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among other properties. In substructural logics, typically premises are not composed into sets, but rather they are composed into more fine-grained structures, such as trees or multisets (sets that distinguish multiple occurrences of elements) or sequences of formulae. For example, in linear logic, since contraction fails, the premises must be composed in something at least as fine-grained as multisets.
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There are numerous ways to compose premises (and in the multiple-conclusion case, conclusions as well). One way is to collect them into a set. But since e.g. {a,a} = {a} we have contraction for free if premises are sets. We also have associativity and permutation (or commutativity) for free as well,
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Substructural logics are a relatively young field. The first conference on the topic was held in
October 1990 in Tübingen, as "Logics with Restricted Structural Rules". During the conference, Kosta Došen proposed the term "substructural logics", which is now in use today.
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The above are basic examples of structural rules. It is not that these rules are contentious, when applied in conventional propositional calculus. They occur naturally in proof theory, and were first noticed there (before receiving a name).
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of the sequent, denoted Γ, initially conceived of as a string (sequence) of propositions. The standard interpretation of this string is as
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style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the
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114:, exchange or associativity. Two of the more significant substructural logics are
640:{\displaystyle \Gamma ,{\mathcal {A}},{\mathcal {B}},\Delta \vdash {\mathcal {C}}}
469:{\displaystyle \Gamma ,{\mathcal {A}},{\mathcal {A}},\Delta \vdash {\mathcal {C}}}
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Galatos, Nikolaos, Peter Jipsen, Tomasz
Kowalski, and Hiroakira Ono (2007),
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operation, the formal setting-up of sequent theory normally includes
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574:{\displaystyle \Gamma ,{\mathcal {A}},\Delta \vdash {\mathcal {C}}}
521:{\displaystyle \Gamma ,{\mathcal {A}},\Delta \vdash {\mathcal {C}}}
394:{\displaystyle {\mathcal {A}},{\mathcal {B}}\vdash {\mathcal {C}}}
344:{\displaystyle {\mathcal {B}},{\mathcal {A}}\vdash {\mathcal {C}}}
219:{\displaystyle {\mathcal {A}},{\mathcal {B}}\vdash {\mathcal {C}}}
1025:
737:
Residuated
Lattices. An Algebraic Glimpse at Substructural Logics
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for rewriting the sequent Γ accordingly—for example for deducing
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There are further structural rules corresponding to the
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merely leaves out the latter rule, on the ground that
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43:but its sources remain unclear because it lacks
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662:is clearly irrelevant to the conclusion.
74:Learn how and when to remove this message
167:Here the structural rules are rules for
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722:An Introduction to Substructural Logics
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156:{\displaystyle \Gamma \vdash \Sigma }
134:, one writes each line of a proof as
94:is a logic lacking one of the usual
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417:properties of conjunction: from
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713:Substructural Logics: A Primer
656:relevant (or relevance) logics
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258:Σ to be a single proposition
229:as the sequent notation for
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958:Ontology (computer science)
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851:Intuitionistic type theory
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694:Substructural type system
584:one can deduce, for any
29:This article includes a
856:Constructive set theory
292:Since conjunction is a
282:{\displaystyle \vdash }
254:Here we are taking the
58:more precise citations.
772:"Substructural logics"
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841:Constructive analysis
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836:Intuitionistic logic
766:at Wikimedia Commons
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179:: we expect to read
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104:intuitionistic logic
1097:Non-classical logic
1092:Substructural logic
1071:Non-monotonic logic
820:Non-classical logic
764:Substructural logic
670:Premise composition
92:substructural logic
1066:Intermediate logic
846:Heyting arithmetic
719:G. Restall (2000)
699:Residuated lattice
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31:list of references
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1061:Inquisitive logic
1056:Dynamic semantics
1009:Three-state logic
963:Ontology language
762:Media related to
745:978-0-444-52141-5
710:F. Paoli (2002),
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730:Further reading
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262:(which is the
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981:Three-valued
922:Linear logic
903:
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739:, Elsevier,
736:
725:, Routledge.
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652:Linear logic
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120:linear logic
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50:Please help
42:
1021:Four-valued
991:Łukasiewicz
986:Four-valued
973:Many-valued
950:Description
940:Dialetheism
298:associative
294:commutative
177:conjunction
112:contraction
106:), such as
56:introducing
1086:Categories
879:Fuzzy rule
705:References
532:Also from
408:idempotent
1033:IEEE 1164
884:Fuzzy set
716:, Kluwer.
628:⊢
625:Δ
599:Γ
562:⊢
559:Δ
543:Γ
509:⊢
506:Δ
490:Γ
457:⊢
454:Δ
428:Γ
414:monotonic
382:⊢
332:⊢
277:⊢
207:⊢
169:rewriting
151:Σ
148:⊢
145:Γ
108:weakening
100:classical
98:(e.g. of
64:June 2016
688:See also
126:Examples
1026:Verilog
778:(ed.).
679:History
245:implies
52:improve
1049:Others
743:
866:Fuzzy
774:. In
354:from
130:In a
88:logic
37:, or
1038:VHDL
741:ISBN
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256:RHS
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