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<Paper uid="P97-1044">
  <Title>Maximal Incrementality in Linear Categorial Deduction</Title>
  <Section position="4" start_page="344" end_page="344" type="intro">
    <SectionTitle>
2 Implicational Linear Logic
</SectionTitle>
    <Paragraph position="0"> Linear logic is an example of a &amp;quot;resource-sensitive&amp;quot; logic, requiring that each assumption ('resource') is used precisely once in any deduction. For the implicational fragment, the set of formulae ~ are defined by 5 r ::= A \[ ~'o-~- (with A a nonempty set of atomic types). A natural deduction formulation requires the elimination and introduction rules in (1), which correspond semantically to steps of functional application and abstraction, respectively.</Paragraph>
    <Paragraph position="2"> The proof (2) (which omits lambda terms) illustrates that 'hypothetical reasoning' in proofs (i.e. the use of additional assumptions that are later discharged or cancelled, such as Z here) is driven by the presence of higher-order formulae (such as Xo-(yc-z) here).</Paragraph>
    <Paragraph position="4"> Various type-logical categorial formalisms (or strictly their implicational fragments) differ from the above system only in imposing further restrictions on resource usage. For example, the associative Lambek calculus imposes a linear order over formulae, in which context, implication divides into two cases, (usually written \ and /) depending on whether the argument type appears to the left or right of the functor. Then, formulae may combine only if they are adjacent and in the appropriate left-right order. The non-associative Lambek calculus (Lambek, 1961) sets the further requirement that types combine under some fixed initial bracketting. Such weaker systems can be implemented by combining implicational linear logic with a labelling system whose labels are structured objects that record relevant resource information, i.e. of sequencing and/or bracketting, and then using this information in restricting permitted inferences to only those that satisfy the resource requirements of the weaker logic.</Paragraph>
  </Section>
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