501 relations: A priori and a posteriori, A System of Logic, Abductive reasoning, Absorption law, Accuracy and precision, Ad hoc hypothesis, Affine logic, Affirming the consequent, Aleph number, Algebraic normal form, Alpha recursion theory, Ambiguity, Analysis, Analytic–synthetic distinction, Antecedent (logic), Anti-psychologism, Antinomy, Antisymmetric relation, Argument, Argument map, Argumentation theory, Arithmetical set, Association for Symbolic Logic, Asymmetric relation, Atomic formula, Atomic sentence, Attacking Faulty Reasoning, Axiom, Axiomatic system, Baralipton, Baroco, Begriffsschrift, Belief, Belief bias, Bias, Biconditional elimination, Biconditional introduction, Bijection, Bijection, injection and surjection, Binary relation, Boolean algebra, Boolean algebra (structure), Boolean algebras canonically defined, Boolean conjunctive query, Boolean domain, Boolean expression, Boolean function, Boolean ring, Boolean-valued function, Boolean-valued model, ..., Bracket (mathematics), Bunched logic, Canonical normal form, Cantor's diagonal argument, Cantor's theorem, Cardinal number, Cardinality of the continuum, Categorical logic, Categorical proposition, Categories (Aristotle), Causality, Charles Sanders Peirce bibliography, Church–Turing thesis, Classical logic, Clause (logic), Codomain, Cognitive bias, Commutativity of conjunction, Complement (set theory), Complete Boolean algebra, Completeness (logic), Composition of relations, Computability logic, Computability theory, Computable function, Computation, Conceptualism, Conditional proof, Conditioned disjunction, Confirmation bias, Congruence relation, Conjunction elimination, Conjunction introduction, Consequent, Consistency, Constructible universe, Constructive dilemma, Constructivist epistemology, Contingency (philosophy), Continuum hypothesis, Contradiction, Contraposition (traditional logic), Contrary (logic), Conventionalism, Converse (logic), Converse implication, Converse nonimplication, Converse relation, Countable set, Counterpart theory, Covering relation, Credibility, Critical pedagogy, Critical reading, Critical thinking, Cyclic order, De Interpretatione, De Morgan's laws, Decidability (logic), Decidophobia, Decision problem, Decision theory, Decision-making, Deductive closure, Deductive reasoning, Definition, Deflationary theory of truth, Degree of truth, Dense order, Denying the antecedent, Deontic logic, Dependence relation, Dependency relation, Derivative algebra (abstract algebra), Description, Description logic, Destructive dilemma, Deviant logic, Dialetheism, Directed set, Disjoint sets, Disjoint union, Disjunction elimination, Disjunction introduction, Disjunctive syllogism, Domain of a function, Domain of discourse, Double negation, Double negative, Doxastic logic, Effective method, Element (mathematics), Emotional reasoning, Empty domain, Empty set, End term, Enthymeme, Entscheidungsproblem, Enumeration, Epistemic modal logic, Equivalence relation, Euclidean relation, Evidence, Exclusive or, Existential fallacy, Expert, Explanation, Explanatory power, Extension (predicate logic), Extensionality, Fact, Fallacy, Fictionalism, Field of sets, Finitary relation, Finite set, First-order logic, First-order predicate, Forcing (mathematics), Forcing (recursion theory), Formal fallacy, Formal language, Formal proof, Formal system, Formalism (philosophy), Formation rule, Free Boolean algebra, Free logic, Free variables and bound variables, Function (mathematics), Function composition, Functional completeness, Fuzzy logic, Game semantics, Game theory, Gödel's completeness theorem, Gödel's incompleteness theorems, Gödel, Escher, Bach, Georg Cantor's first set theory article, Halting problem, Higher-order logic, Higher-order thinking, History of logic, History of the Church–Turing thesis, Hypothetical syllogism, Idempotence, Idempotency of entailment, Identity (mathematics), Identity (philosophy), Illuminationism, Immediate inference, Implicant, Independence (mathematical logic), Index of logic articles, Index set, Inductive reasoning, Inference, Infinitary logic, Infinite set, Infinity, Informal logic, Inquiry, Intension, Intensional logic, Intermediate logic, Interpretability, Interpretability logic, Interpretation (logic), Interpretive discussion, Intersection (set theory), Intransitivity, Introduction to Mathematical Philosophy, Intuitionistic logic, Inverse (logic), Inverse function, Involution (mathematics), Journal of Logic, Language and Information, Journal of Philosophical Logic, Judgment (mathematical logic), Lambda calculus, Language, Truth, and Logic, Large cardinal, Law of excluded middle, Law of identity, Law of noncontradiction, Laws of Form, Löwenheim–Skolem theorem, Linear logic, Linguistics and Philosophy, List of Boolean algebra topics, List of fallacies, List of first-order theories, List of logicians, List of mathematical logic topics, List of paradoxes, List of philosophers of language, List of rules of inference, List of set theory topics, List of undecidable problems, Lists of mathematics topics, Literal (mathematical logic), Logic, Logic alphabet, Logic gate, Logic programming, Logic redundancy, Logical atomism, Logical biconditional, Logical conjunction, Logical connective, Logical consequence, Logical constant, Logical disjunction, Logical equivalence, Logical form, Logical holism, Logical matrix, Logical NOR, Logical reasoning, Logical truth, Logicism, Many-valued logic, Map (mathematics), Material conditional, Material nonimplication, Mathematical fallacy, Mathematical logic, Mathematics, Meaning (linguistics), Meaning (non-linguistic), Metalanguage, Metalogic, Metasyntactic variable, Metatheorem, Metatheory, Metavariable, Middle term, Minimal logic, Modal algebra, Modal fictionalism, Modal logic, Model theory, Modus ponendo tollens, Modus ponens, Modus tollens, Monadic Boolean algebra, Monadic predicate calculus, Monotonicity of entailment, Morse–Kelley set theory, Multiset, Naive set theory, Name, Narrative logic, Natural deduction, Natural language, Necessity and sufficiency, Negation, Negation normal form, Nominalism, Non-classical logic, Non-monotonic logic, Non-standard model, Noncommutative logic, Nonfirstorderizability, Novum Organum, Object language, Object theory, Obversion, Occam's razor, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Open formula, Opinion, Ordered pair, Organon, Outline of discrete mathematics, Outline of mathematics, Outline of philosophy, Paraconsistent logic, Paradox, Parity function, Partial equivalence relation, Partial function, Partially ordered set, Partition of a set, Peirce's law, Philosophical logic, Philosophy, Philosophy of Arithmetic, Philosophy of logic, Plural quantification, Pointed set, Polish Logic, Polish notation, Polylogism, Polysyllogism, Port-Royal Logic, Possible world, Post correspondence problem, Post's theorem, Posterior Analytics, Power set, Practical syllogism, Pragmatism, Precision questioning, Predicate (mathematical logic), Predicate variable, Preintuitionism, Premise, Preorder, Presupposition, Prewellordering, Primitive recursive function, Principia Mathematica, Principle of bivalence, Principle of explosion, Principles of Mathematical Logic, Prior Analytics, Probability, Probability theory, Product term, Projection (set theory), Proof by exhaustion, Proof theory, Propaganda, Propaganda techniques, Proposition, Propositional calculus, Propositional formula, Propositional function, Propositional variable, Provability logic, Prudence, Pseudophilosophy, Psychologism, Q.E.D., Quantifier (logic), Quantum logic, Quasitransitive relation, Ramism, Range (mathematics), Reason, Recursion (computer science), Recursive language, Recursive set, Recursively enumerable language, Recursively enumerable set, Reductio ad absurdum, Reduction (recursion theory), Reference, Reflexive relation, Relation algebra, Relative term, Relevance, Relevance logic, Residuated Boolean algebra, Resolution (logic), Rhetoric, Rhetoric (Aristotle), Rigour, Rule of inference, Russell's paradox, Satisfiability, Second-order predicate, Self-reference, Semantic theory of truth, Semantics, Sentence (mathematical logic), Sequence, Sequent, Sequential logic, Serial relation, Set (mathematics), Set theory, Sheffer stroke, Simple theorems in the algebra of sets, Singleton (mathematics), Singular term, Skolem's paradox, Socratic method, Socratic questioning, Sophist, Sophistical Refutations, Sorites paradox, Soundness, Source credibility, Source criticism, Square of opposition, Statement (logic), Strict conditional, Strict logic, Structure (mathematical logic), Subset, Substitution (logic), Substructural logic, Sum of Logic, Surjective function, Syllogism, Syllogistic fallacy, Symbol (formal), Symmetric Boolean function, Symmetric relation, Syntax (logic), Tautology (logic), Temporal logic, Term logic, Ternary relation, Tetralemma, The Art of Being Right, The Foundations of Arithmetic, Theorem, Theory, Theory of justification, Tolerant sequence, Topical logic, Topics (Aristotle), Tractatus Logico-Philosophicus, Transitive relation, Transposition (logic), Trichotomy (mathematics), Trivialism, True quantified Boolean formula, Truth, Truth condition, Truth function, Truth table, Truth value, Tuple, Turing degree, Turing machine, Two-element Boolean algebra, Type theory, Type–token distinction, Ultrafinitism, Uncountable set, Unification (computer science), Union (set theory), Universal generalization, Universal instantiation, Universal set, Unordered pair, Use–mention distinction, Vagueness, Validity, Variable (mathematics), Venn diagram, Von Neumann–Bernays–Gödel set theory, Well-formed formula, Well-founded relation, Zermelo set theory, Zermelo–Fraenkel set theory, Zeroth-order logic. 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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, Ratiocinative and Inductive is an 1843 book by English philosopher John Stuart Mill.
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.
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.
Precision is a description of random errors, a measure of statistical variability.
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 is a substructural logic whose proof theory rejects the structural rule of contraction.
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.
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.
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.
In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals \alpha.
Ambiguity is a type of meaning in which several interpretations are plausible.
Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it.
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.
An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause.
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 (Greek ἀντί, antí, "against, in opposition to", and νόμος, nómos, "law") refers to a real or apparent mutual incompatibility of two laws.
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.
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.
In informal logic and philosophy, an argument map or argument diagram is a visual representation of the structure of an argument.
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.
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.
The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic.
In mathematics, an asymmetric relation is a binary relation on a set X where.
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.
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 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.
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.
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.
In classical logic, Baralipton is a mnemonic word used to identify a form of syllogism.
In classical logic, baroco is a mnemonic word used to memorize a syllogism.
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 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 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 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 is the name of two valid rules of inference of propositional logic.
In propositional logic, biconditional introduction is a valid rule of inference.
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.
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.
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.
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.
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.
Boolean algebra is a mathematically rich branch of abstract algebra.
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.
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true.
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.
In mathematics and logic, a (finitary) Boolean function (or switching function) is a function of the form ƒ: Bk → B, where B.
In mathematics, a Boolean ring R is a ring for which x2.
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.
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory.
In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses, square brackets, braces, and angle brackets ⟨.
Bunched logic is a variety of substructural logic proposed by Peter O'Hearn and David Pym.
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.
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.
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.
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.
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 is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic.
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).
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 (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.
This Charles Sanders Peirce bibliography consolidates numerous references to Charles Sanders Peirce's writings, including letters, manuscripts, publications, and Nachlass.
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 (or standard logic) is an intensively studied and widely used class of formal logics.
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).
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.
A cognitive bias is a systematic pattern of deviation from norm or rationality in judgment.
In propositional logic, the commutativity of conjunction is a valid argument form and truth-functional tautology.
In set theory, the complement of a set refers to elements not in.
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound).
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.
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 (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, 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 functions are the basic objects of study in computability theory.
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 is a philosophical theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind.
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.
In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary logical connective introduced by Church.
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.
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.
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 (often abbreviated simply as conjunction and also called and introduction) is a valid rule of inference of propositional logic.
A consequent is the second half of a hypothetical proposition.
In classical deductive logic, a consistent theory is one that does not contain a contradiction.
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 is a valid rule of inference of propositional logic.
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.
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).
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions.
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 is the relationship between two propositions when they cannot both be true (although both may be false).
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.
In logic, the converse of a categorical or implicational statement is the result of reversing its two parts.
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).
In logic, converse nonimplication is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).
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.
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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.
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 comprises the objective and subjective components of the believability of a source or message.
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 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 is the objective analysis of facts to form a judgment.
In mathematics, a cyclic order is a way to arrange a set of objects in a circle.
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.
In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference.
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 is, according to Princeton University philosopher Walter Kaufmann, a fear of making decisions.
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 (or the theory of choice) is the study of the reasoning underlying an agent's choices.
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 is a property of a set of objects (usually the objects in question are statements).
Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).
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.
In standard mathematics, propositions can typically be considered unambiguously true or false.
In mathematics, a partial order or total order.
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 is the field of philosophical logic that is concerned with obligation, permission, and related concepts.
In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, and reflexive; i.e. a finite tolerance relation.
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 is the pattern of narrative development that aims to make vivid a place, an object, a character, or a group.
Description logics (DL) are a family of formal knowledge representation languages.
Destructive dilemma is the name of a valid rule of inference of propositional logic.
Philosopher Susan Haack uses the term "deviant logic" to describe certain non-classical systems of logic.
Dialetheism is the view that there are statements which are both true and false.
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.
In mathematics, two sets are said to be disjoint sets if they have no element in common.
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.
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 or addition (also called or introduction) is a rule of inference of propositional logic and almost every other deduction system.
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.
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.
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.
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.
A double negative is a grammatical construction occurring when two forms of negation are used in the same sentence.
Doxastic logic is a type of logic concerned with reasoning about beliefs.
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.
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.
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.
In first-order logic the empty domain is the empty set having no members.
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.
The end terms in a categorical syllogism are the major term and the minor term (not the middle term).
An enthymeme (ἐνθύμημα, enthumēma) is a rhetorical syllogism (a three-part deductive argument) used in oratorical practice.
In mathematics and computer science, the Entscheidungsproblem (German for "decision problem") is a challenge posed by David Hilbert in 1928.
An enumeration is a complete, ordered listing of all the items in a collection.
Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge.
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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, broadly construed, is anything presented in support of an assertion.
Exclusive or or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).
The existential fallacy, or existential instantiation, is a formal fallacy.
An expert is someone who has a prolonged or intense experience through practice and education in a particular field.
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 is the ability of a hypothesis or theory to effectively explain the subject matter it pertains to.
The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate.
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties.
A fact is a statement that is consistent with reality or can be proven with evidence.
A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves" in the construction of an argument.
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").
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.
In mathematics, a finitary relation has a finite number of "places".
In mathematics, a finite set is a set that has a finite number of elements.
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.
In mathematical logic, a first-order predicate is a predicate that takes only individual(s) constants or variables as argument(s).
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results.
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.
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.
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.
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.
A formal system is the name of a logic system usually defined in the mathematical way.
The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy.
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.
In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that.
A free logic is a logic with fewer existential presuppositions than classical logic.
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.
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
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 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 (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 is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers".
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 are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.
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 was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.
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.
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, known as higher order thinking skills (HOTS), is a concept of education reform based on learning taxonomies (such as Bloom's taxonomy).
The history of logic deals with the study of the development of the science of valid inference (logic).
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.
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 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 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.
In mathematics an identity is an equality relation A.
In philosophy, identity, from ("sameness"), is the relation each thing bears only to itself.
Illuminationist or ishraqi philosophy is a type of Islamic philosophy introduced by Shahab al-Din Suhrawardi in the twelfth century CE.
An immediate inference is an inference which can be made from only one statement or proposition.
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.
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.
In mathematics, an index set is a set whose members label (or index) members of another set.
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.
Inferences are steps in reasoning, moving from premises to logical consequences.
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.
In set theory, an infinite set is a set that is not a finite set.
Infinity (symbol) is a concept describing something without any bound or larger than any natural number.
Informal logic, intuitively, refers to the principles of logic and logical thought outside of a formal setting.
An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem.
In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase, or another symbol.
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).
In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic.
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.
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.
An interpretation is an assignment of meaning to the symbols of a formal language.
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.
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.
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations.
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, 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.
In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence.
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.
In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.
The Journal of Logic, Language and Information is the official journal of the European Association for Logic, Language and Information.
The Journal of Philosophical Logic is a peer-reviewed scientific journal founded in 1972.
In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in the metalanguage.
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 is a 1936 work of philosophy by Alfred Jules Ayer.
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.
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.
In logic, the law of identity states that each thing is identical with itself.
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 (hereinafter LoF) is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy.
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 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 is a peer reviewed journal addressing "structure and meaning in natural language".
This is a list of topics around Boolean algebra and propositional logic.
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.
In mathematical logic, a first-order theory is given by a set of axioms in some language.
A logician is a person whose topic of scholarly study is logic.
This is a list of mathematical logic topics, by Wikipedia page.
This is a list of paradoxes, grouped thematically.
This is a list of philosophers of language.
This is a list of rules of inference, logical laws that relate to mathematical formulae.
This page is a list of articles related to set theory.
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.
This article itemizes the various lists of mathematics topics.
In mathematical logic, a literal is an atomic formula (atom) or its negation.
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.
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.
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 is a type of programming paradigm which is largely based on formal logic.
Logic redundancy occurs in a digital gate network containing circuitry that does not affect the static logic function.
Logical atomism is a philosophical belief that originated in the early 20th century with the development of analytic philosophy.
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.
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.
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 (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.
In logic, a logical constant of a language \mathcal is a symbol that has the same semantic value under every interpretation of \mathcal.
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.
In logic, statements p and q are logically equivalent if they have the same logical content.
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 is the belief that the world operates in such a way that no part can be known without the whole being known first.
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.
In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or.
Informally, two kinds of logical reasoning can be distinguished in addition to formal deduction: induction and abduction.
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature.
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.
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.
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.
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 or abjunction (Latin ab.
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy.
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
In linguistics, meaning is the information or concepts that a sender intends to convey, or does convey, in communication with a receiver.
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.
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined.
Metalogic is the study of the metatheory of logic.
A metasyntactic variable is a specific word or set of words identified as a placeholder in computer science and specifically computer programming.
In logic, a metatheorem is a statement about a formal system proven in a metalanguage.
A metatheory or meta-theory is a theory whose subject matter is some theory.
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.
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, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson (1936).
In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that.
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 is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality.
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 (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic.
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.
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.
In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.
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 is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions.
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).
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.
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
A name is a term used for identification.
Narrative logic describes any logical process of narrative analysis used by readers or viewers to understand and draw conclusions from narratives.
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.
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.
In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements.
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.
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.
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 logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic.
A non-monotonic logic is a formal logic whose consequence relation is not monotonic.
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 is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus.
In formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in particular theories in first-order logic.
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.
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 is a theory in philosophy and mathematical logic concerning objects and the statements that can be made about objects.
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 (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.
"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.
An open formula is a formula that contains at least one free variable.
An opinion is a judgment, viewpoint, or statement that is not conclusive.
In mathematics, an ordered pair (a, b) is a pair of objects.
The Organon (Greek: Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logic.
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
Mathematics is a field of study that investigates topics including number, space, structure, and change.
The following outline is provided as an overview of and topical guide to philosophy: Philosophy – study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way.
A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion.
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.
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.
In mathematics, a partial function from X to Y (written as or) is a function, for some subset X ′ of X.
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.
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.
In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce.
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 (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 (PA; Philosophie der Arithmetik.) is an 1891 book by Edmund Husserl.
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.
In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular, values.
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 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 1920–1939.
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 is the belief that different groups of people reason in fundamentally different ways (coined from Greek poly.
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, 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.
In philosophy and logic, the concept of a possible world is used to express modal claims.
The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946.
In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.
The Posterior Analytics (Ἀναλυτικὰ Ὕστερα; Analytica Posteriora) is a text from Aristotle's Organon that deals with demonstration, definition, and scientific knowledge.
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,.
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 is a philosophical tradition that began in the United States around 1870.
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.
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.
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).
In the mathematical philosophy, the pre-intuitionists were a small but influential group who informally shared similar philosophies on the nature of mathematics.
A premise or premiss is a statement that an argument claims will induce or justify a conclusion.
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.
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.
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).
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).
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.
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.
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 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.
The Prior Analytics (Ἀναλυτικὰ Πρότερα; Analytica Priora) is Aristotle's work on deductive reasoning, which is known as his syllogistic.
Probability is the measure of the likelihood that an event will occur.
Probability theory is the branch of mathematics concerned with probability.
In Boolean logic, a product term is a conjunction of literals, where each literal is either a variable or its negation.
In set theory, a projection is one of two closely related types of functions or operations, namely.
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 is a major branchAccording to Wang (1981), pp.
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.
Common media for transmitting propaganda messages include news reports, government reports, historical revision, junk science, books, leaflets, movies, social media, radio, television, and posters.
The term proposition has a broad use in contemporary analytic philosophy.
Propositional calculus is a branch of logic.
In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value.
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.
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 is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'.
Prudence (prudentia, contracted from providentia meaning "seeing ahead, sagacity") is the ability to govern and discipline oneself by the use of reason.
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 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. (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.
In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula.
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account.
Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics.
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.
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 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 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).
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.
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.
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.
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.
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.
In computability theory, many reducibility relations (also called reductions, reducibilities, and notions of reducibility) are studied.
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.
In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.
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 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, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related.
In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra.
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 is the art of discourse, wherein a writer or speaker strives to inform, persuade, or motivate particular audiences in specific situations.
Aristotle's Rhetoric (Rhētorikḗ; Ars Rhetorica) is an ancient Greek treatise on the art of persuasion, dating from the 4th century BC.
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.
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).
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.
In mathematical logic, satisfiability and validity are elementary concepts of semantics.
In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument.
Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself.
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 (from σημαντικός sēmantikós, "significant") is the linguistic and philosophical study of meaning, in language, programming languages, formal logics, and semiotics.
In mathematical logic, a sentence of a predicate logic is a boolean-valued well-formed formula with no free variables.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
In mathematical logic, a sequent is a very general kind of conditional assertion.
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.
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).
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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".
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.
In mathematics, a singleton, also known as a unit set, is a set with exactly one element.
A singular term is a paradigmatic referring device in a language.
In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem.
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 (or Socratic maieutics) was named after Socrates, who was a philosopher in c. 470 BCE–c.
A sophist (σοφιστής, sophistes) was a specific kind of teacher in ancient Greece, in the fifth and fourth centuries BC.
Sophistical Refutations (Σοφιστικοὶ Ἔλεγχοι; De Sophisticis Elenchis) is a text in Aristotle's Organon in which he identified thirteen fallacies.
The sorites paradox (sometimes known as the paradox of the heap) is a paradox that arises from vague predicates.
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 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 (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.
The square of opposition is a diagram representing the relations between the four basic categorical propositions.
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.
In logic, a strict conditional is a conditional governed by a modal operator, that is, a logical connective of modal logic.
Strict logic is essentially synonymous with relevant logic, though it can be characterized proof-theoretically as.
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.
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 is a fundamental concept in 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.
The Summa Logicae ("Sum of Logic") is a textbook on logic by William of Ockham.
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).
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 fallacies are formal fallacies that occur in syllogisms.
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.
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.
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.
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them.
In logic, a tautology (from the Greek word ταυτολογία) is a formula or assertion that is true in every possible interpretation.
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time.
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.
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three.
The tetralemma is a figure that features prominently in the logic of India.
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 (Die Grundlagen der Arithmetik) is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic.
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.
A theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking.
Theory of justification is a part of epistemology that attempts to understand the justification of propositions and beliefs.
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 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.
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).
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.
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.
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.
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.
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.
In computational complexity theory, the language TQBF is a formal language consisting of the true quantified Boolean formulas.
Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard.
In semantics and pragmatics, a truth condition is the condition under which a sentence is true.
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.
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).
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.
In mathematics, a tuple is a finite ordered list (sequence) of elements.
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.
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.
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.
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.
The type–token distinction is used in disciplines such as logic, linguistics, metalogic, typography, and computer programming to clarify what words mean.
In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism, is a form of finitism.
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
In logic and computer science, unification is an algorithmic process of solving equations between symbolic expressions.
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
In predicate logic, generalization (also universal generalization or universal introduction, GEN) is a valid inference rule.
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.
In set theory, a universal set is a set which contains all objects, including itself.
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.
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.
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.
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.
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.
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.
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).
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.
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, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern 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 is first-order logic without variables or quantifiers.
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