Substructural logic

2025-10-27T13:50:31Z

2A02:A03F:63D6:D901:2992:7620:BD5B:D58D:

{{Short description|Branch of non-classical logic}}
{{no footnotes|date=June 2016 }}

In [[logic]], a '''substructural logic''' is a logic lacking one of the usual [[structural rule]]s (e.g. of [[classical logic|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|relevance logic]] and [[linear logic]].

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

:<math>\Gamma\vdash\Sigma</math>.

Here the structural rules are rules for [[rewriting]] the [[Sides of an equation|LHS]] of the sequent, denoted Γ, initially conceived of as a finite [[sequence (mathematics)|string]] (sequence)  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]] <math>\vdash</math>.

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 ''[[Monotonicity of entailment|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.

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).

== 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 [[rooted tree|tree]]s or [[multiset]]s (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 ==
Substructural logic 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==
{{reflist}}
*{{cite book
| last = Paoli
| first = Francesco
| title = Substructural Logics: A Primer
| url = https://books.google.com/books?id=RkPsCAAAQBAJ
| year = 2002
| publisher = Springer Science & Business Media
| location = Dordrecht
| series = Trends in Logic
| volume = 13
| isbn = 978-90-481-6014-3
| doi = 10.1007/978-94-017-3179-9
}}
*{{cite book
| last = Restall
| first = Greg
| author-link = Greg Restall
| title = An Introduction to Substructural Logics
| url = https://books.google.com/books?id=NQTm_bRupAgC
| year = 2000
| publisher = Routledge
| location = London and New York
| isbn = 0-415-21533-1
}}

== Further reading ==
*{{cite book
| last1 = Galatos
| first1 = Nikolaos
| last2 = Jipsen
| first2 = Peter
| last3 = Kowalski
| first3 = Tomasz
| last4 = Ono
| first4 = Hiroakira
| title = Residuated Lattices: An Algebraic Glimpse at Substructural Logics
| year = 2007
| publisher = Elsevier
| location = Amsterdam
| isbn = 978-0-444-52141-5
| series = Studies in Logic and Practical Reasoning
| volume = 2
}}

==External links==
*{{Commonscat-inline}}
*{{cite SEP |url-id=logic-substructural |title=Substructural logics |last=Restall |first=Greg}}

{{Non-classical logic}}

[[Category:Substructural logic| ]]
[[Category:Non-classical logic]]

Cut rule

2025-10-27T09:54:14Z

2A02:A03F:63D6:D901:2992:7620:BD5B:D58D:

{{Short description|Inference rule}}
{{onesource|date=July 2017}}
In [[mathematical logic]], the '''cut rule''' is an [[inference rule]] of [[sequent calculus]]. It is a generalisation of the classical [[modus ponens]] inference rule. The meaning of the cut rule is that, if a formula ''A'' appears as a conclusion in one proof and a hypothesis in another, then another proof in which the formula ''A'' does not appear can be deduced. This applies to cases of [[modus ponens]], such as how instances of ''man'' are eliminated from ''Every man is mortal, [[Socrates]] is a man'' to deduce ''Socrates is mortal''.

== Formal notation ==
The cut rule is normally written in formal notation in sequent calculus as :
:
: <math>
\begin{array}{c}\Gamma \vdash A, \Delta \quad \Gamma', A \vdash \Delta' \\ \hline \Gamma, \Gamma' \vdash \Delta, \Delta'\end{array} </math>cut<ref>{{Cite web |title=cut rule in nLab |url=https://ncatlab.org/nlab/show/cut+rule |access-date=2024-10-22 |website=ncatlab.org |language=en}}</ref>

== Elimination ==
The cut rule is the subject of an important theorem, the [[cut-elimination theorem]]. This states that any sequent that has a proof in the sequent calculus making use of the cut rule also has a cut-free proof, that is, a proof that does not make use of the cut rule.

==References==
{{reflist}}

[[Category:Rules of inference]]
[[Category:Logical calculi]]

{{Mathlogic-stub}}

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Substructural logic

Cut rule