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# Outline of logic

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. 

## A priori and a posteriori

The Latin phrases a priori ("from the earlier") and a posteriori ("from the latter") are philosophical terms of art popularized by Immanuel Kant's Critique of Pure Reason (first published in 1781, second edition in 1787), one of the most influential works in the history of philosophy.

## A System of Logic

A System of Logic, Ratiocinative and Inductive is an 1843 book by English philosopher John Stuart Mill.

## Abductive reasoning

Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference which starts with an observation or set of observations then seeks to find the simplest and most likely explanation.

## Absorption law

In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.

## Accuracy and precision

Precision is a description of random errors, a measure of statistical variability.

## Ad hoc hypothesis

In science and philosophy, an ad hoc hypothesis is a hypothesis added to a theory in order to save it from being falsified.

## Affine logic

Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.

## Affirming the consequent

Affirming the consequent, sometimes called converse error, fallacy of the converse or confusion of necessity and sufficiency, is a formal fallacy of inferring the converse from the original statement.

## Aleph number

In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.

## Algebraic normal form

In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), Zhegalkin normal form, or Reed–Muller expansion is a way of writing logical formulas in one of three subforms.

## Alpha recursion theory

In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals \alpha.

## Ambiguity

Ambiguity is a type of meaning in which several interpretations are plausible.

## Analysis

Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it.

## Analytic–synthetic distinction

The analytic–synthetic distinction (also called the analytic–synthetic dichotomy) is a semantic distinction, used primarily in philosophy to distinguish propositions (in particular, statements that are affirmative subject–predicate judgments) into two types: analytic propositions and synthetic propositions.

## Antecedent (logic)

An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause.

## Anti-psychologism

In logic, anti-psychologism (also logical objectivism or logical realism) is a theory about the nature of logical truth, that it does not depend upon the contents of human ideas but exists independent of human ideas.

## Antinomy

Antinomy (Greek ἀντί, antí, "against, in opposition to", and νόμος, nómos, "law") refers to a real or apparent mutual incompatibility of two laws.

## Antisymmetric relation

In mathematics, a binary relation R on a set X is anti-symmetric if there is no pair of distinct elements of X each of which is related by R to the other.

## Argument

In logic and philosophy, an argument is a series of statements typically used to persuade someone of something or to present reasons for accepting a conclusion.

## Argument map

In informal logic and philosophy, an argument map or argument diagram is a visual representation of the structure of an argument.

## Argumentation theory

Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be reached through logical reasoning; that is, claims based, soundly or not, on premises.

## Arithmetical set

In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic.

## Association for Symbolic Logic

The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic.

## Asymmetric relation

In mathematics, an asymmetric relation is a binary relation on a set X where.

## Atomic formula

In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas.

## Atomic sentence

In logic, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences.

## Attacking Faulty Reasoning

Attacking Faulty Reasoning is a textbook on logical fallacies by T. Edward Damer that has been used for many years in a number of college courses on logic, critical thinking, argumentation, and philosophy.

## Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

## Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.

## Baralipton

In classical logic, Baralipton is a mnemonic word used to identify a form of syllogism.

## Baroco

In classical logic, baroco is a mnemonic word used to memorize a syllogism.

## Begriffsschrift

Begriffsschrift (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.

## Belief

Belief is the state of mind in which a person thinks something to be the case with or without there being empirical evidence to prove that something is the case with factual certainty.

## Belief bias

Belief bias is the tendency to judge the strength of arguments based on the plausibility of their conclusion rather than how strongly they support that conclusion.

## Bias

Bias is disproportionate weight in favour of or against one thing, person, or group compared with another, usually in a way considered to be unfair.

## Biconditional elimination

Biconditional elimination is the name of two valid rules of inference of propositional logic.

## Biconditional introduction

In propositional logic, biconditional introduction is a valid rule of inference.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Bijection, injection and surjection

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Boolean algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.

## Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

## Boolean algebras canonically defined

Boolean algebra is a mathematically rich branch of abstract algebra.

## Boolean conjunctive query

In the theory of relational databases, a Boolean conjunctive query is a conjunctive query without distinguished predicates, i.e., a query in the form R_1(t_1) \wedge \cdots \wedge R_n(t_n), where each R_i is a relation symbol and each t_i is a tuple of variables and constants; the number of elements in t_i is equal to the arity of R_i.

## Boolean domain

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true.

## Boolean expression

In computer science, a Boolean expression is an expression in a programming language that produces a Boolean value when evaluated, i.e. one of true or false.

## Boolean function

In mathematics and logic, a (finitary) Boolean function (or switching function) is a function of the form ƒ: Bk → B, where B.

## Boolean ring

In mathematics, a Boolean ring R is a ring for which x2.

## Boolean-valued function

A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f: X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B.

## Boolean-valued model

In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory.

## Bracket (mathematics)

In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses, square brackets, braces, and angle brackets ⟨.

## Bunched logic

Bunched logic is a variety of substructural logic proposed by Peter O'Hearn and David Pym.

## Canonical normal form

In Boolean algebra, any Boolean function can be put into the canonical disjunctive normal form (CDNF) or minterm canonical form and its dual canonical conjunctive normal form (CCNF) or maxterm canonical form.

## Cantor's diagonal argument

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.

## Cantor's theorem

In elementary set theory, Cantor's theorem is a fundamental result that states that, for any set A, the set of all subsets of A (the power set of A, denoted by \mathcal(A)) has a strictly greater cardinality than A itself.

## Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

## Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.

## Categorical logic

Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic.

## Categorical proposition

In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the predicate term).

## Categories (Aristotle)

The Categories (Greek Κατηγορίαι Katēgoriai; Latin Categoriae) is a text from Aristotle's Organon that enumerates all the possible kinds of things that can be the subject or the predicate of a proposition.

## Causality

Causality (also referred to as causation, or cause and effect) is what connects one process (the cause) with another process or state (the effect), where the first is partly responsible for the second, and the second is partly dependent on the first.

## Charles Sanders Peirce bibliography

This Charles Sanders Peirce bibliography consolidates numerous references to Charles Sanders Peirce's writings, including letters, manuscripts, publications, and Nachlass.

## Church–Turing thesis

In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a hypothesis about the nature of computable functions.

## Classical logic

Classical logic (or standard logic) is an intensively studied and widely used class of formal logics.

## Clause (logic)

In logic, a clause is an expression formed from a finite collection of literals (atoms or their negations) that is true either whenever at least one of the literals that form it is true (a disjunctive clause, the most common use of the term), or when all of the literals that form it are true (a conjunctive clause, a less common use of the term).

## Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall.

## Cognitive bias

A cognitive bias is a systematic pattern of deviation from norm or rationality in judgment.

## Commutativity of conjunction

In propositional logic, the commutativity of conjunction is a valid argument form and truth-functional tautology.

## Complement (set theory)

In set theory, the complement of a set refers to elements not in.

## Complete Boolean algebra

In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound).

## Completeness (logic)

In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete.

## Composition of relations

In the mathematics of binary relations, the composition relations is a concept of forming a new relation from two given relations R and S. The composition of relations is called relative multiplication in the calculus of relations.

## Computability logic

Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth.

## Computability theory

Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

## Computable function

Computable functions are the basic objects of study in computability theory.

## Computation

Computation is any type of calculation that includes both arithmetical and non-arithmetical steps and follows a well-defined model, for example an algorithm.

## Conceptualism

Conceptualism is a philosophical theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind.

## Conditional proof

A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.

## Conditioned disjunction

In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary logical connective introduced by Church.

## Confirmation bias

Confirmation bias, also called confirmatory bias or myside bias,David Perkins, a professor and researcher at the Harvard Graduate School of Education, coined the term "myside bias" referring to a preference for "my" side of an issue.

## Congruence relation

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure.

## Conjunction elimination

In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, or simplification) is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction A and B is true, then A is true, and B is true.

## Conjunction introduction

Conjunction introduction (often abbreviated simply as conjunction and also called and introduction) is a valid rule of inference of propositional logic.

## Consequent

A consequent is the second half of a hypothetical proposition.

## Consistency

In classical deductive logic, a consistent theory is one that does not contain a contradiction.

## Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets.

## Constructive dilemma

Constructive dilemma is a valid rule of inference of propositional logic.

## Constructivist epistemology

Constructivist epistemology is a branch in philosophy of science maintaining that scientific knowledge is constructed by the scientific community, who seek to measure and construct models of the natural world.

## Contingency (philosophy)

In philosophy and logic, contingency is the status of propositions that are neither true under every possible valuation (i.e. tautologies) nor false under every possible valuation (i.e. contradictions).

## Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.

## Contradiction

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions.

## Contraposition (traditional logic)

In traditional logic, contraposition is a form of immediate inference in which a proposition is inferred from another and where the former has for its subject the contradictory of the original logical proposition's predicate.

## Contrary (logic)

Contrary is the relationship between two propositions when they cannot both be true (although both may be false).

## Conventionalism

Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on (explicit or implicit) agreements in society, rather than on external reality.

## Converse (logic)

In logic, the converse of a categorical or implicational statement is the result of reversing its two parts.

## Converse implication

Converse implication is the converse of implication, written ←. That is to say; that for any two propositions P and Q, if Q implies P, then P is the converse implication of Q. It is written P \leftarrow Q, but may also be notated P \subset Q, or "Bpq" (in Bocheński notation).

## Converse nonimplication

In logic, converse nonimplication is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).

## Converse relation

In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Counterpart theory

In philosophy, specifically in the area of modal metaphysics, counterpart theory is an alternative to standard (Kripkean) possible-worlds semantics for interpreting quantified modal logic.

## Covering relation

In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours.

## Credibility

Credibility comprises the objective and subjective components of the believability of a source or message.

## Critical pedagogy

Critical pedagogy is a philosophy of education and social movement that has developed and applied concepts from critical theory and related traditions to the field of education and the study of culture.

## Critical reading

Critical reading is a form of language analysis that does not take the given text at face value, but involves a deeper examination of the claims put forth as well as the supporting points and possible counterarguments.

## Critical thinking

Critical thinking is the objective analysis of facts to form a judgment.

## Cyclic order

In mathematics, a cyclic order is a way to arrange a set of objects in a circle.

## De Interpretatione

De Interpretatione or On Interpretation (Greek: Περὶ Ἑρμηνείας, Peri Hermeneias) is the second text from Aristotle's Organon and is among the earliest surviving philosophical works in the Western tradition to deal with the relationship between language and logic in a comprehensive, explicit, and formal way.

## De Morgan's laws

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference.

## Decidability (logic)

In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a boolean true or false value that is correct (instead of looping indefinitely, crashing, returning "don't know" or returning a wrong answer).

## Decidophobia

Decidophobia is, according to Princeton University philosopher Walter Kaufmann, a fear of making decisions.

## Decision problem

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values.

## Decision theory

Decision theory (or the theory of choice) is the study of the reasoning underlying an agent's choices.

## Decision-making

In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several alternative possibilities.

## Deductive closure

Deductive closure is a property of a set of objects (usually the objects in question are statements).

## Deductive reasoning

Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.

## Definition

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).

## Deflationary theory of truth

In philosophy and logic, a deflationary theory of truth is one of a family of theories that all have in common the claim that assertions of predicate truth of a statement do not attribute a property called "truth" to such a statement.

## Degree of truth

In standard mathematics, propositions can typically be considered unambiguously true or false.

## Dense order

In mathematics, a partial order or total order.

## Denying the antecedent

Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement.

## Deontic logic

Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts.

## Dependence relation

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.

## Dependency relation

In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, and reflexive; i.e. a finite tolerance relation.

## Derivative algebra (abstract algebra)

In abstract algebra, a derivative algebra is an algebraic structure of the signature where is a Boolean algebra and D is a unary operator, the derivative operator, satisfying the identities.

## Description

Description is the pattern of narrative development that aims to make vivid a place, an object, a character, or a group.

## Description logic

Description logics (DL) are a family of formal knowledge representation languages.

## Destructive dilemma

Destructive dilemma is the name of a valid rule of inference of propositional logic.

## Deviant logic

Philosopher Susan Haack uses the term "deviant logic" to describe certain non-classical systems of logic.

## Dialetheism

Dialetheism is the view that there are statements which are both true and false.

## Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound.

## Disjoint sets

In mathematics, two sets are said to be disjoint sets if they have no element in common.

## Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in.

## Disjunction elimination

In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.

## Disjunction introduction

Disjunction introduction or addition (also called or introduction) is a rule of inference of propositional logic and almost every other deduction system.

## Disjunctive syllogism

In classical logic, disjunctive syllogism (historically known as modus tollendo ponens (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Domain of discourse

In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.

## 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-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.

## Double negative

A double negative is a grammatical construction occurring when two forms of negation are used in the same sentence.

## Doxastic logic

Doxastic logic is a type of logic concerned with reasoning about beliefs.

## Effective method

In logic, mathematics and computer science, especially metalogic and computability theory, an effective methodHunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971 or effective procedure is a procedure for solving a problem from a specific class.

## Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

## Emotional reasoning

Emotional reasoning is a cognitive process by which a person concludes that his/her emotional reaction proves something is true, regardless of the observed evidence.

## Empty domain

In first-order logic the empty domain is the empty set having no members.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## End term

The end terms in a categorical syllogism are the major term and the minor term (not the middle term).

## Enthymeme

An enthymeme (ἐνθύμημα, enthumēma) is a rhetorical syllogism (a three-part deductive argument) used in oratorical practice.

## Entscheidungsproblem

In mathematics and computer science, the Entscheidungsproblem (German for "decision problem") is a challenge posed by David Hilbert in 1928.

## Enumeration

An enumeration is a complete, ordered listing of all the items in a collection.

## Epistemic modal logic

Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that formalizes "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other.".

## Evidence

Evidence, broadly construed, is anything presented in support of an assertion.

## Exclusive or

Exclusive or or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).

## Existential fallacy

The existential fallacy, or existential instantiation, is a formal fallacy.

## Expert

An expert is someone who has a prolonged or intense experience through practice and education in a particular field.

## Explanation

An explanation is a set of statements usually constructed to describe a set of facts which clarifies the causes, context, and consequences of those facts.

## Explanatory power

Explanatory power is the ability of a hypothesis or theory to effectively explain the subject matter it pertains to.

## Extension (predicate logic)

The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate.

## Extensionality

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties.

## Fact

A fact is a statement that is consistent with reality or can be proven with evidence.

## Fallacy

A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves" in the construction of an argument.

## Fictionalism

Fictionalism is the view in philosophy according to which statements that appear to be descriptions of the world should not be construed as such, but should instead be understood as cases of "make believe", of pretending to treat something as literally true (a "useful fiction").

## Field of sets

In mathematics a field of sets is a pair \langle X, \mathcal \rangle where X is a set and \mathcal is an algebra over X i.e., a non-empty subset of the power set of X closed under the intersection and union of pairs of sets and under complements of individual sets.

## Finitary relation

In mathematics, a finitary relation has a finite number of "places".

## Finite set

In mathematics, a finite set is a set that has a finite number of elements.

## First-order logic

First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

## First-order predicate

In mathematical logic, a first-order predicate is a predicate that takes only individual(s) constants or variables as argument(s).

## Forcing (mathematics)

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results.

## Forcing (recursion theory)

Forcing in recursion theory is a modification of Paul Cohen's original set theoretic technique of forcing to deal with the effective concerns in recursion theory.

## Formal fallacy

In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur (Latin for "it does not follow") is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic.

## Formal language

In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it.

## Formal proof

A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference.

## Formal system

A formal system is the name of a logic system usually defined in the mathematical way.

## Formalism (philosophy)

The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy.

## Formation rule

In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language.

## Free Boolean algebra

In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that.

## Free logic

A free logic is a logic with fewer existential presuppositions than classical logic.

## Free variables and bound variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Functional completeness

In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression.

## Fuzzy logic

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1.

## Game semantics

Game semantics (dialogische Logik, translated as dialogical logic) is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes.

## Game theory

Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers".

## Gödel's completeness theorem

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

## Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.

## Gödel, Escher, Bach

Gödel, Escher, Bach: An Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter.

## Georg Cantor's first set theory article

Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.

## Halting problem

In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running (i.e., halt) or continue to run forever.

## Higher-order logic

In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics.

## Higher-order thinking

Higher-order thinking, known as higher order thinking skills (HOTS), is a concept of education reform based on learning taxonomies (such as Bloom's taxonomy).

## History of logic

The history of logic deals with the study of the development of the science of valid inference (logic).

## History of the Church–Turing thesis

The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable.

## Hypothetical syllogism

In classical logic, hypothetical syllogism is a valid argument form which is a syllogism having a conditional statement for one or both of its premises.

## Idempotence

Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.

## Idempotency of entailment

Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one.

## Identity (mathematics)

In mathematics an identity is an equality relation A.

## Identity (philosophy)

In philosophy, identity, from ("sameness"), is the relation each thing bears only to itself.

## Illuminationism

Illuminationist or ishraqi philosophy is a type of Islamic philosophy introduced by Shahab al-Din Suhrawardi in the twelfth century CE.

## Immediate inference

An immediate inference is an inference which can be made from only one statement or proposition.

## Implicant

In Boolean logic, an implicant is a "covering" (sum term or product term) of one or more minterms in a sum of products (or maxterms in product of sums) of a Boolean function.

## Independence (mathematical logic)

In mathematical logic, independence refers to the unprovability of a sentence from other sentences.

No description.

## Index set

In mathematics, an index set is a set whose members label (or index) members of another set.

## Inductive reasoning

Inductive reasoning (as opposed to ''deductive'' reasoning or ''abductive'' reasoning) is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion.

## Inference

Inferences are steps in reasoning, moving from premises to logical consequences.

## Infinitary logic

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.

## Infinite set

In set theory, an infinite set is a set that is not a finite set.

## Infinity

Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

## Informal logic

Informal logic, intuitively, refers to the principles of logic and logical thought outside of a formal setting.

## Inquiry

An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem.

## Intension

In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase, or another symbol.

## Intensional logic

Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (extensions), by additional quantifiers that range over terms that may have such individuals as their value (intensions).

## Intermediate logic

In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic.

## Interpretability

In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.

## Interpretability logic

Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities.

## Interpretation (logic)

An interpretation is an assignment of meaning to the symbols of a formal language.

## Interpretive discussion

An interpretive discussion is a discussion in which participants explore and/or resolve interpretations often pertaining to texts of any medium containing significant ambiguity in meaning.

## Intersection (set theory)

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

## Intransitivity

In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations.

## Introduction to Mathematical Philosophy

Introduction to Mathematical Philosophy is a book by Bertrand Russell, published in 1919, written in part to exposit in a less technical way the main ideas of his and Whitehead's Principia Mathematica (1910–13), including the theory of descriptions.

## Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof.

## Inverse (logic)

In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence.

## Inverse function

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.

## Involution (mathematics)

In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.

## Journal of Logic, Language and Information

The Journal of Logic, Language and Information is the official journal of the European Association for Logic, Language and Information.

## Journal of Philosophical Logic

The Journal of Philosophical Logic is a peer-reviewed scientific journal founded in 1972.

## Judgment (mathematical logic)

In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in the metalanguage.

## Lambda calculus

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.

## Language, Truth, and Logic

Language, Truth, and Logic is a 1936 work of philosophy by Alfred Jules Ayer.

## Large cardinal

In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.

## Law of excluded middle

In logic, the law of excluded middle (or the principle of excluded middle) states that for any proposition, either that proposition is true or its negation is true.

## Law of identity

In logic, the law of identity states that each thing is identical with itself.

## Law of noncontradiction

In classical logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory statements cannot both be true in the same sense at the same time, e.g. the two propositions "A is B" and "A is not B" are mutually exclusive.

## Laws of Form

Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy.

## Löwenheim–Skolem theorem

In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.

## 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.

## Linguistics and Philosophy

Linguistics and Philosophy is a peer reviewed journal addressing "structure and meaning in natural language".

## List of Boolean algebra topics

This is a list of topics around Boolean algebra and propositional logic.

## List of fallacies

In reasoning to argue a claim, a fallacy is reasoning that is evaluated as logically incorrect and that undermines the logical validity of the argument and permits its recognition as unsound.

## List of first-order theories

In mathematical logic, a first-order theory is given by a set of axioms in some language.

## List of logicians

A logician is a person whose topic of scholarly study is logic.

## List of mathematical logic topics

This is a list of mathematical logic topics, by Wikipedia page.

## List of paradoxes

This is a list of paradoxes, grouped thematically.

## List of philosophers of language

This is a list of philosophers of language.

## List of rules of inference

This is a list of rules of inference, logical laws that relate to mathematical formulae.

## List of set theory topics

This page is a list of articles related to set theory.

## List of undecidable problems

In computability theory, an undecidable problem is a type of computational problem that requires a yes/no answer, but where there cannot possibly be any computer program that always gives the correct answer; that is, any possible program would sometimes give the wrong answer or run forever without giving any answer.

## Lists of mathematics topics

This article itemizes the various lists of mathematics topics.

## Literal (mathematical logic)

In mathematical logic, a literal is an atomic formula (atom) or its negation.

## Logic

Logic (from the logikḗ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.

## Logic alphabet

The logic alphabet, also called the X-stem Logic Alphabet (XLA), constitutes an iconic set of symbols that systematically represents the sixteen possible binary truth functions of logic.

## Logic gate

In electronics, a logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more binary inputs and produces a single binary output.

## Logic programming

Logic programming is a type of programming paradigm which is largely based on formal logic.

## Logic redundancy

Logic redundancy occurs in a digital gate network containing circuitry that does not affect the static logic function.

## Logical atomism

Logical atomism is a philosophical belief that originated in the early 20th century with the development of analytic philosophy.

## Logical biconditional

In logic and mathematics, the logical biconditional (sometimes known as the material biconditional) is the logical connective of two statements asserting "P if and only if Q", where P is an antecedent and Q is a consequent.

## Logical conjunction

In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true.

## Logical connective

In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.

## 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.

## Logical constant

In logic, a logical constant of a language \mathcal is a symbol that has the same semantic value under every interpretation of \mathcal.

## Logical disjunction

In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true.

## Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

## Logical form

In philosophy and mathematics, a logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system.

## Logical holism

Logical holism is the belief that the world operates in such a way that no part can be known without the whole being known first.

## Logical matrix

A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets.

## Logical NOR

In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or.

## Logical reasoning

Informally, two kinds of logical reasoning can be distinguished in addition to formal deduction: induction and abduction.

## Logical truth

Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature.

## Logicism

Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic.

## Many-valued logic

In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values.

## Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

## Material conditional

The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→".

## Material nonimplication

Material nonimplication or abjunction (Latin ab.

## Mathematical fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy.

## Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Meaning (linguistics)

In linguistics, meaning is the information or concepts that a sender intends to convey, or does convey, in communication with a receiver.

## Meaning (non-linguistic)

A non-linguistic meaning is an actual or possible derivation from sentience, which is not associated with signs that have any original or primary intent of communication.

## Metalanguage

Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined.

## Metalogic

Metalogic is the study of the metatheory of logic.

## Metasyntactic variable

A metasyntactic variable is a specific word or set of words identified as a placeholder in computer science and specifically computer programming.

## Metatheorem

In logic, a metatheorem is a statement about a formal system proven in a metalanguage.

## Metatheory

A metatheory or meta-theory is a theory whose subject matter is some theory.

## Metavariable

In logic, a metavariable (also metalinguistic variable or syntactical variable) is a symbol or symbol string which belongs to a metalanguage and stands for elements of some object language.

## Middle term

In logic, a middle term is a term that appears (as a subject or predicate of a categorical proposition) in both premises but not in the conclusion of a categorical syllogism.

## Minimal logic

Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson (1936).

## Modal algebra

In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that.

## Modal fictionalism

Modal fictionalism is a term used in philosophy, and more specifically in the metaphysics of modality, to describe the position that holds that modality can be analysed in terms of a fiction about possible worlds.

## Modal logic

Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality.

## Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

## Modus ponendo tollens

Modus ponendo tollens (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic.

## Modus ponens

In propositional logic, modus ponens (MP; also modus ponendo ponens (Latin for "mode that affirms by affirming") or implication elimination) is a rule of inference.

## Modus tollens

In propositional logic, modus tollens (MT; also modus tollendo tollens (Latin for "mode that denies by denying") or denying the consequent) is a valid argument form and a rule of inference.

## Monadic Boolean algebra

In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature where &lang;A, &middot;, +, ', 0, 1&rang; is a Boolean algebra.

## Monadic predicate calculus

In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols.

## Monotonicity of entailment

Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions.

## Morse–Kelley set theory

In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG).

## Multiset

In mathematics, a multiset (aka bag or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements.

## Naive set theory

Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.

## Name

A name is a term used for identification.

## Narrative logic

Narrative logic describes any logical process of narrative analysis used by readers or viewers to understand and draw conclusions from narratives.

## Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning.

## Natural language

In neuropsychology, linguistics, and the philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation.

## Necessity and sufficiency

In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements.

## 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 (¬P), which is interpreted intuitively as being true when P is false, and false when P is true.

## Negation normal form

In mathematical logic, a formula is in negation normal form if the negation operator (\lnot) is only applied to variables and the only other allowed Boolean operators are conjunction (\land) and disjunction (\lor). Negation normal form is not a canonical form: for example, a \land (b\lor \lnot c) and (a \land b) \lor (a \land \lnot c) are equivalent, and are both in negation normal form.

## Nominalism

In metaphysics, nominalism is a philosophical view which denies the existence of universals and abstract objects, but affirms the existence of general or abstract terms and predicates.

## Non-classical logic

Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic.

## Non-monotonic logic

A non-monotonic logic is a formal logic whose consequence relation is not monotonic.

## Non-standard model

In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).

## Noncommutative logic

Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus.

## Nonfirstorderizability

In formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in particular theories in first-order logic.

## Novum Organum

The Novum Organum, fully Novum Organum Scientiarum ('new instrument of science'), is a philosophical work by Francis Bacon, written in Latin and published in 1620.

## Object language

An object language is a language which is the "object" of study in various fields including logic, linguistics, mathematics, and theoretical computer science.

## Object theory

Object theory is a theory in philosophy and mathematical logic concerning objects and the statements that can be made about objects.

## Obversion

In traditional logic, obversion is a "type of immediate inference in which from a given proposition another proposition is inferred whose subject is the same as the original subject, whose predicate is the contradictory of the original predicate, and whose quality is affirmative if the original proposition's quality was negative and vice versa".

## Occam's razor

Occam's razor (also Ockham's razor or Ocham's razor; Latin: lex parsimoniae "law of parsimony") is the problem-solving principle that, the simplest explanation tends to be the right one.

## On Formally Undecidable Propositions of Principia Mathematica and Related Systems

"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel.

## Open formula

An open formula is a formula that contains at least one free variable.

## Opinion

An opinion is a judgment, viewpoint, or statement that is not conclusive.

## Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

## Organon

The Organon (Greek: Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logic.

## Outline of discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

## Outline of mathematics

Mathematics is a field of study that investigates topics including number, space, structure, and change.

## Outline of philosophy

The following outline is provided as an overview of and topical guide to philosophy: Philosophy &ndash; study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.

## Paraconsistent logic

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way.

## Paradox

A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion.

## Parity function

In Boolean algebra, a parity function is a Boolean function whose value is 1 if and only if the input vector has an odd number of ones.

## Partial equivalence relation

In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) R on a set X is a relation that is symmetric and transitive.

## Partial function

In mathematics, a partial function from X to Y (written as or) is a function, for some subset X ′ of X.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

## Peirce's law

In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce.

## Philosophical logic

Philosophical logic refers to those areas of philosophy in which recognized methods of logic have traditionally been used to solve or advance the discussion of philosophical problems.

## Philosophy

Philosophy (from Greek φιλοσοφία, philosophia, literally "love of wisdom") is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.

## Philosophy of Arithmetic

Philosophy of Arithmetic (PA; Philosophie der Arithmetik.) is an 1891 book by Edmund Husserl.

## Philosophy of logic

Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic.

## Plural quantification

In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular, values.

## Pointed set

In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint.

## Polish Logic

Polish Logic is an anthology of papers by several authors—Kazimierz Ajdukiewicz, Leon Chwistek, Stanislaw Jaskowski, Zbigniew Jordan, Tadeusz Kotarbinski, Stanislaw Lesniewski, Jan Lukasiewicz, Jerzy Słupecki, and Mordchaj Wajsberg—published in 1967 and covering the period 1920&ndash;1939.

## Polish notation

Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators precede their operands, in contrast to reverse Polish notation (RPN) in which operators follow their operands.

## Polylogism

Polylogism is the belief that different groups of people reason in fundamentally different ways (coined from Greek poly.

## Polysyllogism

A polysyllogism (also called multi-premise syllogism, sorites, climax, or gradatio) is a string of any number of propositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with the next proposition, is a premise for the next, and so on.

## Port-Royal Logic

Port-Royal Logic, or Logique de Port-Royal, is the common name of La logique, ou l'art de penser, an important textbook on logic first published anonymously in 1662 by Antoine Arnauld and Pierre Nicole, two prominent members of the Jansenist movement, centered on Port-Royal.

## Possible world

In philosophy and logic, the concept of a possible world is used to express modal claims.

## Post correspondence problem

The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946.

## Post's theorem

In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.

## Posterior Analytics

The Posterior Analytics (Ἀναλυτικὰ Ὕστερα; Analytica Posteriora) is a text from Aristotle's Organon that deals with demonstration, definition, and scientific knowledge.

## Power set

In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.

## Practical syllogism

The practical syllogism is an instance of practical reasoning which takes the form of a syllogism, where the conclusion of the syllogism is an action.

## Pragmatism

Pragmatism is a philosophical tradition that began in the United States around 1870.

## Precision questioning

Precision questioning (PQ), an intellectual toolkit for critical thinking and for problem solving, grew out of a collaboration between Dennis Matthies (1946-) and, while both taught/studied at Stanford University.

## Predicate (mathematical logic)

In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory.

## Predicate variable

In first-order logic, a predicate variable is a predicate letter which can stand for a relation (between terms) but which has not been specifically assigned any particular relation (or meaning).

## Preintuitionism

In the mathematical philosophy, the pre-intuitionists were a small but influential group who informally shared similar philosophies on the nature of mathematics.

## Premise

A premise or premiss is a statement that an argument claims will induce or justify a conclusion.

## Preorder

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.

## Presupposition

In the branch of linguistics known as pragmatics, a presupposition (or PSP) is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse.

## Prewellordering

In set theory, a prewellordering is a binary relation \le that is transitive, total, and wellfounded (more precisely, the relation x\le y\land y\nleq x is wellfounded).

## Primitive recursive function

In computability theory, primitive recursive functions are a class of functions that are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions (µ-recursive functions are also called partial recursive).

## Principia Mathematica

The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913.

## Principle of bivalence

In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.

## Principle of explosion

The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)"), or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction.

## Principles of Mathematical Logic

Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert's and Wilhelm Ackermann's classic text Grundzüge der theoretischen Logik, on elementary mathematical logic.

## Prior Analytics

The Prior Analytics (Ἀναλυτικὰ Πρότερα; Analytica Priora) is Aristotle's work on deductive reasoning, which is known as his syllogistic.

## Probability

Probability is the measure of the likelihood that an event will occur.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

## Product term

In Boolean logic, a product term is a conjunction of literals, where each literal is either a variable or its negation.

## Projection (set theory)

In set theory, a projection is one of two closely related types of functions or operations, namely.

## Proof by exhaustion

Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds.

## Proof theory

Proof theory is a major branchAccording to Wang (1981), pp.

## Propaganda

Propaganda is information that is not objective and is used primarily to influence an audience and further an agenda, often by presenting facts selectively to encourage a particular synthesis or perception, or using loaded language to produce an emotional rather than a rational response to the information that is presented.

## Propaganda techniques

Common media for transmitting propaganda messages include news reports, government reports, historical revision, junk science, books, leaflets, movies, social media, radio, television, and posters.

## Proposition

The term proposition has a broad use in contemporary analytic philosophy.

## Propositional calculus

Propositional calculus is a branch of logic.

## Propositional formula

In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value.

## Propositional function

A propositional function in logic, is a sentence expressed in a way that would assume the value of true or false, except that within the sentence is a variable (x) that is not defined or specified, which leaves the statement undetermined.

## Propositional variable

In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is a variable which can either be true or false.

## Provability logic

Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'.

## Prudence

Prudence (prudentia, contracted from providentia meaning "seeing ahead, sagacity") is the ability to govern and discipline oneself by the use of reason.

## Pseudophilosophy

Pseudophilosophy is a term, often considered derogatory, applied to criticize philosophical ideas or systems which are claimed not to meet an expected set of standards.

## Psychologism

Psychologism is a philosophical position, according to which psychology plays a central role in grounding or explaining some other, non-psychological type of fact or law.

## Q.E.D.

Q.E.D. (also written QED and QED) is an initialism of the Latin phrase quod erat demonstrandum meaning "what was to be demonstrated" or "what was to be shown." Some may also use a less direct translation instead: "thus it has been demonstrated." Traditionally, the phrase is placed in its abbreviated form at the end of a mathematical proof or philosophical argument when the original proposition has been restated exactly, as the conclusion of the demonstration or completion of the proof.

## Quantifier (logic)

In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula.

## Quantum logic

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account.

## Quasitransitive relation

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics.

## Ramism

Ramism was a collection of theories on rhetoric, logic, and pedagogy based on the teachings of Petrus Ramus, a French academic, philosopher, and Huguenot convert, who was murdered during the St. Bartholomew's Day massacre in August 1572.

## Range (mathematics)

In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage.

## Reason

Reason is the capacity for consciously making sense of things, establishing and verifying facts, applying logic, and changing or justifying practices, institutions, and beliefs based on new or existing information.

## Recursion (computer science)

Recursion in computer science is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem (as opposed to iteration).

## Recursive language

In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language.

## Recursive set

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set.

## Recursively enumerable language

In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language.

## Recursively enumerable set

In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if.

## Reductio ad absurdum

In logic, reductio ad absurdum ("reduction to absurdity"; also argumentum ad absurdum, "argument to absurdity") is a form of argument which attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were not true, the result would be absurd or impossible.

## Reduction (recursion theory)

In computability theory, many reducibility relations (also called reductions, reducibilities, and notions of reducibility) are studied.

## Reference

Reference is a relation between objects in which one object designates, or acts as a means by which to connect to or link to, another object.

## Reflexive relation

In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.

## Relation algebra

In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.

## Relative term

A relative term is a term that makes two or more distinct references to objects (which may be the same object, for example in "The Morning Star is the Evening Star").

## Relevance

Relevance is the concept of one topic being connected to another topic in a way that makes it useful to consider the second topic when considering the first.

## Relevance logic

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related.

## Residuated Boolean algebra

In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra.

## Resolution (logic)

In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation theorem-proving technique for sentences in propositional logic and first-order logic.

## Rhetoric

Rhetoric is the art of discourse, wherein a writer or speaker strives to inform, persuade, or motivate particular audiences in specific situations.

## Rhetoric (Aristotle)

Aristotle's Rhetoric (Rhētorikḗ; Ars Rhetorica) is an ancient Greek treatise on the art of persuasion, dating from the 4th century BC.

## Rigour

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

## Rule of inference

In logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions).

## Russell's paradox

In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.

## Satisfiability

In mathematical logic, satisfiability and validity are elementary concepts of semantics.

## Second-order predicate

In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument.

## Self-reference

Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself.

## Semantic theory of truth

A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.

## Semantics

Semantics (from σημαντικός sēmantikós, "significant") is the linguistic and philosophical study of meaning, in language, programming languages, formal logics, and semiotics.

## Sentence (mathematical logic)

In mathematical logic, a sentence of a predicate logic is a boolean-valued well-formed formula with no free variables.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Sequent

In mathematical logic, a sequent is a very general kind of conditional assertion.

## Sequential logic

In digital circuit theory, sequential logic is a type of logic circuit whose output depends not only on the present value of its input signals but on the sequence of past inputs, the input history.

## Serial relation

In set theory, a serial relation is a binary relation R for which every element of the domain has a corresponding range element (∀ x ∃ y x R y).

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Sheffer stroke

In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written ↑, also written | (not to be confused with "||", which is often used to represent disjunction), or Dpq (in Bocheński notation), denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both".

## Simple theorems in the algebra of sets

The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix &cup), intersection (infix &cap), and set complement (postfix ') of sets.

## Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

## Singular term

A singular term is a paradigmatic referring device in a language.

## Skolem's paradox

In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem.

## Socratic method

The Socratic method, also can be known as maieutics, method of elenchus, elenctic method, or Socratic debate, is a form of cooperative argumentative dialogue between individuals, based on asking and answering questions to stimulate critical thinking and to draw out ideas and underlying presumptions.

## Socratic questioning

Socratic questioning (or Socratic maieutics) was named after Socrates, who was a philosopher in c. 470 BCE–c.

## Sophist

A sophist (σοφιστής, sophistes) was a specific kind of teacher in ancient Greece, in the fifth and fourth centuries BC.

## Sophistical Refutations

Sophistical Refutations (Σοφιστικοὶ Ἔλεγχοι; De Sophisticis Elenchis) is a text in Aristotle's Organon in which he identified thirteen fallacies.

## Sorites paradox

The sorites paradox (sometimes known as the paradox of the heap) is a paradox that arises from vague predicates.

## Soundness

In mathematical logic, a logical system has the soundness property if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.

## Source credibility

Source credibility is "a term commonly used to imply a communicator's positive characteristics that affect the receiver's acceptance of a message." Academic studies of this topic began in the 20th century and were given a special emphasis during World War II, when the US government sought to use propaganda to influence public opinion in support of the war effort.

## Source criticism

Source criticism (or information evaluation) is the process of evaluating an information source, i.e. a document, a person, a speech, a fingerprint, a photo, an observation, or anything used in order to obtain knowledge.

## Square of opposition

The square of opposition is a diagram representing the relations between the four basic categorical propositions.

## Statement (logic)

In logic, the term statement is variously understood to mean either: In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement.

## Strict conditional

In logic, a strict conditional is a conditional governed by a modal operator, that is, a logical connective of modal logic.

## Strict logic

Strict logic is essentially synonymous with relevant logic, though it can be characterized proof-theoretically as.

## Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Substitution (logic)

Substitution is a fundamental concept in logic.

## Substructural logic

In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity.

## Sum of Logic

The Summa Logicae ("Sum of Logic") is a textbook on logic by William of Ockham.

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Syllogism

A syllogism (συλλογισμός syllogismos, "conclusion, inference") is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.

## Syllogistic fallacy

Syllogistic fallacies are formal fallacies that occur in syllogisms.

## Symbol (formal)

A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern.

## Symmetric Boolean function

In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the permutation of its input bits, i.e., it depends only on the number of ones in the input.

## Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a. In mathematical notation, this is: Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

## Syntax (logic)

In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them.

## Tautology (logic)

In logic, a tautology (from the Greek word ταυτολογία) is a formula or assertion that is true in every possible interpretation.

## Temporal logic

In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time.

## Term logic

In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century.

## Ternary relation

In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three.

## Tetralemma

The tetralemma is a figure that features prominently in the logic of India.

## The Art of Being Right

The Art of Being Right: 38 Ways to Win an Argument (also Eristic Dialectic: The Art of Winning an Argument; German: Eristische Dialektik: Die Kunst, Recht zu behalten; 1831) is an acidulous and sarcastic treatise written by the German philosopher Arthur Schopenhauer in sardonic deadpan.

## The Foundations of Arithmetic

The Foundations of Arithmetic (Die Grundlagen der Arithmetik) is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic.

## Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.

## Theory

A theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking.

## Theory of justification

Theory of justification is a part of epistemology that attempts to understand the justification of propositions and beliefs.

## Tolerant sequence

In mathematical logic, a tolerant sequence is a sequence of formal theories such that there are consistent extensions of these theories with each S_ interpretable in S_i.

## Topical logic

Topical logic is the logic of topical argument, a branch of rhetoric developed in the Late Antique period from earlier works, such as Aristotle's Topics and Cicero's Topica.

## Topics (Aristotle)

The Topics (Τοπικά; Topica) is the name given to one of Aristotle's six works on logic collectively known as the Organon: The Topics constitutes Aristotle's treatise on the art of dialectic—the invention and discovery of arguments in which the propositions rest upon commonly held opinions or endoxa (ἔνδοξα in Greek).

## Tractatus Logico-Philosophicus

The Tractatus Logico-Philosophicus (TLP) (Latin for "Logico-Philosophical Treatise") is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime.

## Transitive relation

In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.

## Transposition (logic)

In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated.

## Trichotomy (mathematics)

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.

## Trivialism

Trivialism is the logical theory that all statements (also known as propositions) are true and that all contradictions of the form "p and not p" (e.g. the ball is red and not red) are true.

## True quantified Boolean formula

In computational complexity theory, the language TQBF is a formal language consisting of the true quantified Boolean formulas.

## Truth

Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard.

## Truth condition

In semantics and pragmatics, a truth condition is the condition under which a sentence is true.

## Truth function

In logic, a truth function is a function that accepts truth values as input and produces a truth value as output, i.e., the input and output are all truth values.

## Truth table

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables (Enderton, 2001).

## 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.

## Tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements.

## Turing degree

In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set.

## Turing machine

A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules.

## Two-element Boolean algebra

In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain.

## Type theory

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics.

## Type–token distinction

The type–token distinction is used in disciplines such as logic, linguistics, metalogic, typography, and computer programming to clarify what words mean.

## Ultrafinitism

In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism, is a form of finitism.

## Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

## Unification (computer science)

In logic and computer science, unification is an algorithmic process of solving equations between symbolic expressions.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

## Universal generalization

In predicate logic, generalization (also universal generalization or universal introduction, GEN) is a valid inference rule.

## Universal instantiation

In predicate logic universal instantiation (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class.

## Universal set

In set theory, a universal set is a set which contains all objects, including itself.

## Unordered pair

In mathematics, an unordered pair or pair set is a set of the form, i.e. a set having two elements a and b with no particular relation between them.

## Use–mention distinction

The use–mention distinction is a foundational concept of analytic philosophy, according to which it is necessary to make a distinction between using a word (or phrase) and mentioning it,Devitt and Sterelny (1999) pp.

## Vagueness

In analytic philosophy and linguistics, a concept may be considered vague if its extension is deemed lacking in clarity, if there is uncertainty about which objects belong to the concept or which exhibit characteristics that have this predicate (so-called "border-line cases"), or if the Sorites paradox applies to the concept or predicate.

## Validity

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

## Variable (mathematics)

In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.

## Venn diagram

A Venn diagram (also called primary diagram, set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets.

## Von Neumann–Bernays–Gödel set theory

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC).

## Well-formed formula

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.

## Well-founded relation

In mathematics, a binary relation, R, is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is an element m not related by sRm (for instance, "s is not smaller than m") for any s ∈ S. In other words, a relation is well founded if Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.

## Zermelo set theory

Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory.

## Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

## Zeroth-order logic

Zeroth-order logic is first-order logic without variables or quantifiers.

## References

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