98 lines
No EOL
23 KiB
HTML
98 lines
No EOL
23 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>Truth-functional connectives - 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="Propositions generated from other (simple) propositions by means of propositional connectives are compound propositions." name="description" /><meta content="Truth-functional connectives" property="og:title" /><meta content="My Zettelkasten" property="og:site_name" /><meta content="article" property="og:type" /><meta content="Truth_functional_connectives" property="neuron:zettel-id" /><meta content="Truth_functional_connectives" 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">Truth-functional connectives</h1><p>Propositions generated from other (simple) propositions by means of propositional connectives are <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Atomic and molecular propositions"><a href="Atomic_and_molecular_propositions.html">compound propositions</a></span></span>.</p><p>We know that <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Logical indeterminacy"><a href="Logical_indeterminacy.html">logically determinant</a></span></span> propositions express a truth value. When simple propositions are joined with a connective to make a compound proposition they also have a truth value. This is determined by the nature of the connective and the truth value of the constituent propositions. We therefore call connectives of this nature truth <em>functional</em> connectives since the <strong>truth value of the compound is a function of the truth values of its components</strong>.</p><blockquote><p>A propositional connective is used truth-functionally if and only if it is used to generate a compound proposition from one or more propositions in such a way that the truth value of the generated compound is wholly determined by the truth-values of those one or more propositions from which the compound is generated, no matter what the truth values may be.</p></blockquote><p>Each truth-functional connective has a characteristic <strong>truth table</strong>. This discloses the conditions under which the constituent propositions have a given truth value when combined with one or more connectives.</p><p>We shall now review each of the truth-functional connectives in detail.</p><h3 id="conjunction">Conjunction</h3><p>Conjunction is equivalent to the word AND in natural language. We use <span class="math inline">\(\land\)</span> as the symbol for this connective.</p><p>A molecular proposition joining two conjuncts P and Q is true iff both conjuncts are true and false otherwise:</p><pre><code class="language-none">P Q P & Q
|
||
T T T
|
||
T F F
|
||
F T F
|
||
F F F</code></pre><h3 id="disjunction">Disjunction</h3><p>Conjunction is equivalent to the word OR in natural language. We use <code>v</code> as the symbol of this connective.</p><p>A molecular proposition joining two disjuncts P and Q is true if either disjunct is true or if both disjuncts are true and false otherwise. This corresponds to the inclusive sense of OR in natural language.</p><pre><code class="language-none">P Q P ∨ Q
|
||
T T T
|
||
T F T
|
||
F T T
|
||
F F F</code></pre><h3 id="negation">Negation</h3><p>In contrast to the two previous connectives, negation is a unary connective not a binary connective. We use <code>~</code> to symbolise negation. It does not join two or more propositions, it applies to one proposition as a whole. This can be a simple proposition or a complex proposition. It simply negates the truth-value of whichever proposition it is applied to. Hence applied to P, it is true if P is false. And if P is false, it is true when P is true. !</p><pre><code class="language-none">P ~ P
|
||
T F
|
||
F T</code></pre><h3 id="material-conditional-aka-implication">Material conditional (a.k.a implication)</h3><p>The material conditional approximates the meaning expressed in natural language when we say <em>if</em> such-and-such is the case <em>then</em> such-and-such will the case. Another way of expressing the sense of the material conditional is to say that <strong>P</strong> implies <strong>Q.</strong></p><pre><code class="language-none">If it rains today the pavement will be wet.</code></pre><p>We call the proposition that expresses the ‘if’ proposition the <strong>antecedent</strong> and the proposition that expresses the ‘then’ statement the <strong>consequent</strong>. The symbol we use to represent the material conditional is <code>⊃</code> although you may see <code>→</code> used as well.</p><p>The truth table is as follows:</p><pre><code class="language-none">P Q P ⊃ Q
|
||
T T T
|
||
T F F
|
||
F T T
|
||
F F T</code></pre><p>The material conditional is perhaps the least intuitive of the logical connectives. The first case (TT) closely matches what we expect the connective to mean: it has rained so the pavement is wet. The antecedent is true and therefore the consequent is true. This chimes with what we tend to mean by ‘if’ in natural language. In the second case (TF) it also makes sense: the complex proposition is false because it rained and the pavement wasn’t wet: this negates the truth of the expression. The final case (FF) is also straight forward. It didn’t rain therefore the pavement wasn’t wet, thus the overall assertion that rain implies wet pavements is retained.</p><p>FT is less intuitive:</p><pre><code class="language-none">It did not rain today. The pavement was wet.</code></pre><p>To some degree one just has to take these statements as axioms, whether or not they have intuitive sense is a secondary, more philosophical question. The semantic issues arise because we tacitly assume the material conditional to be a causal connective: there is something about the nature of <strong>P</strong> that <em>engenders</em> or <em>brings about</em> <strong>Q</strong> but causality is not a logical concern.</p><p>If we instead just focus on the simple propositions that comprise the truth value it is more plausible. In the case of FT we can say it didn’t rain yet the pavement was wet does not stop the pavement being wet when it rains. The fact that I can pour a beer on the pavement thereby making it wet doesn’t stop or render false the idea that the rain can also make the pavement wet. The same explanation covers the FF case: it hasn’t rained and so the pavement is not wet does not contradict the assertion that when it rains the pavement will be wet.</p><p>Things are elucidated when we look at an equivalent expression of P ⊃ Q, ~P v Q:</p><pre><code class="language-none">P Q ~ P ∨ Q
|
||
T T T
|
||
T F F
|
||
F T T
|
||
F F T</code></pre><p>A disjunction is true whenever either disjunct is true so when both are false the overall expression is false, the same as with FT and FF with the material conditional.</p><h3 id="material-biconditional-aka-equivalence">Material biconditional (a.k.a equivalence)</h3><p>The material biconditional equates to the English expression ‘if and only if’, as a conditional connective it therefore avoids some of the perplexity aroused by its material cousin. In this scenario both antecedent and consequent have to be true for the overall expression to be true. If either is false the complex proposition is false. Other ways of expressing the semantics of this connective is to say that one proposition implies the other or that <strong>P</strong> and <strong>Q</strong> are equivalent.</p><pre><code class="language-none">If and only if James studies every day he will pass the exam.</code></pre><p>There is no possibility in which James passes the exam and has not studied every day. If he studies for three out of the seven days leading up to the exam he will not pass. Alternatively, there is no possibility that James studied every day yet failed the exam. The antecedent and consequent are locked, as indicated by the truth-table:</p><pre><code class="language-none">P Q P ≡ Q
|
||
T T T
|
||
T F F
|
||
F T F
|
||
F F T</code></pre><p>The last condition (FF) maybe requires some explanation: if he has not studied every day then he cannot have passed the exam. Therefore, to say that he will pass iff he studies every day is rendered true.</p><h2 id="combinations-of-truth-functional-connectives">Combinations of truth-functional connectives</h2><hr /><p>So far we have applied connectives to simple propositions. In so doing we generate complex propositions. However propositions and connectives are inherently generative: we can build more complex expressions from less complex parts, using more than one type of connective or several different connectives to make larger complex propositions and express more detailed logical conditions ans statements about the world.</p><p>For example the proposition:</p><pre><code class="language-none">Socrates was either a philosopher or a drinker but he wasn't a politician.</code></pre><p>Can be expressed with greater logical clarity as:</p><pre><code class="language-none">Socrates was a philosopher or Socrates was a drinker and Socrates was not a politician.</code></pre><p>Using P for ‘Socrates was a philosopher’, Q for ‘Socrates was a drinker’ and R for ‘Socrates was a politician’ we can express this symbolically as:</p><pre><code class="language-none">(P v Q) & ~R</code></pre><p>Which has the truth table:</p><pre><code class="language-none">P Q R ( P ∨ Q ) & ~ R
|
||
T T T F
|
||
T T F T
|
||
T F T F
|
||
T F F T
|
||
F T T F
|
||
F T F T
|
||
F F T F
|
||
F F F F</code></pre><p>Let’s walk through each case where S stands for the overall proposition.</p><ol><li>S is false if Socrates was a philosopher, a drinker and a politician.</li><li><strong>S is true if Socrates was a philosopher, a drinker but not a politician.</strong></li><li>S is false if Socrates was a philosopher, a politician but not a drinker.</li><li><strong>S is true if Socrates was a philosopher but not a drinker or politician.</strong></li><li>S is false if Socrates was not a philosopher but was a drinker and politician</li><li><strong>S is true if Socrates was not a philosopher or politician but was a drinker.</strong></li><li>S is false if Socrates was neither a philosopher or drinker but was a politician.</li><li>S is false if Socrates was neither a philosopher, drinker, or politician.</li></ol><p>If we look just at the true cases for simplicity, it becomes obvious that the truth value of the whole is a function of the truth-values of the parts.</p><p>At the highest level of generality the proposition is a conjunction with two disjuncts: <code>P v Q</code> and <code>~R</code> . Therefore, for the proposition to be true both conjuncts must be true. The first conjunct is true just if one of the subordinate disjuncts is true (Socrates is either a philosopher, a drinker, or both). The second conjunct is true just if Socrates is not a politician. Thus there is only one variation for the second conjunct (not being a politician) and two variations for the first conjunct (being a drinker/being a philosopher) hence there are three cases where the overall proposition is true.</p><h3 id="logical-equivalence">Logical equivalence</h3><p>Once we start working with complex propositions with more than one truth-functional connective it becomes clear that the same proposition expressed in natural language can be expressed formally more than one way and thus that in logical terms, both formal expressions are equivalent. We can prove this equivalence by comparing truth tables.</p><p>For example the proposition:</p><pre><code class="language-none">I am going to the shops and the gym.</code></pre><p>Can obviously be expressed formally as:</p><pre><code class="language-none">P & Q</code></pre><p>But also as:</p><pre><code class="language-none">~ (~P v ~Q)</code></pre><p>And we know this because the truth-tables are identical:consistency</p><pre><code class="language-none">P Q P & Q
|
||
T T T
|
||
T F F
|
||
F T F
|
||
F F F</code></pre><pre><code class="language-none">P Q ~ ( ~ P ∨ ~ Q )
|
||
T T T
|
||
T F F
|
||
F T F
|
||
F F F</code></pre><p>Another example of equivalent expressions:</p><pre><code class="language-none">Neither Watson or Sherlock Holmes is fond of criminals</code></pre><p>The first formalisation:</p><pre><code class="language-none">~P & ~Q</code></pre><p>Equivalent to:</p><pre><code class="language-none">~(P v Q)</code></pre><p>Again the truth-tables for verification:</p><pre><code class="language-none">P Q ~ P & ~ Q
|
||
T T F
|
||
T F F
|
||
F T F
|
||
F F T</code></pre><p><code>~P & ~Q</code></p><pre><code class="language-none">P Q ~ ( P ∨ Q )
|
||
T T F
|
||
T F F
|
||
F T F
|
||
F F T</code></pre><h3 id="important-equivalences">Important equivalences</h3><p>The example above is a key equivalence that you will encounter a lot especially when deriving formal proofs. It goes together with another one. We have noted them both below for future reference:</p><pre><code class="language-none">~P & ~Q = ~P v ~Q</code></pre><pre><code class="language-none">~P v ~Q = ~(P & Q)</code></pre><h2 id="enforcing-binary-connectives-through-bracketing">Enforcing binary connectives through bracketing</h2><p>If we had a proposition of the form</p><pre><code class="language-none">Socrates is man, is mortal and a philosopher.</code></pre><p>We could not write this as:</p><pre><code class="language-none">P & Q & R</code></pre><p>This would not be a well-formed proposition because at most truth functional connectives can only connect two simple propositions. It would not be possible to generate truth conditions for this proposition in its current form. Instead we introduce brackets to enforce a binary grouping of simple propositions. In this instance, the placement of the brackets does not affect the accurate interpretation of the truth conditions of the compound, so the following two formalisations are equivalent:</p><pre><code class="language-none">(P & Q) & R
|
||
|
||
P & (Q & R)</code></pre></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> |