63 lines
No EOL
13 KiB
HTML
63 lines
No EOL
13 KiB
HTML
<!DOCTYPE html><html><head><meta content="text/html; charset=utf-8" http-equiv="Content-Type" /><meta content="width=device-width, initial-scale=1" name="viewport" /><!--replace-start-0--><!--replace-start-5--><!--replace-start-8--><title>Soundness - My Zettelkasten</title><!--replace-end-8--><!--replace-end-5--><!--replace-end-0--><link href="https://cdn.jsdelivr.net/npm/fomantic-ui@2.8.7/dist/semantic.min.css" rel="stylesheet" /><link href="https://fonts.googleapis.com/css?family=Merriweather|Libre+Franklin|Roboto+Mono&display=swap" rel="stylesheet" /><!--replace-start-1--><!--replace-start-4--><!--replace-start-7--><link href="https://raw.githubusercontent.com/srid/neuron/master/assets/neuron.svg" rel="icon" /><meta content="Recall that in the definition of deductive validity we do not say: an argument is valid iff if the premises are true and the conclusion is true. We say if it is possible for the premises to be true. This is important: we are not interested in the actual truth of the premises or the conclusion." name="description" /><meta content="Soundness" property="og:title" /><meta content="My Zettelkasten" property="og:site_name" /><meta content="article" property="og:type" /><meta content="Soundness" property="neuron:zettel-id" /><meta content="Soundness" property="neuron:zettel-slug" /><meta content="logic" property="neuron:zettel-tag" /><meta content="propositional-logic" property="neuron:zettel-tag" /><script type="application/ld+json">[]</script><style type="text/css">body{background-color:#eeeeee !important;font-family:"Libre Franklin", serif !important}body .ui.container{font-family:"Libre Franklin", serif !important}body h1, h2, h3, h4, h5, h6, .ui.header, .headerFont{font-family:"Merriweather", sans-serif !important}body code, pre, tt, .monoFont{font-family:"Roboto Mono","SFMono-Regular","Menlo","Monaco","Consolas","Liberation Mono","Courier New", monospace !important}body div.z-index p.info{color:#808080}body div.z-index ul{list-style-type:square;padding-left:1.5em}body div.z-index .uplinks{margin-left:0.29999em}body .zettel-content h1#title-h1{background-color:rgba(33,133,208,0.1)}body nav.bottomPane{background-color:rgba(33,133,208,2.0e-2)}body div#footnotes{border-top-color:#2185d0}body p{line-height:150%}body img{max-width:100%}body .deemphasized{font-size:0.94999em}body .deemphasized:hover{opacity:1}body .deemphasized:not(:hover){opacity:0.69999}body .deemphasized:not(:hover) a{color:#808080 !important}body div.container.universe{padding-top:1em}body div.zettel-view ul{padding-left:1.5em;list-style-type:square}body div.zettel-view .pandoc .highlight{background-color:#ffff00}body div.zettel-view .pandoc .ui.disabled.fitted.checkbox{margin-right:0.29999em;vertical-align:middle}body div.zettel-view .zettel-content .metadata{margin-top:1em}body div.zettel-view .zettel-content .metadata div.date{text-align:center;color:#808080}body div.zettel-view .zettel-content h1{padding-top:0.2em;padding-bottom:0.2em;text-align:center}body div.zettel-view .zettel-content h2{border-bottom:solid 1px #4682b4;margin-bottom:0.5em}body div.zettel-view .zettel-content h3{margin:0px 0px 0.4em 0px}body div.zettel-view .zettel-content h4{opacity:0.8}body div.zettel-view .zettel-content div#footnotes{margin-top:4em;border-top-style:groove;border-top-width:2px;font-size:0.9em}body div.zettel-view .zettel-content div#footnotes ol > li > p:only-of-type{display:inline;margin-right:0.5em}body div.zettel-view .zettel-content aside.footnote-inline{width:30%;padding-left:15px;margin-left:15px;float:right;background-color:#d3d3d3}body div.zettel-view .zettel-content .overflows{overflow:auto}body div.zettel-view .zettel-content code{margin:auto auto auto auto;font-size:100%}body div.zettel-view .zettel-content p code, li code, ol code{padding:0.2em 0.2em 0.2em 0.2em;background-color:#f5f2f0}body div.zettel-view .zettel-content pre{overflow:auto}body div.zettel-view .zettel-content dl dt{font-weight:bold}body div.zettel-view .zettel-content blockquote{background-color:#f9f9f9;border-left:solid 10px #cccccc;margin:1.5em 0px 1.5em 0px;padding:0.5em 10px 0.5em 10px}body div.zettel-view .zettel-content.raw{background-color:#dddddd}body .ui.label.zettel-tag{color:#000000}body .ui.label.zettel-tag a{color:#000000}body nav.bottomPane ul.backlinks > li{padding-bottom:0.4em;list-style-type:disc}body nav.bottomPane ul.context-list > li{list-style-type:lower-roman}body .footer-version img{-webkit-filter:grayscale(100%);-moz-filter:grayscale(100%);-ms-filter:grayscale(100%);-o-filter:grayscale(100%);filter:grayscale(100%)}body .footer-version img:hover{-webkit-filter:grayscale(0%);-moz-filter:grayscale(0%);-ms-filter:grayscale(0%);-o-filter:grayscale(0%);filter:grayscale(0%)}body .footer-version, .footer-version a, .footer-version a:visited{color:#808080}body .footer-version a{font-weight:bold}body .footer-version{margin-top:1em !important;font-size:0.69999em}@media only screen and (max-width: 768px){body div#zettel-container{margin-left:0.4em !important;margin-right:0.4em !important}}body span.zettel-link-container span.zettel-link a{color:#2185d0;font-weight:bold;text-decoration:none}body span.zettel-link-container span.zettel-link a:hover{background-color:rgba(33,133,208,0.1)}body span.zettel-link-container span.extra{color:auto}body span.zettel-link-container.errors{border:solid 1px #ff0000}body span.zettel-link-container.errors span.zettel-link a:hover{text-decoration:none !important;cursor:not-allowed}body [data-tooltip]:after{font-size:0.69999em}body div.tag-tree div.node{font-weight:bold}body div.tag-tree div.node a.inactive{color:#555555}body .tree.flipped{-webkit-transform:rotate(180deg);-moz-transform:rotate(180deg);-ms-transform:rotate(180deg);-o-transform:rotate(180deg);transform:rotate(180deg)}body .tree{overflow:auto}body .tree ul.root{padding-top:0px;margin-top:0px}body .tree ul{position:relative;padding:1em 0px 0px 0px;white-space:nowrap;margin:0px auto 0px auto;text-align:center}body .tree ul::after{content:"";display:table;clear:both}body .tree ul:last-child{padding-bottom:0.1em}body .tree li{display:inline-block;vertical-align:top;text-align:center;list-style-type:none;position:relative;padding:1em 0.5em 0em 0.5em}body .tree li::before{content:"";position:absolute;top:0px;right:50%;border-top:solid 2px #cccccc;width:50%;height:1.19999em}body .tree li::after{content:"";position:absolute;top:0px;right:50%;border-top:solid 2px #cccccc;width:50%;height:1.19999em}body .tree li::after{right:auto;left:50%;border-left:solid 2px #cccccc}body .tree li:only-child{padding-top:0em}body .tree li:only-child::after{display:none}body .tree li:only-child::before{display:none}body .tree li:first-child::before{border-style:none;border-width:0px}body .tree li:first-child::after{border-radius:5px 0px 0px 0px}body .tree li:last-child::after{border-style:none;border-width:0px}body .tree li:last-child::before{border-right:solid 2px #cccccc;border-radius:0px 5px 0px 0px}body .tree ul ul::before{content:"";position:absolute;top:0px;left:50%;border-left:solid 2px #cccccc;width:0px;height:1.19999em}body .tree li div.forest-link{border:solid 2px #cccccc;padding:0.2em 0.29999em 0.2em 0.29999em;text-decoration:none;display:inline-block;border-radius:5px 5px 5px 5px;color:#333333;position:relative;top:2px}body .tree.flipped li div.forest-link{-webkit-transform:rotate(180deg);-moz-transform:rotate(180deg);-ms-transform:rotate(180deg);-o-transform:rotate(180deg);transform:rotate(180deg)}</style><script
|
|
async=""
|
|
id="MathJax-script"
|
|
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
|
|
></script>
|
|
<link
|
|
href="https://cdnjs.cloudflare.com/ajax/libs/prism/1.23.0/themes/prism.min.css"
|
|
rel="stylesheet"
|
|
/><link rel="preconnect" href="https://fonts.googleapis.com" /><link
|
|
rel="preconnect"
|
|
href="https://fonts.gstatic.com"
|
|
crossorigin
|
|
/><link
|
|
href="https://fonts.googleapis.com/css2?family=IBM+Plex+Mono:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&family=IBM+Plex+Sans+Condensed:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&family=IBM+Plex+Sans:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&family=IBM+Plex+Serif:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&display=swap"
|
|
rel="stylesheet"
|
|
/>
|
|
<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.23.0/components/prism-core.min.js"></script>
|
|
<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.23.0/plugins/autoloader/prism-autoloader.min.js"></script>
|
|
<style>
|
|
body .ui.container,
|
|
body ul {
|
|
font-family: "IBM Plex Sans" !important;
|
|
}
|
|
body blockquote {
|
|
border-left-width: 3px !important;
|
|
font-style: italic;
|
|
}
|
|
.headerFont,
|
|
.ui.header,
|
|
body h1,
|
|
h2,
|
|
h3,
|
|
h4,
|
|
h5,
|
|
h6 {
|
|
font-family: "IBM Plex Sans Condensed" !important;
|
|
}
|
|
body p {
|
|
line-height: 1.4;
|
|
}
|
|
.monoFont,
|
|
body code,
|
|
pre,
|
|
tt {
|
|
font-family: "IBM Plex Mono" !important;
|
|
font-size: 12px !important;
|
|
line-height: 1.4 !important;
|
|
}
|
|
</style>
|
|
<!--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">Soundness</h1><p>Recall that in the definition of <a href="Validity_and_entailment.md#validity">deductive validity</a> we do not say: an argument is valid iff if the premises <em>are true</em> and the conclusion <em>is true</em>. We say <em>if it is possible for the premises to be true</em>. This is important: we are not interested in the actual truth of the premises or the conclusion.</p><h4 id="demonstration">Demonstration</h4><p>Therefore this argument is valid:</p><pre><code class="language-none">(P1) Oranges are the same colour as bananas.
|
|
(P2) Bananas are yellow.
|
|
____________________________________________
|
|
(C) Oranges are yellow.</code></pre><p>Of course oranges are not yellow but <em>were</em> (P1) true, then given (P2), the conclusion must also be true.</p><p>This argument is also valid:</p><pre><code class="language-none">(P1) Oranges are the same colour as carrots.
|
|
(P2) Carrots are orange.
|
|
____________________________________________
|
|
(C) Oranges are orange in colour.</code></pre><p>The difference here is that the premises happen to be true and, given that the argument is valid, the conclusion must also be true. What we have defined here is <strong>soundness</strong>: the argument is said to be sound as well as valid. This is an additional and stronger criterion of evaluation.</p><blockquote><p>An argument is sound if and only if it is deductively valid and all its premises are true.</p></blockquote><p>We must not forget that truth alone is not the sole condition for soundness. We can have arguments whose conclusion and premises are all true without the argument being sound:</p><pre><code class="language-none">(P1) London is the capital of the United Kingdom
|
|
(P2) The capital of the United Kingdom is in the southern part of the United Kingdom.
|
|
(P3) Cambridge is not the capital of the United Kingdom
|
|
____________________________________________
|
|
(C) London is south of Cambridge</code></pre><p>All propositions are true here but the argument is not deductively valid: the premises are all true but the conclusion is false.</p><p>We can also have arguments which are valid but which are not sound:</p><pre><code class="language-none">(P1) Vitamin C prevents colds.
|
|
(P2) Vitamin C does not prevent colds.
|
|
____________________________________________
|
|
(C) Vitamin C is harmless</code></pre><p>This argument is valid because we cannot consistently assert the premises and deny the conclusion. In either case, the conclusion can be said to follow from the premises. The problem is that we cannot consistently assert both premises: it is not possible for both propositions to be true at the same time.</p></div></article><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> |