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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">Logical consistency</h1><h2 id="informal-definition">Informal definition</h2><p>A set of propositions is consistent if and only if <strong>it is possible for all the members of the set to be true at the same time</strong>. A set of propositions is inconsistent if and only if it is not consistent.</p><h3 id="demonstration">Demonstration</h3><p>The following set of propositions form an inconsistent set:</p><ol><li>Anyone who takes astrology seriously is a lunatic.</li><li>Alice is my sister and no sister of mine has a lunatic for a husband.</li><li>David is Alice’s husband and he read’s the horoscope column every morning.</li><li>Anyone who reads the horoscope column every morning takes astrology seriously.</li></ol><p>The set is inconsistent because not all of them can be true. If (1), (3), (4) are true, (2) cannot be. If (2), (3),(4) are true, (1) cannot be.</p><h2 id="formal-definition">Formal definition</h2><blockquote><p>A finite set of propositions <span class="math inline">\(\Gamma\)</span> is truth-functionally consistent if and only if there is at least one truth-assignment in which all propositions of <span class="math inline">\(\Gamma\)</span> are true.</p></blockquote><h3 id="informal-expression">Informal expression</h3><pre><code class="language-none">The book is blue or the book is brown
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The book is brown</code></pre><h3 id="formal-expression">Formal expression</h3><p><span class="math display">$$
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\{P \lor Q, Q\}
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$$</span></p><h3 id="truth-table">Truth table</h3><p>$ {P, Q} $ form a consistent set because there is at least one assignment when both propositions are true. In fact there are two (corresponding to each disjunct) but one is sufficient.</p><table class="ui table"><thead><tr><th><span class="math inline">\(P\)</span></th><th><span class="math inline">\(Q\)</span></th><th>$ P \lor Q $</th><th><span class="math inline">\(Q\)</span></th></tr></thead><tbody><tr><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>T</td><td>F</td></tr><tr><td>F</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td><td>F</td></tr></tbody></table><h2 id="derivation">Derivation</h2><blockquote><p>In terms of logical derivation, a finite <span class="math inline">\(\Gamma\)</span> of propositions is <strong>inconsistent</strong> in a system of derivation for propositional logic if and only if a proposition of the form <span class="math inline">\(P \& \lnot P\)</span> is derivable from <span class="math inline">\(\Gamma\)</span>. It is <strong>consistent</strong> just if this is not the case.</p></blockquote><p>In other terms, if you can derive a contradiction from the set, the set is logically inconsistent.</p><p>A <a href="Logical_truth_and_falsity.md#logical-falsity">contradiction</a> has very important consequences for reasoning because if a set of propositions is inconsistent, any other proposition is derivable from it.</p><p><img src="/static/derivation_from_contradiction.png" /></p><p><em>A demonstration of the the consequences of deriving a contradiction in a sequence of reasoning.</em></p><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></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="Syllogism.html">Syllogism</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p>In order to make assertions about the relative <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Logical consistency"><a href="Logical_consistency.html">consistency</a></span></span> or inconsistency of a set of propositions we advance arguments. Consider everyday life: if we are having an argument with someone, we believe that they are wrong. A more logical way to say this is that we believe that their beliefs are inconsistent. In order to change their viewpoint or point out why they are wrong we advance an argument intended to show that belief A conflicts with belief B. Or if C is true, then you cannot believe that D.</p></div></li></ul></li><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Logical_truth_and_falsity.html">Logical truth and falsity</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p>Neither proposition can be true because the truth of the first clause is contradicted by the second. By the principle of <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Logical consistency"><a href="Logical_consistency.html">consistency</a></span></span>, it is not possible for both clauses to be true at once therefore the proposition, overall has the truth value of false.</p></div></li></ul></li><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Logical_possibility_and_necessity.html">Logical possibility and necessity</a></span>
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