In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly...

in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that...

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...

Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving...

in an ill-formed syllogism, in order to make the form valid. see Deduction theorem Johnson-Laird, Phil (30 December 2009). "Deductive reasoning". WIREs...

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...

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...

implies a negative answer to the Entscheidungsproblem. Using the deduction theorem, the Entscheidungsproblem encompasses the more general problem of...

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...

for the consistency result, the cut elimination theorem—the Hauptsatz—directly for natural deduction. For this reason he introduced his alternative system...

can be restated as shown in the following table. Especially, the deduction theorem specific to Hilbert-style logic matches the process of abstraction...

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...

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...

associated normalization theorem establishes that every derivation in natural deduction can be transformed into normal form. Natural deduction is a system of formal...

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...

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories...

way to decompose the resources used by components of a system. The deduction theorem of classical logic relates conjunction and implication: A ∧ B ⊢ C...

tautology (or theorem). Gentzen style. Every line is a conditional tautology (or theorem) with zero or more conditions on the left. Natural deduction. Every...

completeness theorem is outlined below. First, using the compactness theorem and the deduction theorem, we may reduce the completeness theorem to its special...

theorem (proof theory) Deduction theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem...

these axioms, it is possible to form conservative extensions of the deduction theorem that permit the use of additional connectives. These extensions are...

inference with which every theorem can be deduced. The Fitch notation is an influential way of presenting proofs in natural deduction systems. Hurley 2016,...

Blok and Pigozzi exploring the different forms that the well-known deduction theorem of classical propositional calculus and first-order logic takes on...

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...

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...

In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf...

to prove A → C (if A, then C) from the first two premises below: Deduction theorem Logical consequence Propositional calculus Robert L. Causey, Logic...

showing which theorems still do hold in minimal logic, often making implicit use of the valid currying rule and the deduction theorem. By implication...

possible to effectively verify that a purportedly valid deduction is actually a deduction; such deduction systems are called effective. A key property of deductive...

"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...