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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">Validity and entailment</h1><h3 id="informal-definition">Informal definition</h3><p>In order to say whether an argument is ‘good’ or ‘bad’ we must have criteria of evaluation. in logic there are different criteria of evaluation:</p><ul><li><p><strong>Deductive validity</strong></p><p>An <strong>argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false</strong>. Linking to consistency: it is not possible to consistently assert all of the premises but deny the conclusion.</p></li><li><p><strong>Inductive strength</strong></p><p>We do not say that inductive arguments have ‘validity’ because despite inductive premises being true, the conclusion may be falsifiable. Therefore we say inductive ‘strength’ rather than ‘validity’. An argument is inductively strong if and only if the conclusion is probably true given the premises.</p></li></ul><h4 id="demonstration">Demonstration</h4><p>The Socrates demonstration above is an example of deductive validity.</p><p>The following is an example of an argument that is inductively strong:</p><pre><code class="language-none">99% of deaf persons have no musical talent.
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Beethoven was deaf.
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___________________________________________
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Beethoven had no musical talent.</code></pre><p>The test for a strong inductive argument is not whether the conclusion is true, rather it concerns the evidence the premises provide in support of the conclusion.</p><blockquote><p>In propositional logic we are concerned solely with deductive validity or invalidity.</p></blockquote><h3 id="formal-definition">Formal definition</h3><blockquote><p>An argument is truth-functionally valid if and only if there is no truth-assignment on which all the premises are true and the conclusion is false.</p></blockquote><p>Linking this to derivation, we say:</p><blockquote><p>In a system of derivation in propositional logic, an argument is valid if the conclusion of the argument is derivable within the system of derivation from the set consisting of the premises, and invalid otherwise.</p></blockquote><h3 id="demonstration-1">Demonstration</h3><p>The inference from the set <span class="math inline">\({P, P \rightarrow Q}\)</span> to <span class="math inline">\(Q\)</span> is valid</p><h3 id="truth-table">Truth-table</h3><table class="ui table"><thead><tr><th><span class="math inline">\(P\)</span></th><th><span class="math inline">\(Q\)</span></th><th><span class="math inline">\(P \rightarrow Q\)</span></th><th><span class="math inline">\(P\)</span></th><th><span class="math inline">\(Q\)</span></th><th>Assessment</th></tr></thead><tbody><tr><td>T</td><td>T</td><td>T</td><td>T</td><td>T</td><td>Valid</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td><td></td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td><td></td></tr></tbody></table><h2 id="entailment">Entailment</h2><h3 id="informal-definition-1">Informal definition</h3><p>Entailment as a concept is almost identical to validity. We say that a proposition is entailed by a set of propositions if it is not possible for every member of this set to be true and the proposition to be false.</p><p>The difference with validity resides in the fact that the propositions are distinguished in terms of whether they are premises or a conclusion. So, technically, validity is a subclass of entailment. A case of entailment where we distinguish propositions in terms of whether they are premises or conclusions. A proposition may be entailed by a given set without that proposition being the <em>conclusion</em> of the set and where the set is a syllogism.</p><h3 id="formal-definition-1">Formal definition</h3><blockquote><p>A finite set of sentences <span class="math inline">\(\Gamma\)</span> <span class="math inline">\(\vdash\)</span> <span class="math inline">\(P\)</span> if and only if there is no truth-assignment in which every member of <span class="math inline">\(\Gamma\)</span> is true and <span class="math inline">\(P\)</span> is false.</p></blockquote><h4 id="informal-demonstration">Informal demonstration</h4><pre><code class="language-none">It is raining.
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If it is raining then the pavement will be wet.
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The pavement is wet.</code></pre><h4 id="formal-demonstration">Formal demonstration</h4><p><span class="math display">$$
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\{ P, P\rightarrow Q \} \vdash Q
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$$</span></p><h4 id="truth-table-1">Truth-table</h4><table class="ui table"><thead><tr><th><span class="math inline">\(P\)</span></th><th><span class="math inline">\(Q\)</span></th><th><span class="math inline">\(P \rightarrow Q\)</span></th><th><span class="math inline">\(P\)</span></th><th><span class="math inline">\(Q\)</span></th><th>Assessment</th></tr></thead><tbody><tr><td>T</td><td>T</td><td>T</td><td>T</td><td>T</td><td>Valid</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td><td>F</td><td></td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td><td></td></tr></tbody></table></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="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>If an argument contains premises which are logically false than this argument will perforce be valid. This is because one cannot consistently assert the premises and deny the conclusion which is the definition of <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Validity and entailment"><a href="Validity_and_entailment.html">validity</a></span></span>. However the <em>reason</em> why one cannot consistently assert the premises and deny the conclusions is because one cannot consistently assert the premises - they conflict with each other. Furthermore as the argument contains false premises, it cannot be sound.</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></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p>In distinguishing the properties of <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Logical consistency"><a href="Logical_consistency.html">logical consistency</a></span></span> and <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Validity and entailment"><a href="Validity_and_entailment.html">validity</a></span></span> we make tacit use of the notion of <strong>possibility</strong>. This is because when we consider the validity of an argument we are assessing truth-conditions and this consists in asking ourselves what could or could not be the case: were it such that <em>P</em>, then it would be the case that <em>Q</em>. It is important to understand what possibility means in the context of logic and how it differs from what we might mean ordinarily when we use the term.</p></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> |