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Constructive logic is a family of logics where proofs must be constructive (i.e., proving something means one must build or exhibit it, not just argue
Constructive_logic
Various systems of symbolic logic
logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by
Intuitionistic_logic
Philosphical view that existence proofs must be constructive
viewpoint on mathematics. Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
Method of proof in mathematics
idea is explored in the Brouwer–Heyting–Kolmogorov interpretation of constructive logic, the Curry–Howard correspondence between proofs and programs, and
Constructive_proof
Control flow operator in functional programming
call/cc to Peirce's law, which extends intuitionistic logic to non-constructive, classical logic: ((α → β) → α) → α. Here, ((α → β) → α) is the type of
Call-with-current-continuation
Call-with-current-continuation
Subfield of mathematics
logics and constructive mathematics. The study of constructive mathematics includes many different programs with various definitions of constructive.
Mathematical_logic
Rule of inference of propositional logic
Constructive dilemma 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
Constructive_dilemma
Value indicating the relation of a proposition to truth
valuation. Whereas in classical logic truth values form a Boolean algebra, in intuitionistic logic, and more generally, constructive mathematics, the truth values
Truth_value
Symbolic logic system
law of the excluded middle. In comparison, intuitionistic logic, like most constructive logics, only rejects the law of excluded middle. As such, neither
Minimal_logic
Formal statement in logic
turned to relevance logic to supply a connection between the antecedent and consequent of provable conditionals. In a constructive setting, the symmetry
Strict_conditional
Approach in philosophy of mathematics and logic
constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and
Intuitionism
Mathematical theory of data types
framework of a type theory bears a resemblance to intuitionistic, or constructive, logic. Formally, type theory is often cited as an implementation of the
Type_theory
Logical principles
logic', sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic
Law_of_thought
Method of deriving conclusions
introduction, disjunction elimination, constructive dilemma, destructive dilemma, absorption, and De Morgan's laws. First-order logic also employs the logical operators
Rule_of_inference
Alternative foundation of mathematics
predicative versions. However, all versions keep the core design of constructive logic using dependent types. Martin-Löf designed the type theory on the
Intuitionistic_type_theory
Study of mathematical analysis seen through computability theory
Bishop's constructive analysis. Instead, it is the stronger form of constructive analysis developed by Brouwer that provides a counterpart in constructive logic
Computable_analysis
Propositional logic extending intuitionistic logic
In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. A logic is a set of propositional formulas
Intermediate_logic
Logical connective AND
In logic, mathematics and linguistics, and ( ∧ {\displaystyle \wedge } ) is the truth-functional operator of conjunction or logical conjunction. The logical
Logical_conjunction
at least one of the consequents is true. constructive logic A branch of logic that emphasizes the constructive proof of existence, requiring an explicit
Glossary_of_logic
Quality of an algorithm being correct with respect to a specification
constructive logic corresponds to a certain program in the lambda calculus. Converting a proof in this way is called program extraction. Hoare logic is
Correctness (computer science)
Correctness_(computer_science)
System of resource-aware logic
of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been
Linear_logic
Study of correct reasoning
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical
Logic
Russian mathematician
formula F of classical logic into a formula Fc' of intuitionistic (constructive) logic, such that Fc' is deducible in intuitionistic logic if and only if F
Nikolai_Shanin
Axiomatic set theories based on the principles of mathematical constructivism
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language
Constructive_set_theory
Logical principle
Pattern of reasoning in propositional logic Constructive set theory Diaconescu's theorem – Theorem in mathematical logic Dichotomy – Partition into two separate
Law_of_excluded_middle
constructive metatheory without the axiom of choice."[1] Erik Palmgren, Developments in Constructive Nonstandard Analysis, Bulletin of Symbolic Logic
Constructive nonstandard analysis
Constructive_nonstandard_analysis
Branch of mathematics
algebra/min-plus algebra). Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory. Intuitionistic
Mathematical_analysis
Mathematical analysis
, a constructive counter-part of Z F {\displaystyle {\mathsf {ZF}}} . Of course, a direct axiomatization may be studied as well. The base logic of constructive
Constructive_analysis
Type of logical system
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system used in mathematics, philosophy
First-order_logic
2024-08-28 Troelstra, Anne Sjerp (1977a). "Aspects of Constructive Mathematics". Handbook of Mathematical Logic. 90: 973–1052. doi:10.1016/S0049-237X(08)71127-3
Mathematical_object
the L1, L2, … etc. Thus Turing showed how one can associate logic with any constructive ordinal. Solomon Feferman, Turing in the Land of O(z) in "The
Ordinal_logic
In logic, a statement which is always true
In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms
Tautology_(logic)
Rule of logical inference
In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP), also known as modus ponendo ponens (from Latin 'mode that by affirming affirms'), implication
Modus_ponens
Branch of mathematical logic
is a constructive system, although it does not meet the requirements of the program of constructivism because it is a theory in classical logic including
Reverse_mathematics
Term for sexual and gender minorities
Velasco, Kristopher; Paxton, Pamela (2022). "Deconstructed and Constructive Logics: Explaining Inclusive Language Change in Queer Nonprofits, 1998–2016"
Queer
Constructed language
and mathematician Alexander Ollongren of Leiden University, using constructive logic. Freudenthal's book on Lincos discusses it with many technical words
Lincos_language
Programming style in which control is passed explicitly
variation of double-negation embeddings of classical logic into intuitionistic (constructive) logic. Unlike the regular double-negation translation, which
Continuation-passing_style
Axiom of set theory
excluded middle. The principle is thus not available in constructive set theory, where non-classical logic is employed. The situation is different when the principle
Axiom_of_choice
Framework for studying interactive computational tasks through logic
classical logic a special fragment of CoL. Thus CoL is a conservative extension of classical logic. Computability logic is more expressive, constructive and
Computability_logic
Type of logical formula
human(X) → mortal(X) ). Horn clauses play a basic role in constructive logic and computational logic. They are important in automated theorem proving by first-order
Horn_clause
Approach to logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to
Term_logic
Branch of logic
Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes
Propositional_logic
Form of logic that allows quantification over predicates
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic
Second-order_logic
System including an indeterminate value
three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which
Three-valued_logic
School of thought in philosophy of mathematics
development of mathematical logic has been taking and which Russell himself has been forced to enter upon in the more constructive parts of his work. Major
Logicism
formulae satisfy a classical equivalence not generally satisfied in constructive logic: ¬ ¬ A ↔ A . {\displaystyle \neg \neg A\leftrightarrow A.} But there
Harrop_formula
Approach to formal semantics
Games, logic, and constructive sets. CSLI Publications. ISBN 978-1-57586-449-5. Computability Logic Homepage GALOP: Workshop on Games for Logic and Programming
Game_semantics
View that there are statements that are both true and false
dialetheism on the basis that, in traditional systems of logic (e.g., classical logic and intuitionistic logic), every statement becomes a theorem if a contradiction
Dialetheism
Property of sets used in constructive mathematics
is not valid in constructive or intuitionistic logic, and so this separate terminology is mostly used in the set theory of constructive mathematics. In
Inhabited_set
Kind of proof calculus
University Press. ISBN 978-0-19-875141-0. Gallier, Jean (2005). "Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λ-Calculi". Archived
Natural_deduction
"On weak Markov's principle". Mathematical Logic Quarterly (2002), vol 48, issue S1, pp. 59–65. Constructive Mathematics (Stanford Encyclopedia of Philosophy)
Markov's_principle
Field of linguistics related to extraterrestrial life
designed for use in interstellar communication, is based on modern constructive logic – which assures that all expressions are verifiable. At a deeper,
Astrolinguistics
P. J. Scott. What results is essentially an intuitionistic (i.e. constructive logic) theory, its content being clarified by the existence of a free topos
History_of_topos_theory
Branch of mathematical logic
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects,
Proof_theory
mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories
Disjunction and existence properties
Disjunction_and_existence_properties
Mathematical use of "there exists"
In predicate logic, an existential quantification is a type of quantifier which asserts the existence of an object with a given property. It is usually
Existential_quantification
Axiomatization of arithmetic
theories over intuitionistic logic, various instances of P E M {\displaystyle {\mathrm {PEM} }} can be proven in this constructive arithmetic. By disjunction
Heyting_arithmetic
Form of typed lambda calculus
subsequent papers. In his PhD thesis, Berardi defined a cube of constructive logics akin to the lambda cube (these specifications are non-dependent)
Pure_type_system
Mathematical model for deduction or proof systems
arithmetic. Early logic systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun
Formal_system
Class of formal logics
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had
Classical_logic
Existence of values making formula true
In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula x + 3 = y {\displaystyle
Satisfiability
Reasoning for mathematical statements
frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols, along with natural language that usually
Mathematical_proof
Assignment of meaning to the symbols of a formal language
formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard
Interpretation_(logic)
In mathematics, a statement that has been proven
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses
Theorem
Branch of mathematics that studies sets
set. Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic. Yet other systems
Set_theory
Argument whose conclusion must be true if its premises are
In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true
Validity_(logic)
Proof in set theory
diagonalization in a constructive context" (PDF), in Link, Godehard (ed.), One hundred years of Russell's paradox, De Gruyter Series in Logic and its Applications
Cantor's_diagonal_argument
3-volume treatise on mathematics, 1910–1913
logic and to minimise the number of primitive notions, axioms, and inference rules; to precisely express mathematical propositions in symbolic logic using
Principia_Mathematica
Symbol representing a property or relation in logic
In logic, a predicate is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all.
Predicate_(logic)
Basic framework of mathematics
intuitionistic theory of types, Twenty-five years of constructive type theory (Venice,1995). Oxford Logic Guides. Vol. 36. New York: Oxford University Press
Foundations_of_mathematics
Theorem in mathematical logic
regarding this assertion). Similar constructive proofs may be provided for the basic modal logic K, intuitionistic logic and μ-calculus, with similar complexity
Craig_interpolation
Mathematical-logic system based on functions
In mathematical logic, the lambda calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and
Lambda_calculus
German philosopher (1889–1964)
and which Becker attributes to Theaetetus. Becker also showed how a constructive logic that denied unrestricted excluded middle could be used to reconstruct
Oskar_Becker
Study of the scope and nature of logic
Philosophy of logic is the branch of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as
Philosophy_of_logic
Non-contradiction of a theory
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T {\displaystyle T} is consistent if there is no
Consistency
German mathematician and philosopher (1915–1994)
protophysics of time and space. He developed constructive logic, constructive type theory and constructive analysis. Lorenzen's work on calculus Differential
Paul_Lorenzen
Term in logic and deductive reasoning
In logic, soundness can refer to either a property of arguments or a property of formal deductive systems. An argument is sound if (and only if) it is
Soundness
Topics referred to by the same term
that human knowledge is active and constructive Constructionism (disambiguation) Constructive theology Constructive empiricism Deconstructivism, a movement
Constructivism
Whether a decision problem has an effective method to derive the answer
effectively determined. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. A theory (set of sentences
Decidability_(logic)
Area of mathematical logic
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing
Model_theory
Diagram that shows all possible logical relations between a collection of sets
set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses simple
Venn_diagram
Algebraic manipulation of "true" and "false"
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
Boolean_algebra
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Pair of logical equivalences
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
De_Morgan's_laws
Mathematical logic concept
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent
Contraposition
characterization of orders like this are thus weaker (when working using just constructive logic) than alternative axioms of a strict total order, which are often
Pseudo-order
Phenomenon resulting from the superposition of two waves
their phase difference. The resultant wave may have greater amplitude (constructive interference) or lower amplitude (destructive interference) if the two
Wave_interference
Standard system of axiomatic set theory
constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which one is constructing
Zermelo–Fraenkel_set_theory
Statement that is taken to be true
well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics,
Axiom
Logical connective OR
In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated
Logical_disjunction
Overview of and topical guide to logic
Classical logic Computability logic Deontic logic Dependence logic Description logic Deviant logic Doxastic logic Epistemic logic First-order logic Formal
Outline_of_logic
Basic notion of sameness in mathematics
Sambin, Giovanni; Smith, Jan M. (eds.). Twenty Five Years of Constructive Type Theory. Oxford Logic Guides. Vol. 36. Clarendon. pp. 83–111. ISBN 978-0-19-158903-4
Equality_(mathematics)
Components of a mathematical or logical formula
In mathematical logic, a term is an arrangement of dependent/bound symbols that denotes a mathematical object within an expression/formula. In particular
Term_(logic)
Reasoning about equations with free variables
logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses
Algebraic_logic
Concept in logic
In logic and mathematics, statements p {\displaystyle p} and q {\displaystyle q} are said to be logically equivalent if they have the same truth value
Logical_equivalence
Characteristic of some logical systems
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can
Completeness_(logic)
University, 1992. Vickers, S. J., "Topology via Constructive Logic", in Moss and Ginzburg and de Rijke, Logic, Language and Computation Vol II, Proceedings
Steve Vickers (computer scientist)
Steve_Vickers_(computer_scientist)
Sequence of words formed by specific rules
In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet".
Formal_language
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India
History_of_logic
Mathematical set containing no elements
variable", The Journal of Philosophy 91: 430–49. Reprinted in 1998, Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard University Press
Empty_set
CONSTRUCTIVE LOGIC
CONSTRUCTIVE LOGIC
Girl/Female
Tamil
Creation, Construction, Arrangement
Girl/Female
Tamil
Trick, Power, Strategy, Solution by logic, By reasoning
Boy/Male
Indian
Intelligent, Logical
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Creation; Evolution; Construction
Girl/Female
Tamil
Viviktha | விவீகà¯à®¤à®¾Â
Distinguished, Pure, Deep, Logically intelligent
Viviktha | விவீகà¯à®¤à®¾Â
Girl/Female
Hindu, Indian, Marathi
Produce; New Construction
Girl/Female
Indian
Built; Construction; Creative Art; All Creation
Girl/Female
Tamil
Vivikta | விவிகதா
Distinguished, Pure, Deep, Logically intelligent
Vivikta | விவிகதா
Girl/Female
Hindu
Light, Beauty, Prosperity, Rank, Power, Steel construction company
Girl/Female
Tamil
Light, Beauty, Prosperity, Rank, Power, Steel construction company
Girl/Female
Hindu
Trick, Power, Strategy, Solution by logic, By reasoning
Boy/Male
Hindu
Love and kindness, Analytical, Logical
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Sindhi, Tamil, Telugu
Construction; Arrangement; Creative Art; All Creation
Girl/Female
Hindu
Distinguished, Pure, Deep, Logically intelligent
Girl/Female
Hindu
Creation, Construction, Arrangement
Boy/Male
Arabic, Muslim
A Persian Construction Probably from the Arabic Mawla (Master; Leader; Lord)
Girl/Female
Hindu
Creation, Construction, Arrangement
Girl/Female
Tamil
Trick, Power, Strategy, Solution by logic, By reasoning
Boy/Male
Tamil
Full of feathers, Full of logic, Name of sage, Vatsyayan
Girl/Female
Tamil
Creation, Construction, Arrangement
CONSTRUCTIVE LOGIC
CONSTRUCTIVE LOGIC
Boy/Male
Hindu
Cheerful, Pleased, Happy
Boy/Male
Hindu, Indian, Marathi, Sanskrit
Lord Krishna
Girl/Female
Hindu, Indian
Monsoon
Girl/Female
Tamil
Ujjwala | உஜà¯à®œà¯à®µà®²à®¾ /उजà¥à¤µà¤²à¤¾
Bright, Lighted
Boy/Male
Arabic, Muslim
Servant of the Allprovider (Allah); Ibn Hammam was One of those Prominent People with this Name
Girl/Female
Arabic, Australian, Indian, Modern, Tamil
Mother; Mom; Mummy
Boy/Male
Hindu
Victorious
Girl/Female
Indian
Humming of bee
Girl/Female
Australian, French, German, Latin, Spanish
Defender of the Land; Scented; Protector
Boy/Male
Arabic
Helper; Assistant
CONSTRUCTIVE LOGIC
CONSTRUCTIVE LOGIC
CONSTRUCTIVE LOGIC
CONSTRUCTIVE LOGIC
CONSTRUCTIVE LOGIC
a.
Having ability to construct or form; employed in construction; as, to exhibit constructive power.
n.
That which is constructed or formed; an edifice; a fabric.
a.
Derived from, or depending on, construction or interpretation; not directly expressed, but inferred.
a.
Building up; constructive; -- opposed to destructive.
adv.
In a constructive manner; by construction or inference.
n.
An obstructive person or thing.
n.
The act of constructing vaults; a vaulted construction.
n.
Instructive discourse.
a.
Serving or tending to bind or constrict.
n.
The act of fabricating, framing, or constructing; construction; manufacture; as, the fabrication of a bridge, a church, or a government.
a.
Constructive.
n.
The method of construing, interpreting, or explaining a declaration or fact; an attributed sense or meaning; understanding; explanation; interpretation; sense.
a.
Pertaining to a master builder, or to architecture; evincing skill in designing or construction; constructive.
a.
Obstructive.
n.
The act of constructing; construction.
a.
According to interpretation; constructive.
a.
Conveying knowledge; serving to instruct or inform; as, experience furnishes very instructive lessons.
a.
Building; constructing.
a.
Reconstructing; tending to reconstruct; as, a reconstructive policy.
n.
The process or art of constructing; the act of building; erection; the act of devising and forming; fabrication; composition.