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<title>Logics</title>
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<h2>Isabelle's Logics</h2>
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<p>Isabelle can be viewed from two main
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perspectives. On the one hand it may serve as a generic framework for rapid
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prototyping of deductive systems. On the other hand, major existing logics
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like <a href="#isabelle_hol"><em>Isabelle/HOL</em></a>
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provide a theorem proving environment
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ready to use for sizable applications.</p>
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<p>The Isabelle distribution includes a large body of
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object logics and other examples (see the <a href=
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"library/index.html">Isabelle theory library</a>).</p>
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<dl>
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<dt id="isabelle_hol"><a href="library/HOL/index.html">Isabelle/HOL</a></dt>
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<dd>is a version of classical higher-order logic resembling that of the
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<a href="http://www.cl.cam.ac.uk/Research/HVG/HOL/">HOL System</a>. The
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main libraries of the HOL 4 System are now <a href=
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"library/HOL/HOL-Complex/HOL4/index.html">available in Isabelle</a>.</dd>
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<dt><a href=
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"library/HOLCF/index.html">Isabelle/HOLCF</a></dt>
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<dd>adds Scott's Logic for Computable Functions (domain theory) to
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HOL.</dd>
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<dt><a href="library/FOL/index.html">Isabelle/FOL</a></dt>
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<dd>provides basic classical and intuitionistic first-order logic. It is
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polymorphic.</dd>
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<dt><a href="library/ZF/index.html">Isabelle/ZF</a></dt>
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<dd>offers a formulation of Zermelo-Fraenkel set theory on top of FOL.</dd>
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</dl>
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<p><em>Isabelle/HOL</em> is currently the best developed object logic, including an
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extensive library of (concrete) mathematics, and various packages for
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advanced definitional concepts like (co-)inductive sets and types,
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well-founded recursion etc. The distribution also includes some large
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applications, for example correctness proofs of cryptographic protocols
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(<a href="library/HOL/Auth/index.html">HOL/Auth</a>) or communication
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protocols (<a href="library/HOLCF/IOA/index.html">HOLCF/IOA</a>).</p>
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<p><em>Isabelle/ZF</em> provides another starting point for applications, with a
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slightly less developed library. Its definitional packages are similar to
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those of Isabelle/HOL. Untyped ZF provides more advanced constructions for
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sets than simply-typed HOL.</p>
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<p>There are a few minor object logics that may serve as further examples:
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<a href="library/CTT/index.html">CTT</a> is an extensional version of
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Martin-Löf's Type Theory, <a href="library/Cube/index.html">Cube</a> is
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Barendregt's Lambda Cube. There are also some sequent calculus examples under
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<a href="library/Sequents/index.html">Sequents</a>, including modal and
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linear logics. Again see the <a href="library/index.html">Isabelle theory
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library</a> for other examples.</p>
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<h2>Defining Logics</h2>
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<p>Logics are not hard-wired into Isabelle, but
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formulated within Isabelle's meta logic: <em>Isabelle/Pure</em>.
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There are quite a lot of syntactic and deductive tools available in generic
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Isabelle. Thus defining new logics or extending existing ones basically works
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as follows:</p>
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<ol>
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<li>declare concrete syntax (via mixfix grammar and syntax macros)</li>
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<li>declare abstract syntax (as higher-order constants)</li>
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<li>declare inference rules (as meta-logical propositions)</li>
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<li>instantiate generic automatic proof tools (simplifier, classical
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tableau prover etc.)</li>
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<li>manually code special proof procedures (via tacticals or hand-written
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ML)</li>
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</ol>
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<p>The first three steps above are fully declarative and involve no ML
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programming at all. Thus one already gets a decent deductive environment
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based on primitive inferences (by employing the built-in mechanisms of
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<em>Isabelle/Pure</em>, in particular higher-order unification and resolution). For
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sizable applications some degree of automated reasoning is essential.
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Instantiating existing tools like the classical tableau prover involves only
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minimal ML-based setup. One may also write arbitrary proof procedures or even
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theory extension packages in ML, without breaking system soundness (Isabelle
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follows the well-known <em>LCF system approach</em> to achieve a secure
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system).</p>
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</div>
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