Deep inferenceDeep inference names a general idea in structural proof theory that breaks with the classical sequent calculus by generalising the notion of structure to permit inference to occur in contexts of high structural complexity. The term deep inference is generally reserved for proof calculi where the structural complexity is unbounded; in this article we will use non-shallow inference to refer to calculi that have structural complexity greater than the sequent calculus, but not unboundedly so, although this is not at present established terminology.
SequentIn mathematical logic, a sequent is a very general kind of conditional assertion. A sequent may have any number m of condition formulas Ai (called "antecedents") and any number n of asserted formulas Bj (called "succedents" or "consequents"). A sequent is understood to mean that if all of the antecedent conditions are true, then at least one of the consequent formulas is true. This style of conditional assertion is almost always associated with the conceptual framework of sequent calculus.
LogicLogic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.
Structural proof theoryIn mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as consistency, provide decision procedures, and allow mathematical or computational witnesses to be extracted as counterparts to theorems, the kind of task that is more often given to model theory.
Analytic proofIn mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided a proof of the theorem that was free from intuitions concerning lines crossing each other at a point, and so he felt happy calling it analytic (Bolzano 1817).
Method of analytic tableauxIn proof theory, the semantic tableau (tæˈbloʊ,_ˈtæbloʊ; plural: tableaux, also called truth tree) is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed for a logical formula, having at each node a subformula of the original formula to be proved or refuted. Computation constructs this tree and uses it to prove or refute the whole formula. The tableau method can also determine the satisfiability of finite sets of formulas of various logics.
Sequent calculusIn mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than to David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology.
Natural deductionIn logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning. Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system).
Hilbert systemIn mathematical physics, Hilbert system is an infrequently used term for a physical system described by a C*-algebra. In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.
Formal proofIn logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system.
