en>Jerome Kohl: /* Integer notation */ attempt at climbing out of a conceptual hole (see the Talk page)

2014-12-19T01:13:50Z

Integer notation: attempt at climbing out of a conceptual hole (see the Talk page)

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	In [[logic]], a '''substructural logic''' is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as [[monotonicity of entailment\|weakening]], [[idempotency of entailment\|contraction]], exchange or associativity. Two of the more significant substructural logics are [[relevant logic]] and [[linear logic]].

	~~In a [[sequent calculus]], one writes each line of a proof as~~

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	~~Here the structural rules~~ are ~~rules for [[rewriting]] the [[Sides of an equation\|LHS]] Γ~~ of the ~~sequent, initially conceived of~~ as ~~a [[string (computer science)\|string]] of propositions~~. ~~The standard interpretation of this string is as [[Logical conjunction\|conjunction]]: we expect to read~~

	:<~~math~~>~~\mathcal A,\mathcal B \vdash\mathcal C~~<~~/math~~>

	as the ~~sequent notation for~~

	~~:(''A'' '''and''' ''B'') '''implies''' ''C''.~~

	~~Here we~~ are ~~taking the [[Sides of an equation\|RHS]] Σ~~ to be ~~a single proposition ''C'' (which is~~ the ~~[[intuitionistic]] style of sequent); but everything applies equally~~ to ~~the general case, since all the manipulations are taking place~~ to ~~the left~~ of ~~the [[Turnstile (symbol)\|turnstile symbol]~~].

	~~Since conjunction is~~ a ~~[[commutative]] and [[associative]] operation,~~ the ~~formal setting-up~~ of ~~sequent theory normally includes '''structural rules''' for rewriting the sequent Γ accordingly - for example for deducing~~

	~~:<math>\mathcal B~~,~~\mathcal~~ A~~\vdash\mathcal C~~<~~/math~~>

	~~from~~

	:<~~math~~>~~\mathcal~~ A~~,\mathcal B\vdash\mathcal C</math>~~.

	~~There are further structural rules corresponding to~~ the ''[~~[idempotent]~~]'~~' and ''[[monotonic]]'' properties~~ of ~~conjunction: from~~

	~~:<math> \Gamma,\mathcal A,\mathcal A,\Delta\vdash\mathcal C</math>~~

	~~we can deduce~~

	~~:<math> \Gamma,\mathcal A,\Delta\vdash\mathcal C</math>~~.

	~~Also from~~

	:<~~math~~> ~~\Gamma,\mathcal A,\Delta\vdash\mathcal C~~<~~/math~~>

	~~one can deduce,~~ for ~~any~~ '~~'B'',~~

	~~:<math> \Gamma,\mathcal A,\mathcal B,\Delta\vdash\mathcal C</math>~~.

	~~[[Linear logic]], in which duplicated hypotheses 'count' differently from single occurrences, leaves~~ out ~~both~~ of ~~these rules~~, ~~while [[relevant logic\|relevant (or relevance) logic]]s merely leaves out~~ the ~~latter rule, on~~ the ~~ground that ''B'' is clearly irrelevant to~~ the ~~conclusion~~.

	These ~~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)~~.

	~~== Premise composition ==~~
	~~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, 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~~.

	~~== History ==~~
	~~It is 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.~~

	~~== See also ==~~
	* [[Substructural type system]]
	* [[Residuated lattice]]

	~~== References ==~~
	* F. Paoli (2002), ~~''Substructural Logics: A Primer'', Kluwer~~.

	~~== Further reading ==~~
	* Galatos, Nikolaos, Peter Jipsen, Tomasz Kowalski, and ~~Hiroakira Ono (2007), ''Residuated Lattices. An Algebraic Glimpse at Substructural Logics'', Elsevier, ISBN 978-0-444-52141-5~~.

	~~== External links ==~~
	*{{sep entry\|logic-substructural\|Substructural logics\|Greg Restall}}

	~~{{logic}}~~

	~~[[Category:Substructural logic\|*]]~~
	~~[[Category:Non-classical logic]]~~

en>Hyacinth: /* See also */ (List)

2013-11-15T10:20:27Z

en>Double sharp: /* Integer notation */

2012-05-06T06:54:13Z

Integer notation

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en>Jerome Kohl: /* Integer notation */ attempt at climbing out of a conceptual hole (see the Talk page)

en>Hyacinth: /* See also */ (List)

en>Double sharp: /* Integer notation */