Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Affine Logic and {\L}ukasiewicz Logic   The multi-valued logic of {\L}ukasiewicz is a substructural logic that has
been widely studied and has many interesting properties. It is classical, in
the sense that it admits the axiom schema of double negation, [DNE]. However,
our understanding of {\L}ukasiewicz logic can be improved by separating its
classical and intuitionistic aspects. The intuitionistic aspect of
{\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the
commutativity of a weak form of conjunction. This is equivalent to a very
restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed
both as an extension of classical affine logic with [CWC], or as an extension
of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE],
intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic
affine logic by the schema [CWC]. At first glance, intuitionistic affine logic
seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results
such as intuitionistic analogues of De Morgan's laws. However the proofs can be
very intricate. We present these results using derived connectives to clarify
and motivate the proofs and give several applications. We give an analysis of
the applicability to these logics of the well-known methods that use negation
to translate classical logic into intuitionistic logic. The usual proofs of
correctness for these translations make much use of contraction. Nonetheless,
we show that all the usual negative translations are already correct for
intuitionistic {\L}ukasiewicz logic, where only the limited amount of
contraction given by [CWC] is allowed. This is in contrast with affine logic
for which we show, by appeal to results on semantics proved in a companion
paper, that both the Gentzen and the Glivenko translations fail.

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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: eess