus ponens). In relevant logics, if AB is a logical truth, then A is relevant to B, in the sense that A and B share at least one propositional variable.

Semantics for the relevant conditional are obtained by furnishing each Routleys-Meyer model with a ternary relation. In the simplified semantics of Priest and Sylvan 1992 and Restall 1993 and 1995, worlds are divided into normal and non-normal. If w is a normal world, AB is true at w iff at all worlds where A is true, B is true. If w is non-normal, AB is true at w iff for all x, y, such that Rwxy, if A is true at x, B is true at y. If B is true at x but not at y where Rwxy, then BB is not true at w. Then one can show that A → (BB) is not a logical truth. (Validity is defined as truth preservation over normal worlds.) This gives the basic relevant logic, B. Stronger logics, such as the logic R, are obtained by adding constraints on the ternary relation.

There are also versions of world-semantics for relevant logics based on Dunn’s relational semantics for FDE. Then negation is extensional. A conditional connective, now needs to be given both truth and falsity conditions. So we have: AB is true at w iff for all x, y, such that Rwxy, if A is true at x, B is true at y; and AB is false at w iff for some x, y, such that Rwxy, if A is true at x, B is false at y. Adding various constraints on the ternary relation provides stronger logics. However, these logics are not the standard relevant logics developed by Anderson and Belnap. To obtain the standard family of relevant logics, one needs neighbourhood frames. (See Mares 2004.) Further details concerning relevant logics can be found in the article on that topic in this encyclopedia.

Bibliography

For Paraconsistency in general:

  • Priest, G., Routley, R., and Norman, J. (eds.) (1989). Paraconsistent Logic: Essays on the Inconsistent, München: Philosophia Verlag.
  • Priest, G. (2002). “Paraconsistent Logic”, Handbook of Philosophical Logic (Second Edition), Vol. 6, D. Gabbay and F. Guenthner (eds.), Dordrecht: Kluwer Academic Publishers, pp. 287-393.

For Inconsistent but Non-Trivial Theories

  • Brown, B. and G. Priest. (2004). “Chunk and Permeate: A Paraconsistent Inference Strategy. Part 1: The Infinitesimal Calculus”, Journal of Philosophical Logic, 33: 379-388.

On Dialetheism

  • Priest, G. (1987). In Contradiction: A Study of the Transconsistent, Dordrecht: Martinus Nijhoff; second edition, Oxford: Oxford University Press, 2006.
  • Priest, G., J.C. Beall and B. Armour-Garb (eds.) (2004). The Law of Non-Contradiction, Oxford: Oxford University Press.

For Automated Reasoning

  • Belnap, N.D., Jr. (1992). “A Useful Four-valued Logic: How a computer should think”, Entailment: The Logic of Relevance and Necessity, Volume II, A.R. Anderson, N.D. Belnap, Jr, and J.M. Dunn, Princeton: Princeton University Press; first appeared as “A Usuful Four-valued Logic”, Modern Use of Multiple-valued Logic, J.M. Dunn and G. Epstein (eds.), Dordrecht: D. Reidel, 1977, and “How a Computer Should Think”, Comtemporary Aspects of Philosophy, G. Ryle (ed.), Oriel Press, 1977.
  • Besnard, P. and Hunter, A. (eds.) (1998). Handbook of Deasible Reasoning and Uncertainty Management Systems, Volume 2, Reasoning with Actual and Potential Contradictions, Dordrecht: Kluwer Academic Publishers.

For Belief Revision

  • Priest, G. (2001). “Paraconsistent Belief Revision”, Theoria, 67: 214-228.
  • Restall, G. and Slaney, J. (1995). “Realistic Belief Revision”, Proceedings of the Second World Conference in the Fundamentals of Artificial Intelligence, M. De Glas and Z. Pawlak (eds.), Paris: Angkor, pp. 367-378.
  • Tanaka, K. (2005). “The AGM Theory and Inconsistent Belief Change”, Logique et Analyse, 48: 113-150.

For Mathematical Significance

  • Brady, R.T. (1989). “The Non-Triviality of Dialectical Set Theory”, Paraconsistent Logic: Essays on the Inconsistent, G. Priest, R. Routley and J. Norman (eds.), München: Philosophia Verlag, pp. 437-471.
  • Mortensen, C. (1995). Inconsistent Mathematics, Dordrecht: Kluwer Academic Publishers.
  • Priest, G. (2003). “Inconsistent Arithmetic: Issues Technical and Philosophical”, in Trends in Logic: 50 Years of Studia Logica (Studia Logica Library, Volume 21), V. F. Hendricks and J. Malinowski (eds.), Dordrecht: Kluwer Academic Publishers, pp. 273-99.

For a History of ex contradictione quodlibet

  • Sylvan, R. (2000). “A Preliminary Western History of Sociative Logics”, Sociative Logics and Their Applications: Essays by the late Richard Sylvan, D. Hyde and G. Priest (eds.), Aldershot: Ashgate Publishers.

For Modern History of Paraconsistent Logic

  • Arruda, A. (1989). “Aspects of the Historical Development of Paraconsistent Logic”, Paraconsistent Logic: Essays on the Inconsistent, G. Priest, R. Routley and J. Norman (eds.), München: Philosophia Verlag, pp. 99-130.
  • Priest, G. (2007). “Paraconsistency and Dialetheism”, in Handbook of the History of Logic , Volume 8, D. Gabbay and J. Woods (eds.), Amsterdam: North Holland, pp. 129-204.

For the Systems of Paraconsistent Logic in general

  • Brown, B. (2002). “On Paraconsistency”, in A Companion to Philosophical Logic, Dale Jacquette (ed.), Oxford: Blackwell, pp. 628-650.

For Discussive Logic

  • Jaśkowski, S. (1948). “Rachunek zdań dla systemów dedukcyjnych sprzecznych”, Studia Societatis Scientiarun Torunesis (Sectio A), 1 (5): 55-77; an English translation appeared as “Propositional Calculus for Contradictory Deductive Systems”, Studia Logica, 24 (1969): 143-157.
  • Jaśkowski, S. (1949). “O koniunkcji dyskusyjnej w rachunku zdań dla systemów dedukcyjnych sprzecznych”, Studia Societatis Scientiarum Torunensis (Sectio A), 1 (8): 171-172; an English translation appeared as “On the Discussive Conjunction in the Propositional Calculus for Inconsistent Deductive Systems”, Logic and Logical Philosophy, 7 (1999): 57-59.
  • da Costa, N.C.A. and Dubikajtis, L. (1977). “On Jaśkowski’s Discussive Logic”, in Non-Classical Logics, Modal Theory and Computability, A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.), Amsterdam: North-Holland Publishing Company, pp. 37-56.

For Non-Adjunctive Systems

  • Rescher, N. and R. Manor (1970-71). “On Inference from Inconsistent Premises”, Theory and Decision, 1: 179-217.

For Preservationism

  • Schotch, P.K. and R.E. Jennings (1980). “Inference and Necessity”, Journal of Philosophical Logic, 9: 327-340.

For Adaptive Logics

  • Batens, D. (2001). “A General Characterization of Adaptive Logics”, Logique et Analyse, 173-175: 45-68.
  • Batens, D. (2007). “A Universal Logic Approach to Adaptive Logics”, Logica Universalis, 1: 221-242.

For Logics of Formal Inconsistency

  • Carnielli, W.A., M.E. Coniglio and J. Marcos (2007). “Logics of Formal Inconsistency”, Handbook of Philosophical Logic, Volume 14 (Second Edition), D. Gabbay and F. Guenthner (eds.), Berlin: Springer, pp. 15-107.
  • da Costa, N.C.A. (1974). “On the Theory of Inconsistent Formal Systems”, Notre Dame Journal of Formal Logic, 15 (4): 497-510.

For Many-Valued Logics

  • Asenjo, F.G. (1966). “A Calculus of Antinomies”, Notre Dame Journal of Formal Logic, 7: 103-5.
  • Dunn, J.M. (1976). “Intuitive Semantics for First Degree Entailment and Coupled Trees”, Philosophicl Studies, 29: 149-68.
  • Priest, G. (1979). “Logic of Paradox”, Journal of Philosophical Logic, 8: 219-241.

For Relevant Logics

  • Anderson, A. and N. Belnap. (1975). Entailment: The Logic of Relevance and Necessity, Volume 1, Princeton: Princeton University Press.
  • Anderson, A., N. Belnap and J.M. Dunn. (1992). Entailment: The Logic of Relevance and Necessity, Volume 2, Princeton: Princeton University Press.
  • Dunn, J.M. and G. Restall (2002). “Relevance Logic”, Handbook of Philosophical Logic, Volume 6, second edition, D. Gabbay and F. Guenthner (eds.), Dordrecht: Kluwer Academic Publishers, pp. 1-136.
  • Routley, R., Plumwood, V., Meyer, R.K., and Brady, R.T. (1982). Relevant Logics and Their Rivals, Volume 1, Ridgeview: Atascadero.
  • Brady, R.T. (ed.) (2003). Relevant Logics and Their Rivals, Volume 2, Aldershot: Ashgate.

Other Works Cited

  • Arruda, A. (1977). “On the Imaginary Logic of N.A. Vasil’év”, in Non-Classicl Logic, Model Theory and Cpmputability, A. Arruda, N, da Costa and R. Chuanqui (eds.), Amsterdam: North Holland, pp. 3-24.
  • da Costa, N.C.A. and E.H. Alves (1977). “Semantical Analysis of the Calculi Cn”, Notre Dame Journal of Formal Logic, 18 (4): 621-630.
  • Dunne, J.D. (2004). Foundations of Dharmakīrti’s Philosophy, Boston: Wisdom Publications.
  • Fine, K. (1974). “Models for Entailment”, Journal of Philosophical Logic, 3: 347-372.
  • Loparic, A. (1977). “Une etude semantique de quelques calculs propositionnels”, Comptes Rendus Hebdomadaires des Seances de l’Academie des Sciences, 284: 835-838.
  • Łukasiewicz, J. (1951). Atistotle’s Syllogistic: From the Standpoint of Modern Formal Logic, Oxford: Oxford University Press.
  • Mares, E. (2004). “‘Four-Valued’ Semantics for the Relevant Logic R”, Journal of Philosophical Logic, 33: 327-341.
  • Martin, C. (1986). “William’s Machine”, Journal of Philosophy, 83: 564-572.
  • Martin, C. (1987). “Embarrassing Arguments and Surprising Conclusions in the Development Theories of the Conditional in the Twelfth Century”, Gilbert De Poitiers Et Ses Contemporains, J. Jolivet, A. De Libera (eds.), Naples: Bibliopolis, pp. 377-401.
  • Martin, C. (1996). “Impossible Positio as the Foundation of Metaphysics or, Logic on the Scotist Plan?”, Vestigia, Imagines, Verba: Semiotics and Logic in Medieval Theological Texts, C. Marmo (ed.), Turnhout: Brepols, pp. 255-276.
  • Priest, G. and R. Sylvan (1992). “Simplified Semantics for Basic Relevant Logics”, Journal of Philosophical Logic, 21: 217-232.
  • Restall, G. (1993). “Simplified Semantics for Relevant Logics (and some of their rivals)”, Journal of Philosophical Logic, 22: 481-511.
  • Restall, G. (1995). “Four-Valued Semantics for Relevant Logics (and some of their rivals)”, Journal of Philosophical Logic, 24: 139-160.
  • Routley, R. and R. Meyer (1993). “Semantics of Entailment”, Truth, Syntax and Modality, H. Leblanc (ed.), Amsterdam: North Holland, pp. 194-243.
  • Routley, R. and V. Routley (1972). “Semantics of First Degree Entailment”, Noûs, 3: 335-359.
  • Tanaka, K. (2003). “Three Schools of Paraconsistency”, The Australasian Journal of Logic, 1: 28-42.
  • Tillemans, Tom J.F. (1999). Scripture, Logic, Language: Essays on Dharmakīrti and His Tibetan Successors, Boston: Wisdom Publications.
  • Urquhart, A. (1972). “Semantics for Relevant Logics”, Journal of Symbolic Logic, 37: 159-169.

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