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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">Truth-tables</h1><p>We are already familiar with truth-tables from the previous entry on the <em>truth-functional connectives</em> and the relationship between sentences, connectives and the overall truth-value of a sentence. Here we will look in further depth at how to build truth-tables and on their mathematical relation to binary truth-values. We will also look at examples of complex truth-tables for large compound expressions and the systematic steps we follow to derive the truth conditions of compound sentences from their simple constituents.</p><h2 id="formulae-for-constructing-truth-tables">Formulae for constructing truth-tables</h2><p>For any truth-table, the number of rows it will contain is equal to <span class="math inline">\(2n\)</span> where:</p><ul><li><span class="math inline">\(n\)</span> stands for the number of sentences</li><li><span class="math inline">\(2\)</span> is the total number of possible truth values that the sentence may have: true or false.</li></ul><p>When we count the number of sentences, we mean atomic sentences. And we only count each sentence once. Hence for a compound sentence of the form <span class="math inline">\((\sim B \supset C) & (A \equiv B)\)</span>, <span class="math inline">\(B\)</span> occurs twice but there are only three sentences: <span class="math inline">\(A\)</span>, <span class="math inline">\(B\)</span>, and <span class="math inline">\(C\)</span>.</p><p>Thus for the sentence <span class="math inline">\(P & Q\)</span> ,we have two sentences so <span class="math inline">\(n\)</span> is 2 which equals 4 rows (2 x 2):</p><pre><code class="language-none">P Q P & Q
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T T T
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T F F
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F T F
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F F F</code></pre><p>For the sentence <span class="math inline">\((P \lor Q) & R\)</span> we have three sentences so <span class="math inline">\(n\)</span> is 3 which equals 8 rows (2 x 2 x 2):</p><pre><code class="language-none">P Q R ( P ∨ Q ) & R
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T T T T
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T T F F
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T F T T
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T F F F
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F T T T
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F T F F
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F F T F
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F F F F</code></pre><p>For the single sentence <span class="math inline">\(P\)</span> we have one sentence so <span class="math inline">\(n\)</span> is 1 which equals 2 rows (2 x 1):</p><pre><code class="language-none">P P
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T T
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F F</code></pre><p>This tells us how many rows the truth-table should have but it doesn’t tell us what each row should consist in. In other words: how many Ts and Fs it should contain. This is fine with simple truth-tables since we can just alternate each value but for truth-tables with three sentences and more it is easy to make mistakes.</p><p>To simplify this and ensure that we are including the right number of possible truth-values we can extend the formula to <span class="math inline">\(2n^-i\)</span>. This formula tells us how many groups of T and F we should have in each column.</p><p>We can already see that there is a pattern at work by looking at the columns of the truth tables above. If we take the sentence <span class="math inline">\((P \lor Q) & R\)</span> we can see that for each sentence:</p><ul><li><span class="math inline">\(P\)</span> consists in two sets of <span class="math inline">\({\textsf{T,T,T,T}}\)</span> and <span class="math inline">\({\textsf{F,F,F,F}}\)</span> with <strong>four</strong> elements per set</li><li><span class="math inline">\(Q\)</span> consists in four sets of <span class="math inline">\({\textsf{T,T}}\)</span> , <span class="math inline">\({\textsf{F,F}}\)</span>, <span class="math inline">\({\textsf{T,T}}\)</span> , <span class="math inline">\({\textsf{F,F}}\)</span> with <strong>two</strong> elements per set</li><li><span class="math inline">\(R\)</span> consists in eight sets of <span class="math inline">\({\textsf{T}}\)</span>, <span class="math inline">\({\textsf{F}}\)</span>, <span class="math inline">\({\textsf{T}}\)</span>, <span class="math inline">\({\textsf{F}}\)</span>, <span class="math inline">\({\textsf{T}}\)</span>, <span class="math inline">\({\textsf{F}}\)</span>, <span class="math inline">\({\textsf{T}}\)</span>, <span class="math inline">\({\textsf{F}}\)</span> with <strong>one</strong> element per set.</li></ul><p>If we work through the formula we see that it returns 4, 2, 1:</p><p><span class="math display">$$\begin{equation} \begin{split} 2n^-1 = 3 -1 \\ = 2 \\ = 2 \cdot 2 \\ = 4 \end{split} \end{equation}$$</span></p><p><span class="math display">$$
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\\begin{equation} \begin{split} 2n^-2 = 3 - 2 \\ = 1 \\ = 2 \cdot 1 \\ = 2 \end{split} \end{equation}
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$$</span></p><p><span class="math display">$$
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\\begin{equation} \begin{split} 2n^-3 = 3 - 3 \\ = 0 \\ = 2 \cdot 0 \\ = 1 \end{split} \end{equation}
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$$</span></p><h2 id="truth-table-concepts">Truth-table concepts</h2><h3 id="recursion">Recursion</h3><p>When we move to complex truth-tables with more than one connective we realise that truth-tables are recursive. The truth-tables for the truth-functional connectives provide all that we need to determine the truth-values of complex sentences:</p><blockquote><p>The core truth-tables tell us how to determine the truth-value of a molecular sentence given the truth-values of its <a href="Syntax%20of%20sentential%20logic.md">immediate sentential components</a>. And if the immediate sentential components of a molecular sentence are also molecular, we can use the information in the characteristic truth-tables to determine how the truth-value of each immediate component depends n the truth-values of <em>its</em> components and so on.</p></blockquote><h3 id="truth-value-assignment">Truth-value assignment</h3><blockquote><p>A truth-value assignment is an assignment of truth-values (either T or F) to the atomic sentences of SL.</p></blockquote><p>When working on complex truth tables, we use the truth-assignment of atomic sentences to count as the values that we feed into the larger expressions at a higher level of the sentential abstraction.</p><h3 id="partial-assignment">Partial assignment</h3><p>We talk about partial assignments of truth-values when we look at one specific row of the truth-table, independently of the others. The total set of partial assignments comprise all possible truth assignments for the given sentence.</p><h2 id="working-through-complex-truth-tables">Working through complex truth-tables</h2><p>The truth-table below shows all truth-value assignments for the sentence <span class="math inline">\((\sim B \supset C) & (A \equiv B)\)</span> :</p><pre><code class="language-none">A B C ( ~ B ⊃ C ) & ( A ≡ B )
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T T T F T T T T T T T
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T T F F T T F T T T T
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T F T T F T T F T F F
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T F F T F F F F T F F
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F T T F T T T F F F T
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F T F F T T F F F F T
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F F T T F T T T F T F
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F F F T F F F F F T F</code></pre><p>As with algebra we work outwards from each set of brackets. The sequence for manually arriving at the above table would be roughly as follows:</p><ol><li>For each sentence letter, copy the truth value for it in each row.</li><li>Identify the connectives in the atomic sentences and the main overall sentence.</li><li>Work out the truth-values for the smallest connectives and sub-compound sentences. The first should always be negation and then the other atomic connectives.</li><li>Feed-in the truth-values of the atomic sentences as values into the main connective, through a process of elimination you then reach the core truth-assignments:</li></ol></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>
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