A paraconsistent logic is a logical system that attempts to deal nontrivially with contradictions. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or "inconsistency-tolerant") systems of logic. (The term will be used in both ways in this article.) Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...

Inconsistency-tolerant logics have been around 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 Peruvian philosopher Francisco Miró Quesada.^[1] Aristotle (Ancient Greek: AristotelÄ“s 384 BC – March 7, 322 BC) was an ancient Greek philosopher, who studied with Plato and taught Alexander the Great. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... Francisco Miró Quesada is a contemporary Peruvian philosopher that 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. ...

Definition

In classical logic (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 ("from a contradiction, anything follows"), can be expressed formally as Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... (A ∧ ¬A)→ B Ex falso quodlibet, also known as ex contradictione (sequitur) quodlibet or the principle of explosion is the rule of classical logic that states that anything follows from a contradiction. ...

where represents logical consequence. 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. Logical consequence is the relation that holds between a set of sentences and a sentence when the latter follows from the former. ... Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. ... In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ...

It should be emphasized that paraconsistent logics are in general weaker than classical logic; that is, they deem fewer inferences valid. (Strictly speaking, a paraconsistent logic may validate inferences that are classically invalid, though this is rarely the case. The point is that a paraconsistent logic can never be an extension of classical logic, that is, validate everything that classical logic does.) In that sense, then, paraconsistent logic is more "conservative" or "cautious" than classical logic.

Motivation

The 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 in a way that may help to determine how they ought to be revised to regain consistency. Info redirects here; for other uses, see . ...

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 paradoxes such as the Liar and Russell's paradox at face value. It should be noted that 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. Perhaps the most prominent contemporary defender of dialetheism (and paraconsistent logic) is Graham Priest, a philosopher at the University of Melbourne. Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ... Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist. ... In philosophy and logic, the liar paradox encompasses paradoxical statements such as: Analysing the statement I am lying now. ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Frege is contradictory. ... Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... The University of Melbourne   The Old Quad Building, formerly Old Law The University of Melbourne, located in Melbourne, Victoria, is the second oldest university in Australia, after the University of Sydney. ...

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:^[2]

Disjunction introduction
Disjunctive syllogism
Transitivity or "cut"

Though each of these principles have been challenged, the most popular approach is to reject disjunctive syllogism. If one is a dialetheist, it makes perfect sense that disjunctive syllogism should fail. For suppose that both A and ¬A are true but B is not. Then A v B is true, since its left disjunct is true. Thus the premises, A v B and ¬A, are true but the conclusion, B, is not. Disjunction introduction is the logic principle that, if A is true, then its true that either A or B is true. ... A disjunctive syllogism is one valid, simple argument form: A or B If not A Then B In logical operator notation: ¬ where represents the logical assertion. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... The Cut-elimination theorem is the central result establishing the significance of the sequent calculus. ...

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 Argentinian logician F. G. Asenjo in 1966 and later popularized by Priest and others.^[3] Motto: Spanish: En Unión y Libertad (English: In Union and Liberty) Anthem: Himno Nacional Argentino Capital Buenos Aires Largest city Buenos Aires Official language(s) Spanish Government President Democratic Federal Republic Néstor Kirchner Independence - May Revolution - Declared - Recognised from Spain 25 May 1810 9 July 1816 in 1821...

One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one.^[4] The binary relation V relates a formula 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 and disjunction are given as follows (the other logical connectives can be defined in the usual ways): Partial plot of a function f. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ... In logic, WFF is an abbreviation for well-formed formula. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... Negation (i. ... Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... In formal logic, logical connectives, also known as logical connectors and sometimes logical constants, serve to connect statements into more complicated compound statements. ...

• 
• 
• or V(B,1)
• and V(B,0)

In other words:

• not A is true iff A is false
• not A is false iff A is true
• A or B is true iff A is true or B is true
• A or B is false iff A is false and B is false

(Semantic) logical consequence is then defined as truth-preservation: ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

Γ A iff A is true whenever every element of Γ 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 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.^[5] In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ... In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...

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 the those of classical propositional logic.^[6] (LP and classical logic differ only in the inferences 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. note that demorgans laws are also a big part in circut design. ... In mathematical logic, natural deduction is the name given to a class of foundational approaches for two key concepts in logic, propositions and proofs. ... When someone sincerely agrees with an assertion, he or she is claiming that it is the truth. ... In logic, a tautology is a statement which is true by its own definition, and is therefore fundamentally uninformative. ... ...

It must be emphasized that LP is but one of many paraconsistent logics that have been proposed.^[7] 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 just in case it satisfies the following condition: Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ...

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. In symbolic logic, a logical constant is a symbol that has the same semantic value in all models. ...

Paraconsistent logic has signficant overlap with many-valued logic; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent). Multi-valued logics are logical calculi in which there are more than two possible truth values. ...

Intuitionistic logic allows A v ¬A to be false, while paraconsistent logic allows A & ¬A to be true. Thus it seems natural to regard paraconsistent logic as the "dual" of intuitionistic logic. However, intuitionistic logic is a specific logical system while paraconsistent logic encompasses a large class of systems. Accordingly, the "dual" of paraconsistent logic is a specific paraconsistent system called dual-intuitionistic logic (sometimes referred to as Brazilian logic, for historical reasons).^[8] The duality between the two systems is best seen within a sequent calculus framework. While in intuitionistic logic the sequent In mathematics, duality has numerous meanings. ... In proof theory and mathematical logic, the sequent calculus is a widely known deduction system for first-order logic (and propositional logic as a special case of it). ...

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 # which is the dual of intuitionistic implication. Very loosely, A # B can be read as ' A but not B '. However, # is not truth-functional as one might expect a 'but not' operator to be. In logic a truth function is a function generated from sentences of the language. ...

Direct Logic

Direct Logic is a recently published paraconsistent logic ^[9]. Goals for Direct Logic include:

• Formalize a notion of “direct” derivability from Ψ to Φ in terms of Ψ├ Φ.
• Not to blow up in the face of inconsistency.
• Formalize standard logical operators and connectives in terms of ├
• Carry out all computational reasoning. E.g., metatheorems of Direct Logic and the theorems of concurrent computation should be derivable in Direct Logic.
• Support all “natural” deductive reasoning that does not blow up.

Sequences of formulas

Direct Logic makes use of unordered sequences of formulas separated by commas intuitively meaning and (i.e. conjunction).

Basic Rules for ├

The basic rules for ├ are as follows:

├ (Ψ├ Ψ) ; reiteration

├ (Ψ,(Ψ├ Φ) ├ Φ) ; detachment

├ ((├ Ψ) ├ (├ ├ Ψ)) ; recursive tautology

├ ((├ Ψ),├ (Ψ├ Φ) ├ (├ Φ)) ; detachment for tautologies

├ ((Ψ├ Φ) ├ (Ψ, Θ ├ Φ)) ; monotonicity

├ ((Ψ├ Φ,Θ) ├ (Ψ├ Φ)) ; contraction

├ ((Ψ├ Φ),(W├ Θ) ├ (Ψ,W├ Φ,Q)) ; independent derivation

├ ((Ψ├ Φ),(Φ├ Θ) ├ (Ψ├ Θ)) ; transitivity of derivation

Propositional Connectives

The usual logical connectives are defined as follows:

Definition of ∧

The definition of conjunction (∧) is

├ (Ψ∧Φ ≡ Ψ,Φ)

The above justifies:

((Ψ∧Φ)├Θ) ≡ (Ψ,Φ├Θ)

(Φ├ (Ψ∧Θ)) ≡ (Φ├Ψ,Θ)

Definition of ∨

The definition of disjunction (∨) is

├ (Ψ∨Φ ≡ ((¬Ψ)├Φ) ∧ ((¬Φ)├Ψ))))

Note that the provenance of ∨ is derivational as opposed to truth functional.

Definition of ⇒

The definition of implication (⇒) is

├ (Ψ⇒Φ ≡ (Ψ├Φ) ∧ (¬Φ├ ¬Ψ))

Note that the provenance of ⇒ is derivational as opposed to truth functional.

Theorem: ├ (Ψ⇒Φ,Φ⇒Θ ├ Ψ⇒Θ)

Rules for Convenience of Users

Direct logic has the following rules for the convenience of users:

Rule of Direct Indirect Proof

The direct form of indirect proof is the following rule:

├ ((Ψ├ ¬Ψ) ├ ¬Ψ)

Rule by Cases for ∨

The Rule by Cases is a follows:

├ (Ψ∨Φ, (Ψ├Θ), (Φ├Θ) ├ Θ)

Equivalences for ∧ and ∨

(Ψ ∧ Ψ) ≡ Ψ

(Ψ ∧ Φ) ≡(Φ ∧ Ψ)

(Ψ ∧ (Φ ∧ Θ)) ≡ ((Ψ ∧ Φ) ∧ Θ)

(Ψ ∨ Ψ) ≡ Ψ

(Ψ ∨ Φ) ≡ (Φ ∨ Ψ)

(Ψ ∨ (Φ ∨ Θ)) ≡ ((Ψ ∨ Φ) ∨ Θ)

(Ψ ∨ (Φ ∧ Θ)) ≡ ((Ψ ∨ Φ) ∧ (Ψ ∨ Θ))

(Ψ ∧ (Φ ∨ Θ)) ≡ ((Ψ ∧ Φ) ∨ (Ψ ∧ Θ))

¬(Ψ ∧ Φ) ≡ ¬Ψ ∨¬ Φ

¬(Ψ ∨ Φ) ≡ ¬Ψ ∧¬ Φ

Rule for double negation The rule for double negation is:

├ (¬¬Ψ ≡ Ψ)

The above rule produces the following results:

Theorem: ├ (Ψ ∨ ¬Ψ)

Theorem: ├ ((Ψ⇒Φ) ≡ (¬Ψ∨Φ) ≡ (¬Φ⇒¬Ψ))

Caveat

The rules for the convenience of users in Direct Logic have been stated as broadly as possible to make the logic more useful. However, it has not yet been verified that that it does not blow up. If it does, then the rules of convenience for users will have to be adapted.

Relationship of Direct Logic to other Logics

Direct Logic is a restriction of Classical Logic in that it does not support alternation expansion ├ (Ψ ├ (Ψ∨Φ)) or general indirect proof ├ ((Ψ├ Φ),(Ψ├ ¬Φ) ├ ¬Ψ).

However, in all other respects, it is the same as Classical Logic in that all the usual classical results can be proved in Direct Logic, e.g., Gödel's incompleteness theorems and the theory of concurrent computation. Also it does a good job of supporing Natural deduction. In this respect it differs from previous paraconsistent logics in that Direct Logic does not support many seemingly arbitrary different versions. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... Wikiquote has a collection of quotations related to: Edsger Dijkstra In computer science, concurrency is a property of systems which consist of computations that execute overlapped in time, and which may permit the sharing of common resources between those overlapped computations. ... In mathematical logic, natural deduction is the name given to a class of foundational approaches for two key concepts in logic, propositions and proofs. ...

Applications

Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:^[10]

• 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 (see paraconsistent mathematics). 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 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 scientists have utilized paraconsistent logic as a means of coping gracefully with inconsistent information.^[11]
• Deontic logic and metaethics. Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts.

In the main, semantics (from the Greek and in greek letters σημαντικός or in latin letters semantikos, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ... When someone sincerely agrees with an assertion, they are claiming that it is the truth. ... In philosophy and logic, the liar paradox encompasses paradoxical statements such as: Analysing the statement I am lying now. ... Currys paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... Paraconsistent mathematics (sometimes called inconsistent mathematics ) represents an attempt to develop the classical infrastructure of mathematics (e. ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Frege is contradictory. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... Epistemology is an analytic branch of philosophy which studies the nature, origin, and scope of knowledge. ... Belief revision is the process changing beliefs to take into account a new piece of information. ... Knowledge Management or KM is a term applied to techniques used for the systematic collection, transfer, security and management of information within organisations, along with systems designed to help make best use of that knowledge. ... Hondas intelligent humanoid robot Artificial intelligence (AI) is defined as intelligence exhibited by an artificial entity. ... Computer science (informally: CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... Deontic logic, first put forward by Ernst Mally in 1926, is a form of modal logic used to describe and reason about obligation and permission. ... In philosophy, ethics is commonly divided into two branches, normative ethics and meta-ethics. ...

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.^[12] A related objection is that "negation" in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator.^[13] David K. Lewis David Kellogg Lewis (September 28, 1941 – October 14, 2001) is considered to have been one of the leading analytic philosophers of the latter half of the 20th century. ... Negation (i. ... The Square of Opposition is a term from the study of Aristotelian logic or Term Logic in which the logical relationship between various types of sentences is spelled out. ...

Notable figures

Notable figures in the history and/or modern development of paraconsistent logic include:

• Alan Ross Anderson (USA, 1925–1973). One of the founders of relevance logic, a kind of paraconsistent logic.
• F. G. Asenjo (Argentina)
• Diderik Batens (Belgium)
• Nuel Belnap (USA, b. 1930). Worked with Anderson on relevance 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)
• Newton da Costa (Brazil, b. 1929). One of the first to develop formal systems of paraconsistent logic.
• Itala M. L. D'Ottaviano (Brazil)
• J. Michael Dunn (USA). An important figure in relevance logic.
• Stanisław Jaśkowski (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 (USA/Australia)
• Chris Mortensen (Australia). Has written extensively on paraconsistent mathematics.
• Val Plumwood [formerly 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). Coined the term paraconsistent logic.
• Peter Schotch (Canada)
• B. H. Slater (Australia). Another articulate critic of paraconsistent logic.
• Richard Sylvan [formerly Routley] (New Zealand/Australia, 1935–1996). Important figure in relevance logic and a frequent collaborator with Plumwood and Priest.

Alan Ross Anderson, born 1925, was an American logician and professor of philosophy at Yale University and the University of Pittsburgh. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ... Nuel D. Belnap Jr. ... Jean-Yves Béziau (born January 15, 1965 in Orléans, France) is a professor and researcher at the Institute of Logic of the University of Neuchâtel, Switzerland. ... Newton Carneiro Affonso da Costa (born on 16 September in 1929 in Curitiba, Brazil), Professor Emeritus, is a Brazilian mathematician, logician, and philosopher of international reputation. ... Stanisław Jaśkowski Stanisław Jaśkowski (1906–1965) was a Polish logician who made important contributions to proof theory and semantics. ... David K. Lewis David Kellogg Lewis (September 28, 1941 – October 14, 2001) is considered to have been one of the leading analytic philosophers of the latter half of the 20th century. ... Jan Łukasiewicz (born 21 December 1878 - 13 February 1956) was a mathematician born in Lwów, Galicia (now Lviv, Ukraine). ... Paraconsistent mathematics (sometimes called inconsistent mathematics ) represents an attempt to develop the classical infrastructure of mathematics (e. ... Val Plumwood (born 1939), formerly Val Routley, is an Australian ecofeminist intellectual and activist, who has been prominent in the development of radical ecosophy since the early 1970s. ... Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St. ... Francisco Miró Quesada is a contemporary Peruvian philosopher that 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. ... Richard Sylvan, born as Richard Routley (1935 - 16 June 1996) was a philosopher and writer. ...

Notes

1. ↑ Priest (2002), p. 288 and §3.3.
2. ↑ See the article on the principle of explosion for more on this.
3. ↑ Priest (2002), p. 306.
4. ↑ LP is also commonly presented as a many-valued logic with three truth values (true, false, and both).
5. ↑ See, for example, Priest (2002), §5.
6. ↑ See Priest (2002), p. 310.
7. ↑ Surveys of various approaches to paraconsistent logic can be found in Bremer (2005) and Priest (2002).
8. ↑ See Aoyama (2004).
9. ↑ Hewitt (2006)
10. ↑ Most of these are discussed in Bremer (2005) and Priest (2002).
11. ↑ See, for example, the articles in Bertossi et al. (2004).
12. ↑ See Lewis (1982).
13. ↑ See Slater (1995), Béziau (2000).

(A ∧ ¬A)→ B Ex falso quodlibet, also known as ex contradictione (sequitur) quodlibet or the principle of explosion is the rule of classical logic that states that anything follows from a contradiction. ... Multi-valued logics are logical calculi in which there are more than two possible truth values. ...

Resources

• Aoyama, Hiroshi (2004). "LK, LJ, Dual Intuitionistic Logic, and Quantum Logic". Notre Dame Journal of Formal Logic 45 (4): 193–213.
• Bertossi, Leopoldo et al., eds. (2004). Inconsistency Tolerance, Berlin: Springer. ISBN 3540242600.
• Béziau, Jean-Yves (2000). “What is Paraconsistent Logic?” In D. Batens et al. (eds.) Frontiers of Paraconsistent Logic, p. 95-111, Baldock: Research Studies Press. ISBN 0863802532.
• Bremer, Manuel (2005). An Introduction to Paraconsistent Logics, Frankfurt: Peter Lang. ISBN 3631534132.
• Brown, Bryson (2002). “On Paraconsistency.” In Dale Jacquette (ed.) A Companion to Philosophical Logic, p. 628-650, Malden, Massachusetts: Blackwell Publishers. ISBN 0631216715.
• Hewitt, Carl (2006). "The repeated demise of logic programming and why it will be reincarnated". Papers from the AAAI Spring Symposium What Went Wrong and Why: Lessons from AI Research and Applications (Technical Report SS-06-08): 2–9.
• Lewis, David [1982] (1998). “Logic for Equivocators” Papers in Philosophical Logic, p. 97–110, Cambridge: Cambridge University Press. ISBN 0521587883.
• Priest, Graham (2002). “Paraconsistent Logic.” In D. Gabbay and F. Guenthner (eds.) Handbook of Philosophical Logic, Volume 6, 2nd ed., p. 287-393, The Netherlands: Kluwer Academic Publishers. ISBN 1402005830.
• Priest, Graham and Tanaka, Koji (2001). Paraconsistent Logic. Stanford Encyclopedia of Philosophy (Winter 2004 edition). URL accessed on February 24, 2006.
• Slater, B. H. (1995). "Paraconsistent Logics?". Journal of Philosophical Logic 24: 233–254.
• Woods, John (2003). Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences, Cambridge: Cambridge University Press. ISBN 0521009340.