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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Interpretability</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">For the machine learning concept, see <a href="Interpretability_(machine_learning)" class="mw-redirect" title="Interpretability (machine learning)">Interpretability (machine learning)</a>.</div>
<div role="note" class="hatnote navigation-not-searchable">For the deep learning concept, see <a href="Mechanistic_Interpretability" class="mw-redirect" title="Mechanistic Interpretability">Mechanistic Interpretability</a>.</div>
<p>In <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, <b>interpretability</b> is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.
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<div class="mw-heading mw-heading2"><h2 id="Informal_definition">Informal definition</h2></div>
<p>Assume <i>T</i> and <i>S</i> are <a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">formal theories</a>. Slightly simplified, <i>T</i> is said to be <i>interpretable</i> in <i>S</i> if and only if the language of <i>T</i> can be translated into the <a href="Language" title="Language">language</a> of <i>S</i> in such a way that <i>S</i> proves the translation of every <a href="Theorem" title="Theorem">theorem</a> of <i>T</i>. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of <a href="Well-formed_formula" title="Well-formed formula">formulas</a>.
</p><p>This concept, together with <a href="Weak_interpretability" title="Weak interpretability">weak interpretability</a>, was introduced by <a href="Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> in 1953. Three other related concepts are <a href="Cointerpretability" title="Cointerpretability">cointerpretability</a>, <a href="Tolerance_(in_logic)" class="mw-redirect" title="Tolerance (in logic)">logical tolerance</a>, and cotolerance, introduced by <a href="Giorgi_Japaridze" title="Giorgi Japaridze">Giorgi Japaridze</a> in 1992–93.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Conservative_extension" title="Conservative extension">Conservative extension</a></li>
<li><a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation (logic)</a></li>
<li><a href="Interpretation_(model_theory)" title="Interpretation (model theory)">Interpretation (model theory)</a></li>
<li><a href="Interpretability_logic" title="Interpretability logic">Interpretability logic</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>Japaridze, G., and De Jongh, D. (1998) "The logic of provability" in Buss, S., ed., <i>Handbook of Proof Theory</i>. North-Holland: 476–546.</li>
<li><a href="Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, <a href="Andrzej_Mostowski" title="Andrzej Mostowski">Andrzej Mostowski</a>, and <a href="Raphael_Robinson" class="mw-redirect" title="Raphael Robinson">Raphael Robinson</a> (1953) <i>Undecidable Theories</i>. North-Holland.</li></ul>
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This article is issued from <a class="external text" title="Last edited on 2025-08-03" href="https://en.wikipedia.org/wiki/?title=Interpretability&oldid=1303976814">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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