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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 trees</h1><h2 id="rationale">Rationale</h2><p>Like <span class="zettel-link-container errors"><span class="zettel-link" title="Wiki-link does not refer to any existing zettel"><a>Truth-tables</a></span></span>, truth-trees are a means of graphically representing the logical relationships that may obtain between propositions. Truth-trees and truth-tables complement each other and which method you choose depends on which logical property you are seeking to derive.</p><p>Whilst truth-tables have the benefit of being exhaustive - every possible truth assignment is factored into the representation - their complexity grows exponentially with each additional proposition they contain. This can make manually constructing truth tables long-winded and prone to mistakes.</p><p>Truth-trees are less onerous but they lack the exhaustive scope of a truth-table. They are more targeted and are best used for demonstrating <em>that something is the case</em> rather than <em>all the possible states that could be the case</em>. For example, a truth tree will tell us that a set <em>S is logically consistent</em> whereas a truth-table will tell us that <em>S is consistent on the following three assignments.</em></p><h2 id="logical-consistency">Logical consistency</h2><p>Recall that a set of propositions is logically or truth-functionally <span class="zettel-link-container errors"><span class="zettel-link" title="Wiki-link does not refer to any existing zettel"><a>Consistency</a></span></span> just if there is at least one assignment of truth conditions which results in all members of the set being true. To identify consistency for a set of three propositions via the truth table approach we would need to construct a truth table with <span class="math inline">\(2^3\)</span> (8) rows. Assume that this set is consistent on one partial assignment only. This means that 87.5% of our rows are redundant, they are not required to prove the consistency of the set. However we can only know this and we can only be sure of consistency once we have gone through the process of generating an assignment for each row.</p><p>Truth trees allow us to reduce the amount of work required and go straight to the assignment that proves consistency, disregarding the rest which are irrelevant.</p><h2 id="truth-tree-structure-and-key-terms">Truth tree structure and key terms</h2><p><strong>When using a truth tree to derive logical consistency, the goal is to determine whether there is a truth-value assignment on which all of the sentences of a set are true. If the set is consistent we should be able to derive a partial assignment from the tree that demonstrates consistency.</strong></p><p>Each truth tree begins with a series of sentences one on top of the other in a column. We call the sentences that comprise the initial column <strong>set members</strong>. In constructing the tree, we work downwards from the initial column decomposing set members into their atomic constituents. We a call an atomic sentence that has been decomposed a <strong>literal.</strong> A literal will either be an atomic sentence or the negation of an atomic sentence. If one of the set members is already a literal, there is no need to decompose it; it can remain as it is.</p><p>Once every set member has been decomposed the truth tree is complete. It can then be interpreted in order to derive logical consistency or inconsistency. If the set is consistent, we are able to derive the partial assignment(s) that demonstrate consistency.</p><p>The rules for decomposing compound sentences match the truth conditions of the logical connectives. There are rules for every possible connective and the negation of every possible connec
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F F T F</code></pre><p><strong>Any time there is an open tree with a closed branch it will be the case that the negated sentences of the closed branch will appear both as</strong> <span class="math inline">\(S\)</span> and <span class="math inline">\(\sim S\)</span> i<strong>n the resultant assignment.</strong></p><p>Invoking the truth-table highlights the differences between the two techniques. The values that are derived when we interpret a truth tree are not the truth-functions of the set members but the truth-values for when they are simultaneously true. With truth-tables in contrast, we are deriving the truth functions for every possible truth-value assignment. In other words the values derived from a truth tree correspond to the left hand side of the truth table not the right hand side.</p><h3 id="second-example">Second example</h3><p>The following is a truth tree for the set <span class="math inline">\({A & \sim B, C, \sim A \lor \sim B }\)</span>.</p><p><img alt="basic-closed-tree 1.svg" src="/static/basic-closed-tree%201.svg" /></p><h3 id="interpretation-1">Interpretation</h3><ul><li>The two molecular set members are decomposed. The disjunction (line 3) results in a branching tree. The conjunction (line 1) results in the continuation of the trunk.</li><li>Both branches are completed making it a completed tree. As each branch is closed this is a closed tree.</li></ul><p>As this is a closed tree, the set is not truth-functionally consistent. This is confirmed by the truth table where there is no partial assignment where all set members are true.</p><pre><code class="language-none">A B C A & ~ B C ~ A ∨ ~ C
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F F F F F T</code></pre><h2 id="truth-tree-decomposition-rules">Truth tree decomposition rules</h2><hr /><p>So far we have encountered the decomposition rules for conjunction (<code>&D</code>) and disjunction (<code>vD</code>). We will now list all the rules. We will see that for each rule, the decomposition either branches or does not branch which is to say that each rule either has the shape of a conjunction or a disjunction (however the permitted values of the specific disjuncts/conjuncts obviously differ in each case). Moreover there is a parallel rule for the decomposition of the negation of each of the main connectives and these rules rely on logical equivalences</p><h3 id="negated-negation-decomposition-d">Negated negation decomposition: <code>~~D</code></h3><p><img alt="negated-negation-decomposition-rule 2.svg" src="/static/negated-negation-decomposition-rule%202.svg" /></p><p>Truth passes only if <span class="math inline">\(P\)</span> is true</p><h3 id="conjunction-decomposition-d">Conjunction decomposition: <code>&D</code></h3><p><img alt="conjunction-decomposition-rule.svg" src="/static/conjunction-decomposition-rule.svg" /></p><p>Truth passes only <span class="math inline">\(P\)</span> and <span class="math inline">\(Q\)</span> are both true.</p><h3 id="negated-conjunction-decomposition-d">Negated Conjunction decomposition: <code>~&D</code></h3><p><img alt="negated-conjunction-decomposition-rule.svg" src="/static/negated-conjunction-decomposition-rule.svg" /></p><p>Truth passes if either <span class="math inline">\(\sim P\)</span> or <span class="math inline">\(\sim Q\)</span> is true. This rule is a consequence of the equivalence between <span class="math inline">\(\sim (P & Q)\)</span> and <span class="math inline">\(\sim P \lor \sim Q\)</span> , the first of DeMorgan’s Laws.</p><h3 id="disjunction-decomposition-vd">Disjunction decomposition: <code>vD</code></h3><p><img alt="disjunction-decomposition-rule.svg" src="/static/disjunction-decomposition-rule.svg" /></p><p>Truth passes if either <span class="math inline">\(P\)</span>or <span class="math inline">\(Q\)</span> are true.</p><h3 id="negated-disjunction-decomposition-vd">Negated Disjunction decomposition: <code>~vD</code></h3><p><img alt="negated-disjunction-decomposition-rule.svg" src="/static/negated-disjunction-decomposition-rule.svg" /></p><p>Truth passes if both <span class="math inline">\(P\)</span> and <span class="math inline">\(Q\)</span> are false. This rule is a consequence of the equivalence between <span class="math inline">\(\sim (P \lor Q)\)</span> and <span class="math inline">\(\sim P & \sim Q\)</span>, the second of DeMorgan’s Laws.</p><h3 id="conditional-decomposition-d">Conditional decomposition: <code>⊃D</code></h3><p><img alt="conditional-decomposition-rule.svg" src="/static/conditional-decomposition-rule.svg" /></p><p>Truth passes if either <span class="math inline">\(\sim P\)</span> or <span class="math inline">\(Q\)</span> are true. This rule is a consequence of the equivalence between <span class="math inline">\(P \supset Q\)</span> and <span class="math inline">\(\sim P \lor Q\)</span> therefore this branch has the shape of a disjunction with <span class="math inline">\(\sim P\)</span> , <span class="math inline">\(Q\)</span> as its disjuncts.</p><h3 id="negated-conditional-decomposition-d">Negated Conditional decomposition: <code>~⊃D</code></h3><p>Truth passes if both <span class="math inline">\(P\)</span> and <span class="math inline">\(\sim Q\)</span> are true. This is a consequence of the equivalence between <span class="math inline">\(\sim (P \supset Q)\)</span> and <span class="math inline">\(P & \sim Q\)</span>.</p><p><img alt="negated-conditional-decomposition-rule.svg" src="/static/negated-conditional-decomposition-rule.svg" /></p><h3 id="biconditional-decomposition-d">Biconditional decomposition: <code>≡D</code></h3><p><img alt="biconditional-decomposition-rule.drawio(1).svg" src="/static/biconditional-decomposition-rule.drawio%281%29.svg" /></p><p>Truth passes if ei
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