This section introduces the rules of the sequent calculus LK as introduced by Gentzen in 1934. (LK, unintuitively, stands for "klassische Prädikatenlogik".) A (formal) proof in this calculus is a sequence of sequents, where each of the sequents is derivable from sequents appearing earlier in the sequence by using one of the rules below.
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In sequent calculus, the completeness of atomic initial sequents states that initial sequents (where is an arbitrary formula) can be derived from only atomic initial sequents (where is an atomic formula). This theorem plays a role analogous to eta expansion in lambda calculus, and dual to cut- elimination and beta reduction. Typically it can be established by induction on the structure of , much more easily than cut-elimination.
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The general notion of sequent introduced here can be specialized in various ways. A sequent is said to be an intuitionistic sequent if there is at most one formula in the succedent (although multi-succedent calculi for intuitionistic logic are also possible). More precisely, the restriction of the general sequent calculus to single-succedent-formula sequents, with the same inference rules as for general sequents, constitutes an intuitionistic sequent calculus. (This restricted sequent calculus is denoted LJ.) Similarly, one can obtain calculi for dual-intuitionistic logic (a type of paraconsistent logic) by requiring that sequents be singular in the antecedent.
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Gentzen's discharging annotations used to internalise hypothetical judgments can be avoided by representing proofs as a tree of sequents Γ ⊢A instead of a tree of A true judgments.
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However, states that the assertion symbol in Gentzen-system sequents, which he denotes as ' ⇒ ', is part of the object language, not the metalanguage., defines sequents to have the form U ⇒ V for (possibly non-empty) sets of formulas U and V. Then he writes: : "Intuitively, a sequent represents 'provable from' in the sense that the formulas in U are assumptions for the set of formulas V that are to be proved. The symbol ⇒ is similar to the symbol ⊢ in Hilbert systems, except that ⇒ is part of the object language of the deductive system being formalized, while ⊢ is a metalanguage notation used to reason about deductive systems." According to Prawitz (1965): "The calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction.".
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There is some freedom of choice regarding the technical details of how sequents and structural rules are formalized. As long as every derivation in LK can be effectively transformed to a derivation using the new rules and vice versa, the modified rules may still be called LK. First of all, as mentioned above, the sequents can be viewed to consist of sets or multisets. In this case, the rules for permuting and (when using sets) contracting formulae are obsolete. The rule of weakening will become admissible, when the axiom (I) is changed, such that any sequent of the form \Gamma , A \vdash A , \Delta can be concluded.
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Historically, sequents have been introduced by Gerhard Gentzen in order to specify his famous sequent calculus., . In his German publication he used the word "Sequenz". However, in English, the word "sequence" is already used as a translation to the German "Folge" and appears quite frequently in mathematics.
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The approach was introduced by G. Japaridze inG.Japaridze, “Introduction to cirquent calculus and abstract resource semantics”. Journal of Logic and Computation 16 (2006), pp. 489–532. as an alternative proof theory capable of “taming” various nontrivial fragments of his computability logic, which had otherwise resisted all axiomatization attempts within the traditional proof-theoretic frameworks.G.Japaridze, “The taming of recurrences in computability logic through cirquent calculus, Part I”. Archive for Mathematical Logic 52 (2013), pages 173-212. G.Japaridze, “The taming of recurrences in computability logic through cirquent calculus, Part II” Archive for Mathematical Logic 52 (2013), pages 213–259. The origin of the term “cirquent” is CIRcuit+seQUENT, as the simplest form of cirquents, while resembling circuits rather than formulas, can be thought of as collections of one-sided sequents (for instance, sequents of a given level of a Gentzen-style proof tree) where some sequents may have shared elements. Cirquent for the "two out of three" combination of resources, inexpressible in linear logic The basic version of cirquent calculus inG.
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Hypersequent calculi based on intuitionistic or single-succedent sequents have been used successfully to capture a large class of intermediate logics, i.e., extensions of intuitionistic propositional logic. Since the hypersequents in this setting are based on single-succedent sequents, they have the following form: : \Gamma_1 \Rightarrow A_1 \mid \dots \mid \Gamma_n \Rightarrow A_n The standard formula interpretation for such an hypersequent is : (\bigwedge\Gamma_1 \to A_1) \lor \dots \lor (\bigwedge\Gamma_n \to A_n) Most hypersequent calculi for intermediate logics include the single-succedent versions of the propositional rules given above, a selection of the structural rules. The characteristics of a particular intermediate logic are mostly captured using a number of additional structural rules. E.g.
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In 2006G. Japaridze, "Introduction to cirquent calculus and abstract resource semantics". Journal of Logic and Computation 16 (2006), pages 489-532. Japaridze conceived cirquent calculus as a proof-theoretic approach that manipulates graph-style constructs, termed cirquents, instead of the more traditional and less general tree-like constructs such as formulas or sequents.
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In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgment or sequents directly. Structural rules often mimic intended meta-theoretic properties of the logic. Logics that deny one or more of the structural rules are classified as substructural logics.
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Cirquents can be thought of as collections of sequents with possibly shared elements Cirquent calculus is a proof calculus which manipulates graph-style constructs termed cirquents, as opposed to the traditional tree-style objects such as formulas or sequents. Cirquents come in a variety of forms, but they all share one main characteristic feature, making them different from the more traditional objects of syntactic manipulation. This feature is the ability to explicitly account for possible sharing of subcomponents between different components. For instance, it is possible to write an expression where two subexpressions F and E, while neither one is a subexpression of the other, still have a common occurrence of a subexpression G (as opposed to having two different occurrences of G, one in F and one in E).
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The assertion symbol in sequents originally meant exactly the same as the implication operator. But over time, its meaning has changed to signify provability within a theory rather than semantic truth in all models. In 1934, Gentzen did not define the assertion symbol ' ⊢ ' in a sequent to signify provability. He defined it to mean exactly the same as the implication operator ' ⇒ '.
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And furthermore: "A proof in a calculus of sequents can be looked upon as an instruction on how to construct a corresponding natural deduction."See , for this and further details of interpretation. In other words, the assertion symbol is part of the object language for the sequent calculus, which is a kind of meta-calculus, but simultaneously signifies deducibility in an underlying natural deduction system.
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In many cases, sequents are also assumed to consist of multisets or sets instead of sequences. Thus one disregards the order or even the numbers of occurrences of the formulae. For classical propositional logic this does not yield a problem, since the conclusions that one can draw from a collection of premises do not depend on these data. In substructural logic, however, this may become quite important.
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In the sequent calculus for an intuitionistic logic, the uniform proofs can be characterised as those in which the upward reading performs all right rules before the left rules. Typically, uniform proofs are not complete for the logic i.e., not all sequents or formulas admit a uniform proof, so one considers fragments where they are complete e.g., the hereditary Harrop fragment of Intuitionistic logic.
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Noncommutative logic is sometimes called ordered logic, since it is possible with most proposed noncommutative logics to impose a total or partial order on the formulae in sequents. However this is not fully general since some noncommutative logics do not support such an order, such as Yetter's cyclic linear logic. Although most noncommutative logics do not allow weakening or contraction together with noncommutativity, this restriction is not necessary.
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The above rules can be divided into two major groups: logical and structural ones. Each of the logical rules introduces a new logical formula either on the left or on the right of the turnstile \vdash. In contrast, the structural rules operate on the structure of the sequents, ignoring the exact shape of the formulae. The two exceptions to this general scheme are the axiom of identity (I) and the rule of (Cut).
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Because Metamath has a very generic concept of what a proof is (namely a tree of formulas connected by inference rules) and no specific logic is embedded in the software, Metamath can be used with species of logic as different as Hilbert-style logics or sequents-based logics or even with lambda calculus. However, Metamath provides no direct support for natural deduction systems. As noted earlier, the database nat.mm formalizes natural deduction.
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David N. Yetter proposed a weaker structural rule in place of the exchange rule of linear logic, yielding cyclic linear logic. Sequents of cyclic linear logic form a ring, and so are invariant under rotation, where multipremise rules glue their rings together at the formulae described in the rules. The calculus supports three structural modalities, a self-dual modality allowing exchange, but still linear, and the usual exponentials (? and !) of linear logic, allowing nonlinear structural rules to be used together with exchange.
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The standard semantics of a sequent is an assertion that whenever every A_i is true, at least one B_i will also be true.For explanations of the disjunctive semantics for the right side of sequents, see , , , , and . Thus the empty sequent, having both cedents empty, is false. One way to express this is that a comma to the left of the turnstile should be thought of as an "and", and a comma to the right of the turnstile should be thought of as an (inclusive) "or".
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The intuitive reading of this is obvious: every formula proves itself. Like the cut rule, the axiom of identity is somewhat redundant: the completeness of atomic initial sequents states that the rule can be restricted to atomic formulas without any loss of provability. Observe that all rules have mirror companions, except the ones for implication. This reflects the fact that the usual language of first-order logic does not include the "is not implied by" connective ot\leftarrow that would be the De Morgan dual of implication.
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In either case one can safely apply rules in any order to hereditary sub-formulas of the same polarity. In the case of a right rule applied to a positive formula, or a left rule applied to a negative formula, one may result in invalid sequents e.g., in LK and LJ there is no proof of the sequent B \lor A \implies A \lor B beginning with a right rule. A calculus admits the focusing principle if when an original reduct was provable then the hereditary reducts of the same polarity are also provable.
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The sequent calculus was developed to study the properties of natural deduction systems.Shankar, N., Owre, S., Rushby, J. M., & Stringer- Calvert, D. W. J., PVS Prover Guide 2.4 (Menlo Park: SRI International, November 2001). Instead of working with one formula at a time, it uses sequents, which are expressions of the form :A_1, \ldots, A_n \vdash B_1, \ldots, B_k, where A1, ..., An, B1, ..., Bk are formulas and the turnstile symbol \vdash is used as punctuation to separate the two halves. Intuitively, a sequent expresses the idea that (A_1 \land \cdots\land A_n) implies (B_1\lor\cdots\lor B_k).
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Thus, swapping left for right in a sequent corresponds to negating all of the constituent formulae. This means that a symmetry such as De Morgan's laws, which manifests itself as logical negation on the semantic level, translates directly into a left-right symmetry of sequents — and indeed, the inference rules in sequent calculus for dealing with conjunction (∧) are mirror images of those dealing with disjunction (∨). Many logicians feel that this symmetric presentation offers a deeper insight in the structure of the logic than other styles of proof system, where the classical duality of negation is not as apparent in the rules.
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Finally, sequent calculus generalizes the form of a natural deduction judgment to : A_1, \ldots, A_n \vdash B_1, \ldots, B_k, a syntactic object called a sequent. The formulas on left-hand side of the turnstile are called the antecedent, and the formulas on right-hand side are called the succedent or consequent; together they are called cedents or sequents. Again, A_i and B_i are formulae, and n and k are nonnegative integers, that is, the left-hand-side or the right-hand-side (or neither or both) may be empty. As in natural deduction, theorems are those B where \vdash B is the conclusion of a valid proof.
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For example, \Gamma\vdash\Sigma can be read as asserting that it cannot be the case that every formula in Γ is true and every formula in Σ is false (this is related to the double-negation interpretations of classical intuitionistic logic, such as Glivenko's theorem). In any case, these intuitive readings are only pedagogical. Since formal proofs in proof theory are purely syntactic, the meaning of (the derivation of) a sequent is only given by the properties of the calculus that provides the actual rules of inference. Barring any contradictions in the technically precise definition above we can describe sequents in their introductory logical form.
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In focused systems for classical and Intuitionistic logic, the use of backtracking can be simulated by pseudo-contraction. Let \uparrow and \downarrow denote change of polarity, the former making a formula negative, and the latter positive; and call a formula with an arrow neutral. Recall that \lor is positive, and consider the neutral polarized sequent \downarrow \uparrow \phi \lor \psi \implies \uparrow \phi \lor \psi, which is interpreted as the actual sequent \phi \lor \psi \implies \phi \lor \psi. For neutral sequents such as this, the focused system forces on to make an explicit choice of which formula to focus on, denoted by \langle \, \rangle .
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Copenhagen: Philibert, pp 426–428 who was nearly blind from cataracts in both eyes but perceived men, women, birds, carriages, buildings, tapestries and scaffolding patterns.) Most people affected are elderly with visual impairments, however the phenomenon does not occur only in the elderly or in those with visual impairments; it can also be caused by damage elsewhere in their optic pathway or brain. Bonnet's philosophical system may be outlined as follows. Man is a compound of two distinct substances, mind and body, the one immaterial and the other material. All knowledge originates in sensations; sensations follow (whether as physical effects or merely as sequents Bonnet will not say) vibrations in the nerves appropriate to each; and lastly, the nerves are made to vibrate by external physical stimulus.
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Using ' → ' instead of ' ⊢ ' and ' ⊃ ' instead of ' ⇒ ', he wrote: "The sequent A1, ..., Aμ → B1, ..., Bν signifies, as regards content, exactly the same as the formula (A1 & ... & Aμ) ⊃ (B1 ∨ ... ∨ Bν)".. : 2.4. Die Sequenz A1, ..., Aμ → B1, ..., Bν bedeutet inhaltlich genau dasselbe wie die Formel ::: (A1 & ... & Aμ) ⊃ (B1 ∨ ... ∨ Bν). (Gentzen employed the right-arrow symbol between the antecedents and consequents of sequents. He employed the symbol ' ⊃ ' for the logical implication operator.) In 1939, Hilbert and Bernays stated likewise that a sequent has the same meaning as the corresponding implication formula.. : Für die inhaltliche Deutung ist eine Sequenz ::: A1, ..., Ar → B1, ..., Bs, : worin die Anzahlen r und s von 0 verschieden sind, gleichbedeutend mit der Implikation ::: (A1 & ... & Ar) → (B1 ∨ ... ∨ Bs) In 1944, Alonzo Church emphasized that Gentzen's sequent assertions did not signify provability.
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Propositional calculus is commonly organized as a Hilbert system, whose operations are just those of Boolean algebra and whose theorems are Boolean tautologies, those Boolean terms equal to the Boolean constant 1. Another form is sequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists of propositions called sequents, such as A∨B, A∧C,... \vdash A, B→C,.... The two halves of a sequent are called the antecedent and the succedent respectively. The customary metavariable denoting an antecedent or part thereof is Γ, and for a succedent Δ; thus Γ,A \vdash Δ would denote a sequent whose succedent is a list Δ and whose antecedent is a list Γ with an additional proposition A appended after it. The antecedent is interpreted as the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of the succedent by the antecedent.
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