Paraconsistent logic
Encyclopedia
A paraconsistent logic is a logical system that attempts to deal with contradiction
s in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic
that is concerned with studying and developing paraconsistent (or “inconsistency-tolerant”) systems of logic.
Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle
); however, the term paraconsistent (“beyond the consistent”) was not coined until 1976, by the Peru
vian philosopher Francisco Miró Quesada
.
(as well as intuitionistic logic
and most other logics), contradictions entail
everything. This curious feature, known as the principle of explosion
or ex contradictione sequitur quodlibet (Latin
, “from a contradiction, anything follows”) can be expressed formally as
Which means: if P and its negation ¬P are both assumed to be true, then P is assumed to be true, from which it follows that at least one of the claims P and some other (arbitrary) claim A is true. However, if we know that either P or A is true, and also that P is not true (that ¬P is true) we can conclude that A, which could be anything, is true. Thus if a theory
contains a single inconsistency, it is trivial
—that is, it has every sentence as a theorem. The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.
weaker than classical logic
; that is, they deem fewer propositional inferences valid. The point is that a paraconsistent logic can never be a propositional extension of classical logic, that is, propositionally validate everything that classical logic does. In that sense, then, paraconsistent logic is more conservative or cautious than classical logic. It is due to such conservativeness that paraconsistent languages can be more expressive than their classical counterparts including the hierarchy of metalanguage
s due to Tarski et al. According to Solomon Feferman
[1984]: “…natural language abounds with directly or indirectly self-referential yet apparently harmless expressions—all of which are excluded from the Tarskian framework.” This expressive limitation can be overcome in paraconsistent logic.
in a controlled and discriminating way. The principle of explosion precludes this, and so must be abandoned. In non-paraconsistent logics, there is only one inconsistent theory: the trivial theory that has every sentence as a theorem. Paraconsistent logic makes it possible to distinguish between inconsistent theories and to reason with them. Sometimes it is possible to revise a theory to make it consistent. In other cases (e.g., large software systems) it is currently impossible to attain consistency.
Some philosophers take a more radical approach, holding that some contradictions are true, and thus a theory’s being inconsistent is not always an indication that it is incorrect. This view, known as dialetheism
, is motivated by several considerations, most notably an inclination to take certain paradox
es such as the Liar
and Russell’s paradox at face value. Not all advocates of paraconsistent logic are dialetheists. On the other hand, being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise having to accept everything as true (i.e. trivialism
).
Though each of these principles has been challenged, the most popular approach among logicians is to reject disjunctive syllogism. If one is a dialetheist, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ¬ A, then A is excluded, so the only way A ∨ B could be true would be if B were true. However, if A and ¬ A can both be true at the same time, then this reasoning fails.
Another approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. The disjunction (A ∨ B) is defined as ¬ (¬A ∧ ¬B). In this approach all of the rules of natural deduction
hold, except for proof by contradiction and disjunction introduction; moreover,
does not mean necessarily that 
, which is also a difference from natural deduction. Also, the following usual Boolean properties hold: excluded middle and (for conjunction and disjunction) associativity
, commutativity
, distributivity
, De Morgan’s laws, and idempotence
. Furthermore, by defining the implication (A → B) as ¬ (A ∧ ¬B), there is a Two-Way Deduction Theorem allowing implications to be easily proved. Carl Hewitt
favours this approach, claiming that having the usual Boolean properties, Natural Deduction, and Deduction Theorem are huge advantages in software engineering
.
Yet another approach is to do both simultaneously. In many systems of relevant logic, as well as linear logic
, there are two separate disjunctive connectives. One allows disjunction introduction, and one allows disjunctive syllogism. Of course, this has the disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them.
The three principles below, when taken together, also entail explosion, so at least one must be abandoned:
Both reductio ad absurdum and the rule of weakening have been challenged in this respect, but without much success. Double negation elimination is challenged, but for unrelated reasons. By removing it alone, while upholding the other two one may still be able to prove all negative propositions from a contradiction.
logician F. G. Asenjo in 1966 and later popularized by Priest
and others.
One way of presenting the semantics for LP is to replace the usual functional
valuation with a relational
one. The binary relation
relates a formula
to a truth value:
means that 
is true, and 
means that 
is false. A formula must be assigned at least one truth value, but there is no requirement that it be assigned at most one truth value. The semantic clauses for negation
and disjunction are given as follows:
(The other logical connective
s are defined in terms of negation and disjunction as usual.)
Or to put the same point less symbolically:
(Semantic) logical consequence is then defined as truth-preservation:
Now consider a valuation
such that 
and 
but it is not the case that 
. It is easy to check that this valuation constitutes a counterexample
to both explosion and disjunctive syllogism. However, it is also a counterexample to modus ponens
for the material conditional
of LP. For this reason, proponents of LP usually advocate expanding the system to include a stronger conditional connective that is not definable in terms of negation and disjunction.
As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as De Morgan’s laws and the usual introduction and elimination rules
for negation, conjunction
, and disjunction. Surprisingly, the logical truth
s (or tautologies
) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the inference
s they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as FDE (“First-Degree Entailment”). Unlike LP, FDE contains no logical truths.
It must be emphasized that LP is but one of many paraconsistent logics that have been proposed. It is presented here merely as an illustration of how a paraconsistent logic can work.
. A logic is relevant iff
it satisfies the following condition:
It follows that a relevance logic cannot have (p ∧ ¬p) → q as a theorem, and thus (on reasonable assumptions) cannot validate the inference from {p, ¬p} to q.
Paraconsistent logic has significant overlap with many-valued logic; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent). Dialetheic logics, which are also many-valued, are paraconsistent, but the converse does not hold.
Intuitionistic logic
allows A ∨ ¬A not to be equivalent to true, while paraconsistent logic allows A ∧ ¬A not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the “dual
” of intuitionistic logic. However, intuitionistic logic is a specific logical system whereas paraconsistent logic encompasses a large class of systems. Accordingly, the dual notion to paraconsistency is called paracompleteness, and the “dual” of intuitionistic logic (a specific paracomplete logic) is a specific paraconsistent system called anti-intuitionistic or dual-intuitionistic logic (sometimes referred to as Brazilian logic, for historical reasons). The duality between the two systems is best seen within a sequent calculus
framework. While in intuitionistic logic the sequent
is not derivable, in dual-intuitionistic logic
is not derivable. Similarly, in intuitionistic logic the sequent
is not derivable, while in dual-intuitionistic logic
is not derivable. Dual-intuitionistic logic contains a connective # known as pseudo-difference which is the dual of intuitionistic implication. Very loosely, can be read as “A but not B”. However, # is not truth-functional as one might expect a ‘but not’ operator to be; similarly, the intuitionistic implication operator cannot be treated like "". Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as
A full account of the duality between paraconsistent and intuitionistic logic, including an explanation on why dual-intuitionistic and paraconsistent logics do not coincide, can be found in Brunner and Carnielli (2005).
Others, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true. A related objection is that “negation” in paraconsistent logic is not really negation
; it is merely a subcontrary
-forming operator.
with Bayesian inference
and the Dempster-Shafer theory
, allowing that no non-tautological belief is completely (100%) irrefutable because it must be based upon incomplete, abstracted, interpreted, likely unconfirmed, potentially uninformed, and possibly incorrect knowledge (of course, this very assumption, if non-tautological, entails its own refutability, if by "refutable" we mean "not completely [100%] irrefutable"). These systems effectively give up several logical principles in practice without rejecting them in theory.
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
s in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
that is concerned with studying and developing paraconsistent (or “inconsistency-tolerant”) systems of logic.
Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
); however, the term paraconsistent (“beyond the consistent”) was not coined until 1976, by the Peru
Peru
Peru , officially the Republic of Peru , is a country in western South America. It is bordered on the north by Ecuador and Colombia, on the east by Brazil, on the southeast by Bolivia, on the south by Chile, and on the west by the Pacific Ocean....
vian philosopher Francisco Miró Quesada
Francisco Miró Quesada
Francisco Miró Quesada Cantuarias is a contemporary Peruvian philosopher who disputes the summary of human nature on the basis that any collective assumption of human nature would be unfulfilling and leave the public with a negative result. He made his debut in 1941 with Sentido del movimiento...
.
Definition
In classical logicClassical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
(as well as intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
and most other logics), contradictions entail
Entailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
everything. This curious feature, known as the principle of explosion
Principle of explosion
The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
or ex contradictione sequitur quodlibet (Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
, “from a contradiction, anything follows”) can be expressed formally as
 |
Premise |
 |
conjunctive elimination |
 |
weakening |
 |
conjunctive elimination |
 |
disjunctive syllogism |
Which means: if P and its negation ¬P are both assumed to be true, then P is assumed to be true, from which it follows that at least one of the claims P and some other (arbitrary) claim A is true. However, if we know that either P or A is true, and also that P is not true (that ¬P is true) we can conclude that A, which could be anything, is true. Thus if a theory
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...
contains a single inconsistency, it is trivial
Trivialism
Trivialism is the theory that every proposition is true. A consequence of trivialism is that all statements, including all contradictions of the form "p and not p" , are true.- Further reading :***...
—that is, it has every sentence as a theorem. The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.
Paraconsistent logics are propositionally weaker than classical logic
Paraconsistent logics are propositionallyPropositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
weaker than classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
; that is, they deem fewer propositional inferences valid. The point is that a paraconsistent logic can never be a propositional extension of classical logic, that is, propositionally validate everything that classical logic does. In that sense, then, paraconsistent logic is more conservative or cautious than classical logic. It is due to such conservativeness that paraconsistent languages can be more expressive than their classical counterparts including the hierarchy of metalanguage
Metalanguage
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined. In logic and linguistics, a metalanguage is a language used to make statements about statements in another language...
s due to Tarski et al. According to Solomon Feferman
Solomon Feferman
Solomon Feferman is an American philosopher and mathematician with major works in mathematical logic.He was born in New York City, New York, and received his Ph.D. in 1957 from the University of California, Berkeley under Alfred Tarski...
[1984]: “…natural language abounds with directly or indirectly self-referential yet apparently harmless expressions—all of which are excluded from the Tarskian framework.” This expressive limitation can be overcome in paraconsistent logic.
Motivation
The primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent informationInformation
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
in a controlled and discriminating way. The principle of explosion precludes this, and so must be abandoned. In non-paraconsistent logics, there is only one inconsistent theory: the trivial theory that has every sentence as a theorem. Paraconsistent logic makes it possible to distinguish between inconsistent theories and to reason with them. Sometimes it is possible to revise a theory to make it consistent. In other cases (e.g., large software systems) it is currently impossible to attain consistency.
Some philosophers take a more radical approach, holding that some contradictions are true, and thus a theory’s being inconsistent is not always an indication that it is incorrect. This view, known as dialetheism
Dialetheism
Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...
, is motivated by several considerations, most notably an inclination to take certain paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es such as the Liar
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
and Russell’s paradox at face value. Not all advocates of paraconsistent logic are dialetheists. On the other hand, being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise having to accept everything as true (i.e. trivialism
Trivialism
Trivialism is the theory that every proposition is true. A consequence of trivialism is that all statements, including all contradictions of the form "p and not p" , are true.- Further reading :***...
).
Tradeoff
Paraconsistency does not come for free: it involves a tradeoff. In particular, abandoning the principle of explosion requires one to abandon at least one of the following three very intuitive principles:| Disjunction introduction Disjunction introduction Disjunction introduction or Addition is a valid, simple argument form in logic:or in logical operator notation: A \vdash A \or B The argument form has one premise, A, and an unrelated proposition, B... |
 |
|---|---|
| Disjunctive syllogism Disjunctive syllogism A disjunctive syllogism, also known as disjunction-elimination and or-elimination , and historically known as modus tollendo ponens,, is a classically valid, simple argument form:where \vdash represents the logical assertion.... |
 |
| Transitivity Transitive relation In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c.... or “cut” |
 |
Though each of these principles has been challenged, the most popular approach among logicians is to reject disjunctive syllogism. If one is a dialetheist, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ¬ A, then A is excluded, so the only way A ∨ B could be true would be if B were true. However, if A and ¬ A can both be true at the same time, then this reasoning fails.
Another approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. The disjunction (A ∨ B) is defined as ¬ (¬A ∧ ¬B). In this approach all of the rules of natural deduction
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...
hold, except for proof by contradiction and disjunction introduction; moreover,
does not mean necessarily that , which is also a difference from natural deduction. Also, the following usual Boolean properties hold: excluded middle and (for conjunction and disjunction) associativityAssociativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
, commutativity
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
, distributivity
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
, De Morgan’s laws, and idempotence
Idempotence
Idempotence is the property of certain operations in mathematics and computer science, that they can be applied multiple times without changing the result beyond the initial application...
. Furthermore, by defining the implication (A → B) as ¬ (A ∧ ¬B), there is a Two-Way Deduction Theorem allowing implications to be easily proved. Carl Hewitt
Carl Hewitt
Carl Hewitt is Board Chair of the International Society for Inconsistency Robustness. He has been a Visiting Professor at Stanford University and the University of Keio. In 2000, he became emeritus in the EECS department at MIT....
favours this approach, claiming that having the usual Boolean properties, Natural Deduction, and Deduction Theorem are huge advantages in software engineering
Software engineering
Software Engineering is the application of a systematic, disciplined, quantifiable approach to the development, operation, and maintenance of software, and the study of these approaches; that is, the application of engineering to software...
.
Yet another approach is to do both simultaneously. In many systems of relevant logic, as well as linear logic
Linear logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter...
, there are two separate disjunctive connectives. One allows disjunction introduction, and one allows disjunctive syllogism. Of course, this has the disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them.
The three principles below, when taken together, also entail explosion, so at least one must be abandoned:
| Reductio ad absurdum Reductio ad absurdum In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction... |
 |
|---|---|
| Rule of weakening |  |
| Double negation elimination |  |
Both reductio ad absurdum and the rule of weakening have been challenged in this respect, but without much success. Double negation elimination is challenged, but for unrelated reasons. By removing it alone, while upholding the other two one may still be able to prove all negative propositions from a contradiction.
A simple paraconsistent logic
Perhaps the most well-known system of paraconsistent logic is the simple system known as LP (“Logic of Paradox”), first proposed by the ArgentinianArgentina
Argentina , officially the Argentine Republic , is the second largest country in South America by land area, after Brazil. It is constituted as a federation of 23 provinces and an autonomous city, Buenos Aires...
logician F. G. Asenjo in 1966 and later popularized by Priest
Graham Priest
Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
and others.
One way of presenting the semantics for LP is to replace the usual functional
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
valuation with a relational
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
one. The binary relation
relates a formulaWell-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
to a truth value:
means that is true, and means that is false. A formula must be assigned at least one truth value, but there is no requirement that it be assigned at most one truth value. The semantic clauses for negationNegation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
and disjunction are given as follows:
(The other logical connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s are defined in terms of negation and disjunction as usual.)
Or to put the same point less symbolically:
- not A is true if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
A is false - not A is false if and only if A is true
- A or B is true if and only if A is true or B is true
- A or B is false if and only if A is false and B is false
(Semantic) logical consequence is then defined as truth-preservation:
-
if and only if 
is true whenever every element of 
is true.
Now consider a valuation
such that and but it is not the case that . It is easy to check that this valuation constitutes a counterexampleCounterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
to both explosion and disjunctive syllogism. However, it is also a counterexample to modus ponens
Modus ponens
In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
for the material conditional
Material conditional
The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a conditional...
of LP. For this reason, proponents of LP usually advocate expanding the system to include a stronger conditional connective that is not definable in terms of negation and disjunction.
As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as De Morgan’s laws and the usual introduction and elimination rules
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...
for negation, conjunction
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
, and disjunction. Surprisingly, the logical truth
Logical truth
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.Logical...
s (or tautologies
Tautology (logic)
In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the inference
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
s they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as FDE (“First-Degree Entailment”). Unlike LP, FDE contains no logical truths.
It must be emphasized that LP is but one of many paraconsistent logics that have been proposed. It is presented here merely as an illustration of how a paraconsistent logic can work.
Relation to other logics
One important type of paraconsistent logic is relevance logicRelevance logic
Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
. A logic is relevant iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
it satisfies the following condition:
- if A → B is a theorem, then A and B share a non-logical constant.
It follows that a relevance logic cannot have (p ∧ ¬p) → q as a theorem, and thus (on reasonable assumptions) cannot validate the inference from {p, ¬p} to q.
Paraconsistent logic has significant overlap with many-valued logic; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent). Dialetheic logics, which are also many-valued, are paraconsistent, but the converse does not hold.
Intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
allows A ∨ ¬A not to be equivalent to true, while paraconsistent logic allows A ∧ ¬A not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the “dual
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
” of intuitionistic logic. However, intuitionistic logic is a specific logical system whereas paraconsistent logic encompasses a large class of systems. Accordingly, the dual notion to paraconsistency is called paracompleteness, and the “dual” of intuitionistic logic (a specific paracomplete logic) is a specific paraconsistent system called anti-intuitionistic or dual-intuitionistic logic (sometimes referred to as Brazilian logic, for historical reasons). The duality between the two systems is best seen within a sequent calculus
Sequent calculus
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934 as a tool for studying natural deduction in...
framework. While in intuitionistic logic the sequent
is not derivable, in dual-intuitionistic logic
is not derivable. Similarly, in intuitionistic logic the sequent
is not derivable, while in dual-intuitionistic logic
is not derivable. Dual-intuitionistic logic contains a connective # known as pseudo-difference which is the dual of intuitionistic implication. Very loosely, can be read as “A but not B”. However, # is not truth-functional as one might expect a ‘but not’ operator to be; similarly, the intuitionistic implication operator cannot be treated like "". Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as
A full account of the duality between paraconsistent and intuitionistic logic, including an explanation on why dual-intuitionistic and paraconsistent logics do not coincide, can be found in Brunner and Carnielli (2005).
Applications
Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:- SemanticsSemanticsSemantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
. Paraconsistent logic has been proposed as means of providing a simple and intuitive formal account of truthTruthTruth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
that does not fall prey to paradoxes such as the LiarLiar paradoxIn philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
. However, such systems must also avoid Curry’s paradox, which is much more difficult as it does not essentially involve negation. - Set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and the foundations of mathematicsFoundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
(see paraconsistent mathematicsParaconsistent mathematicsParaconsistent mathematics represents an attempt to develop the classical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic...
). Some believe that paraconsistent logic has significant ramifications with respect to the significance of Russell’s paradox and Gödel’s incompleteness theorems. - Epistemology and belief revisionBelief revisionBelief revision is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational agents....
. Paraconsistent logic has been proposed as a means of reasoning with and revising inconsistent theories and belief systems. - Knowledge managementKnowledge managementKnowledge management comprises a range of strategies and practices used in an organization to identify, create, represent, distribute, and enable adoption of insights and experiences...
and artificial intelligenceArtificial intelligenceArtificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...
. Some computer scientistComputer scientistA computer scientist is a scientist who has acquired knowledge of computer science, the study of the theoretical foundations of information and computation and their application in computer systems....
s have utilized paraconsistent logic as a means of coping gracefully with inconsistent information. - Deontic logicDeontic logicDeontic logic is the field of logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts...
and metaethics. Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts. - Software engineeringSoftware engineeringSoftware Engineering is the application of a systematic, disciplined, quantifiable approach to the development, operation, and maintenance of software, and the study of these approaches; that is, the application of engineering to software...
. Paraconsistent logic has been proposed as a means for dealing with the pervasive inconsistencies among the documentationDocumentationDocumentation is a term used in several different ways. Generally, documentation refers to the process of providing evidence.Modules of Documentation are Helpful...
, use cases, and codeSource codeIn computer science, source code is text written using the format and syntax of the programming language that it is being written in. Such a language is specially designed to facilitate the work of computer programmers, who specify the actions to be performed by a computer mostly by writing source...
of large software systems. - ElectronicsElectronicsElectronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
design routinely uses a four valued logicFour valued logicIn logic, a four-valued logic is used to model signal values in digital circuits: the four values are Z, X and the boolean values 1 and 0. Z stands for high impedance or open circuit, while X stands for "unknown"...
, with “hi-impedance (z)” and “don’t care (x)” playing similar roles to “don’t know” and “both true and false” respectively, in addition to True and False. This logic was developed independently of Philosophical logics.
Criticism
Some philosophers have argued against paraconsistent logic on the ground that the counterintuitiveness of giving up any of the three principles above outweighs any counterintuitiveness that the principle of explosion might have.Others, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true. A related objection is that “negation” in paraconsistent logic is not really negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
; it is merely a subcontrary
Square of opposition
In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...
-forming operator.
Alternatives
Approaches exist that allow for resolution of inconsistent beliefs without violating any of the intuitive logical principles. Most such systems use multi-valued logicMulti-valued logic
In logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...
with Bayesian inference
Bayesian inference
In statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...
and the Dempster-Shafer theory
Dempster-Shafer theory
The Dempster–Shafer theory is a mathematical theory of evidence. It allows one to combine evidence from different sources and arrive at a degree of belief that takes into account all the available evidence. The theory was first developed by Arthur P...
, allowing that no non-tautological belief is completely (100%) irrefutable because it must be based upon incomplete, abstracted, interpreted, likely unconfirmed, potentially uninformed, and possibly incorrect knowledge (of course, this very assumption, if non-tautological, entails its own refutability, if by "refutable" we mean "not completely [100%] irrefutable"). These systems effectively give up several logical principles in practice without rejecting them in theory.
Notable figures
Notable figures in the history and/or modern development of paraconsistent logic include:- Alan Ross AndersonAlan Ross AndersonAlan Ross Anderson was an American logician and professor of philosophy at Yale University and the University of Pittsburgh....
(USA, 1925–1973). One of the founders of relevance logicRelevance logicRelevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
, a kind of paraconsistent logic. - F. G. Asenjo (ArgentinaArgentinaArgentina , officially the Argentine Republic , is the second largest country in South America by land area, after Brazil. It is constituted as a federation of 23 provinces and an autonomous city, Buenos Aires...
) - Diderik BatensDiderik BatensDiderik Batens is a Belgian logician and epistemologist at the University of Ghent, known chiefly for his work on adaptive and paraconsistent logics. His epistemological views may be broadly characterized as fallibilist.- External links :*...
(BelgiumBelgiumBelgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...
) - Nuel BelnapNuel BelnapNuel D. Belnap, Jr. is an American logician and philosopher who has made many important contributions to the philosophy of logic, temporal logic, and structural proof theory. He has taught at the University of Pittsburgh since 1961; before that he was at Yale University. His best known work is...
(USA, b. 1930). Worked with Anderson on relevance logic. - Jean-Yves BéziauJean-Yves BéziauJean-Yves Béziau is a professor and researcher of the Brazilian Research Council - CNPq - at the Federal University of Ceara, Brazil. Béziau is a dual citizen of France and Switzerland...
(FranceFranceThe French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
/SwitzerlandSwitzerlandSwitzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
, b. 1965). Has written extensively on the general structural features and philosophical foundations of paraconsistent logics. - Ross Brady (AustraliaAustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
) - Bryson Brown (CanadaCanadaCanada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
) - Walter CarnielliWalter CarnielliWalter Alexandre Carnielli is a Brazilian mathematician, logician, and philosopher, full professor of Logic at the . With a Bachelor and a Ms.C. degree in mathematics at the in Campinas he obtained his Ph.D. in 1984 in the same university under the supervision of Newton C...
(BrazilBrazilBrazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...
). The developer of the possible-translations semantics, a new semantics which makes paraconsistent logics applicable and philosophically understood. - Newton da CostaNewton da CostaNewton Carneiro Affonso da Costa is a Brazilian mathematician, logician, and philosopher. He studied engineering and mathematics at the Federal University of Paraná in Curitiba and the title of his 1961 Ph.D...
(BrazilBrazilBrazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...
, b. 1929). One of the first to develop formal systems of paraconsistent logic. - Itala M. L. D’Ottaviano (BrazilBrazilBrazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...
) - J. Michael Dunn (USA). An important figure in relevance logic.
- Stanisław Jaśkowski (PolandPolandPoland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
). One of the first to develop formal systems of paraconsistent logic. - R. E. Jennings (CanadaCanadaCanada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
) - David Kellogg LewisDavid Kellogg LewisDavid Kellogg Lewis was an American philosopher. Lewis taught briefly at UCLA and then at Princeton from 1970 until his death. He is also closely associated with Australia, whose philosophical community he visited almost annually for more than thirty years...
(USA, 1941–2001). Articulate critic of paraconsistent logic. - Jan Łukasiewicz (PolandPolandPoland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
, 1878–1956) - Robert K. Meyer (USA/AustraliaAustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
) - Chris Mortensen (AustraliaAustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
). Has written extensively on paraconsistent mathematicsParaconsistent mathematicsParaconsistent mathematics represents an attempt to develop the classical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic...
. - Lorenzo PeñaLorenzo PeñaLorenzo Peña is a Spanish philosopher, lawyer, logician and political thinker. His rationalism is a neo-Leibnizian approach both in metaphysics and law.-Life:Lorenzo Peña was born in Alicante, Spain, on August 29, 1944...
(SpainSpainSpain , officially the Kingdom of Spain languages]] under the European Charter for Regional or Minority Languages. In each of these, Spain's official name is as follows:;;;;;;), is a country and member state of the European Union located in southwestern Europe on the Iberian Peninsula...
, b. 1944). Has developed an original line of paraconsistent logic, gradualistic logic (also known as transitive logic, TL), akin to Fuzzy LogicFuzzy logicFuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
. - Val PlumwoodVal PlumwoodVal Plumwood , formerly Val Routley, was an Australian ecofeminist intellectual and activist, who was prominent in the development of radical ecosophy from the early 1970s through the remainder of the 20th century....
[formerly Routley] (AustraliaAustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
, b. 1939). Frequent collaborator with Sylvan. - Graham PriestGraham PriestGraham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
(AustraliaAustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
). Perhaps the most prominent advocate of paraconsistent logic in the world today. - Francisco Miró QuesadaFrancisco Miró QuesadaFrancisco Miró Quesada Cantuarias is a contemporary Peruvian philosopher who disputes the summary of human nature on the basis that any collective assumption of human nature would be unfulfilling and leave the public with a negative result. He made his debut in 1941 with Sentido del movimiento...
(PeruPeruPeru , officially the Republic of Peru , is a country in western South America. It is bordered on the north by Ecuador and Colombia, on the east by Brazil, on the southeast by Bolivia, on the south by Chile, and on the west by the Pacific Ocean....
). Coined the term paraconsistent logic. - Peter Schotch (CanadaCanadaCanada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
) - B. H. Slater (AustraliaAustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
). Another articulate critic of paraconsistent logic. - Richard SylvanRichard SylvanRichard Sylvan was a philosopher, logician, and environmentalist.- Biography :Sylvan was born Francis Richard Routley in Levin, New Zealand, and his early work is cited with this surname...
[formerly Routley] (New ZealandNew ZealandNew Zealand is an island country in the south-western Pacific Ocean comprising two main landmasses and numerous smaller islands. The country is situated some east of Australia across the Tasman Sea, and roughly south of the Pacific island nations of New Caledonia, Fiji, and Tonga...
/AustraliaAustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
, 1935–1996). Important figure in relevance logic and a frequent collaborator with Plumwood and Priest. - Nicolai A. VasilievNicolai A. VasilievNicolai Alexandrovich Vasiliev , also Vasil'ev, Vassilieff, Wassilieff was a Russian logician, philosopher, psychologist, poet, the forerunner of paraconsistent and multi-valued logics.-Early years:...
(RussiaRussiaRussia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
, 1880–1940). First to construct logic tolerant to contradiction (1910).
See also
- Deviant logicDeviant logicPhilosopher Susan Haack uses the term "deviant logic" to describe certain non-classical systems of logic. In these logics,* the set of well-formed formulas generated equals the set of well-formed formulas generated by classical logic....
- Formal logicFormal logicClassical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
- Probability logic
- Synthetic logic
- Table of logic symbolsTable of logic symbolsIn logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many common symbols together with their name, pronunciation and related...