Proof by example

<h2 id="valid-cases-of-proof-by-example">Valid cases of proof by example</h2>
<p>In certain circumstances, examples can suffice as <a href="/facts/Logically_valid/oMGkywGC">logically valid</a> proof.
</p>
<h3>Proofs of existential statements</h3>
<p>In some scenarios, an argument by example may be valid if it leads from a singular premise to an <i>existential</i> conclusion (i.e. proving that a claim is true for at least one case, instead of for all cases). For example:
</p>
Socrates is wise.
Therefore, someone is wise.
<p>(or)
</p>
I've seen a person steal.
Therefore, (some) people can steal.
<p>These examples outline the informal version of the logical rule known as <a href="/facts/List_of_rules_of_inference/kmoqX2vy">existential introduction</a>, also known as <i>particularisation</i> or <i>existential generalization</i>:
</p>
Existential Introduction

φ
              (
              β
              
                /
              
              α
              )
            
            _
          
        
        
        
      
    
    {\displaystyle {\underline {\varphi (\beta /\alpha )}}\,\!}

∃
        α
        
        φ
        
        
      
    
    {\displaystyle \exists \alpha \,\varphi \,\!}

<p>(where 
  
    
      
        φ
        (
        β
        
          /
        
        α
        )
      
    
    {\displaystyle \varphi (\beta /\alpha )}
  
 denotes the <a href="/facts/Formula_(logic)/YpTVlzuj">formula</a> formed by substituting all <a href="/facts/Free_occurrence/ndWnXXYs">free occurrences</a> of the variable 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 in 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 by 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
.)
</p><p>Likewise, finding a <a href="/facts/Counterexample/dnThYpHW">counterexample</a> disproves (proves the <a href="/facts/Logical_negation/q5AQBvBj">negation</a> of) a universal conclusion. This is used in a <a href="/facts/Proof_by_contradiction/vHtFzFQ5">proof by contradiction</a>.
</p>
<h3>Exhaustive proofs</h3>
<p class="note">Main article: <a href="/facts/Proof_by_exhaustion/bTuwLV7R">Proof by exhaustion</a></p>
<p>Examples also constitute valid, if <a href="/facts/Mathematical_elegance/HYwRap01">inelegant</a>, proof, when it has <i>also</i> been demonstrated that the examples treated cover all possible cases.
</p><p>In mathematics, proof by example can also be used to refer to attempts to illustrate a claim by proving cases of the claim, with the understanding that these cases contain key ideas which can be generalized into a full-fledged proof.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Affirming_the_consequent/dK0zDWFB">Affirming the consequent</a></li>
<li><a href="/facts/Anecdotal_evidence/r8mQyKXW">Anecdotal evidence</a></li>
<li><a href="/facts/Bayesian_probability/Cvn7WWfG">Bayesian probability</a></li>
<li><a href="/facts/Counterexample/dnThYpHW">Counterexample</a></li>
<li><a href="/facts/Hand-waving/SsMMAaHY">Hand-waving</a></li>
<li><a href="/facts/Inductive_reasoning/EJ992JJ9">Inductive reasoning</a>
<ul><li><a href="/facts/Problem_of_induction/4u6Lt35z">Problem of induction</a></li></ul></li>
<li><a href="/facts/Modus_ponens/Wq4NcmrG">Modus ponens</a></li>
<li><a href="/facts/Proof_by_construction/0QhhRjhW">Proof by construction</a></li>
<li><a href="/facts/Proof_by_intimidation/q40z7pcA">Proof by intimidation</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Benjamin Matschke: Valid proofs by example in mathematics (arXiv)</li></ul>

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

<ol>
<li id="fn:1"><p>"Logical fallacies". www.auburn.edu. Archived from the original on 31 July 2002. <a href="https://web.archive.org/web/20020731165349/http://www.auburn.edu/~marchjl/fallacies.htm" target="_blank">https://web.archive.org/web/20020731165349/http://www.auburn.edu/~marchjl/fallacies.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Proof by example – Grammarist". Grammarist. Retrieved 2019-12-01. <a href="https://grammarist.com/rhetoric/proof-by-example/" target="_blank">https://grammarist.com/rhetoric/proof-by-example/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Proof by example – Grammarist". Grammarist. Retrieved 2019-12-01. <a href="https://grammarist.com/rhetoric/proof-by-example/" target="_blank">https://grammarist.com/rhetoric/proof-by-example/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"What is Proof By Example?". Simplicable. Retrieved 2019-12-01. <a href="https://simplicable.com/new/proof-by-example" target="_blank">https://simplicable.com/new/proof-by-example</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Proofs". www.mit.edu. Retrieved 2019-12-01. <a href="http://www.mit.edu/people/dmredish/wwwMLRF/links/Humor/Math_Proofs.html" target="_blank">http://www.mit.edu/people/dmredish/wwwMLRF/links/Humor/Math_Proofs.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Proof by example open-in-new

Proof by example