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Metatheorem in mathematical logic
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly
Deduction_theorem
Fundamental theorem in mathematical logic
the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense
Gödel's_completeness_theorem
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Theorem in formal logic
in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that
Cut-elimination_theorem
Axiom used in logic and philosophy
intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type
Peirce's_law
Form of reasoning
in an ill-formed syllogism, in order to make the form valid. see Deduction theorem Johnson-Laird, Phil (30 December 2009). "Deductive reasoning". WIREs
Deductive_reasoning
Type of formal logic
R))\to ((P\to Q)\to (P\to R))} ** for deduction theorem (note: {t,b}→{f} = {f} follows from the deduction theorem) ¬ ( P → Q ) → P {\displaystyle \lnot
Paraconsistent_logic
Impossible task in computing
implies a negative answer to the Entscheidungsproblem. Using the deduction theorem, the Entscheidungsproblem encompasses the more general problem of
Entscheidungsproblem
Relationship between programs and proofs
can be restated as shown in the following table. Especially, the deduction theorem specific to Hilbert-style logic matches the process of abstraction
Curry–Howard_correspondence
Rule of inference in predicate logic
Proof: In this proof, universal generalization was used in step 8. The deduction theorem was applicable in steps 10 and 11 because the formulas being moved
Universal_generalization
(proof theory) Deduction theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory)
List_of_theorems
Kind of proof calculus
for the consistency result, the cut elimination theorem—the Hauptsatz—directly for natural deduction. For this reason he introduced his alternative system
Natural_deduction
System of formal deduction in logic
these axioms, it is possible to form conservative extensions of the deduction theorem that permit the use of additional connectives. These extensions are
Hilbert_system
Statement in a metalanguage
that the same basic thought (e.g. deduction theorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly
Judgment_(mathematical_logic)
Version of classical propositional calculus that uses only one connective
completeness theorem is outlined below. First, using the compactness theorem and the deduction theorem, we may reduce the completeness theorem to its special
Implicational propositional calculus
Implicational_propositional_calculus
Style of formal logical argumentation
tautology (or theorem). Gentzen style. Every line is a conditional tautology (or theorem) with zero or more conditions on the left. Natural deduction. Every
Sequent_calculus
Set of sentences in a formal language
language with deduction rules. An element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the
Theory_(mathematical_logic)
Algebraic structure used in logic
identities in Heyting algebras. In practice, one frequently uses the deduction theorem in such proofs. Since for any a and b in a Heyting algebra H we have
Heyting_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
Branch of logic
way to decompose the resources used by components of a system. The deduction theorem of classical logic relates conjunction and implication: A ∧ B ⊢ C
Bunched_logic
Symbolic logic system
showing which theorems still do hold in minimal logic, often making implicit use of the valid currying rule and the deduction theorem. By implication
Minimal_logic
Logical connective
conditional and the logical consequence relation is given by the deduction theorem. Γ ∪ { A } ⊢ B {\displaystyle \Gamma \cup \{A\}\vdash B\;} if and
Material_conditional
category theory and related mathematics Deduction Theorem McLarty, Colin (1992). "§17.3 The fundamental theorem". Elementary Categories, Elementary Toposes
Fundamental theorem of topos theory
Fundamental_theorem_of_topos_theory
In mathematics, a statement that has been proven
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
System of formal mathematical logic
to have the same value in both after the replacement is done. The Deduction Theorem for Q0 shows that proofs from hypotheses using Rule R′ can be converted
Q0_(mathematical_logic)
Aspect of mathematical logic
Blok and Pigozzi exploring the different forms that the well-known deduction theorem of classical propositional calculus and first-order logic takes on
Abstract_algebraic_logic
Type of logical system
possible to effectively verify that a purportedly valid deduction is actually a deduction; such deduction systems are called effective. A key property of deductive
First-order_logic
Interactive theorem prover software
Catalogues Digital Math by Category: Tactic Provers Automated Deduction Systems and Groups Theorem Proving and Automated Reasoning Systems Database of Existing
Proof_assistant
Method of deriving conclusions
times. Various formalisms are used to express logical systems. Natural deduction systems employ many intuitive rules of inference to reflect how people
Rule_of_inference
Subfield of computer science and logic
deduction. John Pollock's OSCAR system is an example of an automated argumentation system that is more specific than being just an automated theorem prover
Automated_reasoning
1895 allegorical dialogue by Lewis Carroll
the Tortoise Said to Achilles public domain audiobook at LibriVox Deduction theorem Homunculus argument Münchhausen trilemma Paradox Regress argument
What the Tortoise Said to Achilles
What_the_Tortoise_Said_to_Achilles
Type of formal logic
JSTOR 2269159. S2CID 250349611. Ruth C. Barcan (December 1946). "The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication"
Modal_logic
Logic statement about a formal system proven in a metalanguage
be proved.[citation needed] Examples of metatheorems include: The deduction theorem for first-order logic says that a sentence of the form φ→ψ is provable
Metatheorem
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Logical proof involving antecedents and consequents
sequent assertions did not signify provability. "Employment of the deduction theorem as primitive or derived rule must not, however, be confused with the
Sequent
Concept in mathematics
corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically
Corollary
Formal proof
to prove A → C (if A, then C) from the first two premises below: Deduction theorem Logical consequence Propositional calculus Robert L. Causey, Logic
Conditional_proof
Graphical aid for deriving some concepts in combinatorics
dots and dividers) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve a variety of counting problems, such as how many
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Theorem in mathematical logic
compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important
Compactness_theorem
itself. deduction The process of reasoning from one or more statements (premises) to reach a logically certain conclusion. deduction theorem A theorem stating
Glossary_of_logic
Theory of logic to account for observations from quantum theory
Likewise, quantum logic with the orthomodular law falsifies the deduction theorem. Quantum logic admits no reasonable material conditional; any connective
Quantum_logic
{\displaystyle {\underline {\lnot \varphi }}} ψ {\displaystyle \psi } Deduction theorem (or Conditional Introduction) φ ⊢ ψ _ {\displaystyle {\underline {\varphi
List_of_rules_of_inference
Logical formalism using combinators instead of variables
A\to B} , then X , A ⊬ B {\displaystyle X,A\not \vdash B} by the deduction theorem, thus the deductive closure of X ∪ { A } {\displaystyle X\cup \{A\}}
Combinatory_logic
Tolerant sequence Cotolerant sequence Deduction theorem Cirquent calculus Nonconstructive proof Existence theorem Intuitionistic logic Intuitionistic type
List of mathematical logic topics
List_of_mathematical_logic_topics
Proof assistant and programming language
Sebastian (2021). "The Lean 4 Theorem Prover and Programming Language". In Platzer, André; Sutcliffe, Geoff (eds.). Automated Deduction – CADE 28. Lecture Notes
Lean_(proof_assistant)
Branch of mathematical logic
proof-theoretic semantics, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications
Proof_theory
Theorem in set theory
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there
Schröder–Bernstein_theorem
Every set is smaller than its power set
question marks, boxes, or other symbols. In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle
Cantor's_theorem
Problem in computer science
Minsky notes: ...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness [of] the state diagram
Halting_problem
Statement that is taken to be true
cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident
Axiom
associated normalization theorem establishes that every derivation in natural deduction can be transformed into normal form. Natural deduction is a system of formal
Normal form (natural deduction)
Normal_form_(natural_deduction)
Summary of a mathematical proof
gives a sketch of a proof of the first of Gödel's incompleteness theorems. This theorem applies to any formal theory that satisfies certain technical hypotheses
Proof sketch for Gödel's first incompleteness theorem
Proof_sketch_for_Gödel's_first_incompleteness_theorem
SNARK, (SRI's New Automated Reasoning Kit), is a theorem prover for multi-sorted first-order logic intended for applications in artificial intelligence
SNARK_(theorem_prover)
Subfield of mathematics
finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it
Mathematical_logic
Theorem that arithmetical truth cannot be defined in arithmetic
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Establishment of a theorem using inference from the axioms
Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof. The theorem is a syntactic consequence of all the well-formed
Formal_proof
Theorem for proving more complex theorems
also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however
Lemma_(mathematics)
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
Mathematical proposition equivalent to the axiom of choice
the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space
Zorn's_lemma
Reasoning for mathematical statements
The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic
Mathematical_proof
Term in logic and deductive reasoning
validity (or the weaker property, truth). If the system allows Hilbert-style deduction, it requires only verifying the validity of the axioms and one rule of
Soundness
Mathematical proof expressed visually
either assumed, or follows from the preceding statements by a rule of deduction, which is itself assumed. Benson, Steve; Addington, Susan; Arshavsky,
Proof_without_words
SPASS is an automated theorem prover for first-order logic with equality developed at the Max Planck Institute for Computer Science and using the superposition
SPASS
On linear-time algorithms for graph logic
In the study of graph algorithms, Courcelle's theorem is the statement that every graph property definable in the monadic second-order logic of graphs
Courcelle's_theorem
American mathematician (1937–2025)
used to help students learn logic by interactively constructing natural deduction proofs. Source code of TPS is available on the Internet Archive. A list
Peter_B._Andrews
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
Counting polynomial roots in an interval
derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval
Sturm's_theorem
Basic framework of mathematics
generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical
Foundations_of_mathematics
Branch of mathematics that studies sets
uncountable, that is, one cannot put all real numbers in a list. This theorem is proved using Cantor's first uncountability proof, which differs from
Set_theory
Area of mathematical logic
It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems) that a theory has a model if and only if it
Model_theory
Epistemological view centered on reason
the intuition and deduction. Some go further to include ethical truths into the category of things knowable by intuition and deduction. Furthermore, some
Rationalism
Notation system for natural deductive logic
natural deduction proofs as sequences of justified steps. Both methods use inference rules derived from Gentzen's 1934/1935 natural deduction system,
Suppes–Lemmon_notation
Mathematical construction
include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization
Ultraproduct
Higher-order logic (HOL) automated theorem prover
The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As a Logic for Computable Functions
Isabelle_(proof_assistant)
Formal language and associated computer program
archiving and verifying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic,
Metamath
Logical problem studied in computer science
range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing.
Satisfiability modulo theories
Satisfiability_modulo_theories
Undecidability of equality of real numbers
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ln 2
Richardson's_theorem
Mathematical model for deduction or proof systems
formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the
Formal_system
Mathematical proof at least partially generated by computer
of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
Computer-assisted_proof
Steps in reasoning
premises to logical consequences. Inference is traditionally divided into deduction and induction, a distinction that dates at least to Aristotle (300s BC)
Inference
Measure of algorithmic complexity
impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program P computing a
Kolmogorov_complexity
Ability of a computing system to simulate Turing machines
precise logical rules of deduction that could be performed by a machine. Soon it became clear that a small set of deduction rules are enough to produce
Turing_completeness
Automated theorem proofer
an automated theorem prover for first-order and equational logic developed by William McCune. Prover9 is the successor of the Otter theorem prover also
Prover9
Yes-or-no question that cannot ever be solved by a computer
are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This
Undecidable_problem
theorem Goodstein's theorem Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's
List_of_mathematical_proofs
Branch of logic
self-evidently true, and theorems are proved by applying deduction rules to these axioms. See § Syntactic proof via axioms. Natural deduction is a syntactic method
Propositional_logic
Form of logic that allows quantification over predicates
effective deduction system for standard semantics could be used to produce a recursively enumerable completion of Peano arithmetic, which Gödel's theorem shows
Second-order_logic
1956 computer program written by Allen Newell, Herbert A. Simon and Cliff Shaw
artificial intelligence program". Logic Theorist proved 38 of the first 52 theorems in chapter two of Whitehead and Bertrand Russell's Principia Mathematica
Logic_Theorist
Non-contradiction of a theory
incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies
Consistency
Mathematical term; concerning axioms used to derive theorems
for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical
Axiomatic_system
Mathematical logic concept
arithmetic and that its consistency is therefore less controversial. Gentzen's theorem is concerned with first-order arithmetic: the theory of the natural numbers
Gentzen's_consistency_proof
Proof that is not easily verified by hand
substituting experiment for deduction: …if we accept the [Four-Color Theorem] as a theorem, we are committed to changing the sense of "theorem", or, more to the
Non-surveyable_proof
American philosopher
Strict Implication", Journal of Symbolic Logic (JSL, 1946), "The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication"
Ruth_Barcan_Marcus
Kind of non-classical logic
113–128, doi:10.2307/2268750, JSTOR 2268750 Moh, Shaw-kwei (1950), "The Deduction Theorems and Two New Logical Systems", Methodos, 2: 56–75 Moh Shaw-Kwei, 1950
Relevance_logic
In logic, a statement which is always true
Wittgenstein proposed that statements that can be deduced by logical deduction are tautological (empty of meaning), as well as being analytic truths
Tautology_(logic)
been proposed. A proof is a deduction whose premises are known truths. A proof of the Pythagorean theorem is a deduction that might use several premises
Argument–deduction–proof distinctions
Argument–deduction–proof_distinctions
Inference seeking the simplest and most likely explanation
operator for the subjective Bayes' theorem is denoted " ϕ ~ {\displaystyle {\widetilde {\phi \,}}} ", and subjective deduction is denoted " ⊚ {\displaystyle
Abductive_reasoning
French mathematician, astronomer, and geophysicist (1713–1765)
to confirm Newton's deduction of the figure of the Earth. In that context, Clairaut deduced what is now known as Clairaut's theorem. He also tackled the
Alexis_Clairaut
Principle relating to fluid dynamics
universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa. In the above
Bernoulli's_principle
French mathematician (1908–1931)
introduced recursive functions. Herbrand's theorem refers to either of two completely different theorems. One is a result from his doctoral thesis in
Jacques_Herbrand
DEDUCTION THEOREM
DEDUCTION THEOREM
Boy/Male
Indian
Education
Girl/Female
Tamil
Education
Girl/Female
Tamil
Samarpana | ஸமரà¯à®ªà®£
Dedication
Samarpana | ஸமரà¯à®ªà®£
Girl/Female
Hindu, Indian
Dedication
Girl/Female
Hindu, Indian, Tamil
Education
Girl/Female
Tamil
Education
Boy/Male
Indian
Dedication, Offer
Girl/Female
Tamil
Pranidhaana | பà¯à®°à®¨à¯€à®¤à®¾à®¨à®¾
Dedication
Pranidhaana | பà¯à®°à®¨à¯€à®¤à®¾à®¨à®¾
Boy/Male
Arabic, Muslim
Education
Girl/Female
Indian, Marathi
Education
Girl/Female
Tamil
Modesty, Education
Boy/Male
Muslim
Dedication, Offer
Boy/Male
Tamil
Education
Girl/Female
Indian, Telugu
Good Education
Boy/Male
Arabic, Muslim
Education; Instruction
Girl/Female
Hindu
Modesty, Education
Girl/Female
Hindu
Education
Girl/Female
Hindu
Dedication
Girl/Female
Indian
Education
Girl/Female
Hindu
Dedication
DEDUCTION THEOREM
DEDUCTION THEOREM
Girl/Female
Hindu, Indian, Sanskrit
The Earth Abundant; Free from Dust; Clean; Pure
Female
Hebrew
(חֵרוּתָה) Variant form of Hebrew Cherut, CHERUTA means "freedom."
Boy/Male
Sikh
Heart of Guru
Boy/Male
Greek English
Place name in Greece.
Boy/Male
Muslim/Islamic
Rivulet stream
Boy/Male
Tamil
Male
English
English surname transferred to forename use, from the Norman French baronial name d'Airelle, DARRELL means "from Airelle."
Boy/Male
Tamil
Divider
Girl/Female
Arabic
Star
Boy/Male
Sikh
Renowned Love, Love of one who will go
DEDUCTION THEOREM
DEDUCTION THEOREM
DEDUCTION THEOREM
DEDUCTION THEOREM
DEDUCTION THEOREM
n.
The act or process of inferring by deduction or induction.
adv.
By deduction.
n.
The action by which the parts of the body are drawn towards its axis]; -- opposed to abduction.
n.
A process of demonstration in which a general truth is gathered from an examination of particular cases, one of which is known to be true, the examination being so conducted that each case is made to depend on the preceding one; -- called also successive induction.
n.
Inference; deduction; thing deduced.
n.
The act or process of educating; the result of educating, as determined by the knowledge skill, or discipline of character, acquired; also, the act or process of training by a prescribed or customary course of study or discipline; as, an education for the bar or the pulpit; he has finished his education.
n.
A devoting or setting aside for any particular purpose; as, a dedication of lands to public use.
n.
That which seduces, or is adapted to seduce; means of leading astray; as, the seductions of wealth.
n.
The act of reducing, or state of being reduced; conversion to a given state or condition; diminution; conquest; as, the reduction of a body to powder; the reduction of things to order; the reduction of the expenses of government; the reduction of a rebellious province.
n.
Act of deducting or taking away; subtraction; as, the deduction of the subtrahend from the minuend.
a.
Of or pertaining to deduction; capable of being deduced from premises; deducible.
n.
The act of setting apart or consecrating to a divine Being, or to a sacred use, often with religious solemnities; solemn appropriation; as, the dedication of Solomon's temple.
n.
Reduction.
n.
That which is deducted; the part taken away; abatement; as, a deduction from the yearly rent.
n.
A logical deduction.
n.
The amount abated; that which is taken away by way of reduction; deduction; decrease; a rebate or discount allowed.
n.
The wrongful, and usually the forcible, carrying off of a human being; as, the abduction of a child, the abduction of an heiress.
n.
Subtraction; deduction.
v. t.
The act, process, or result of reducing; as, the reduction of iron from its ores; the reduction of aldehyde from alcohol.
n.
The act of detecting; the laying open what was concealed or hidden; discovery; as, the detection of a thief; the detection of fraud, forgery, or a plot.