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<!--replace-end-7--><!--replace-end-4--><!--replace-end-1--></head><body><div class="ui fluid container universe"><!--replace-start-2--><!--replace-start-3--><!--replace-start-6--><div class="ui text container" id="zettel-container" style="position: relative"><div class="zettel-view"><article class="ui raised attached segment zettel-content"><div class="pandoc"><h1 id="title-h1">Negation Elimination</h1><p>Like the <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Negation Introduction"><a href="Negation_Introduction.html">introduction</a></span></span> rule for negation, the elimination rule also works by deriving a contradiction. It is basically <em>Negation Introduction</em> in reverse. Instead of starting the sub-proof with a true proposition from which you derive a contradiction, you start with the negation of a proposition, derive a contradiction and then assert the positive of the negated proposition you started out with.</p><p><img src="/static/negate-elim.png" /></p></div></article><nav class="ui attached segment deemphasized backlinksPane" id="neuron-backlinks-pane"><h3 class="ui header">Backlinks</h3><ul class="backlinks"><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Strategies_for_constructing_proofs.html">Strategies for constructing proofs in propositional logic</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc">We will do this by assuming the negation of what we want to prove (<span class="math inline">\(\lnot (\lnot A \lor \lnot B)\)</span>) and then apply <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Negation Elimination"><a href="Negation_Elimination.html">Negation Elimination</a></span></span> to get <span class="math inline">\(\lnot A \lor \lnot B\)</span>.</div></li><li class="item"><div class="pandoc">This requires us to derive a contradiction. We get this on lines 23 and 24. This requires as previous steps that we have two sub-proofs that use <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Negation Elimination"><a href="Negation_Elimination.html">Negation Elimination</a></span></span> to release <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span></div></li></ul></li><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Logical_consistency.html">Logical consistency</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p>Here we want to derive some proposition <span class="math inline">\(Q\)</span>. If we can derive a contradiction from its negation as an assumption then, by the <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Negation Elimination"><a href="Negation_Elimination.html">negation elimination</a></span></span>) rule, we can assert <span class="math inline">\(Q\)</span>. This is why contradictions should be avoided in arguments, they ‘prove’ everything which, by association, undermines any particular premise you are trying to assert.</p></div></li></ul></li><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Formal_proofs_in_propositional_logic.html">Formal proofs in propositional logic</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Negation Elimination"><a href="Negation_Elimination.html">Negation Elimination</a></span></span></div></li></ul></li></ul></nav><nav class="ui attached segment deemphasized bottomPane" id="neuron-tags-pane"><div><span class="ui basic label zettel-tag" title="Tag">logic</span><span class="ui basic label zettel-tag" title="Tag">propositional-logic</span></div></nav><nav class="ui bottom attached icon compact inverted menu blue" id="neuron-nav-bar"><!--replace-start-9--><!--replace-end-9--><a class="right item" href="impulse.html" title="Open Impulse"><i class="wave square icon"></i></a></nav></div></div><!--replace-end-6--><!--replace-end-3--><!--replace-end-2--><div class="ui center aligned container footer-version"><div class="ui tiny image"><a href="https://neuron.zettel.page"><img alt="logo" src="https://raw.githubusercontent.com/srid/neuron/master/assets/neuron.svg" title="Generated by Neuron 1.9.35.3" /></a></div></div></div></body></html> |