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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, a substructural logic is a logic lacking one of the usual
structural rule 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 ...
s (e.g. of classical and
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
), such as weakening,
contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...
, exchange or associativity. Two of the more significant substructural logics are
relevance logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but ...
and
linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also ...
.


Examples

In a
sequent calculus In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology i ...
, one writes each line of a proof as :\Gamma\vdash\Sigma. Here the structural rules are rules for
rewriting In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduc ...
the LHS of the sequent, denoted Γ, initially conceived of as a string (sequence) of propositions. The standard interpretation of this string is as
conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...
: we expect to read :\mathcal A,\mathcal B \vdash\mathcal C as the sequent notation for :(''A'' and ''B'') implies ''C''. Here we are taking the RHS Σ to be a single proposition ''C'' (which is the
intuitionistic In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
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 \vdash. Since conjunction is a
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
operation, the formal setting-up of sequent theory normally includes structural rules for rewriting the sequent Γ accordingly—for example for deducing :\mathcal B,\mathcal A\vdash\mathcal C from :\mathcal A,\mathcal B\vdash\mathcal C. There are further structural rules corresponding to the ''
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
'' and ''
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
'' properties of conjunction: from : \Gamma,\mathcal A,\mathcal A,\Delta\vdash\mathcal C we can deduce : \Gamma,\mathcal A,\Delta\vdash\mathcal C. Also from : \Gamma,\mathcal A,\Delta\vdash\mathcal C one can deduce, for any ''B'', : \Gamma,\mathcal A,\mathcal B,\Delta\vdash\mathcal C.
Linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also ...
, in which duplicated hypotheses 'count' differently from single occurrences, leaves out both of these rules, while relevant (or relevance) logics 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. = 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

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.


See also

* Substructural type system * Residuated lattice


Notes


References

* F. Paoli (2002),
Substructural Logics: A Primer
', Kluwer. * G. Restall (2000)
An Introduction to Substructural Logics
', Routledge.


Further reading

* Galatos, Nikolaos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono (2007), ''Residuated Lattices. An Algebraic Glimpse at Substructural Logics'', Elsevier, .


External links

* * {{Non-classical logic Non-classical logic