HOME

TheInfoList



OR:

Paraconsistent logic is a type of
non-classical logic Non-classical logics (and sometimes alternative logics or non-Aristotelian logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this ...
that allows for the coexistence of contradictory statements without leading to a logical explosion where anything can be proven true. Specifically, paraconsistent logic is the subfield of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
that is concerned with studying and developing "inconsistency-tolerant" systems of logic, purposefully excluding the principle of explosion. Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
); however, the term ''paraconsistent'' ("beside the consistent") was first coined in 1976, by the
Peru Peru, officially the Republic of Peru, is a country in western South America. It is bordered in the north by Ecuador and Colombia, in the east by Brazil, in the southeast by Bolivia, in the south by Chile, and in the south and west by the Pac ...
vian
philosopher Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
Francisco Miró Quesada Cantuarias. The study of paraconsistent logic has been dubbed paraconsistency, which encompasses the school of dialetheism.


Definition

In
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
(as well as intuitionistic logic and most other logics), contradictions entail everything. This feature, known as the principle of explosion or ''ex contradictione sequitur quodlibet'' (
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
, "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 of the two claims ''P'' and (some arbitrary) ''A'', at least one is true. Therefore, ''P'' or ''A'' is true. However, if we know that either ''P'' or ''A'' is true, and also that ''P'' is false (that ¬''P'' is true) we can conclude that ''A'', which could be anything, is true. Thus if a
theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
contains a single inconsistency, the theory 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.


Comparison with classical logic

The entailment relations of paraconsistent logics are propositionally ''weaker'' than
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
; 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 every entailment that classical logic does. In some 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 metalanguages due to
Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
and others. According to Solomon Feferman: "natural language abounds with directly or indirectly
self-referential Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural language, natural or formal languages, ...
yet apparently harmless expressions—all of which are excluded from the Tarskian framework." This expressive limitation can be overcome in paraconsistent logic.


Motivation

A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent
information Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...
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. Research into paraconsistent logic has also led to the establishment of the philosophical school of dialetheism (most notably advocated by Graham Priest), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues. Being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing trivialism, i.e. accepting that all contradictions (and equivalently all statements) are true. However, the study of paraconsistent logics does not necessarily entail a dialetheist viewpoint. For example, one need not commit to either the existence of true theories or true contradictions, but would rather prefer a weaker standard like empirical adequacy, as proposed by Bas van Fraassen.


Philosophy

In classical logic, Aristotle's three laws, namely, the excluded middle (''p'' or ¬''p''), non-contradiction ¬ (''p'' ∧ ¬''p'') and identity (''p'' iff ''p''), are regarded as the same, due to the inter-definition of the connectives. Moreover, traditionally contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory entails all possible consequences) are assumed inseparable, granted that negation is available. These views may be philosophically challenged, precisely on the grounds that they fail to distinguish between contradictoriness and other forms of inconsistency. On the other hand, it is possible to derive triviality from the 'conflict' between consistency and contradictions, once these notions have been properly distinguished. The very notions of consistency and inconsistency may be furthermore internalized at the object language level.


Tradeoffs

Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires one to abandon at least one of the following two principles: Both of these principles have been challenged. One approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. In this approach, rules of natural deduction hold, except for disjunction introduction and excluded middle; moreover, inference A⊢B does not necessarily mean entailment A⇒B. Also, the following usual Boolean properties hold: double negation as well as associativity, commutativity, distributivity, De Morgan, and idempotence inferences (for conjunction and disjunction). Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A. Another approach is to reject disjunctive syllogism. From the perspective of dialetheism, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ''¬ A'', then ''A'' is excluded and ''B'' can be inferred from ''A ∨ B''. However, if ''A'' may hold as well as ''¬A'', then the argument for the inference is weakened. 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. Furthermore, the rule of proof of negation (below) just by itself is inconsistency non-robust in the sense that the negation of every proposition can be proved from a contradiction. Strictly speaking, having just the rule above is paraconsistent because it is not the case that ''every'' proposition can be proved from a contradiction. However, if the rule double negation elimination (\neg \neg A \vdash A) is added as well, then every proposition can be proved from a contradiction. Double negation elimination does not hold for intuitionistic logic.


Logic of Paradox

One example of paraconsistent logic is the system known as LP ("Logic of Paradox"), first proposed by the Argentinian logician Florencio González Asenjo in 1966 and later popularized by
Priest A priest is a religious leader authorized to perform the sacred rituals of a religion, especially as a mediatory agent between humans and one or more deity, deities. They also have the authority or power to administer religious rites; in parti ...
and others. One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one. The binary relation V\, relates a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
to a
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...
: V(A,1)\, means that A\, is true, and V(A,0)\, means that A\, 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 In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
and
disjunction In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...
are given as follows: * V( \neg A,1) \Leftrightarrow V(A,0) * V( \neg A,0) \Leftrightarrow V(A,1) * V(A \lor B,1) \Leftrightarrow V(A,1) \text V(B,1) * V(A \lor B,0) \Leftrightarrow V(A,0) \text V(B,0) (The other logical connectives 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 if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''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 Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more stat ...
is then defined as truth-preservation: : \Gamma\vDash A if and only if A\, is true whenever every element of \Gamma\, is true. Now consider a valuation V\, such that V(A,1)\, and V(A,0)\, but it is not the case that V(B,1)\,. 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 The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. M ...
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 truths (or tautologies) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the ''
inference Inferences are steps in logical reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinct ...
s'' they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as first-degree entailment (FDE). Unlike LP, FDE contains no logical truths. LP is only 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 logic. A logic is ''relevant'' if 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 to ''q''. Paraconsistent logic has significant overlap with
many-valued logic Many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's Term logic, logical calculus, there were only two possible values (i.e., "true" and ...
; 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. The ideal 3-valued paraconsistent logic given below becomes the logic RM3 when the contrapositive is added. 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 : \vdash A \lor \neg A is not derivable, in dual-intuitionistic logic : A \land \neg A \vdash is not derivable. Similarly, in intuitionistic logic the sequent : \neg \neg A \vdash A is not derivable, while in dual-intuitionistic logic : A \vdash \neg \neg A 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). These other logics avoid explosion: implicational propositional calculus, positive propositional calculus, equivalential calculus and minimal logic. The latter, minimal logic, is both paraconsistent and paracomplete (a subsystem of intuitionistic logic). The other three simply do not allow one to express a contradiction to begin with since they lack the ability to form negations.


An ideal three-valued paraconsistent logic

Here is an example of a three-valued logic which is paraconsistent and ''ideal'' as defined in "Ideal Paraconsistent Logics" by O. Arieli, A. Avron, and A. Zamansky, especially pages 22–23. The three truth-values are: ''t'' (true only), ''b'' (both true and false), and ''f'' (false only). A formula is true if its truth-value is either ''t'' or ''b'' for the valuation being used. A formula is a tautology of paraconsistent logic if it is true in every valuation which maps atomic propositions to . Every tautology of paraconsistent logic is also a tautology of classical logic. For a valuation, the set of true formulas is closed under modus ponens and the deduction theorem. Any tautology of classical logic which contains no negations is also a tautology of paraconsistent logic (by merging ''b'' into ''t''). This logic is sometimes referred to as "Pac" or "LFI1".


Included

Some tautologies of paraconsistent logic are: * All axiom schemas for paraconsistent logic: :P \to (Q \to P) ** for deduction theorem and ?→ = :(P \to (Q \to R)) \to ((P \to Q) \to (P \to R)) ** for deduction theorem (note: → = follows from the deduction theorem) :\lnot (P \to Q) \to P ** →? = :\lnot (P \to Q) \to \lnot Q ** ?→ = :P \to (\lnot Q \to \lnot (P \to Q)) ** → = :\lnot \lnot P \to P ** ~ = :P \to \lnot \lnot P ** ~ = (note: ~ = and ~ = follow from the way the truth-values are encoded) :P \to (P \lor Q) ** v? = :Q \to (P \lor Q) ** ?v = :\lnot (P \lor Q) \to \lnot P ** v? = :\lnot (P \lor Q) \to \lnot Q ** ?v = :(P \to R) \to ((Q \to R) \to ((P \lor Q) \to R)) ** v = :\lnot P \to (\lnot Q \to \lnot (P \lor Q)) ** v = :(P \land Q) \to P ** &? = :(P \land Q) \to Q ** ?& = :\lnot P \to \lnot (P \land Q) ** &? = :\lnot Q \to \lnot (P \land Q) ** ?& = :(\lnot P \to R) \to ((\lnot Q \to R) \to (\lnot (P \land Q) \to R)) ** & = :P \to (Q \to (P \land Q)) ** & = :(P \to Q) \to ((\lnot P \to Q) \to Q) ** ? is the union of with * Some other theorem schemas: :P \to P :(\lnot P \to P) \to P :((P \to Q) \to P) \to P :P \lor \lnot P :\lnot (P \land \lnot P) :(\lnot P \to Q) \to (P \lor Q) :((\lnot P \to Q) \to Q) \to (((P \land \lnot P) \to Q) \to (P \to Q)) ** every truth-value is either ''t'', ''b'', or ''f''. :((P \to Q) \to R) \to (Q \to R)


Excluded

Some tautologies of classical logic which are ''not'' tautologies of paraconsistent logic are: :\lnot P \to (P \to Q) ** no explosion in paraconsistent logic :(\lnot P \to Q) \to ((\lnot P \to \lnot Q) \to P) :(P \to Q) \to ((P \to \lnot Q) \to \lnot P) :(P \lor Q) \to (\lnot P \to Q) ** disjunctive syllogism fails in paraconsistent logic :(P \to Q) \to (\lnot Q \to \lnot P) ** contrapositive fails in paraconsistent logic :(\lnot P \to \lnot Q) \to (Q \to P) :((\lnot P \to Q) \to Q) \to (P \to Q) :(P \land \lnot P) \to (Q \land \lnot Q) ** not all contradictions are equivalent in paraconsistent logic :(P \to Q) \to (\lnot Q \to (P \to R)) :((P \to Q) \to R) \to (\lnot P \to R) :((\lnot P \to R) \to R) \to (((P \to Q) \to R) \to R) ** counter-factual for →? = (inconsistent with ''b''→''f'' = ''f'')


Strategy

Suppose we are faced with a contradictory set of premises Γ and wish to avoid being reduced to triviality. In classical logic, the only method one can use is to reject one or more of the premises in Γ. In paraconsistent logic, we may try to compartmentalize the contradiction. That is, weaken the logic so that Γ→''X'' is no longer a tautology provided the propositional variable ''X'' does not appear in Γ. However, we do not want to weaken the logic any more than is necessary for that purpose. So we wish to retain modus ponens and the deduction theorem as well as the axioms which are the introduction and elimination rules for the logical connectives (where possible). To this end, we add a third truth-value ''b'' which will be employed within the compartment containing the contradiction. We make ''b'' a fixed point of all the logical connectives. : b = \lnot b = (b \to b) = (b \lor b) = (b \land b) We must make ''b'' a kind of truth (in addition to ''t'') because otherwise there would be no tautologies at all. To ensure that modus ponens works, we must have : (b \to f) = f , that is, to ensure that a true hypothesis and a true implication lead to a true conclusion, we must have that a not-true (''f'') conclusion and a true (''t'' or ''b'') hypothesis yield a not-true implication. If all the propositional variables in Γ are assigned the value ''b'', then Γ itself will have the value ''b''. If we give ''X'' the value ''f'', then : (\Gamma \to X) = (b \to f) = f . So Γ→''X'' will not be a tautology. Limitations: (1) There must not be constants for the truth values because that would defeat the purpose of paraconsistent logic. Having ''b'' would change the language from that of classical logic. Having ''t'' or ''f'' would allow the explosion again because : \lnot t \to X or f \to X would be tautologies. Note that ''b'' is not a fixed point of those constants since ''b'' ≠ ''t'' and ''b'' ≠ ''f''. (2) This logic's ability to contain contradictions applies only to contradictions among particularized premises, not to contradictions among axiom schemas. (3) The loss of disjunctive syllogism may result in insufficient commitment to developing the 'correct' alternative, possibly crippling mathematics. (4) To establish that a formula Γ is equivalent to Δ in the sense that either can be substituted for the other wherever they appear as a subformula, one must show :(\Gamma \to \Delta) \land (\Delta \to \Gamma) \land (\lnot \Gamma \to \lnot \Delta) \land (\lnot \Delta \to \lnot \Gamma). This is more difficult than in classical logic because the contrapositives do not necessarily follow.


Applications

Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:Most of these are discussed in Bremer (2005) and Priest (2002). *
Semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
: Paraconsistent logic has been proposed as means of providing a simple and intuitive formal account of
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
that does not fall prey to paradoxes such as the Liar. However, such systems must also avoid Curry's paradox, which is much more difficult as it does not essentially involve negation. *
Set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
and the
foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
*
Epistemology Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowle ...
and
belief revision Belief revision (also called belief change) is the process of changing beliefs to take into account a new piece of information. The formal logic, logical formalization of belief revision is researched in philosophy, in databases, and in artifici ...
: Paraconsistent logic has been proposed as a means of reasoning with and revising inconsistent theories and belief systems. *
Knowledge management Knowledge management (KM) is the set of procedures for producing, disseminating, utilizing, and overseeing an organization's knowledge and data. It alludes to a multidisciplinary strategy that maximizes knowledge utilization to accomplish organ ...
and
artificial intelligence Artificial intelligence (AI) is the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of re ...
: Some
computer scientist A computer scientist is a scientist who specializes in the academic study of computer science. Computer scientists typically work on the theoretical side of computation. Although computer scientists can also focus their work and research on ...
s have utilized paraconsistent logic as a means of coping gracefully with inconsistent or contradictory information. Mathematical framework and rules of paraconsistent logic have been proposed as the
activation function The activation function of a node in an artificial neural network is a function that calculates the output of the node based on its individual inputs and their weights. Nontrivial problems can be solved using only a few nodes if the activation f ...
of an
artificial neuron An artificial neuron is a mathematical function conceived as a model of a biological neuron in a neural network. The artificial neuron is the elementary unit of an ''artificial neural network''. The design of the artificial neuron was inspired ...
in order to build a
neural network A neural network is a group of interconnected units called neurons that send signals to one another. Neurons can be either biological cells or signal pathways. While individual neurons are simple, many of them together in a network can perfor ...
for
function approximation In general, a function approximation problem asks us to select a function (mathematics), function among a that closely matches ("approximates") a in a task-specific way. The need for function approximations arises in many branches of applied ...
, model identification, and control with success. * Deontic logic and
metaethics In metaphilosophy and ethics, metaethics is the study of the nature, scope, ground, and meaning of moral judgment, ethical belief, or values. It is one of the three branches of ethics generally studied by philosophers, the others being normativ ...
: Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts. *
Software engineering Software engineering is a branch of both computer science and engineering focused on designing, developing, testing, and maintaining Application software, software applications. It involves applying engineering design process, engineering principl ...
: Paraconsistent logic has been proposed as a means for dealing with the pervasive inconsistencies among the
documentation Documentation is any communicable material that is used to describe, explain or instruct regarding some attributes of an object, system or procedure, such as its parts, assembly, installation, maintenance, and use. As a form of knowledge managem ...
,
use cases In both software and systems engineering, a use case is a structured description of a system’s behavior as it responds to requests from external actors, aiming to achieve a specific goal. It is used to define and validate functional requireme ...
, and
code In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communicati ...
of large software systems.Hewitt (2008b)Hewitt (2008a) *
Expert system In artificial intelligence (AI), an expert system is a computer system emulating the decision-making ability of a human expert. Expert systems are designed to solve complex problems by reasoning through bodies of knowledge, represented mainly as ...
. The Para-analyzer algorithm based on paraconsistent annotated logic by 2-value annotations (PAL2v), also called paraconsistent annotated evidential logic (PAL ''E''t), derived from paraconsistent logic, has been used in decision-making systems, such as to support medical diagnosis. *
Electronics Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...
design routinely uses a four-valued logic, 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. *
Control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...
: A model reference control built with recurrent paraconsistent neural network for a rotary inverted pendulum presented better robustness and lower control effort compared to a classical well tuned pole placement controller. *
Digital filter In signal processing, a digital filter is a system that performs mathematical operations on a Sampling (signal processing), sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other ma ...
: PAL2v Filter Algorithm, using a paraconsistent artificial neural cell of learning by contradiction extraction (PANLctx) in the composition of a paraconsistent analysis network (PANnet), based on the PAL2V rules and equations, can be used as an estimator, average extractor, filtering and in signal treatment for industrial automation and robotics. *
Contradiction In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
Extractor. A recurrent algorithm based on the PAL2v rules and equations has been used to extract contradictions in a set of statistical data. *
Quantum physics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
*
Black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
physics *
Hawking radiation Hawking radiation is black-body radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974. The radiation was not predicted by previous models which assumed that onc ...
*
Quantum computing A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using s ...
*
Spintronics Spintronics (a portmanteau meaning spin transport electronics), also known as spin electronics, is the study of the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge, in solid-st ...
*
Quantum entanglement Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic o ...
* Quantum coupling *
Uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...


Criticism

Logic, as it is classically understood, rests on three main rules (
Laws of Thought The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally the ...
): The
Law of Identity In logic, the law of identity states that each thing is identical with itself. It is the first of the traditional three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are b ...
(''LOI''), the Law of Non-Contradiction (''LNC''), and the
Law of the Excluded Middle In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and th ...
(''LEM''). Paraconsistent logic deviates from classical logic by refusing to accept ''LNC''. However, the ''LNC'' can be seen as closely interconnected with the ''LOI'' as well as the ''LEM'':    ''LoI'' states that ''A'' is ''A'' (''A''≡''A''). This means that ''A'' is distinct from its opposite or negation (''not A'', or ¬''A''). In classical logic this distinction is supported by the fact that when ''A'' is true, its opposite is not. However, without the ''LNC'', both ''A'' and ''not A'' can be true (''A''∧¬''A''), which blurs their distinction. And without distinction, it becomes challenging to define identity. Dropping the ''LNC'' thus runs risk to also eliminate the ''LoI''.    ''LEM'' states that either ''A'' or ''not A'' are true (''A''∨¬''A''). However, without the ''LNC'', both ''A'' and ''not A'' can be true (''A''∧¬''A''). Dropping the ''LNC'' thus runs risk to also eliminate the ''LEM'' Hence, dropping the ''LNC'' in a careless manner risks losing both the ''LOI'' and ''LEM'' as well. And dropping ''all'' three classical laws does not just change the ''kind'' of logic—it leaves us without any functional system of logic altogether. Loss of ''all'' logic eliminates the possibility of structured reasoning, A careless paraconsistent logic therefore might run risk of disapproving of any means of thinking other than chaos. Paraconsistent logic aims to evade this danger using careful and precise technical definitions. As a consequence, most criticism of paraconsistent logic also tends to be highly technical in nature (e.g. surrounding questions such as whether a paradox can be true). However, even on a highly technical level, paraconsistent logic can be challenging to argue against. It is obvious that paraconsistent logic leads to contradictions. However, the paraconsistent logician embraces contradictions, including any contradictions that are a part or the result of paraconsistent logic. As a consequence, much of the critique has focused on the applicability and comparative effectiveness of paraconsistent logic. This is an important debate since embracing paraconsistent logic comes at the risk of losing a large amount of
theorems In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
that form the basis of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
. Logician Stewart Shapiro aimed to make a case for paraconsistent logic as part of his argument for a pluralistic view of logic (the view that different logics are equally appropriate, or equally correct). He found that a case could be made that either, intuitonistic logic as the "One True Logic", or a pluralism of intuitonistic logic and
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
is interesting and fruitful. However, when it comes to paraconsistent logic, he found "no examples that are ... compelling (at least to me)". In "Saving Truth from Paradox", Hartry Field examines the value of paraconsistent logic as a solution to paradoxa. Field argues for a view that avoids both truth gluts (where a statement can be both true and false) and truth gaps (where a statement is neither true nor false). One of Field's concerns is the problem of a paraconsistent
metatheory A metatheory or meta-theory is a theory on a subject matter that is a theory in itself. Analyses or descriptions of an existing theory would be considered meta-theories. For mathematics and mathematical logic, a metatheory is a mathematical theo ...
: If the logic itself allows contradictions to be true, then the metatheory that describes or governs the logic might also have to be paraconsistent. If the metatheory is paraconsistent, then the justification of the logic (why we should accept it) might be suspect, because any argument made within a paraconsistent framework could potentially be both valid and invalid. This creates a challenge for proponents of paraconsistent logic to explain how their logic can be justified without falling into paradox or losing explanatory power. Stewart Shapiro expressed similar concerns: "there are certain notions and concepts that the dialetheist invokes (informally), but which she cannot adequately express, unless the meta-theory is (completely) consistent. The insistence on a consistent meta-theory would undermine the key aspect of dialetheism" In his book "In Contradiction", which argues in favor of paraconsistent dialetheism, Graham Priest admits to metatheoretic difficulties: "Is there a metatheory for paraconsistent logics that is acceptable in paraconsistent terms? The answer to this question is not at all obvious." Littmann and Keith Simmons argued that dialetheist theory is unintelligible: "Once we realize that the theory includes not only the statement '(L) is both true and false' but also the statement '(L) isn't both true and false' we may feel at a loss." Some philosophers have argued against dialetheism on the grounds 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 In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
''; it is merely a subcontrary-forming operator.See Slater (1995), Béziau (2000).


Alternatives

Approaches exist that allow for resolution of inconsistent beliefs without violating any of the intuitive logical principles. Most such systems use
multi-valued logic Many-valued logic (also multi- or multiple-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 (i.e., "true" and "false") ...
with
Bayesian inference Bayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian infer ...
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 00%irrefutable").


Notable figures

Notable figures in the history and/or modern development of paraconsistent logic include: *
Alan Ross Anderson Alan Ross Anderson (1925–1973) was an American logician and professor of philosophy at Yale University and the University of Pittsburgh. A frequent collaborator with Nuel Belnap, Anderson was instrumental in the development of relevance ...
(United States, 1925–1973). One of the founders of relevance logic, a kind of paraconsistent logic. * Florencio González Asenjo (
Argentina Argentina, officially the Argentine Republic, is a country in the southern half of South America. It covers an area of , making it the List of South American countries by area, second-largest country in South America after Brazil, the fourt ...
, 1927-2013) * Diderik Batens (Belgium) *
Nuel Belnap Nuel Dinsmore Belnap Jr. (; May 1, 1930 – June 12, 2024) was an American logician and philosopher who has made contributions to the philosophy of logic, temporal logic, and structural proof theory. He taught at the University of Pittsburgh ...
(United States, b. 1930) developed logical connectives of a four-valued logic. * Jean-Yves Béziau (France/Switzerland, b. 1965). Has written extensively on the general structural features and philosophical foundations of paraconsistent logics. * Ross Brady (Australia) * Bryson Brown (Canada) * Walter Carnielli (
Brazil Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
). The developer of the ''possible-translations semantics'', a new semantics which makes paraconsistent logics applicable and philosophically understood. * Newton da Costa (
Brazil Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
, 1929-2024). One of the first to develop formal systems of paraconsistent logic. * Itala M. L. D'Ottaviano (
Brazil Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
) * J. Michael Dunn (United States). An important figure in relevance logic. * Carl Hewitt * Stanisław Jaśkowski (
Poland Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
). One of the first to develop formal systems of paraconsistent logic. * R. E. Jennings (Canada) *
David Kellogg Lewis David (; , "beloved one") was a king of ancient Israel and Judah and the third king of the United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Aramaic-inscribed stone erected by a king of Aram-Dama ...
(USA, 1941–2001). Articulate critic of paraconsistent logic. *
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. His work centred on philosophical logic, mathematical logic and history of logi ...
(
Poland Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
, 1878–1956) * Robert K. Meyer (United States/Australia) * Chris Mortensen (Australia). Has written extensively on paraconsistent mathematics. * Lorenzo Peña (Spain, b. 1944). Has developed an original line of paraconsistent logic, gradualistic logic (also known as ''transitive logic'', TL), akin to
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
. * Val Plumwood ormerly Routley(Australia, b. 1939). Frequent collaborator with Sylvan. * Graham Priest (Australia). Perhaps the most prominent advocate of paraconsistent logic in the world today. * Francisco Miró Quesada (
Peru Peru, officially the Republic of Peru, is a country in western South America. It is bordered in the north by Ecuador and Colombia, in the east by Brazil, in the southeast by Bolivia, in the south by Chile, and in the south and west by the Pac ...
). Coined the term ''paraconsistent logic''. * B. H. Slater (Australia). Another articulate critic of paraconsistent logic. * Richard Sylvan ormerly Routley(New Zealand/Australia, 1935–1996). Important figure in relevance logic and a frequent collaborator with Plumwood and Priest. * Nicolai A. Vasiliev (Russia, 1880–1940). First to construct logic tolerant to contradiction (1910).


See also

*
Deviant logic Deviant logic is a type of logic incompatible with classical logic. Philosopher 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 ...
*
Formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
*
Fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
*
Probability logic Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A diffi ...
* Intuitionistic logic *
Table of logic symbols In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subs ...


Notes


Resources

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


External links

* * *
"World Congress on Paraconsistency, Ghent 1997, Juquehy 2000, Toulouse, 2003, Melbourne 2008, Kolkata, 2014"

Paraconsistent First-Order Logic with infinite hierarchy levels of contradiction LP#. Axiomatical system HST#, as paraconsistent generalization of Hrbacek set theory HST
* O. Arieli, A. Avron, A. Zamansky
"Ideal Paraconsistent Logics"
{{Authority control Belief revision Non-classical logic Philosophical logic Systems of formal logic