Paraconsistent analysis
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A paraconsistent logic is an attempt at a
logical system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic which reject 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); however, the term ''paraconsistent'' ("beside the consistent") was first coined in 1976, by the Peruvian
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
Francisco Miró Quesada Cantuarias Francisco Miró Quesada Cantuarias (21 December 1918 – 11 June 2019) was a Peruvian philosopher, journalist and politician. In his works he discusses the belief in "human nature" on the basis that any collective assumption about such a natu ...
. 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 class ...
(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, "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 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.


Comparison with classical logic

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 class ...
; 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 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
metalanguage In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quot ...
s due to Alfred Tarski et al. According to Solomon Feferman 984 "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

A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent information 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 In philosophy of science, constructive empiricism is a form of empiricism. While it is sometimes referred to as an empiricist form of structuralism, its main proponent, Bas van Fraassen, has consistently distinguished between the two views. Overvie ...
, 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 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 In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not (not ...
as well as associativity,
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, distributivity,
De Morgan De Morgan or de Morgan is a surname, and may refer to: * Augustus De Morgan (1806–1871), British mathematician and logician. ** De Morgan's laws (or De Morgan's theorem), a set of rules from propositional logic. ** The De Morgan Medal, a trien ...
, 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 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. Although the logic has also be ...
, 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 by contradiction (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 In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not (not ...
(\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.


Example

One well-known system 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 and others. One way of presenting the semantics for LP is to replace the usual
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
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 betwee ...
to a truth value: 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 complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
and
disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
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 ''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) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...
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 In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
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 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 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 distinction that in ...
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 In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes ...
. 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; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent).
Dialetheic logic Dialetheism (from Greek 'twice' and 'truth') is the view that there are statements that are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true ...
s, 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
" 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 In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ...
,
positive propositional calculus This article contains a list of sample Hilbert-style deductive systems for propositional logics. Classical propositional calculus systems Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent ...
,
equivalential calculus This article contains a list of sample Hilbert-style deductive systems for propositional logics. Classical propositional calculus systems Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent ...
and
minimal logic Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It is an intuitionistic and paraconsistent logic, that rejects both the law of the excluded middle as well as the principle of explosion (' ...
. 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 In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
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: Paraconsistent logic has been proposed as means of providing a simple and intuitive formal account of truth 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 and the foundations of mathematics * Epistemology and belief revision: Paraconsistent logic has been proposed as a means of reasoning with and revising inconsistent theories and belief systems. * Knowledge management and artificial intelligence: Some
computer scientist A computer scientist is a person who is trained in the academic study of computer science. Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus (al ...
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 of an artificial neuron in order to build a
neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
for function approximation,
model identification In statistics, identifiability is a property which a model must satisfy for precise inference to be possible. A model is identifiable if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining ...
, and
control Control may refer to: Basic meanings Economics and business * Control (management), an element of management * Control, an element of management accounting * Comptroller (or controller), a senior financial officer in an organization * Controlling ...
with success. * Deontic logic and metaethics: Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts. * Software engineering: 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 manageme ...
,
use cases In software and systems engineering, the phrase use case is a polyseme with two senses: # A usage scenario for a piece of software; often used in the plural to suggest situations where a piece of software may be useful. # A potential scenario i ...
, 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 communication ...
of large
software systems A software system is a system of intercommunicating components based on software forming part of a computer system (a combination of hardware and software). It "consists of a number of separate programs, configuration files, which are used to ...
.Hewitt (2008b)Hewitt (2008a) * Electronics 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. *
Quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
*
Black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
physics * Hawking radiation *
Quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
* Spintronics * Quantum entanglement * Quantum coupling * Uncertainty principle


Criticism

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 complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
''; it is merely a
subcontrary An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" ( ...
-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 with
Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, a ...
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"). 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 Anderson (United States, 1925–1973). One of the founders of relevance logic, a kind of paraconsistent logic. * Florencio González Asenjo ( Argentina, 1927-2013) *
Diderik Batens Diderik Batens (born 15 November 1944), 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 ...
(Belgium) * Nuel Belnap (United States, b. 1930) developed logical connectives of a four-valued logic. *
Jean-Yves Béziau Jean-Yves Béziau (; born January 15, 1965, in Orléans, France) is a professor and researcher of the Brazilian Research Council (CNPq) at the University of Brazil in Rio de Janeiro. Career Béziau works in the field of logic—in particular, ...
(France/Switzerland, b. 1965). Has written extensively on the general structural features and philosophical foundations of paraconsistent logics. *
Ross Brady Ross or ROSS may refer to: People * Clan Ross, a Highland Scottish clan * Ross (name), including a list of people with the surname or given name Ross, as well as the meaning * Earl of Ross, a peerage of Scotland Places * RoSS, the Republic of Sout ...
(Australia) * Bryson Brown (Canada) *
Walter Carnielli Walter Alexandre Carnielli (born 11 January 1952 in Campinas, Brazil) is a Brazilian mathematician, logician, and philosopher, full professor of Logic at thState University of Campinas (UNICAMP) With Bachelor and Ms.C. degrees in mathematics at th ...
( Brazil). The developer of the ''possible-translations semantics'', a new semantics which makes paraconsistent logics applicable and philosophically understood. *
Newton da Costa Newton Carneiro Affonso da Costa (born 16 September 1929 in Curitiba, Brazil) 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 ...
( Brazil, b. 1929). One of the first to develop formal systems of paraconsistent logic. * Itala M. L. D'Ottaviano ( Brazil) *
J. Michael Dunn J. Michael Dunn (June 19, 1941 – April 5, 2021) was Oscar Ewing Professor Emeritus of Philosophy, Professor Emeritus of Informatics and Computer Science, was twice chair of the Philosophy Department, was Executive Associate Dean of the College o ...
(United States). An important figure in relevance logic. * Carl Hewitt *
Stanisław Jaśkowski Stanisław Jaśkowski (22 April 1906, in Warsaw – 16 November 1965, in Warsaw) was a Polish logician who made important contributions to proof theory and formal semantics. He was a student of Jan Łukasiewicz and a member of the Lwów–War ...
( Poland). One of the first to develop formal systems of paraconsistent logic. * R. E. Jennings (Canada) * David Kellogg Lewis (USA, 1941–2001). Articulate critic of paraconsistent logic. * Jan Łukasiewicz ( Poland, 1878–1956) *
Robert K. Meyer The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honou ...
(United States/Australia) * Chris Mortensen (Australia). Has written extensively on paraconsistent mathematics. *
Lorenzo Peña Lorenzo Peña (born August 29, 1944) 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, 194 ...
(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 Val Plumwood (11 August 1939 – 29 February 2008) was an Australian philosopher and ecofeminist known for her work on anthropocentrism. From the 1970s she played a central role in the development of radical ecosophy. Working mostly as an indepe ...
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 Francisco is the Spanish and Portuguese form of the masculine given name ''Franciscus''. Nicknames In Spanish, people with the name Francisco are sometimes nicknamed "Paco". San Francisco de Asís was known as ''Pater Comunitatis'' (father of ...
( Peru). Coined the term ''paraconsistent logic''. *
B. H. Slater B is the second letter of the Latin alphabet. B may also refer to: Science, technology, and mathematics Astronomy * Astronomical objects in the List of dark nebulae#Barnard objects, Barnard list of dark nebulae (abbreviation B) * Latitude (''b ...
(Australia). Another articulate critic of paraconsistent logic. *
Richard Sylvan Richard Sylvan (13 December 1935 – 16 June 1996) was a New Zealand–born philosopher, logician, and environmentalist. Biography Sylvan was born Francis Richard Routley in Levin, New Zealand, and his early work is cited with this surname. H ...
ormerly Routley(New Zealand/Australia, 1935–1996). Important figure in relevance logic and a frequent collaborator with Plumwood and Priest. *
Nicolai A. Vasiliev Nicolai Alexandrovich Vasiliev (russian: Николай Александрович Васильев), also Vasil'ev, Vassilieff, Wassilieff (December 31, 1940), was a Russian logician, philosopher, psychologist, poet. He was a forerunner of Paracons ...
(Russia, 1880–1940). First to construct logic tolerant to contradiction (1910).


See also

* Deviant logic * Formal logic *
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"
{{Non-classical logic Belief revision Non-classical logic Philosophical logic Systems of formal logic