A type of standard-form categorical proposition, asserting that all members of the subject category are included in the predicate category; symbolized as "All S are P".[1][2]
A form of reasoning characterized by drawing a conclusion based on the best available explanation for a set of premises. Often used in hypothesis formation.
A logical rule stating that if a proposition implies another, then adding any additional conjunction to the first proposition does not change the implication. Symbolized as .
The process or result of generalization by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose.
A formula of the form (∀α)(∀β)(Abst(α) = Abst(β) ↔ Equ(α, β)), where Abst is an abstraction operator mapping the type of entities ranged over by α and β to objects, and “Equ” is an equivalence relation on the type of entities ranged over by α and β.[6] For instance, Hume's principle, and Basic Law V.
A rule of inference in formal logic where from any proposition, a disjunction can be formed by disjoining it with any other proposition. Symbolized as .
The first part of a conditional statement, the "if" clause, which specifies a condition for the consequent.
anti-extension
In set theory and logic, the complement of the extension of a concept or predicate, consisting of all objects that do not fall under the concept.[11][12][13]
antilogism
A syllogism with three premises leading to a contradiction, showing the inconsistency of the premises.[14][15][16]
A property of some binary operations in which the grouping of operations does not affect the result. For example, in arithmetic, addition and multiplication are associative.
An isomorphism from a mathematical object to itself, preserving all the structure of the object. In logic, it often refers to symmetries within logical structures.
axiological logic
A branch of logic that deals with the study of value, including ethical and aesthetic values, often in the context of modal logic.[22][23][24]
A statement or proposition that is accepted as true without proof, serving as a starting point for further reasoning and arguments.
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bad company objection
An objection to mathematical abstractionism that points out the difficulty in distinguishing between legitimate and illegitimate forms of abstraction, particularly in the context of Frege's Basic Law V and the paradoxes it generates.[25][26][27]
barbara
A standard form of categorical syllogism in Aristotelian logic, where all three propositions (major premise, minor premise, and conclusion) are universal affirmatives, symbolized as AAA. The form is: All M are P, All S are M, therefore All S are P.[28][29][30]
A principle in modal logic that asserts the interchangeability of quantification and possibility: necessarily, if there exists something, then there necessarily exists something.
A principle proposed by Gottlob Frege in his attempt to reduce arithmetic to logic, stating that the extension of a concept is determined by the objects falling under the concept. It leads to Russell's paradox.
The Brouwer-Heyting-Kolmogorov interpretation, a constructivist interpretation of intuitionistic logic, where the truth of a statement is equated with the existence of a proof for it.
A systematic deviation from neutrality, objectivity, or fairness, often resulting from a particular tendency or inclination, especially in statistical or cognitive contexts.
A function that is both injective (no two elements of the domain map to the same element of the codomain) and surjective (every element of the codomain is mapped to by some element of the domain), establishing a one-to-one correspondence between the domain and codomain.
A function that takes two arguments. In logic and mathematics, this is often a function that combines two values to produce a third value, such as addition or multiplication in arithmetic.
An area of algebra in which the values of the variables are the truth values true and false, typically used in computer science, logic, and mathematical logic.
Boolean negation
A form of negation where the negation of a non-true proposition is true, and the negation of a non-false proposition is false.[34][35][36]
An operator used in logic and computer science that performs logical operations on its operands, such as AND, OR, and NOT.
borderline case
A situation or instance that falls at the boundary between categories or classifications, often challenging strict definitions or distinctions.[37][38][39][40]
A quantifier that operates within a specific domain or set, as opposed to an unbounded or universal quantifier that applies to all elements of a particular type.
A type of quantifier in formal logic that allows for the expression of dependencies between different quantified variables, representing more complex relationships than can be expressed with standard linear quantification.
Brouwerian modal logic
A form of modal logic that incorporates principles of intuitionism, as developed by L.E.J. Brouwer, focusing on the notion of possibility grounded in constructivist or intuitionist mathematics.[41][42][43]
Buridan's sophismata
A collection of paradoxes and logical exercises attributed to the medieval philosopher Jean Buridan, designed to challenge logical and linguistic intuitions.[44][45][46]
A problem in computability theory that seeks the Turing machine with the largest possible behavior (e.g., producing the most output, running the longest) among all Turing machines of a certain size, illustrating limits of computability.
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Caesar problem
A problem in the philosophy of language and logic regarding the applicability of mathematical concepts to non-mathematical objects, famously illustrated by Gottlob Frege's question of whether the concept of being a successor in number applies to Julius Caesar.[47][48][49]
A sentence formulated to express the empirical content of a theory in logical positivism, named after Rudolf Carnap and Frank P. Ramsey, aimed at separating theoretical terms from observational terms.
A proposition that asserts or denies that all or some of the members of one category are included in another category, fundamental in syllogistic reasoning.
A form of deductive reasoning in Aristotelian logic consisting of three categorical propositions that involve three terms and deduce a conclusion from two premises.
category
In mathematics and logic, a collection of objects and morphisms between them that satisfies certain axioms, fundamental to category theory.
category theory
A branch of mathematics that deals with abstract algebraic structures and relationships between them, providing a unifying framework for various areas of mathematics.
causal logic
A branch of logic concerned with the study of causal relationships, including the representation and reasoning about causes and effects.[50][51]
causal modal logic
An extension of modal logic that includes modalities for necessity and possibility along with causal relations, allowing for the formal analysis of causal statements.[52][53]
A theorem establishing the undecidability of certain decision problems in logic, such as the Entscheidungsproblem, proving that there is no consistent, complete, and effectively calculable logic.
A hypothesis proposing that any function that can be naturally regarded as computable by a human being can be computed by a Turing machine, thereby defining the limits of what can be computed.
A form of argument presenting two alternatives, both leading to the same conclusion, often used in classical rhetoric and logic to demonstrate inevitability.
The traditional framework of logic based on principles of bivalence, non-contradiction, and excluded middle, primarily focusing on propositional and predicate logic.
classical reductio ad absurdum
A stronger form of reductio ad absurdum,[56] where instead of only deriving from showing that leads to a contradiction, one can also derive from showing that leads to a contradiction.
coextensive
Having the same scope or range, especially referring to two terms or concepts that apply to the same set of objects.[57][58]
A function or expression in combinatory logic that acts on arguments to produce results without the need for variable bindings.
combinatorialism
Combinatorialism is the view that any arbitrary combination of elements constitutes a legitimate mathematical structure, whether that structure is definable or not.[63][64][65]
A concept in philosophy and mathematics referring to an actual infinity that is considered as a completed whole, contrasting with potential infinities that are indefinitely extendable.
A classification of decision problems based on their inherent computational complexity, grouping problems that can be solved within similar resource constraints.
The principle in semantics that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them.
A statement in logic that is formed by combining two or more statements with logical connectives, allowing for the construction of more complex statements from simpler ones.[67][68]
A method in logic for proving a conditional statement by assuming the antecedent and showing that the consequent follows.
conditionalization
The conditional obtained by taking the conjunction of the premises of the argument as antecedent and the conclusion of the argument as consequent. For instance, the conditionalization of modus ponens, , is the formula , called pseudo modus ponens.[6]
A property of a graph in which there is a path between any two vertices, or a property of a topological space in which it cannot be divided into two disjoint nonempty open sets.
A relation between sets of sentences or propositions, where the truth of the first set (the premises) necessitates the truth of the second set (the consequences).
A form of argument where, given two conditional statements and evidence that at least one of their antecedents is true, one can conclude that at least one of the consequents is true.
A branch of logic that emphasizes the constructive proof of existence, requiring an explicit construction of an object to assert its existence rather than relying on indirect arguments.
A philosophy of mathematics that requires mathematical objects to be constructible and computable, rejecting non-constructive proofs such as those involving the law of excluded middle in its full generality.
A proof that demonstrates the existence of a mathematical object by providing a method to construct it explicitly, as opposed to proving indirectly by contradiction.
A logical principle that states that a conditional statement is logically equivalent to its contrapositive, transforming "If P, then Q" into "If not Q, then not P".
The philosophical doctrine that the truth or falsity of a statement is determined by how it relates to the world and whether it accurately describes (corresponds with) that world.
1. (broadly) An example that disproves a statement or proposition, showing that it is not universally true.
2. (to an argument form) A counterexample to an argument form, or sequent, is an argument in the same logical form where the premises are clearly true but the conclusion is clearly false, showing that the form is invalid, since it lacks semantic validity.[73]
A conditional statement (if...then...) concerning an event that did not actually happen but is considered for the sake of argument.
counterfactual logic
A branch of logic that studies counterfactual conditionals and their implications, often used in philosophical discussions about causality and decision theory.[74][75][76]
countermodel
A countermodel of an argument is a model in which the premises are true and the conclusion false, showing that the argument is not valid.[77][78][73]
counternecessary conditional
A conditional statement that considers a situation against a necessarily true backdrop, exploring implications in hypothetical scenarios that contradict necessary truths; also known as counterpossible.[79]
A philosophical theory proposed by Lewis that addresses the semantics of modal logic, suggesting that objects in possible worlds have counterparts in other possible worlds.
A principle in mathematics and logic that defines a function based on the values it takes on smaller arguments, essential for defining functions like factorials and other recursive functions.
A paradox in logic that arises when considering a statement that asserts its own unprovability, leading to contradictions in certain systems of formal logic.
A procedure in proof theory that systematically removes cuts from a proof, simplifying it and showing that any result that can be proved with cuts can also be proved without them.
A theory for which there exists a decision procedure, meaning that for any statement within the theory, it is possible to algorithmically determine whether the statement is true or false within the theory.
An algorithm or systematic method that can decide whether given statements are theorems (true) or non-theorems (false) in a logical system or mathematical theory.
Referring to the way a statement attributes a property to a noun phrase as a whole, often contrasted with de re, which attributes a property to the thing itself.
A theorem stating that if a statement can be derived from a set of premises together with another statement, then the conclusion can be derived from the premises alone by adding the statement as a conditional.
A theory of truth that argues the role of the term "true" is merely to allow the expression of propositions that cannot be expressed otherwise, without implying a substantive property of truth.
degree-theoretic semantics
An approach in semantics where the truth of sentences is measured in degrees, rather than as strictly true or false, applicable in fuzzy logic and some theories of vagueness.[87][88][89]
The principle stating that the negation of a conjunction is equivalent to the disjunction of the negations, and vice versa, reflecting the duality between the logical operators AND and OR.
Two transformation rules stating that the negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations.
A branch of modal logic concerned with obligation, permission, and related modalities.
derivable rule
A rule that can be inferred from the axioms and inference rules of a logical system, as opposed to being an axiom or primitive rule of the system.[93][94][95]
designated value
A semantic value that plays the same role for logical consequence as the role played by truth in classical logic, so that, for the consequence to hold, a designated value must be assigned to the conclusion if it is assigned to the premises.[96][97][98]
A lemma used in the proof of Gödel's incompleteness theorems, stating that for any formula with one free variable, there exists a sentence that asserts its own unprovability.
A relation is directed if, for every pair of elements, there is a third element related to both, ensuring a kind of coherence or consistency within the set.
discharge
The act of eliminating an assumption in a logical derivation, often by using it to prove a conditional statement.[102][103][104]
disjunct
One of the component propositions in a disjunction, each of which is an alternative to the others.[71]
A theory of truth that focuses on the disquotation principle, which suggests that the function of the truth predicate is to remove quotation marks to form equivalent sentences.
A property of predicates in logic that allows them to be applied to each element of a subject class individually rather than to the class as a whole.[106][107]
A symbol () used in logic to denote semantic entailment or logical consequence, indicating that the truth of some propositions necessitates the truth of another.
A branch of modal logic that deals with the logic of belief, modeling the beliefs of rational agents.
dual
The dual of a truth-table is obtained by interchanging the truth values "true" and "false" (or 0 and 1) throughout the table. Connectives are dual if their truth-tables are dual: conjunction and disjunction are dual, and negation is self-dual.[110] The dual of a formula is obtained by replacing each connective by its dual,[110][111] e.g., for a formula containing only conjunction, disjunction, and negation (such as a formula in disjunctive normal form), its dual is the result of replacing each conjunction with a disjunction, and each disjunction with a conjunction. (For a formula in disjunctive normal form, its dual is a formula in conjunctive normal form.)[112][113]
dynamic modal logic
A branch of modal logic that studies necessary and possible connections between events.[114][115]
A method or process that guarantees a solution to a particular problem or class of problems, typically through a finite number of steps that can be precisely followed.
A function for which there exists an algorithm or mechanical procedure that can compute the function's value for any valid input in a finite amount of time.
effectively decidable relation
A binary relation for which there exists a mechanical method to determine, for any given pair of elements, whether the relation holds between them.[116][117]
effectively decidable theory
A theory in which there exists an algorithm capable of determining whether any given statement within the theory is true or false.[118][78]
A process in logical deduction where quantifiers are removed from logical expressions while preserving equivalence, often used in the theory of real closed fields.
A rule in logical inference that allows the derivation of simpler formulas from more complex ones, often by removing logical connectives or quantifiers.
empty concept
A concept that does not have any instantiation in reality or does not refer to any existing object or group of objects.[119][120]
The decision problem, a challenge posed by David Hilbert asking for an algorithm to determine the truth or falsity of any given mathematical statement. The problem was proven to be unsolvable by Alan Turing and Alonzo Church.
A self-referential paradox involving a statement made by Epimenides, a Cretan, who stated that all Cretans are liars, leading to a logical contradiction if taken to be true.
A branch of modal logic that deals with reasoning about knowledge and belief, using modalities to express what is known and what is believed.
epistemic paradox
A paradox arising from basic intuitions regarding knowledge, belief, or related epistemic notions. For instance, the knower paradox and the Fitch paradox.
A paradox presented by Eubulides of Miletus, including the liar paradox, which involves a statement declaring itself to be false, creating a contradiction.
The principle that for any proposition, either that proposition is true or its negation is true, with no middle ground.
exclusion negation
In three-valued logic, form of negation that strictly excludes the possibility of something being true, as opposed to constructive negation which asserts the truth of an opposite proposition.[125][126]
A logical rule that allows one to infer the existence of a particular individual from a statement asserting the existence of such an individual generically.
A quantifier used in predicate logic to indicate that there exists at least one member of the domain for which the predicate holds true.
existential variable
A variable in predicate logic that is bound by an existential quantifier, representing an unspecified member of the domain that satisfies the predicate.[129][130]
The collection of objects or entities to which a term or concept applies, contrasted with its intension, which refers to the properties or characteristics defining those objects or entities.
A form of logic where the truth of sentences and arguments depends solely on the extension of the terms involved, disregarding their intension or conceptual content.
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factivity
The property of certain expressions or verbs that implies the truth of the propositions they refer to, often used in the context of knowledge and belief.[134][135]
An entity or set of conditions that makes a proposition false. In metaphysics, it's considered the counterpart to a truthmaker, which validates the truth of a proposition.
A symbol used in logic to represent falsity or a contradiction, often denoted as .
"Fido"-Fido principle
The principle in philosophy of language suggesting that the meaning of a word is the object it refers to, exemplified by the idea that the meaning of "Fido" is the dog Fido itself.[136]
Pertaining to methods or processes that involve a finite number of steps or elements.
finitary arithmetic
An approach to arithmetic focusing on finitary methods, avoiding infinities and emphasizing constructions that can be completed in a finite number of steps.[137]
finitary formal system
A formal system in which all operations, proofs, and expressions are finitary, relying only on objects that can be constructed or demonstrated in a finite number of steps.[138][139]
The property of certain mathematical or logical systems where every relevant feature or property can be determined by examining only a finite part of the system.
A philosophical view that rejects the existence of infinite entities and infinite processes, emphasizing only those quantities or procedures that are finite.
first-degree entailment (FDE)
A logical system that allows for the existence of both true and false atomic propositions but does not require every proposition to be either true or false, rejecting the law of the excluded middle for certain propositions.[143][144]
In mathematics and logic, a value or element that is mapped to itself by a particular function or operation.
forced march sorites
A type of sorites paradox involving a series of incremental steps or changes that lead to a contradiction, challenging the precision of vague predicates by forcing a march from one end of a spectrum to another.[145][146]
A set of strings of symbols that are constructed according to specific syntactic rules, used in mathematics, computer science, and formal logic to precisely define expressions without ambiguity.
An expression in a formal language that can be evaluated as true or false within a given interpretation, often involving variables and logical connectives.
A theory in linguistics and logic that uses frames—conceptual structures for representing stereotypical situations—as a means of understanding how language conveys meaning.
A sequence (typically of natural numbers) where each term is chosen freely, not determined by any rule or algorithm, often used in discussions of constructivism and intuitionism.
A form of logic that allows for terms that do not denote any existing object, differing from classical logic by not requiring every term to refer to something in the domain of discourse.
A result in logic and mathematics demonstrating that arithmetic can be derived from logic through the introduction of the concept of a successor and the use of second-order quantification.
A relation between sets that associates every element of a first set with exactly one element of a second set, often represented as a mapping from elements of one set to elements of another.
The erroneous belief that if an event occurs more frequently than normal during the past, it is less likely to happen in the future (or vice versa), often arising in contexts of gambling and misinterpretation of statistics.
game-theoretic semantics
An approach to semantics that interprets the meaning of linguistic expressions through the outcomes of certain idealized games played between a verifier and a falsifier, emphasizing the interactive process of establishing truth or falsehood.[150]
A type of sentence that raises issues in the philosophy of language and logic regarding context-dependence, referential opacity, and the limits of formal semantic analysis. Named after philosophers Peter Geach and David Kaplan.
In logic and linguistics, a quantifier that can express more complex relationships than standard quantifiers like "all" or "some," allowing for the expression of concepts like "most," "many," and "few."
A result in logic stating that if a formula is provable in classical logic, then its double negation is provable in intuitionistic logic, establishing a connection between the two logics.
A method of encoding mathematical and logical symbols and expressions as natural numbers, introduced by Kurt Gödel as part of his incompleteness theorems.
A self-referential sentence constructed in formal systems to demonstrate Gödel's incompleteness theorems, asserting its own unprovability within the system.
A form of intuitionistic logic that includes a principle of maximal elements, allowing for the expression of certain intermediate truth values between true and false.
A theorem proving that in any consistent formal system that is capable of expressing basic arithmetic, there are propositions that cannot be proven or disproven within the system.
A theorem establishing that no consistent system capable of doing arithmetic can prove its own consistency, building on the first incompleteness theorem.
An argument concerning the semantics of reference and truth, challenging the coherence of theories that attempt to distinguish between facts and true propositions in a fine-grained manner.
A paradox related to self-reference and linguistic categories, particularly whether the word "heterological," meaning not applicable to itself, applies to itself.
The decision 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. Proven to be undecidable by Alan Turing.
In logic and philosophy, the notion that the introduction and elimination rules for a logical connective should be in balance, ensuring that the connective does not introduce more than it can eliminate, and vice versa.
An informal fallacy in which a conclusion is not logically justified by sufficient or unbiased evidence; drawing a general conclusion from a too-small sample size.
A system in which entities are ranked one above the other based on certain criteria, often used in the context of sets, classes, or organizational structures. In logic, an important one is Tarski's hierarchy. In set theory, an important one is the cumulative hierarchy.
A form of logic that extends first-order logic by allowing quantification over predicates and possibly other higher-order entities, not just individuals.
Vagueness about the application of the concept of vagueness itself, particularly in the context of predicates that are borderline cases of borderline cases.[37][155]
higher-order variable
A variable in higher-order logic that represents a function, predicate, or relation, rather than an individual object.[156][157]
An ambitious project proposed by David Hilbert to provide a solid foundation for all of mathematics by formalizing it and proving it consistent using finitary methods.
The idea that systems and their properties should be analyzed as wholes, not just as a collection of parts, often discussed in the context of meaning, knowledge, and the philosophy of science.
A structure-preserving map between two algebraic structures of the same type, such as groups, rings, or vector spaces, that respects the operations of the structures.
horn
Refers to either of the two alternatives presented by a dilemma.[158][159]
The principle that the number of objects in one collection is equal to the number of objects in another collection if and only if there is a one-to-one correspondence between the two collections.
A type of modal logic that incorporates additional syntactic elements to refer directly to worlds in its models, allowing for more expressive power than standard modal logics.
A form of logical argument consisting of three propositions: two conditional statements and a conclusion that infers a relationship between the antecedent of the first conditional and the consequent of the second.
A property of certain operations in which applying the operation multiple times has the same effect as applying it once. For example, the union of a set with itself is the set itself.
A logical relation where the truth of one statement (the antecedent) brings about the truth of another statement (the consequent).
implicit definition
A definition that specifies an entity or concept not by direct enumeration of its properties but by its relations to other entities or concepts.[131][160]
A description that does not uniquely identify a single individual or entity but refers to any member of a class that satisfies a certain condition.
indefinite extensibility
The concept that certain collections (such as the set of all sets) cannot be comprehensively listed because any attempt to enumerate them leads to the possibility of generating new members.[161][162][163]
A finding in logic and mathematics that a particular statement cannot be proven or disproven within a given system, assuming the system's axioms are consistent.
A logic that extends first-order logic to allow for more nuanced expressions of quantifier scope and dependence, particularly in contexts of game-theoretical semantics.
Inition proposed by W.V.O. Quine, suggesting that no unique translation between languages can be determined solely by empirical evidence, due to the underdetermination of theories by data.
A conditional statement used to express factual implications or predictions about real situations, as opposed to counterfactual or hypothetical statements.
A method of proof in which the negation of the statement to be proven is assumed, and a contradiction is derived, thereby proving the original statement by contradiction.
A method used in formal logic and mathematics to prove properties of all well-formed formulas by showing they hold for basic formulas and are preserved under the operations that generate new formulas.[6]
A flaw in reasoning that occurs in natural language arguments due to ambiguity, irrelevance, or other factors outside the formal structure of the argument.
The inherent content or essential properties and meanings of a concept or term, as opposed to its extension, which refers to the range of things it applies to.
A logic that deals with the intensional aspects of meaning, such as belief, necessity, and possibility, distinguishing between logically equivalent expressions that have different modal properties.
Any logical system that falls between intuitionistic logic and classical logic in strength, allowing for distinctions not permissible in classical logic.
A result stating that if a formula is provable, then there exists a formula containing only the non-logical symbols common to and such that and are both provable.
A philosophy of mathematics that denies the reality of the mathematical infinite and the completeness of mathematical truth, requiring constructive proofs.
A system of logic that reflects the principles of intuitionism, rejecting the law of excluded middle and requiring more constructive proofs of existence.
An operation that is its own inverse, meaning applying it twice returns to the original state.
I-proposition
In traditional logic, a particular affirmative categorical proposition, stating that some members of the subject class are members of the predicate class.[2][168]
A fictional scenario used in logic puzzles where inhabitants are either knights, who always tell the truth, or knaves, who always lie, posing challenges to deductive reasoning.
A bijective (one-to-one and onto) correspondence between two structures that preserves the operations and relations of the structures, indicating they have the same form or structure.
A logical connective in propositional logic, equivalent to the nor operator, that is true if and only if both propositions it connects are false. It denies the joint assertion of both propositions.
Logical connectives defined using Kleene's three-valued logic, which includes a third truth value (undefined or unknown) in addition to true and false, accommodating indeterminate propositions.[172]
A paradox arising from the assumption that if a statement is true, then it is possible to know that it is true, leading to contradictions in certain epistemic frameworks.
A logic developed to handle higher-order quantification and modalities, reflecting discussions on the foundations of mathematics by Kreisel and Putnam.
A framework for interpreting modal logic through the use of possible worlds, developed by Saul Kripke, allowing for the formal analysis of necessity, possibility, and other modal notions.
A mathematical structure used in modal logic and computer science to model systems that can be in various states and transition between them, forming the basis for Kripke semantics.
A formal system in mathematical logic and computer science for expressing computation based on function abstraction and application, using variable binding and substitution.
A self-referential paradox involving a statement that declares itself to be false, leading to a contradiction if it is either true or false.
liar sentence
A sentence that asserts its own falsity, such as "This sentence is false," which creates the basis for the liar paradox.[173]
limitation result
A result that establishes a boundary or limit on what can be achieved within a particular logical or mathematical system, often related to incompleteness or undecidability.[6]
A subfield of logic that emphasizes the concept of resources, where logical operations consume their arguments, differing from classical logic's treatment of assumptions as reusable.
A total order on a set where every pair of elements is comparable, meaning for any two elements, one is either greater than, less than, or equal to the other.
Another term for linear order, emphasizing the arrangement of elements in a sequence where each is comparable to the others in a single, unambiguous way.
A paradox in modal logic that arises from attempting to formalize a statement's provability within the system, leading to conclusions that appear counterintuitive or self-contradictory.
A theorem in mathematical logic that provides conditions under which a statement about its own provability is provable, related to Gödel's incompleteness theorems.
A physical device implementing a Boolean function, used in digital circuits to perform logical operations on one or more binary inputs to produce a single binary output.
A logical system that allows for some contradictions to be true, challenging the traditional law of non-contradiction and exploring the consequences of paradoxical statements.
A branch of logic that deals with the study of relations, including their properties, composition, and inversion, and how they interact with logical operators.
The philosophical position that logical truths do not correspond to an independent reality but are instead products of human conventions, language, or thought processes.[6]
A symbol or word used in logic to connect propositions or sentences, forming more complex expressions that convey relationships such as conjunction, disjunction, and negation.
A relationship between statements where the truth of one or more premises necessitates the truth of a conclusion, based on the logical structure of the statements. See semantic consequence and syntactic consequence.
A symbol in logic that has the same meaning in all interpretations, such as connectives and quantifiers, as opposed to variables whose interpretations can vary.
The relationship between statements that are true under exactly the same conditions, allowing them to be substituted for one another in logical proofs.
logical falsehood
A statement that is false under all possible interpretations, also known as a contradiction.[176]
A symbol or function in logic that applies to one or more propositions, producing another proposition that expresses a logical operation such as negation, conjunction, or disjunction.
A statement or group of statements that lead to a contradiction or a situation that defies intuition, often highlighting limitations or problems within the logical system. Sometimes distinguished from semantic paradox.
The property of an argument wherein if the premises are true, the conclusion necessarily follows, due to the structure of the argument rather than the specific nature of the premises or conclusion.
A theorem in mathematical logic that states any countable theory with an infinite model has models of all infinite cardinalities, highlighting the limitations of first-order logic in controlling the sizes of its models.
A variant of first-order logic that allows for multiple domains of discourse, with variables and quantifiers distinguished by the sort or type of objects they range over.
A logical system that extends beyond classical two-valued true/false logic to include additional truth values, accommodating indeterminacy, uncertainty, or levels of truth.
A principle in constructive mathematics stating that if it is impossible for a mathematical object not to have a certain property, then there exists an object with that property.
Describing propositions that are true under exactly the same conditions or have the same truth value across all possible worlds.
mathematical abstractionism
A philosophical stance that views mathematical entities as abstractions from physical objects or properties, rather than as inherently existing objects.[180][181]
A set of formulas in a logical system that is consistent (no contradictions can be derived from it) and maximal (no additional formulas can be added without causing inconsistency).
In philosophy, especially in discussions of language, "mention" involves referencing a word or phrase itself rather than employing it for its semantic content. This typically occurs when discussing the word as a linguistic entity. In the use-mention distinction, "mention" is signified by the use of quotation marks or other indicators that the words are subjects of discussion rather than tools for communication. For example, in the sentence "The word 'books' consists of five letters," "books" is mentioned, not used.
mere possibilia
Hypothetical or possible entities that do not actually exist but could exist under different circumstances.[186]
The view that modal statements (about possibility and necessity) can be treated as useful fictions without committing to the existence of possible worlds.
A branch of logic that deals with modalities such as necessity, possibility, and related concepts, often formalized through the use of modal operators.
A rule of inference that allows one to derive a conclusion from a conditional statement and the negation of its consequent, formalized as if and , then .
molecule
In logic and philosophy, often used metaphorically to refer to a compound entity or concept that is made up of simpler, atomic parts.[194]
A variant of first-order logic restricted to predicates that take only one argument, focusing on properties of individual objects rather than relations between them.
A morphism in category theory that is left-cancellable, meaning if two compositions with it are equal, then the other morphisms must be equal, akin to an injective function in set theory.
A type of logic in which adding new premises to a set does not decrease the set of conclusions that can be derived, ensuring that conclusions are preserved under the addition of new information.
The property of a function or process that preserves order, in logic, referring to systems where conclusions derived from a set of premises are not invalidated by adding more premises.
A logical operation meaning "not and"; it produces a true result for all input combinations except the case where all inputs are true. It is a fundamental operation since any logical function can be constructed using only NAND operations.
A relation that involves n elements, where n is a natural number, extending the concept of binary relations to relations between more than two entities.
A system of logical inference that attempts to mirror the intuitive ways humans reason, consisting of a set of inference rules for introducing and eliminating logical connectives.
A rule in natural deduction that allows for the introduction of negation into a proof, typically by deriving a contradiction from the assumption that the negation is false.
A way of expressing logical formulas where negation is only applied directly to atomic propositions, and the only other allowed connectives are conjunction and disjunction.
negative proposition
A proposition that asserts the non-existence or absence of something, or denies some property of an object.[201][202]
A philosophical stance revisiting Frege's logicism with the aim of grounding mathematics, particularly arithmetic and analysis, in logic through the use of Hume's Principle and other axioms.
A movement in the philosophy of mathematics seeking to revive logicism, the project of founding mathematics on logic, through new insights and approaches, particularly in the wake of criticisms of traditional logicism.
A system of set theory proposed by W.V. Quine with a distinctive axiom schema intended to avoid the paradoxes of naïve set theory while allowing a universal set.
The philosophical view that abstract concepts, general terms, or universals have no independent existence but exist only as names or labels for groups of individual objects.
non-alethic modal logic
A form of modal logic that deals with modes of truth beyond the alethic modes of necessity and possibility, such as deontic (duty and permission) or epistemic (knowledge and belief) modalities.[203][204]
Any logical system that diverges from the principles of classical logic, including intuitionistic logic, many-valued logics, modal logics, and others that challenge classical assumptions or introduce new principles.
A logical system in which the order of application of operations affects the outcome, contrasting with classical logic where operations like conjunction and disjunction are commutative.
The complexity class NP, consisting of decision problems for which a 'yes' answer can be verified by a deterministic Turing machine in polynomial time, given the correct certificate or witness.
A theoretical model of computation that, at each step, can make a 'choice' from multiple possibilities, allowing it to explore many possible branches of execution simultaneously.
non-standard logic
Logics that diverge from or extend classical logic, including non-classical logics, many-valued logics, and modal logics, among others.[205][206] Also called non-classical logics.
A model of a theory that satisfies the axioms of the theory but has properties not intended by the original formulation, often revealing the theory's consistency or independence results.
In logic, a standardized way of expressing logical formulas, such as conjunctive normal form (CNF) or disjunctive normal form (DNF), to facilitate analysis or computation.
A class of modal logics that include the necessitation rule and the distribution axiom, allowing for the derivation of necessary truths from given axioms and rules of inference.
A complexity class (nondeterministic polynomial time) that includes decision problems for which a 'yes' answer can be verified in polynomial time by a deterministic Turing machine.
A class of decision problems in NP for which any problem in NP can be reduced to it in polynomial time, and whose solution can be verified in polynomial time; considered among the hardest problems in NP.
numerical quantifier
A quantifier that specifies the exact number of instances for which a predicate holds within a domain of discourse, such as 'exactly three', 'at least five'.[207][208][209]
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object language
The language that is being studied, analyzed, or used to construct statements in a logical system, as opposed to the metalanguage used to talk about the object language.[210][211]
objectual quantifier
A type of quantifier in formal logic that quantifies over objects in the domain of discourse, as opposed to substituting variables with constants or other expressions.[212][213]
In traditional logic, the process of negating the predicate of a categorical proposition and changing its quality (affirmative to negative, or vice versa) without altering its truth value.
A function or mapping from one set to another where each element of the first set is associated with a unique element of the second set; also known as injective.
A function or mapping from one set to another where every element of the second set is associated with at least one element of the first set; also known as surjective.
A formula in a formal language that contains free variables, meaning it cannot be determined as true or false until the variables are bound or specified.
open pair
A paradox about a pair of statements that deny each other.[216][217]
An expression in a formal language that contains free variables, which does not denote a specific object or truth value until the variables are instantiated.
o-proposition
In traditional logic, a particular negative categorical proposition, stating that some members of the subject class are not members of the predicate class.[2][218]
A logical connective (disjunction) that links propositions in a way that the compound proposition is true if at least one of the linked propositions is true.
Paradoxes that arise from the counterintuitive consequences of the material conditional, especially when the antecedent is false or when there is no causal or necessary connection between the antecedent and consequent.
Also known as begging the question, an informal fallacy where the conclusion of an argument is assumed in one of the premises.
Philonian conditional
Another term for the material conditional, emphasizing its use in propositional logic to represent "if...then..." statements without implying a causal relation.[227]
The study of the more abstract or theoretical aspects of logic, often concerning questions about reference, modality, quantification, and the structure of propositions and arguments.
A branch of philosophy that examines the nature and scope of logic, including the assumptions, methodologies, and implications of various logical systems.
Quantification over multiple objects or entities considered together, extending beyond singular quantification to express statements about sets or groups.
A prefix notation for logic and arithmetic where operators precede their operands, eliminating the need for parentheses to indicate order of operation.
polyadic first-order logic
First-order logic extended to include predicates with more than one argument, allowing for the expression of relations between multiple objects.[228][229]
possibility
A modality indicating that a proposition may be true, even if it is not actually true; the capacity for some state of affairs to occur.[230][231]
A hypothetical total way things might have been or could be, used in modal logic to analyze possibility, necessity, and other modal concepts.
Post consistency
A theory is "Post consistent" (or absolutely consistent) if and only if there is at least one statement in the language of the theory that is not a theorem; otherwise, it is "Post inconsistent".[232][233]
A function or relation that asserts a property about or a relationship between individuals or objects in a domain of discourse.
predicate functor
In logic, a symbol that represents a function from individuals or tuples of individuals to truth values, essentially a generalization of a predicate.[234]
A logical system that combines elements of predicate logic with the concept of functors, allowing for a more expressive representation of properties and relations.
A way of writing mathematical and logical expressions where the operator precedes its operands, facilitating unambiguous interpretation without parentheses.
A form of recursion where a function is defined in terms of itself, using simpler cases, with a base case to stop the recursion.
primitive recursive function
A function computable by a primitive recursive algorithm, representing a class of functions that can be defined by initial functions and operations of composition and primitive recursion.[188]
primitive recursive relation
A relation that can be defined by primitive recursive functions, characterizing a subset of computable relations.[188]
A logical system that incorporates probabilistic elements to deal with uncertainty, extending classical logic to handle degrees of belief or likelihood.
A logical or mathematical argument that demonstrates the truth of a statement or theorem, based on axioms, definitions, and previously established theorems.
The branch of mathematical logic that studies the structure and properties of mathematical proofs, aiming to understand and formalize the process of mathematical reasoning.
A mental state expressed by verbs such as believe, desire, hope, and know, followed by a proposition, reflecting an individual's attitude towards the truth of the proposition.
The branch of logic that deals with propositions as units and uses propositional connectives to construct complex statements, focusing on the truth-values of propositions.
A branch of modal logic concerned with the properties of provability and modalities that express notions of necessity as provability within a formal system.
provability predicate
A predicate, often called "Bew", that expresses the concept of a statement being provable within a given formal system.[242]
An argument by Hilary Putnam challenging the conventional understanding of reference and truth, suggesting that semantic externalism leads to radical skepticism about the meanings of terms and the contents of thoughts.[247]
An extension of modal logic that includes quantifiers such as "all" and "some", allowing for expressions involving necessity or possibility applied to individuals or properties quantitatively.[248]
A logical operator that specifies the quantity of specimens in the domain of discourse that satisfy an open formula, such as "all", "some", or "exists".
A logical fallacy involving the incorrect interchange of the position of two quantifiers, or a quantifier and a modal operator, leading to invalid conclusions.
A non-classical logic that attempts to capture the peculiarities of quantum mechanics, challenging traditional logical principles such as the law of excluded middle and distributivity.
Quine's dictum
The principle that "To be is to be the value of a variable", emphasizing ontological commitment in terms of quantification and the variables of quantified theory.[250][251][252]
A Latin phrase meaning "which was to be demonstrated", traditionally used at the end of a mathematical proof or logical argument to signify its completion.
An extension of the simple theory of types that includes a hierarchy of levels, allowing for the distinction between objects and functions at different orders to avoid paradoxes such as Russell's paradox.
A criterion for evaluating the acceptability of conditional statements in terms of belief revision: if adding the antecedent to one's stock of beliefs requires adding the consequent for consistency, then the conditional is accepted.
A definition of a function, set, or other mathematical object that is defined in terms of itself, using a base case and a rule for generating subsequent elements.
A relation defined on a set where the relation is specified in terms of itself, allowing for the construction of complex relational structures from simpler ones.
recursively axiomatizable theory
A theory for which there exists a recursive set of axioms that can generate all theorems of the theory through logical deduction.[253][254][255]
A property of expressions wherein substituting a co-referential term does not necessarily preserve truth, typically occurring in intensional contexts like belief reports.
A theoretical model of computation that uses a set of registers to store numbers and a program of instructions to perform calculations, serving as an alternative to the Turing machine model.
An approach to interpreting logical languages where the meaning of sentences is defined in terms of relations between possible worlds or states of affairs, commonly used in modal and temporal logics.
relative consistency proof
A proof showing that if a mathematical system is consistent, then an extension of by adding new axioms is also consistent, used to compare the foundational strength of different theories.[6]
A non-classical logic that seeks to capture the notion that the premises of a valid argument must be relevant to the conclusion, avoiding paradoxes of material implication.
representation
A n+1-ary predicate Prepresents an n-ary function f if, and only if, it is the case that: is true if, and only if, . Similarly, a unary predicate Prepresents a set S if, and only if, it is the case that: Px is true if, and only if, x is a member of S.[6]
A program in mathematical logic that seeks to determine which axioms are necessary to prove theorems of mathematics by proving theorems from the weakest possible systems.
revision theory of truth
A theory proposing a non-classical approach to the concept of truth, suggesting that truth values of propositions can be revised in light of paradoxes, notably the liar paradox.[259]
A term that refers to the same object in all possible worlds where that object exists, used in discussions of necessity and identity across possible worlds.
A paradox in deontic logic arising from imperatives that imply counterintuitive obligations, demonstrating challenges in formalizing moral and ethical reasoning.
A rule in formal logic allowing for the substitution of equivalent expressions within logical proofs, maintaining the validity of the argument.
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salve veritate
A Latin phrase meaning "with truth unchanged", used in logic and philosophy to indicate a transformation or substitution that preserves the truth value of statements.[260]
In model theory, the relation between a structure and a sentence where the structure makes the sentence true, according to the interpretation of the sentence's symbols in that structure.[261]
The property of a logical formula if there exists at least one interpretation under which the formula is true.
schema
A template or pattern representing a class of similar statements or propositions, often used in the formulation of logical axioms and rules.[262][263][264]
A theory of truth based on the correspondence between statements and facts or states of affairs in the world, emphasizing the role of meaning and interpretation.[265]
semantic consequence
The relation that holds between a set of premises and a conclusion formulated within a certain formal language, such that, for this language, there is no possible interpretation that evaluates the premises as true and the conclusion as false.[266][267][268]
A paradox that arises due to some peculiarity of semantic concepts, such as truth, falsity, and definability, as distinguished from logical or set-theoretical concepts. The distinction between semantical and logical paradoxes is controversial and was invented by Ramsey.[269][270]
A method of proof in logic that uses a tree structure to systematically explore the truth or falsity of logical expressions by breaking them down into simpler components.
The study of meaning in language, including the interpretation of words, phrases, sentences, and texts, and the study of the principles that govern the assignment of meanings.
semi-decidable theory
A theory for which there exists an algorithm that can enumerate all its theorems, but there may not be an algorithm to decide non-theorems.[278]
sense
The aspect of meaning that pertains to the inherent content or connotation of an expression, distinct from its reference or denotation.[256]
Another term for propositional logic, focusing on the logical relationships between whole sentences or propositions rather than their internal structure.
An ordered list of objects or terms, each of which is identified by its position in the list, used in mathematics and logic to define functions, sets, and series.
In sequent calculus, a formal representation of a logical deduction, consisting of a sequence of formulas that precede a turnstile and a sequence of formulas that follow it, indicating premises and conclusion.
A formal system for deriving logical entailments, representing deductions as sequences of formulas, and emphasizing the structural rules of logical derivation.
The property of a relation where for every element in the domain, there exists an element in the codomain that is related to another element in a specific way, particularly in the context of binary relations and modal logic.
A modal logic system characterized by the axioms that necessitate reflexivity and transitivity for the accessibility relation, implying that if something is necessary, then it is necessarily necessary.
sharpening
The process of making a vague or imprecise concept more precise or clearly defined, often discussed in the context of semantic vagueness.[123][279][280]
A logical operation equivalent to the nand (not and) function; it is functionally complete, meaning all other logical operations can be constructed from it.
In logic and algebra, a set of symbols along with their arities, defining the kinds of operations, functions, and relations considered in a structure or theory.[281]
simple type theory
A type theory that divides objects into a simple hierarchy of objects, classes of objects, classes of classes of objects, etc. The adjective "simple" is used to contrast it with ramified type theory, which further stratifies these simple types into orders.[282]
single turnstile
A symbol used in logic () to denote syntactic entailment, indicating that a formula or set of formulas derives or proves another formula within a formal system.[283]
A term in logic that refers to a single object or entity, distinguishing it from general terms that may refer to classes of objects or properties.
situation
In logic and philosophy, a set of circumstances or a state of affairs to which truth-values of statements are relative, often used in situation semantics.[284]
An approach to semantics that analyzes meaning in terms of situations, rather than attempting to account for meaning solely in terms of truth conditions at possible worlds.
A technique in first-order logic for eliminating existential quantifiers by introducing Skolem functions, used in the process of converting formulas to a standard form.
A theorem stating that if a first-order theory has an infinite model, then it has models of every infinite cardinality, highlighting the flexibility of first-order semantics.
A way of expressing first-order logic formulas where all existential quantifiers are moved inside and replaced by Skolem functions, leaving only universal quantifiers at the front.
The apparent paradox arising from the Skolem-Lowenheim theorem, where countable models can be found for theories that intuitively require uncountably many objects, challenging notions of absolute size in set theory.
An argument aiming to show that all true statements refer to the same "fact" or "entity", raising questions about the correspondence theory of truth and the nature of facts.
An informal fallacy or rhetorical argument suggesting that a relatively small first step or minor decision will lead to a chain of related events culminating in a significant (often negative) outcome, without sufficient justification for such inevitability.
A fundamental theorem in the theory of computable functions that provides a method for constructing a specific computable function from a given computable function, highlighting the universality and flexibility of computable functions.
sophism
An argument or form of reasoning deemed fallacious, misleading, or deceptive, historically associated with the Sophists in ancient Greece, who were known for their rhetorical skill and relativistic views on truth and morality.[285]
sophisma
A puzzle or paradoxical question that challenges conventional wisdom or logical reasoning, often used in medieval logic to teach students about logical fallacies and the complexities of language.[286]
A paradox arising from vague predicates and the problem of heap, illustrating how a series of seemingly acceptable premises can lead to a paradoxical or absurd conclusion.
sorites series
A sequence of propositions associated with the sorites paradox, each adding a small amount to the previous one, challenging the boundary between truth and falsehood for vague concepts.
An argument where the logical structure ensures the truth of the conclusion if the premises are true, and where the premises are indeed true, making the argument both valid and sound.
A diagram representing the logical relationships between the four types of categorical propositions (A, E, I, O) in traditional logic, showing their contradictions, contraries, subcontraries, and subalternations.
A definition that assigns a meaning to a word for the first time, or proposes a new meaning for an existing word, without claiming to capture an already established usage.
A conditional statement interpreted in terms of necessity, such that the truth of the antecedent necessarily implies the truth of the consequent, unlike the material conditional.
A relation between propositions where the truth of the first (the antecedent) necessarily brings about the truth of the second (the consequent), often associated with modal logic.
strong completeness
The property of a logical system where if a formula is semantically valid (true in all interpretations), then it is syntactically derivable within the system.[287][288]
A form of mathematical induction that allows one to assume the proposition for all smaller instances simultaneously when proving it for any given instance.
Strong paraconsistency is the view that there are possible worlds where contradictions are true, or where some statements are both true and false. Compare weak paraconsistency, the view that true contradictions, and worlds that contain them, are merely a formal tool used to study reasoning.[293]
strongly connected
A relation R is strongly connected (or total) if and only if, for all x and y, either Rxy, or Ryx.[294]
In logic, especially in proof theory, a rule that concerns the manipulation of the components of sequents or deductions without reference to their internal logical structure, such as contraction, weakening, and exchange.
subaltern
In traditional syllogistic logic, a term describing the relationship between two categorical propositions where the truth of the first (the universal) implies the truth of the second (the particular), but not vice versa.[2]
The logical relationship between a universal statement and its corresponding particular statement, where the truth of the universal necessitates the truth of the particular.
In traditional logic, a pair of particular statements (I and O propositions) that cannot both be false together, though both can be true under the square of opposition.
A subset of a language that uses a restricted vocabulary or simpler grammatical structures, often for a specific purpose or domain.
sublogic
A logical system that is a subset of a more comprehensive logic, retaining some but not all of the operations and principles of the larger system.[295]
The act of replacing a variable or expression within a logical formula with another, maintaining logical consistency.
substitution-instance
The well-formed formula which results from a given well-formed formula by replacing one or more of the variables occurring in the well-formed formula throughout by some other well-formed formulas, it being understood that each variable so replaced is replaced by the same well-formed formula wherever it occurs.[72][296]
substitutional quantifier
A type of quantifier interpreted as ranging over expressions or names rather than over objects directly, used in certain theories of reference and meaning.[297]
A way of writing expressions where operators follow their operands, also known as reverse Polish notation, used in some calculators and programming languages for its efficiency.
A task that consists of an infinite sequence of operations completed in a finite amount of time, often discussed in the context of philosophical paradoxes and theoretical physics.
supertrue
A term used in certain theories of truth, such as supervaluational semantics, to describe propositions that remain true across all precisifications or interpretations of vague terms.[298]
supervaluational semantics
A semantic theory designed to handle vagueness by considering multiple precisifications of vague terms, with a proposition deemed supertrue if it is true under all precisifications.[276]
In medieval logic, a relation between an expression and the object or concept that the expression is being used to talk about, where the supposition of the expression need not be its literal reference.
A function from one set to another where every element of the target set is mapped to by at least one element of the domain set, also known as an onto function.
A form of deductive reasoning consisting of a major premise, a minor premise, and a conclusion, traditionally used in Aristotelian logic to infer relationships between categories.
syllogistic figure
The form of a syllogism, determined by the position of the middle term in its premises, categorized into four figures that structure the syllogistic argument differently.[299]
syllogistic mood
The type of a syllogism, defined by the nature of its premises (universal affirmative, universal negative, particular affirmative, particular negative) and how they combine to form a conclusion.[300]
syllogistic terms
The three terms in a syllogism: the major term (predicate of the conclusion), the minor term (subject of the conclusion), and the middle term (appears in both premises but not in the conclusion). See also barbara.
symmetry
A property of binary relations where if one element is related to another, then the second is related to the first, such as the relation of equality.[301]
Terms that do not stand for objects or have a reference by themselves but contribute to the meaning of expressions in which they occur, such as conjunctions, prepositions, and quantifiers.
The set of rules, principles, and processes that govern the structure of sentences in a given language, distinguishing between correct and incorrect forms of expression.
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Tarskian hierarchy
A hierarchical structure of languages proposed by Alfred Tarski to avoid paradoxes in semantic theories, where each level of language can only reference levels below it, preventing self-reference.[210]
A theorem stating that truth cannot be consistently defined within the same language it applies to, requiring a meta-language for a definition of truth to avoid paradoxes.
A paradox where, from "the temperature is ninety" and "the temperature is rising", it is concluded that "ninety is rising", which seems invalid but can actually be valid under some formalization schemes.
temporal modal logic
A branch of modal logic that deals with modalities related to time, such as 'always', 'sometimes', and 'never', allowing for reasoning about temporal aspects of propositions.[303]
An approach to logic focusing on the relations between terms in propositions and the inferences that can be drawn from them, characteristic of Aristotelian logic.
A statement or proposition that has been formally proven on the basis of previously established statements or axioms within a logical or mathematical system.
A coherent set of propositions or statements, especially one that forms a comprehensive explanation of some aspect of the natural world or an abstract concept.
A logical system that introduces a third truth value (such as 'unknown', 'indeterminate', or 'both true and false') in addition to the classic binary true and false values.
An instance of a type, such as a particular occurrence of a word or phrase, in contrast to the abstract concept or category it represents.
tolerant
In the theory of vagueness, a predicate is considered tolerant if, and only if, small changes in the relevant underlying properties of an object do not affect the justice with which the predicate applies to it. Thus, the predicate "bald" is tolerant, since one hair more or less does not transform a clear instance of baldness into a clear instance of non-baldness.[123]
tonk
A fictional logical connective introduced to illustrate the importance of preserving inference rules in defining logical operators, showing that arbitrary rules can lead to absurdity.[305][306][10]
In logic, a symbol (⊤) representing the highest or maximal element in a lattice or order, often used to denote a tautology or universally true proposition in propositional logic.
A concept in category theory generalizing set theory concepts within a more abstract framework, allowing for the definition of mathematical structures in different contexts.
The study of toposes, which are categories that behave like the category of sets and provide a foundation for much of mathematics, allowing for generalized notions of computation and logic.
The smallest transitive relation that contains a given relation, effectively adding the minimum necessary elements to make the original relation transitive.
A property of a relation where if the relation holds between A and B, and between B and C, then it also holds between A and C, ensuring a kind of consistency or continuity in the relation across elements.
A translation is a function from the expressions of one language to the expressions of another language. Translations are typically intended to preserve either the meanings or the truth conditions of the translated expressions.
A concept in modal logic and metaphysics concerning the identity of individuals across different possible worlds, addressing questions of persistence and change.
trichotomy
A relation R is trichotomous (or comparable) if and only if, for any objects x and y, either Rxy, or Ryx, or x = y.[307]
A law in order theory and mathematics stating that for any two elements in a certain set, exactly one of three relationships (greater than, less than, or equal to) must hold.
The state or quality of being trivial, in logic and mathematics, often referring to statements, propositions, or problems that are oversimplified or of little interest or importance.
A concept in logic and philosophy concerning the property of statements, beliefs, or propositions corresponding to reality or fact, or being in accord with the actual state of affairs.
A function that takes truth values as input and produces a truth value as output, used in logic to model the truth conditions of logical connectives.
truth-functional
Pertaining to an operator or connective in logic whose output truth value depends solely on the input truth values, without regard to the content of the propositions involved.[308]
A predicate that assigns the property of being true to propositions, often discussed in relation to Tarski's semantic conception of truth and the liar paradox.
A table used in logic to show the truth value of a compound statement for every possible combination of truth values of its components, instrumental in analyzing logical expressions.
truth-teller
The converse of the liar paradox, a statement that asserts its own truth, raising questions about self-reference and the nature of truth.[310]
The value indicating the truth or falsity of a proposition or statement, typically represented as true or false in classical logic, but possibly more varied in many-valued logics.
truth-value gap
A situation where a statement or proposition cannot be assigned a traditional truth value of true or false, often due to vagueness or undefined terms.[311]
truth-value glut
A condition in which a statement or proposition is paradoxically both true and false simultaneously, associated with dialetheism and contradictions.[311]
The Tarski schema for defining truth, stating that 'P' is true if and only if P, where 'P' is a placeholder for a proposition and P is the proposition itself.
A logical fallacy that attempts to discredit an opponent's position by asserting the opponent's failure to act consistently with that position, essentially accusing them of hypocrisy.
A function that can be calculated by a Turing machine, representing the class of functions that are computable in principle, according to the Church–Turing thesis.
A symbol used in logic () to denote syntactic entailment or provability, indicating that the statement or set of statements to the right is a logical consequence of the statements to the left within a given formal system.
type
1. (In type theory.) A category or class of entities that share certain characteristics, used in logic and mathematics to distinguish between different kinds of objects, expressions, or variables, preventing certain kinds of logical paradoxes.
A framework in mathematical logic and computer science that uses types to classify expressions and objects, aiming to avoid paradoxes like Russell's paradox by organizing objects into hierarchies or levels and restricting operations to objects of the same type.
A function that operates on a single input or argument, common in mathematics and logic for representing operations like negation or the absolute value function.
A rule of inference in predicate logic that allows for the derivation of a specific statement about an individual from a general statement that applies to all members of a category.
A rule of inference in predicate logic that allows for the generalization of a statement to all members of a category if the statement is shown to hold for an arbitrary but specific individual.
universal proposition
A statement in logic that asserts something about all members of a certain category, typically formulated using a universal quantifier.
The set of all objects, individuals, or values that are relevant in a particular logical or mathematical discussion, serving as the domain over which quantifiers range.
An argument that is either invalid in its logical form or contains at least one false premise, and therefore does not guarantee the truth of its conclusion.
use
In philosophy, particularly in the analysis of language, "use" refers to the actual application of a word or a phrase in a sentence to convey meaning. In the use-mention distinction, "use" involves employing words to refer to things, actions, qualities, or concepts in the world. For example, in the sentence "I enjoy reading books," the word "books" is used to refer to objects that can be read; it is not merely mentioned.
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vacuous quantifier
A vacuous quantifier is one that doesn't bind any variables, such as the second quantifier in .[312]
vagueness
The characteristic of terms, concepts, or propositions that lack clear boundaries or precision in meaning, leading to indeterminate or borderline cases.[37]
1. A deductive argument whose structure ensures that if all the premises are true, then the conclusion must also be true, demonstrating logical validity.
3. In proof-theoretic semantics, a formula that is either an explicit rule of inference of a system, or that does not allow one to prove anything that could not be proved using the explicit rules of inference.[313]
The Latin word for "or", used in logic as a name for ∨, the descending wedge symbol. The symbol is used to denote a disjunction that is inclusive, meaning at least one of the disjuncts must be true for the whole expression to be true.
The verity (or "degree of truth") of a statement is the semantic value of that statement within degree-theoretic semantics, which assigns degrees between 0 and 1 to statements.[317]
Verum (Latin for "true") is another name for the symbol,[318] which represents a primitive, necessarily true statement, and is sometimes considered a nullaryconnective.
A principle against definitions or arguments that are circular, ensuring that the thing being defined is not used in its own definition or premise in a way that presupposes its conclusion.
The property of a logical system where if a statement is semantically valid (true under all interpretations), then there is a proof of the statement within the system.[319]
weak counterexample
Within intuitionistic logic and intuitionistic mathematics, a weak counterexample is a situation in which we have no positive evidence for the (intuitionistic) truth of some instance of the law of excluded middle, .[320][321]
A principle in intuitionistic logic stating that for any proposition P, either P is provable or not-P is provable, but not necessarily both, reflecting a more nuanced view of truth than the classical law of excluded middle.
A form of mathematical induction that only assumes the truth of the statement for the immediately preceding case to prove its truth for any natural number, as opposed to strong induction, which assumes the statement for all smaller numbers.
A form of negation in some non-classical logics where the negation of a proposition does not assert the truth of the opposite proposition but rather the absence of truth of the original proposition.
weak paraconsistency
Weak paraconsistency is the view that true contradictions, and worlds that contain them, are merely a formal tool used to study reasoning. Compare strong paraconsistency, the view that there are possible worlds where contradictions are true, or where some statements are both true and false.[293]
A rule in both propositional and predicate logic allowing the addition of propositions to a derivation without affecting its validity, reflecting the idea that if something follows from a set of premises, it also follows from any larger set of premises.
A string of symbols in a formal language that follows the syntactic rules of the language, making it a meaningful or grammatically correct expression within the context of that system.
wff
Short for well-formed formula. Pronounced "woof",[322][323][324][325] or sometimes "wiff",[326][327][328] "weff",[329][330] or "whiff".[331] (All sources supported "woof". The sources cited for "wiff", "weff", and "whiff" gave these pronunciations as alternatives to "woof". Gensler[322] gives "wood" and "woofer" as examples of how to pronounce the vowel in "woof".) Plural "wffs".[322]
A paradox involving an infinite sequence of sentences, each of which states that all following sentences in the sequence are false. Unlike the liar paradox, it does not rely on self-reference, raising questions about the nature of paradoxes and infinity.
A series of paradoxes proposed by the ancient Greek philosopher Zeno of Elea to challenge the coherence of the concepts of plurality, motion, and the continuum, including the famous paradoxes of Achilles and the tortoise, and the dichotomy.
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