intuitionistic

The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus that makes use of the cut rule also possesses a cut-free proof ...

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Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventionally the story of modern proof theory is seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's problem of reducing all mathematics to a finitist formal system, then his incompleteness theorems showed that was unattainable. All of this work was carried out with the pr ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Suppose P is any predicate in two variables that doesn't use the symbol B. Then in the formal language of the Zermelo-Fraenkel axioms, the axiom schema reads: or in words: If, given any set X, there is a unique set Y such that P holds for X and Y, then, given any set A, there is a set B such that, given any set C, C is a member of B if and only if there is a set D such that D is a member of A ...

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Axiom schema of replacement, Axiom schema of replacement - Statement, Axiom schema of replacement - Example applications, Axiom schema of replacement - History and philosophy, Axiom schema of replacement - Relation to the axiom schema of specification

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The presentation of natural deduction so far has concentrated on the nature of propositions without giving a formal definition of a proof. To formalise the notion of proof, we alter the presentation of hypothetical derivations slightly. We label the antecedents with proof variables (from some countable set V of variables), and decorate the succedent with the actual proof. The antecedents or hypotheses are separated from the succedent by means of a turnstile (⊢). This modification sometimes goes under the name of localised ...

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Natural deduction, Natural deduction - Judgements and propositions, Natural deduction - Introduction and elimination, Natural deduction - Hypothetical derivations, Natural deduction - Consistency completeness and normal forms, Natural deduction - First and higher order extensions, Natural deduction - Proofs and type-theory, Natural deduction - Classical and modal logics, Natural deduction - Comparison with other foundational approaches, Natural deduction - Sequent calculus

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The logical connectives are re-examined in this resource-interpretation; each connective splits into multiplicative and additive versions, which correspond to simultaneous and alternative presence, respectively. To motivate the connectives, let us use the example of a vending machine. Multiplicative conjunction, also called tensor (written ⊗), denotes simultaneous occurrence of resources. For example, if I insert 50 cents into the vending machine, then the vending machine simultaneouslySee also:

Linear logic, Linear logic - Linear connectives, Linear logic - Flavours of linear logic

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Main article: Ordinal analysis Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Main article: Tableau systems Tableau systems apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Linear logic has many restrictions and variants. The primary axis of variation is along the classical/intuitionistic divide. Classical linear logic (CLL) is the original linear logic as proposed by Girard. In CLL every connective has a dual. The following is a two-sided presentation of CLL as a sequent calculus: Linear implication is definable in terms of linear negation and multiplicative disjunction in CLL: A ⊸ B ≡ ¬ A ⅋ B. This is familiar from other classical logics: for example, th ...

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Linear logic, Linear logic - Linear connectives, Linear logic - Flavours of linear logic

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The ordinal number ω2 = ω + ω (using the modern definition due to von Neumann) is the first ordinal that cannot be constructed without replacement. The axiom of infinity asserts the existence of the infinite sequence ω = {0, 1 ,2 ,...}, and only this sequence. One would like to define ω2 to be the sequence {ω, ω + 1, ω + 2,...}, but in general classes of ordinals need not be sets (the class of all ordinals is not a set, for example). Replacement allows one to replace each finite number n in ω with the corresponding ω + n< ...

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Axiom schema of replacement, Axiom schema of replacement - Statement, Axiom schema of replacement - Example applications, Axiom schema of replacement - History and philosophy, Axiom schema of replacement - Relation to the axiom schema of specification

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Most of the applications to which replacement might naïvely be put in fact do not require it. For example, suppose that f is a function from a set S to a set T. Then we may construct a functional predicate F such that F(x) = f(x) whenever x is a member of S, letting F(x) be anything we like otherwise (it won't matter for this application). Then given a subset A of S, applying the axiom schema of replacement to F constructs the image f ...

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Axiom schema of replacement, Axiom schema of replacement - Statement, Axiom schema of replacement - Example applications, Axiom schema of replacement - History and philosophy, Axiom schema of replacement - Relation to the axiom schema of specification

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Main article: Structural proof theory Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz; the definition is slightly more complex, we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Gira ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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The three most well known proof calculi are: The Hilbert-style calculi The natural deduction calculus The sequent calculus To say these are proof calculi, rather than proof systems, is to say they are flexible frameworks for the study of many kinds of logical consequence relations. Each of these can formalise propositional or predicate logics of either the classical or intuitionistic flavour, or almost any modal logic studied, many substructural logics, such as relevance logic or lin ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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A theory is said to be consistent if falsehood is not provable (from no assumptions) and is complete if every theorem is provable using the inference rules of the logic. These are statements about the entire logic, and are usually tied to some notion of a model. However, there are local notions of consistency and completeness that are purely syntactic checks on the inference rules, and require no appeals to models. The first of these is local consistency, also known as local reducibility, which says that any derivation containing an introduc ...

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Natural deduction, Natural deduction - Judgements and propositions, Natural deduction - Introduction and elimination, Natural deduction - Hypothetical derivations, Natural deduction - Consistency completeness and normal forms, Natural deduction - First and higher order extensions, Natural deduction - Proofs and type-theory, Natural deduction - Classical and modal logics, Natural deduction - Comparison with other foundational approaches, Natural deduction - Sequent calculus

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A pervasive operation in mathematical logic is reasoning from assumptions. For example, consider the following derivation: A ∧ (B ∧ C) true ----------------- ∧E2 B ∧ C true ----------- ∧E1 B true This derivation does not established the truth of B as such; rather, it establishes the following fact: If A ∧ (B ∧ C) is true then B is true. In logic, one says "assuming A ∧ (B ∧ C) is true, we show that B is trueSee also:

Natural deduction, Natural deduction - Judgements and propositions, Natural deduction - Introduction and elimination, Natural deduction - Hypothetical derivations, Natural deduction - Consistency completeness and normal forms, Natural deduction - First and higher order extensions, Natural deduction - Proofs and type-theory, Natural deduction - Classical and modal logics, Natural deduction - Comparison with other foundational approaches, Natural deduction - Sequent calculus

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Now we discuss the "A true" judgement. Inference rules that introduce a logical connective in the conclusion are known as introduction rules. To introduce conjunctions, i.e., to conclude "A and B true" for propositions A and B, one requires evidence for "A true" and "B true". As an inference rule: It must be understood that in such rules the objects are propositions. That is, the above rule is really an ...

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Natural deduction, Natural deduction - Judgements and propositions, Natural deduction - Introduction and elimination, Natural deduction - Hypothetical derivations, Natural deduction - Consistency completeness and normal forms, Natural deduction - First and higher order extensions, Natural deduction - Proofs and type-theory, Natural deduction - Classical and modal logics, Natural deduction - Comparison with other foundational approaches, Natural deduction - Sequent calculus

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The logic of the earlier section is an example of a single-sorted logic, i.e., a logic with a single kind of object: propositions. Many extensions of this simple framework have been proposed; in this section we will extend it with a second sort of individuals or terms. More precisely, we will add a new kind of judgement, "t is a term" (or "t term") where t is schematic. We shall fix a countable set V of variables, another countable set F of function symbols, an ...

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Natural deduction, Natural deduction - Judgements and propositions, Natural deduction - Introduction and elimination, Natural deduction - Hypothetical derivations, Natural deduction - Consistency completeness and normal forms, Natural deduction - First and higher order extensions, Natural deduction - Proofs and type-theory, Natural deduction - Classical and modal logics, Natural deduction - Comparison with other foundational approaches, Natural deduction - Sequent calculus

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For simplicity, the logics presented so far have been intuitionistic. Classical logic extends intuitionistic logic with an additional axiom or principle of excluded middle: For any proposition p, the proposition p ∨ ¬ p is true. This statement is not obviously either an introduction or an elimination; indeed, it involves two distinct connectives. Gentzen's original treatment of excluded middle prescribed one of the following three (equivalent) formulations, which were already present in analogous forms in the systems of Hilbert and Heyt ...

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Natural deduction, Natural deduction - Judgements and propositions, Natural deduction - Introduction and elimination, Natural deduction - Hypothetical derivations, Natural deduction - Consistency completeness and normal forms, Natural deduction - First and higher order extensions, Natural deduction - Proofs and type-theory, Natural deduction - Classical and modal logics, Natural deduction - Comparison with other foundational approaches, Natural deduction - Sequent calculus

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A judgement is something that is knowable, that is, an object of knowledge. It is evident if one in fact knows it. Thus "it is raining" is a judgement, which is evident for the one who knows that it is actually raining; in this case one may readily find evidence for the judgement by looking outside the window or stepping out of the house. In mathematical logic however, evidence is often not as directly observable, but rather deduced from more basic evident judgements. The process of deduction is what constitutes a proof; ...

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Natural deduction, Natural deduction - Judgements and propositions, Natural deduction - Introduction and elimination, Natural deduction - Hypothetical derivations, Natural deduction - Consistency completeness and normal forms, Natural deduction - First and higher order extensions, Natural deduction - Proofs and type-theory, Natural deduction - Classical and modal logics, Natural deduction - Comparison with other foundational approaches, Natural deduction - Sequent calculus

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However, the proofs used in everyday mathematical practice are almost never like the formal proofs in proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, and given enough time and patience. For most mathematicians, writing a fully formal proof would have all the drawbacks of programming in machine code. Formal proofs are constructed, with the help of computers, in automated theorem proving. Significantly, these proofs can be checked automatically by com ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Main article: Consistency proof As we have discussed, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable Pi-0-1 sentences) are finitarily true; once so grounded w ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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