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Paraconsistent Logic
The development of paraconsistent logic was initiated in order
to challenge the logical principle that anything follows from
contradictory premises, ex contradictione quodlibet (ECQ). Let
be a relation of
logical consequence, defined either semantically or
prooftheoretically. Let us say that
is explosive iff for every formula
A and B, {A , ~A}
B. Classical logic,
intuitionistic logic, and most other standard logics are explosive. A
logic is said to be paraconsistent iff its relation of logical
consequence is not explosive.
The modern history of paraconsistent logic is relatively short. Yet
the subject has already been shown to be an important development in
logic for many reasons. These involve the motivations for the subject,
its philosophical implications and its applications. In the first half
of this article, we will review some of these. In the second, we will
give some idea of the basic technical constructions involved in
paraconsistent logics. Further discussion can be found in the
references given at the end of the article.
A most telling reason for paraconsistent logic is the fact that there
are theories which are inconsistent but nontrivial. Clearly, once we
admit the existence of such theories, their underlying logics must be
paraconsistent. Examples of inconsistent but nontrivial 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. Hence, whatever inference mechanism it was that underlay it, this
must have been paraconsistent.
The importance of paraconsistent logic also follows if, more
contentiously, but as some people have argued, there are true
contradictions (dialetheias), i.e., there are sentences, A,
such that both A and ~A are true. If there are
dialetheias then some inferences of the form {A , ~A}
B must fail. For only true conclusions follow validly from
the true premises. Hence logic has to be paraconsistent. A plausible
example of 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.
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
theoremprovers, 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.
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 A_{1},…, A_{n}.
But they are also aware that no book of any complexity contains only
truths. So they rationally believe ~(A_{1}
&…& A_{n}) 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.
Other applications of paraconsistent logic concern theories of
mathematical significance. Examples of such theories are 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
truthconditions. Now, prima facie at least, truth is a
predicate characterised by the Tarski Tscheme:
T(A) ↔
A,
where A is a sentence and A is its
name. But given any standard means of selfreference, e.g.,
arithmetisation, one can construct a sentence, B, which means
that ~T(B). The Tscheme 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:
∃y∀x(x ∈ y
↔ A)
∀x(x ∈
y ↔ x ∈ z) → 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 ‘y
y’ for A in the
Comprehension Schema and instantiating the existential quantifier to an
arbitrary such object ‘r’ gives:
∀y(y
∈ r ↔ y
y)
So, instantiating the universal quantifier to ‘r’
gives:
r ∈ r ↔ r
r
It then follows that r ∈ r & r
r.
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
fundamental intuitions about these notions are respected. The
contradictions may be allowed to arise, but these need not infect the
rest of the theory.
Paraconsistent logic also has important philosophical ramifications.
One example of this concerns 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.
The foregoing discussion indicates some of the motivations for
paraconsistent logic, its applications and implications. We will now
indicate some of the main approaches to paraconsistency. There are many
different paraconsistent logics. Most of them can be defined in terms
of a semantics which allows both A and ~A to hold in
an interpretation. Validity is then defined in terms of the
preservation of holding in an interpretation, and so ECQ fails. We will
illustrate this with four kinds of propositional paraconsistent logics:
nonadjunctive, nontruthfunctional,
manyvalued, and relevant. (Paraconsistent quantified
logics are straightforward extensions of these.) In all the following
systems, not only ECQ fails, but so does the Disjunctive Syllogism
(DS), defined as the following inference rule: {A, ~A
B}
B. In particular, then, if one defines
the material conditional, A ⊃ B, as ~A
B (as usual) then modus
ponens for this fails.
Let us start with nonadjunctive systems, so called because the
inference from A and B to A & B
fails. The first of these to be produced was also the first formal
paraconsistent logic. This was Jaskowski's discussive (or
discursive) logic. In a discourse, each participant
puts forward some information, beliefs, or opinions. What is true in a
discourse is the sum of opinions given by participants. Each
participant's opinions are taken to be selfconsistent, but may be
inconsistent with those of others. To formalise this idea, take an
interpretation, I, to be one for a standard modal logic, say
S5. Each participant's belief set is the set of sentences true
in a possible world in I. Thus, A holds in I
iff A holds at some world in I. Clearly, one
may have both A and ~A (but not A &
~A) holding in an interpretation. Since modus ponens
for ⊃ fails, Jaskowski introduced a connective he called discussive
implication, ⊃_{d}, defined as
(A ⊃ B). It is easy to check that
in S5 discussive implication satisfies modus ponens.
The study of nontruthfunctional systems was initiated by da Costa
(who has also produced several other kinds of system). The main idea
here was to maintain the apparatus of some positive logic, say
classical or intuitionistic, but to allow negation in an interpretation
to behave nontruthfunctionally. Thus, take an interpretation to be a
function which maps formulas to 1 or 0; &,
,
and → behave in the usual (classical) way, but the
value of ~A is independent of that of A. In
particular, both may take the value 1. Negation has no significant
properties under these semantics. Various properties of negation may be
obtained by adding further constraints on interpretations. If we add
the requirements that, for any A, either A or
~A must take the value 1 (giving the Law of Excluded Middle)
and that whenever ~~A takes the value 1, so does A,
we obtain the core of da Costa's systems
C_{i} , for finite i. If we start
with an appropriate semantics for positive intuitionist logic, and
proceed in the same way, we obtain da Costa's logic
C_{ω}. If we write A° for
~(A & ~A) then it is natural to take it as
expressing the consistency of A. Further postulates
constraining how A° behaves differentiate between the
C_{i} systems for finite i.
Perhaps the simplest way of generating a paraconsistent logic, first
proposed by Asenjo, is to use a manyvalued logic, that is, a logic
with more than two truth values. The formulas which hold in a
manyvalued interpretations are those which have a value said to be
designated. A manyvalued logic will therefore be
paraconsistent if it allows both a formula and its negation to be
designated. The simplest strategy is to use three truth values:
true (only) and false (only), which function as in
classical logic, and both truth and false (which, naturally,
is a fixed point for negation). Both varieties of truth are designated.
This is the approach of the paraconsistent logic LP. If one
adds a fourth value, neither true nor false, which behaves in
an appropriate way, one obtains Dunn's semantics for First Degree
Entailment, which is a fragment of
relevant logics.
If one takes the truth values to be the real numbers
between 0 and 1, with a suitable set of designated values, the logic
will be a natural paraconsistent fuzzy logic.
Relevant logics were pioneered by Anderson and Belnap. Worldsemantics
for them were developed by R. and V.Routley and Meyer. In an
RoutleysMeyer interpretation for such logics, conjunction and
disjunction behave in the usual way. But each world, w, has
an associate world, w*; and ~A is true at w
iff A is false, not at w, but w*. Thus, if
A is true at w, but false at w*, 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 concern with relevant logics is not so much with negation as
with a conditional connective, → (satisfying modus
ponens). Semantics for this are obtained by furnishing each
interpretation with a ternary relation, R. In the
simplified semantics of Priest, Sylvan and Restall, worlds are divided
into normal and nonnormal. If w is a normal world, A
→ B is true at w iff at all worlds where
A is true, B is true. If w is nonnormal,
A → B is true at w iff for all
x, y, such that Rwxy, if A is true
at x, B is true at y. (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 worldsemantics for relevant logics based on Dunn's
fourvalued semantics for First Degree Entailment. In these, an evaluation
is a relation between a formula and {true, false} rather than a function.
Then negation is extensional. For a conditional connective, it seems natural
to define it as: A → B is true at w iff for all
x, y, such that Rwxy, if A
is true at x, B is true at y; and
A → B 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. Further details concerning
relevant logics
can be found in the article on that topic in this encyclopedia.
 Priest, G., Routley, R., and Norman, J. (eds.) Paraconsistent
Logic: Essays on the Inconsistent, Philosophia Verlag,
München, 1989.
 Priest, G. "Paraconsistent Logic", Handbook of Philosophical
Logic (Second Edition), Vol. 6, D. Gabbay and F. Guenthner (eds.),
Kluwer Academic Publishers, Dordrecht, pp. 287393, 2002.
 Priest, G. "Logic of Paradox", Journal of Philosophical
Logic, Vol. 8, pp. 219241, 1979.
 Priest, G. In Contradiction: A Study of the
Transconsistent, Martinus Nijhoff, Dordrecht, 1987
(Second Edition, Oxford University Press, Oxford, forthcoming).
 Belnap, N.D., Jr. "A Useful Fourvalued Logic: How a computer
should think", Entailment: The Logic of Relevance and
Necessity, Vol II, A.R. Anderson, N.D. Belnap, Jr, and J.M. Dunn,
Princeton University Press, 1992, first appeared as "A Usuful
Fourvalued Logic", Modern Use of Multiplevalued Logic, J.M.
Dunn and G. Epstein (eds.), D.Reidel Publishing Company, Dordrecht,
1977, and "How a Computer Should Think", Comtemporary Aspects of
Philosophy, G. Ryle (ed.), Oriel Press, 1977.
 Besnard, P. and Hunter, A. (eds.) Handbook of Deasible Reasoning and
Uncertainty Management Systems, Vol. 2, Reasoning with Actual
and Potential Contradictions, Kluwer Academic Publishers, Dordrecht,
1998.
 Priest, G. "Paraconsistent Belief Revision",
Theoria, Vol. 67, pp. 214228, 2001.
 Restall, G. and Slaney, J. "Realistic Belief Revision",
Proceedings of the Second World Conference on Foundations of
Artificial Intelligence, pp. 367378, 1995.
 Tanaka, K. "The AGM Theory and Inconsistent Belief Change",
Logique et Analyse, Vol. 47, 2004, forthcoming.
 Mortensen, C. Inconsistent Mathematics,
Kluwer Academic Publishers, Dordrecht, 1995.
 Priest, G. "Inconsistent Arithmetic: Issues Technical and
Philosophical", Trends in Logic: 50 Years of Studia Logica
(Studia Logica Library, Vol. 21), V. F. Hendricks and
J. Malinowski (eds.), Kluwer Academic Publishers,
pp. 27399, 2003.
 Jaskowski, S. "Propositional Calculus for Contradictory Deductive
Systems", Studia Logica, Vol. 24, pp. 143157, 1969, first
published as "Rachunek zdah dla systemow dedukcyjnych sprzecznych",
Studia Societatis Scientiarun Torunesis, Sectio A, Vol. 1, No.
5, pp. 5577, 1948.
 da Costa, N.C.A. and Dubikajtis, L. "On Jaskowski's Discussive
Logic", NonClassical Logics, Modal Theory and Computability,
A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.), NorthHolland
Publishing Company, Amsterdam, pp.3756, 1977.
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Journal of Philosophical Logic, Vol. IX, pp. 327340,
1980.
 da Costa, N.C.A. "On the Theory of Inconsistent Formal Systems",
Notre Dame Journal of Formal Logic, Vol. 15, No. 4, pp.
497510, 1974.
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Calculi Cn", Notre Dame Journal of Formal Logic, Vol. 18,
No. 4, pp. 621630, 1977.
 Loparic, A. "Une etude semantique de quelques calculs
propositionnels", Comptes Rendus Hebdomadaires des Seances de
l'Academic des Sciences, Paris 284, pp. 835838, 1977.
 Asenjo, F.G. "A Calculus of Antinomies", Notre Dame Journal of
Formal Logic, Vol. 7, pp. 1035, 1966.
 Dunn, J.M. "Intuitive Semantics for First Degree Entailment and
Coupled Trees", Philosophicl Studies, Vol. 29, pp. 14968,
1976.
 Kotas, J. and da Costa, N. "On the Problem of Jaskowski and the
Logics of Łukasiewicz", NonClassical Logic, Model Theory and
Computability, A.I. Arruda, N.C.A da Costa, and R. Chuaqui (eds.),
North Holland Publishing Company, Amsterdam, pp. 12739, 1977.
 Priest, G. "Fuzzy Relevant Logic", Paraconsistency: the Logical
Way to the Inconsistent, W.Carnielli et al. (eds.),
Marcel Dekker, pp. 261274, 2002.
 Dunn, J.M. and Restall, G. "Relevance Logic", Handbook of
Philosophical Logic (Second Edition), Vol. 6,
D. Gabbay and F. Guenthner (eds.), Kluwer Academic Publishers,
Dordrecht, pp. 1136, 2002.
 Mares, E. "'FourValued' Semantics for the Relevant Logic R",
Journal of Philosophical Logic, Vol. 33, pp. 327341,
2004.
 Restall, G. "Simplified Semantics for Relevant Logics (and some of
their rivals)", Journal of Philosophical Logic, Vol. 22, pp.
481511, 1993.
 Restall, G. "FourValued Semantics for Relevant Logics
(and some of their rivals)", Journal of Philosophical Logic,
Vol. 24, pp. 139160, 1995.
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Relevant Logics and Their Rivals, Vol. 1, Atascadero, Ridgeview, CA,
1982.
 Brady, R.T. (ed.) Relevant Logics and Their Rivals, Vol. 2,
Ashgate, Aldershot, 2003.
[Please contact the authors with suggestions.]
dialetheism [dialethism] 
logic: manyvalued 
logic: relevance 
logic: substructural 
mathematics: inconsistent 
Sorites paradox
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Stanford Encyclopedia of Philosophy