Outline of logic

Logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...

formal science Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, gam ...

reason Reason is the capacity of Consciousness, consciously applying logic by Logical consequence, drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activ ...

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...

natural language In neuropsychology, linguistics, and 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. Natural languag ...

fallacies A fallacy is the use of Validity (logic), invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual ...

paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...

valid Validity or Valid may refer to: Science/mathematics/statistics: * Validity (logic), a property of a logical argument * Scientific: ** Internal validity, the validity of causal inferences within scientific studies, usually based on experiments ** ...

inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that ...

arguments An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

Philosophy of logic Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their applicatio ...

A priori and a posteriori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...

Definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...

Description Description is the pattern of narrative development that aims to make vivid a place, object, character, or group. Description is one of four rhetorical modes (also known as ''modes of discourse''), along with exposition, argumentation, and na ...

Entailment Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...

Identity (philosophy) In philosophy, identity (from , "sameness") is the relation each thing bears only to itself. The notion of identity gives rise to many philosophical problems, including the identity of indiscernibles (if ''x'' and ''y'' share all their propert ...

Inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that ...

Logical form In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambig ...

Logical implication Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...

Logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whi ...

Necessity Necessary or necessity may refer to: * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by a particular moment * Necessary and sufficient condition, in logic, something that is ...

Material conditional The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...

Meaning (linguistic) Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...

Meaning (non-linguistic) Non-linguistic (or pre-linguistic) meaning is a type of meaning not mediated or perceived through linguistic signs. In linguistics, the concept is used in discussions about whether such meaning is different from meaning expressed through languag ...

Paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

list A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby uni ...

Possible world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their ...

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. Examples of presuppositions includ ...

Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

Reason Reason is the capacity of Consciousness, consciously applying logic by Logical consequence, drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activ ...

Reasoning Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...

Reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a '' name'' ...

Semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and compu ...

Strict conditional In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the neces ...

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. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of ...

Truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...

Truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or ''false''). Computing In some prog ...

Validity Validity or Valid may refer to: Science/mathematics/statistics: * Validity (logic), a property of a logical argument * Scientific: ** Internal validity, the validity of causal inferences within scientific studies, usually based on experiments ...

Affine logic Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. It can also be characterized as linear logic with weakening. The name "affine logic" is associated with linear logic, to which it differs by al ...

Alethic logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...

Aristotelian logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...

Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

Buddhist logic Buddhist logico-epistemology is a term used in Western scholarship for ''pramāṇa-vāda'' (doctrine of proof) and ''Hetu-vidya'' (science of causes). Pramāṇa-vāda is an epistemological study of the nature of knowledge; Hetu-vidya is a syste ...

Bunched logic Bunched logic is a variety of substructural logic proposed by Peter O'Hearn and David Pym. Bunched logic provides primitives for reasoning about ''resource composition'', which aid in the compositional analysis of computer and other systems. It ha ...

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. It was introduced and so named by ...

Deontic logic Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. I ...

Deviant logic Deviant logic is a type of logic incompatible with classical logic. Philosopher Susan Haack uses the term ''deviant logic'' to describe certain non-classical systems of logic. In these logics: * the set of well-formed formulas generated equals ...

Doxastic logic Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a d ...

Epistemic logic Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applic ...

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

Formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

Free logic A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter proper ...

Fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and complet ...

Higher-order logic 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 logics with their standard semantics are more expres ...

Informal logic Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. J ...

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 individual ...

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. In particular, system ...

Linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also ...

Many-valued logic Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...

Mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

Narrative logic A narrative, story, or tale is any account of a series of related events or experiences, whether nonfictional ( memoir, biography, news report, documentary, travelogue, etc.) or fictional (fairy tale, fable, legend, thriller, novel, etc.). ...

Non-Aristotelian 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. There are several ways in which this is done, including by way of ...

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. There are several ways in which this is done, including by way of ...

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. Its sequent calculus relies on the structure of order varietie ...

Ordered logic 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 logics with their standard semantics are more express ...

Paraconsistent logic A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syst ...

Philosophical logic Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...

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

Propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

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. They may be viewed as a family of substructural or modal logics. It is generally, but ...

Strict logic In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive ...

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. Two of the more significant substructural logics are ...

Symbolic logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal s ...

Temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...

Term logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, t ...

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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Gree ...

Traditional logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...

Zeroth-order logic Zeroth-order logic is first-order logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus,. but an alternative definition extends propositional logic by adding constan ...

Informal logic Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. J ...

Critical thinking Critical thinking is the analysis of available facts, evidence, observations, and arguments to form a judgement. The subject is complex; several different definitions exist, which generally include the rational, skeptical, and unbiased an ...

Argumentation theory Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be supported or undermined by premises through logical reasoning. With historical origins in logic, dialectic, and rhetoric, argumentation theory, includ ...

Argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...

Argument map An argument map or argument diagram is a visual representation of the structure of an argument. An argument map typically includes the key components of the argument, traditionally called the '' conclusion'' and the ''premises'', also called ''con ...

Accuracy and precision Accuracy and precision are two measures of '' observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their '' true value'', while ''precision'' is how close the measurements are to each ot ...

Ambiguity Ambiguity is the type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty. It is thus an attribute of any idea or statement w ...

Analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

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. It explains 60 of the most commonl ...

Belief A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take ...

Bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...

Cognitive bias A cognitive bias is a systematic pattern of deviation from norm or rationality in judgment. Individuals create their own "subjective reality" from their perception of the input. An individual's construction of reality, not the objective input, ...

Confirmation bias Confirmation bias is the tendency to search for, interpret, favor, and recall information in a way that confirms or supports one's prior beliefs or values. People display this bias when they select information that supports their views, ignorin ...

Credibility Credibility comprises the objective and subjective components of the believability of a source or message. Credibility dates back to Aristotle theory of Rhetoric. Aristotle defines rhetoric as the ability to see what is possibly persuasive in ...

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

Decidophobia Walter Arnold Kaufmann (July 1, 1921 – September 4, 1980) was a German-American philosopher, translator, and poet. A prolific author, he wrote extensively on a broad range of subjects, such as authenticity and death, moral philosophy and e ...

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 possible alternative options. It could be either ra ...

Emotional reasoning Emotional reasoning is a cognitive process by which an individual concludes that their emotional reaction proves something is true, despite contrary empirical evidence. Emotional reasoning creates an 'emotional truth', which may be in direct con ...

Evidence Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, eviden ...

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. It may establish rules or laws, and may clarify the existing rules or laws in relat ...

Explanatory power Explanatory power is the ability of a hypothesis or theory to explain the subject matter effectively to which it pertains. Its opposite is ''explanatory impotence''. In the past, various criteria or measures for explanatory power have been prop ...

Fallacy A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was int ...

Inquiry An inquiry (also spelled as enquiry in British English) is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ...

Narrative logic A narrative, story, or tale is any account of a series of related events or experiences, whether nonfictional ( memoir, biography, news report, documentary, travelogue, etc.) or fictional (fairy tale, fable, legend, thriller, novel, etc.). ...

Occam's razor Occam's razor, Ockham's razor, or Ocham's razor ( la, novacula Occami), also known as the principle of parsimony or the law of parsimony ( la, lex parsimoniae), is the problem-solving principle that "entities should not be multiplied beyond neces ...

Opinion An opinion is a judgment, viewpoint, or statement that is not conclusive, rather than facts, which are true statements. Definition A given opinion may deal with subjective matters in which there is no conclusive finding, or it may deal with ...

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. Aristotle Aristotle discusses the notion of the practical syllogism within his treatise on et ...

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

Propaganda Propaganda is communication that is primarily used to influence or persuade an audience to further an agenda, which may not be objective and may be selectively presenting facts to encourage a particular synthesis or perception, or using loa ...

Prudence Prudence ( la, prudentia, contracted from meaning "seeing ahead, sagacity") is the ability to govern and discipline oneself by the use of reason. It is classically considered to be a virtue, and in particular one of the four Cardinal virtue ...

Reasoning Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...

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. The concept of relevance is studied in many different fields, including cognitive sc ...

Rhetoric Rhetoric () is the art of persuasion, which along with grammar and logic (or dialectic), is one of the three ancient arts of discourse. Rhetoric aims to study the techniques writers or speakers utilize to inform, persuade, or motivate par ...

Rigour Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as m ...

Socratic questioning Socratic questioning (or Socratic maieutics) was named after Socrates. He used an educational method that focused on discovering answers by asking questions from his students. According to Plato, who was one of his students, Socrates believed t ...

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 Wo ...

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. In relation to a given p ...

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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Gree ...

Conceptualism In metaphysics, conceptualism is a theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. Intermediate between nominalism and realism, the conceptualist view approaches the metaphysical c ...

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. Unspoken rules play a key role in the philosophy's structur ...

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

Formalism (philosophy) The term ''formalism'' describes an emphasis on form over content or meaning in the arts, literature, or philosophy. A practitioner of formalism is called a ''formalist''. A formalist, with respect to some discipline, holds that there is no trans ...

Illuminationist philosophy Illuminationism ( Persian حكمت اشراق ''hekmat-e eshrāq'', Arabic: حكمة الإشراق ''ḥikmat al-ishrāq'', both meaning "Wisdom of the Rising Light"), also known as ''Ishrāqiyyun'' or simply ''Ishrāqi'' ( Persian اشراق, Ar ...

Logical atomism Logical atomism is a philosophical view that originated in the early 20th century with the development of analytic philosophy. Its principal exponent was the British philosopher Bertrand Russell. It is also widely held that the early works of his ...

Logical holism In Philosophy, 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. Theoretical holism is a theory in philosophy of science, that a theory of science can only be underst ...

Logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or a ...

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. The theory comes in two versions: S ...

Nominalism In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universalsthings th ...

Polylogism Polylogism is the belief that different groups of people reason in fundamentally different ways (coined from Greek ''poly'' 'many' + ''logos'' 'logic'). The term is attributed to Ludwig von Mises, who used it to refer to Nazism, Marxism and other ...

Pragmatism Pragmatism is a philosophical tradition that considers words and thought as tools and instruments for prediction, problem solving, and action, and rejects the idea that the function of thought is to describe, represent, or mirror reality. ...

Sophism A sophist ( el, σοφιστής, sophistes) was a teacher in ancient Greece in the fifth and fourth centuries BC. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics, and mathematics. They taught ...

Fallacy A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was int ...

list A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby uni ...

Formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of s ...

Propositional variable In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propos ...

Predicate variable In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predi ...

Literal Literal may refer to: * Interpretation of legal concepts: ** Strict constructionism ** The plain meaning rule (a.k.a. "literal rule") * Literal (mathematical logic), certain logical roles taken by propositions * Literal (computer programmin ...

Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

Brackets A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or ' ...

Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

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). Definition Converse nonimplication is notated P \nleftarrow Q, or P \not \sub ...

Exclusive or Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...

Logical biconditional In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as t ...

Logical conjunction In logic, mathematics and linguistics, And (\wedge) 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. The logical connective that represents thi ...

Logical disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...

Material nonimplication Material nonimplication or abjunction ( Latin ''ab'' = "from", ''junctio'' =–"joining") is the negation of material implication. That is to say that for any two propositions P and Q, the material nonimplication from P to Q is true if ...

Negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...

Atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) 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 subform ...

Open sentence An open formula is a formula that contains at least one free variable. An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like ''true'' or ...

Proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

Analytic proposition Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * A ...

Axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...

Clause (logic) In logic, a clause is a propositional formula formed from a finite collection of literals (atoms or their negations) and logical connectives. A clause is true either whenever at least one of the literals that form it is true (a disjunctive clause ...

Contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle' ...

Logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whi ...

Propositional formula In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional fo ...

Rule of inference In the philosophy of 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). For example, the rule of ...

Sentence (mathematical logic) :''This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence (linguistics), for a less technical introductory article see Statement (logic).'' In mathematical logic, a s ...

Sequent In mathematical logic, a sequent is a very general kind of conditional assertion. : A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n. A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of ass ...

Statement (logic) In logic, the term statement is variously understood to mean either: #a meaningful declarative sentence that is true or false, or #a proposition. Which is the ''assertion'' that is made by (i.e., the meaning of) a true or false declarative se ...

Subalternation Subalternation is an immediate inference which is only made between A (All S are P) and I (Some S are P) categorical propositions and between E (No S are P or originally, No S is P) and O (Some S are not P or originally, Not every S is P) categorica ...

Theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

Rule of inference In the philosophy of 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). For example, the rule of ...

list A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby uni ...

Constructive dilemma Constructive dilemmaCopi and Cohen is a valid rule of inference of propositional logic. It is the inference that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''P'' or ''R'' is true, then either ''Q or S'' has to be true. In sum, if ...

Conversion (logic) In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...

De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...

Destructive dilemma Destructive dilemmaMoore and Parker is the name of a valid rule of inference of propositional logic. It is the inference that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''Q'' is false or ''S'' is false, then either ''P'' or ''R'' ...

Double negation elimination In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not (no ...

Hypothetical syllogism In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises. An example in English: :If I do not wake up, then I cannot go to work. :If I cannot go to work, th ...

Law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...

Modus tollens In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens'' ...

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

Formal proof In logic and mathematics, 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 sequ ...

Metalanguage In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quota ...

Metalinguistic variable 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. For instance, in the sentence :''Let A and B be two sent ...

Deductive system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...

Metatheorem In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheor ...

Metatheory A metatheory or meta-theory is a theory whose subject matter is theory itself, aiming to describe existing theory in a systematic way. In mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. ...

Interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...

Propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

Clause (logic) In logic, a clause is a propositional formula formed from a finite collection of literals (atoms or their negations) and logical connectives. A clause is true either whenever at least one of the literals that form it is true (a disjunctive clause ...

Deductive closure In mathematical logic, a set of logical formulae is deductively closed if it contains every formula that can be logically deduced from , formally: if always implies . If is a set of formulae, the deductive closure of is its smallest super ...

Distributive property In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic ...

Entailment Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...

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.. ("Complete set of logical connectives").. (" ...

Literal (mathematical logic) In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution (logic) ...

Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

Open sentence An open formula is a formula that contains at least one free variable. An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like ''true'' or ...

Propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

Propositional formula In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional fo ...

Propositional variable In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propos ...

Rule of inference In the philosophy of 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). For example, the rule of ...

Strict conditional In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the neces ...

Substitution instance Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions. To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols by other expressions. T ...

Truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...

Zeroth-order logic Zeroth-order logic is first-order logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus,. but an alternative definition extends propositional logic by adding constan ...

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

list A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby uni ...

Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

Introduction to Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

Free Boolean algebra In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called ''generators'', such that: #Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean opera ...

Residuated Boolean algebra In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alph ...

Modal algebra In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that *\langle A,\land,\lor,-,0,1\rangle is a Boolean algebra, *\Box is a unary operation on ''A'' satisfying \Box1=1 and \Box(x\land y)=\Box x\land\Box ...

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: # 0D = 0 # ''x'' ...

De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...

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: * The entire formula is purely tr ...

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 tupl ...

Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...

Boolean expression In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean con ...

Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean a ...

Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ...

Boolean-valued function A Boolean-valued function (sometimes called a Predicate (logic), predicate or a proposition) is a function (mathematics), function of the type f : X → B, where X is an arbitrary Set (mathematics), set and where B is a Boolean domain, i.e. a gene ...

Parity function In Boolean algebra, a parity function is a Boolean function whose value is one if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR function. The parity function is notable for its ...

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.. ("Complete set of logical connectives").. (" ...

Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

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. Matrix representation ...

Truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...

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

Atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) 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 subform ...

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. Overview The dom ...

Empty domain In first-order logic the empty domain is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid. Interpretations with an empty domain are shown to be a trivial ...

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

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 (symbol) that specifies places in an expression where substitution may take place and is ...

Predicate (mathematical logic) In logic, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula P(a), the symbol P is a predicate which applies to the individual constant a. Similarly, in the formula R(a,b), R is a predicat ...

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

Predicate variable In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predi ...

Second-order predicate In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument. Compare higher-order predicate. The idea of second order predication was introduced by the German mathematician and philosopher Freg ...

Sentence (mathematical logic) :''This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence (linguistics), for a less technical introductory article see Statement (logic).'' In mathematical logic, a s ...

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 ...

Mathematical relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...

Antisymmetric relation In mathematics, a binary relation R on a set X is antisymmetric if there is no pair of ''distinct'' elements of X each of which is related by R to the other. More formally, R is antisymmetric precisely if for all a, b \in X, \text \,aRb\, \te ...

Asymmetric relation In mathematics, an asymmetric relation is a binary relation R on a set X where for all a, b \in X, if a is related to b then b is ''not'' related to a. Formal definition A binary relation on X is any subset R of X \times X. Given a, b \in X ...

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 ''ma ...

Binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

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 in the sense that algebraic operations done ...

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. For example, the converse of the relation 'child of' is the relation 'parent&nb ...

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. The covering relation is commonly used to graphically expr ...

Directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...

Equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

Homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...

Idempotence Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pla ...

Involutive relation In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution i ...

Partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...

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. A poset consists of a set together with a binar ...

Preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...

Reflexive relation In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...

Serial relation In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation. Bert ...

Surjective function In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

Symmetric relation A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X ...

Transitive relation In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A hom ...

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

Mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

Set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

list A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby uni ...

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 ''ma ...

Binary set 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, where = . In contrast, an ordered pair (''a'', ''b'') has ''a'' as its first e ...

Cantor's diagonal argument In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a ...

Cantor's first uncountability proof Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is unc ...

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. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \ma ...

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. The cardinality of a finite set is a natural number: the number of elements in the set. T ...

Codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...

Complement (set theory) In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the ...

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

Continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent ...

Countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...

Disjoint sets In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. ...

Disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...

Domain of a function In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is ...

Effective enumeration In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...

Element (mathematics) In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets ...

Empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

Enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration ( ...

Extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...

Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...

Forcing (mathematics) In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from ...

Function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...

Generalized continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent t ...

Index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consis ...

Infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only ...

Intension In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or ano ...

Intersection (set theory) In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is writ ...

Inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...

Large cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...

Löwenheim–Skolem theorem In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-orde ...

Map (mathematics) In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distin ...

Multiset In mathematics, a multiset (or 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. The number of instances given for each element is called the multiplicity of that ...

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 closel ...

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

Ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...

Partition of a set In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every pa ...

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. Maps between pointed sets (X, x_0) and (Y, y_0) – called based ma ...

Power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is ...

Projection (set theory) In set theory, a projection is one of two closely related types of functions or operations, namely: * A set-theoretic operation typified by the ''j''th projection map, written \mathrm_, that takes an element \vec = (x_1,\ \ldots,\ x_j,\ \ldots,\ ...

Proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

Range of a function In mathematics, the range of a function may refer to either of two closely related concepts: * The codomain of the function * The image of the function Given two sets and , a binary relation between and is a (total) function (from to ) ...

Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contain ...

Sequence (mathematics) In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

Set (mathematics) A set is the mathematical model for a collection of different things; a set contains ''elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or e ...

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 operator: ∪), intersection (infix operator: ∩), and set complement ( postfix ') of sets. These properties assume the existence of ...

Singleton (mathematics) In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, t ...

Skolem paradox In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and t ...

Subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

Superset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

Tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

Uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...

Union (set theory) In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of z ...

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–choice set theory (ZFC). NBG introduces the notion of class, which is a coll ...

Zermelo set theory Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It ...

Zermelo–Fraenkel set theory In set theory, 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 ...

metatheory A metatheory or meta-theory is a theory whose subject matter is theory itself, aiming to describe existing theory in a systematic way. In mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. ...

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 ...

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. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of ...

Consistency In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...

Decidability (logic) In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems ...

Deductive system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...

Interpretation (logic) An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning unt ...

Church's thesis Church's is a high-end footwear manufacturer that was founded in 1873, by Thomas Church, in Northampton, England. In 1999 the company came under the control of Italian luxury fashion house Prada in a US$170 million deal. History Between the ...

Effective method In logic, mathematics and computer science, especially metalogic and computability theory, an effective method Hunter, Geoffrey, ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', University of California Press, 1971 ...

Formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...

Independence (mathematical logic) In mathematical logic, independence is the unprovability of a sentence from other sentences. A sentence σ is independent of a given first-order theory ''T'' if ''T'' neither proves nor refutes σ; that is, it is impossible to prove σ from ''T' ...

Löwenheim–Skolem theorem In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-orde ...

Metalanguage In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quota ...

Metatheorem In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheor ...

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. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other ...

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 a word (or phrase) and it.Devitt and Sterelny (1999) pp. 40–1W.V. Quine (1940) p. 24 Many philos ...

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. A formal language can be ...

Axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...

Deductive system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...

Formal proof In logic and mathematics, 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 sequ ...

Formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...

Formal theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

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. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of ...

Transformation rules In the philosophy of 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). For example, the rule of ...

Model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

Interpretation (logic) An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning unt ...

Truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or ''false''). Computing In some prog ...

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

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 thesis about the nature of co ...

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. It was introduced and so named by ...

Computable function Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can d ...

Computation Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm). Mechanical or electronic devices (or, historically, people) that perform computations are known as '' computers''. An esp ...

Decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whethe ...

Effective method In logic, mathematics and computer science, especially metalogic and computability theory, an effective method Hunter, Geoffrey, ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', University of California Press, 1971 ...

Enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration ( ...

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, or continue to run forever. Alan Turing proved in 1936 that a ...

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. It is a universal model of computation t ...

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 so ...

Primitive recursive function In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined ...

Recursion (computer science) In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves ...

Recursive set In computability theory, a set of natural numbers is called computable, recursive, 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 ...

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 ...

Recursively enumerable set In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...

Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...

Formal semantics (natural language) Formal semantics is the study of grammatical meaning in natural languages using formal tools from logic and theoretical computer science. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of lan ...

Alternative semantics Alternative semantics (or Hamblin semantics) is a framework in formal semantics and logic. In alternative semantics, expressions denote ''alternative sets'', understood as sets of objects of the same semantic type. For instance, while the word "L ...

Categorial grammar Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and sema ...

Combinatory categorial grammar Combinatory categorial grammar (CCG) is an efficiently parsable, yet linguistically expressive grammar formalism. It has a transparent interface between surface syntax and underlying semantic representation, including predicate–argument structure ...

Dynamic semantics Dynamic semantics is a framework in logic and natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynam ...

Inquisitive semantics Inquisitive semantics is a framework in logic and natural language semantics. In inquisitive semantics, the semantic content of a sentence captures both the information that the sentence conveys and the issue that it raises. The framework provides ...

Montague grammar __notoc__ Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on mathematical logic, especially higher-order predicate logic and lambda calculus, and makes use ...

Compositionality In semantics, mathematical logic and related disciplines, the principle of compositionality is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. ...

Generalized quantifier In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases. For example, the generalized quantifier ''every boy'' denotes the set of sets of ...

Mereology In logic, philosophy and related fields, mereology ( (root: , ''mere-'', 'part') and the suffix ''-logy'', 'study, discussion, science') is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation be ...

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. Examples of presuppositions includ ...

Propositional attitudes A propositional attitude is a mental state held by an agent toward a proposition. Linguistically, propositional attitudes are denoted by a verb (e.g. "believed") governing an embedded "that" clause, for example, 'Sally believed that she had won ...

Scope (formal semantics) In formal semantics, the scope of a semantic operator is the semantic object to which it applies. For instance, in the sentence "''Paulina doesn't drink beer but she does drink wine''," the proposition that Paulina drinks beer occurs within the sco ...

Law of the excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...

Double negation elimination In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not (no ...

Law of noncontradiction In 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 propositions cannot both be true in the same sense at the sa ...

Principle of explosion In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (, 'from falsehood, anything ollows; or ), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a ...

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. This property can be captured by a structural rule called contraction, and in ...

logical operator In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

Term logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, t ...

Baroco In Aristotelian logic, baroco is a mnemonic word used to memorize a syllogism. Specifically, it has the first proposition universal and affirmative, but the second and third particular and negative, and the middle term the attribute in the two fir ...

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. A logic satisfying this principle is called ...

Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

Boolean-valued function A Boolean-valued function (sometimes called a Predicate (logic), predicate or a proposition) is a function (mathematics), function of the type f : X → B, where X is an arbitrary Set (mathematics), set and where B is a Boolean domain, i.e. a gene ...

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''). The study of arguments ...

End term A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of Argument, logical argument that applies deductive reasoning to arrive at a Logical consequence, conclusion based on two propositions that are as ...

Enthymeme An enthymeme ( el, ἐνθύμημα, ''enthýmēma'') is a form of rational appeal, or deductive argument. It is also known as a rhetorical syllogism and is used in oratorical practice. While the syllogism is used in dialectic, or the art of logi ...

Law of contraries In term logic (a branch of philosophical logic), the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate ''De Interpret ...

Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

Octagon of Prophecies In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...

Organon The ''Organon'' ( grc, Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logical analysis and dialectic. The name ''Organon'' was given by Aristotle's followers, the Peripatetics. The si ...

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, ...

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 J ...

Premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...

Prior Analytics The ''Prior Analytics'' ( grc-gre, Ἀναλυτικὰ Πρότερα; la, Analytica Priora) is a work by Aristotle on reasoning, known as his syllogistic, composed around 350 BCE. Being one of the six extant Aristotelian writings on logic ...

Sorites paradox The sorites paradox (; sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a sing ...

Square of opposition In term logic (a branch of philosophical logic), the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate '' On Interpr ...

Sum of Logic The ''Summa Logicae'' ("Sum of Logic") is a textbook on logic by William of Ockham. It was written around 1323. Systematically, it resembles other works of medieval logic, organised under the basic headings of the Aristotelian Predicables, Cate ...

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

Truth function In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly on ...