False (logic)

<h2 id="in-classical-logic-and-boolean-logic">In classical logic and Boolean logic</h2>
<p>In <a href="/facts/Boolean_logic/VkCToAmg">Boolean logic</a>, each variable denotes a <a href="/facts/Truth_value/SY4e2w8l">truth value</a> which can be either true (1), or false (0).
</p><p>In a <a href="/facts/Classical_logic/ls62MAms">classical</a> <a href="/facts/Propositional_calculus/MNt5v31V">propositional calculus</a>, each <a href="/facts/Proposition/0877MFMD">proposition</a> will be assigned a truth value of either true or false.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Some systems of classical logic include dedicated symbols for false (0 or 
  
    
      
        ⊥
      
    
    {\displaystyle \bot }
  
), while others instead rely upon formulas such as p ∧ ¬p and ¬(p → p).
</p><p>In both Boolean logic and Classical logic systems, true and false are opposite with respect to <a href="/facts/Negation/q5AQBvBj">negation</a>; the negation of false gives true, and the negation of true gives false.
</p>
<h3><a href="/facts/Truth_table/4KsmArDW">Truth Tables</a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>:</h3>
<h4>Negation (¬):</h4>
<table><tbody><tr><th>                    x              {\displaystyle x}  </th><th>                    ¬        x              {\displaystyle \neg x}  </th></tr><tr><th>true</th><td>false</td></tr><tr><th>false</th><td>true</td></tr></tbody></table>
<p>The negation of false is equivalent to the truth not only in classical logic and Boolean logic, but also in most other logical systems, as explained below.
</p>
<h4>Conjunction (AND ∧):</h4>
<p>False ∧ True = False (False AND anything is False).
</p>
<h4>Disjunction (OR ∨):</h4>
<p>False ∨ True = True (OR is True if at least one operand is True).
</p>
<h4>Implication (→):</h4>
<p>False → True = True (A false premise makes the implication vacuously true).
</p>
<h2 id="false-negation-and-contradiction">False, negation and contradiction</h2>
<p>In most logical systems, <a href="/facts/Negation/q5AQBvBj">negation</a>, <a href="/facts/Material_conditional/hYf4b7AE">material conditional</a> and false are related as:
</p>
¬p ⇔ (p → ⊥)
<p>In fact, this is the definition of negation in some systems,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> such as <a href="/facts/Intuitionistic_logic/zI7ulCnM">intuitionistic logic</a>, and can be proven in propositional calculi where negation is a fundamental connective. Because p → p is usually a theorem or axiom, a consequence is that the negation of false (¬ ⊥) is true.
</p><p>A <a href="/facts/Contradiction/PQHH5a2J">contradiction</a> is the situation that arises when a <a href="/facts/Statement_(logic)/CjpWAyxk">statement</a> that is assumed to be true is shown to <a href="/facts/Entailment/AyDXp4dn">entail</a> false (i.e., φ ⊢ ⊥). Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from ⊢ ¬φ. A statement that entails false itself is sometimes called a contradiction, and contradictions and false are sometimes not distinguished, especially due to the <a href="/facts/Latin/pM3Kge85">Latin</a> term <i>falsum</i> being used in English to denote either, but false is one specific <a href="/facts/Proposition/0877MFMD">proposition</a>.
</p><p>Logical systems may or may not contain the <a href="/facts/Principle_of_explosion/qMyRl2Ya">principle of explosion</a> (<i>ex falso quodlibet</i> in <a href="/facts/Latin/pM3Kge85">Latin</a>), ⊥ ⊢ φ for all φ. By that principle, contradictions and false are equivalent, since each entails the other.
</p>
<h2 id="consistency">Consistency</h2>
<p class="note">Main article: <a href="/facts/Consistency/IHkdnA9M">Consistency</a></p>
<p>A <a href="/facts/Theory_(mathematical_logic)/mYe5gQpI">formal theory</a> using the "
  
    
      
        ⊥
      
    
    {\displaystyle \bot }
  
" connective is defined to be consistent, if and only if the false is not among its <a href="/facts/Theorem/f2Pe1FMD">theorems</a>. In the absence of propositional constants, some substitutes (such as the ones described above) may be used instead to define consistency.
</p>
<h2 id="see-also">See also</h2>

Wikiquote has quotations related to <i>Falsehood</i>.

<ul><li><a href="/facts/Contradiction/PQHH5a2J">Contradiction</a></li>
<li><a href="/facts/Logical_truth/N0bUYhUW">Logical truth</a></li>
<li><a href="/facts/Tautology_(logic)/p82wWMaN">Tautology (logic)</a> (for symbolism of logical truth)</li>
<li><a href="/facts/Truth_table/4KsmArDW">Truth table</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Jennifer Fisher, On the Philosophy of Logic, Thomson Wadsworth, 2007, ISBN 0-495-00888-5, p. 17. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Willard Van Orman Quine, Methods of Logic, 4th ed, Harvard University Press, 1982, ISBN 0-674-57176-2, p. 34. <a href="/wiki/Willard_Van_Orman_Quine" target="_blank">/wiki/Willard_Van_Orman_Quine</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Truth-value | logic". Encyclopedia Britannica. Retrieved 2020-08-15. <a href="https://www.britannica.com/topic/truth-value" target="_blank">https://www.britannica.com/topic/truth-value</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>George Edward Hughes and D.E. Londey, The Elements of Formal Logic, Methuen, 1965, p. 151. <a href="/wiki/George_Edward_Hughes" target="_blank">/wiki/George_Edward_Hughes</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Leon Horsten and Richard Pettigrew, Continuum Companion to Philosophical Logic, Continuum International Publishing Group, 2011, ISBN 1-4411-5423-X, p. 199. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Graham Priest, An Introduction to Non-Classical Logic: From If to Is, 2nd ed, Cambridge University Press, 2008, ISBN 0-521-85433-4, p. 105. <a href="/wiki/Graham_Priest" target="_blank">/wiki/Graham_Priest</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Aristotle (Organon) <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Barnes’ Complete Works of Aristotle (Princeton, 1984, Vol. 1):

De Int.: pp. 25–38; Metaphysics IV: pp. 1588–1595. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Gottlob Frege (1879, Begriffsschrift) <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Alfred Tarski (1930s, Introduction to Logic, Chapter II (Symbolic Logic)) <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Dov M. Gabbay and Franz Guenthner (eds), Handbook of Philosophical Logic, Volume 6, 2nd ed, Springer, 2002, ISBN 1-4020-0583-0, p. 12. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
</ol>

False (logic) open-in-new

False (logic)