Modus tollens

<h2 id="explanation">Explanation</h2>
<p>The form of a <i>modus tollens</i> argument is a mixed <a href="/facts/Hypothetical_syllogism/XMNClcnc">hypothetical syllogism</a>, with two premises and a conclusion:
</p>
If <i>P</i>, then <i>Q</i>.
Not <i>Q</i>.
Therefore, not <i>P</i>.
<p>The first premise is a <a href="/facts/Material_conditional/hYf4b7AE">conditional</a> ("if-then") claim, such as <i>P</i> implies <i>Q</i>. The second premise is an assertion that <i>Q</i>, the <a href="/facts/Consequent/UIzcuaKF">consequent</a> of the conditional claim, is not the case. From these two premises it can be logically concluded that <i>P</i>, the <a href="/facts/Antecedent_(logic)/8VVVqa06">antecedent</a> of the conditional claim, is also not the case.
</p><p>For example:
</p>
If the dog detects an intruder, the dog will bark.
The dog did not bark.
Therefore, no intruder was detected by the dog.
<p>Supposing that the premises are both true (the dog will bark if he or she detects an intruder, and does indeed not bark), it logically follows that no intruder has been detected. This is a valid argument since it is not possible for the conclusion to be false if the premises are true. (It is conceivable that there may have been an intruder that the dog did not detect, but that does not invalidate the argument; the first premise is "if the dog <i>detects</i> an intruder". The thing of importance is that the dog detects or does not detect an intruder, not whether there is one.)
</p><p>Example 1:
</p>
If I am the burglar, then I can crack a safe.
I cannot crack a safe.
Therefore, I am not the burglar.
<p>Example 2: 
</p>
If Rex is a chicken, then he is a bird.
Rex is not a bird.
Therefore, Rex is not a chicken.
<h2 id="relation-to-modus-ponens">Relation to <i>modus ponens</i></h2>
<p>Every use of <i>modus tollens</i> can be converted to a use of <i><a href="/facts/Modus_ponens/Wq4NcmrG">modus ponens</a></i> and one use of <a href="/facts/Transposition_(logic)/lJaaRrxT">transposition</a> to the premise which is a material implication. For example:
</p>
If <i>P</i>, then <i>Q</i>. (premise – material implication)
If not <i>Q</i>, then not <i>P</i>. (derived by transposition)
Not <i>Q</i> . (premise)
Therefore, not <i>P</i>. (derived by <i>modus ponens</i>)
<p>Likewise, every use of <i>modus ponens</i> can be converted to a use of <i>modus tollens</i> and transposition.
</p>
<h2 id="formal-notation">Formal notation</h2>
<p>The <i>modus tollens</i> rule can be stated formally as:
</p>

P
              →
              Q
              ,
              ¬
              Q
            
            
              ∴
              ¬
              P
            
          
        
      
    
    {\displaystyle {\frac {P\to Q,\neg Q}{\therefore \neg P}}}

<p>where 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
 stands for the statement "P implies Q". 
  
    
      
        ¬
        Q
      
    
    {\displaystyle \neg Q}
  
 stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
" and "
  
    
      
        ¬
        Q
      
    
    {\displaystyle \neg Q}
  
" each appear by themselves as a line of a <a href="/facts/Formal_proof/6nEhV4cw">proof</a>, then "
  
    
      
        ¬
        P
      
    
    {\displaystyle \neg P}
  
" can validly be placed on a subsequent line.
</p><p>The <i>modus tollens</i> rule may be written in <a href="/facts/Sequent/7YoB2hYb">sequent</a> notation:
</p>

P
        →
        Q
        ,
        ¬
        Q
        ⊢
        ¬
        P
      
    
    {\displaystyle P\to Q,\neg Q\vdash \neg P}

<p>where 
  
    
      
        ⊢
      
    
    {\displaystyle \vdash }
  
 is a <a href="/facts/Metalogic/zfooQjCo">metalogical</a> symbol meaning that 
  
    
      
        ¬
        P
      
    
    {\displaystyle \neg P}
  
 is a <a href="/facts/Logical_consequence/AyDXp4dn">syntactic consequence</a> of 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
 and 
  
    
      
        ¬
        Q
      
    
    {\displaystyle \neg Q}
  
 in some <a href="/facts/Formal_system/KaeDAjK7">logical system</a>;
</p><p>or as the statement of a functional <a href="/facts/Tautology_(logic)/p82wWMaN">tautology</a> or <a href="/facts/Theorem/f2Pe1FMD">theorem</a> of propositional logic:
</p>

(
        (
        P
        →
        Q
        )
        ∧
        ¬
        Q
        )
        →
        ¬
        P
      
    
    {\displaystyle ((P\to Q)\land \neg Q)\to \neg P}

<p>where 
  
    
      
        P
      
    
    {\displaystyle P}
  
 and 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 are propositions expressed in some <a href="/facts/Formal_system/KaeDAjK7">formal system</a>;
</p><p>or including assumptions:
</p>

Γ
              ⊢
              P
              →
              Q
               
               
               
              Γ
              ⊢
              ¬
              Q
            
            
              Γ
              ⊢
              ¬
              P
            
          
        
      
    
    {\displaystyle {\frac {\Gamma \vdash P\to Q~~~\Gamma \vdash \neg Q}{\Gamma \vdash \neg P}}}

<p>though since the rule does not change the set of assumptions, this is not strictly necessary.
</p><p>More complex rewritings involving <i>modus tollens</i> are often seen, for instance in <a href="/facts/Set_theory/1ptfsvUP">set theory</a>:
</p>

P
        ⊆
        Q
      
    
    {\displaystyle P\subseteq Q}

x
        ∉
        Q
      
    
    {\displaystyle x\notin Q}

∴
        x
        ∉
        P
      
    
    {\displaystyle \therefore x\notin P}

<p>("P is a subset of Q. x is not in Q. Therefore, x is not in P.")
</p><p>Also in first-order <a href="/facts/Predicate_logic/lcISqWIg">predicate logic</a>:
</p>

∀
        x
        :
         
        P
        (
        x
        )
        →
        Q
        (
        x
        )
      
    
    {\displaystyle \forall x:~P(x)\to Q(x)}

¬
        Q
        (
        y
        )
      
    
    {\displaystyle \neg Q(y)}

∴
         
        ¬
        P
        (
        y
        )
      
    
    {\displaystyle \therefore ~\neg P(y)}

<p>("For all x if x is P then x is Q. y is not Q. Therefore, y is not P.")
</p><p>Strictly speaking these are not instances of <i>modus tollens</i>, but they may be derived from <i>modus tollens</i> using a few extra steps.
</p>
<h2 id="justification-via-truth-table">Justification via truth table</h2>
<p>The validity of <i>modus tollens</i> can be clearly demonstrated through a <a href="/facts/Truth_table/4KsmArDW">truth table</a>.
</p>
<table><tbody><tr><th>p</th><th>q</th><th>p → q</th></tr><tr><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td></tr></tbody></table>
<p>In instances of <i>modus tollens</i> we assume as premises that p → q is true and q is false. There is only one line of the truth table—the fourth line—which satisfies these two conditions. In this line, p is false. Therefore, in every instance in which p → q is true and q is false, p must also be false.
</p>
<h2 id="formal-proof">Formal proof</h2>
<h3>Via disjunctive syllogism</h3>
<table><tbody><tr><th><i>Step</i></th><th><i>Proposition</i></th><th><i>Derivation</i></th></tr><tr><td>1</td><td>                    P        →        Q              {\displaystyle P\rightarrow Q}  </td><td>Given</td></tr><tr><td>2</td><td>                    ¬        Q              {\displaystyle \neg Q}  </td><td>Given</td></tr><tr><td>3</td><td>                    ¬        P        ∨        Q              {\displaystyle \neg P\lor Q}  </td><td><a href="/facts/Material_implication_(rule_of_inference)/1WHxa8Se">Material implication</a> (1)</td></tr><tr><td>4</td><td>                    ¬        P              {\displaystyle \neg P}  </td><td><a href="/facts/Disjunctive_syllogism/cfnyp2wY">Disjunctive syllogism</a> (3,2)</td></tr></tbody></table>
<h3>Via <i>reductio ad absurdum</i></h3>
<table><tbody><tr><th><i>Step</i></th><th><i>Proposition</i></th><th><i>Derivation</i></th></tr><tr><td>1</td><td>                    P        →        Q              {\displaystyle P\rightarrow Q}  </td><td>Given</td></tr><tr><td>2</td><td>                    ¬        Q              {\displaystyle \neg Q}  </td><td>Given</td></tr><tr><td>3</td><td>                    P              {\displaystyle P}  </td><td>Assumption</td></tr><tr><td>4</td><td>                    Q              {\displaystyle Q}  </td><td><a href="/facts/Modus_ponens/Wq4NcmrG">Modus ponens</a> (1,3)</td></tr><tr><td>5</td><td>                    Q        ∧        ¬        Q              {\displaystyle Q\land \neg Q}  </td><td><a href="/facts/Conjunction_introduction/XGt4Ie0R">Conjunction introduction</a> (2,4)</td></tr><tr><td>6</td><td>                    ¬        P              {\displaystyle \neg P}  </td><td><i><a href="/facts/Reductio_ad_absurdum/5AJUtVmL">Reductio ad absurdum</a></i> (3,5)</td></tr></tbody></table>
<h3>Via contraposition</h3>
<table><tbody><tr><th><i>Step</i></th><th><i>Proposition</i></th><th><i>Derivation</i></th></tr><tr><td>1</td><td>                    P        →        Q              {\displaystyle P\rightarrow Q}  </td><td>Given</td></tr><tr><td>2</td><td>                    ¬        Q              {\displaystyle \neg Q}  </td><td>Given</td></tr><tr><td>3</td><td>                    ¬        Q        →        ¬        P              {\displaystyle \neg Q\rightarrow \neg P}  </td><td><a href="/facts/Contraposition/ENPumbyv">Contraposition</a> (1)</td></tr><tr><td>4</td><td>                    ¬        P              {\displaystyle \neg P}  </td><td><a href="/facts/Modus_ponens/Wq4NcmrG">Modus ponens</a> (2,3)</td></tr></tbody></table>
<h2 id="correspondence-to-other-mathematical-frameworks">Correspondence to other mathematical frameworks</h2>
<h3>Probability calculus</h3>
<p><i>Modus tollens</i> represents an instance of the <a href="/facts/Law_of_total_probability/rq6NTRn1">law of total probability</a> combined with <a href="/facts/Bayes%27_theorem/wQqFB2Dt">Bayes' theorem</a> expressed as:
</p><p>
  
    
      
        Pr
        (
        P
        )
        =
        Pr
        (
        P
        ∣
        Q
        )
        Pr
        (
        Q
        )
        +
        Pr
        (
        P
        ∣
        ¬
        Q
        )
        Pr
        (
        ¬
        Q
        )
        
        ,
      
    
    {\displaystyle \Pr(P)=\Pr(P\mid Q)\Pr(Q)+\Pr(P\mid \lnot Q)\Pr(\lnot Q)\,,}

</p><p>where the conditionals 
  
    
      
        Pr
        (
        P
        ∣
        Q
        )
      
    
    {\displaystyle \Pr(P\mid Q)}
  
 and 
  
    
      
        Pr
        (
        P
        ∣
        ¬
        Q
        )
      
    
    {\displaystyle \Pr(P\mid \lnot Q)}
  
 are obtained with (the extended form of) <a href="/facts/Bayes%27_theorem/wQqFB2Dt">Bayes' theorem</a> expressed as:
</p><p>
  
    
      
        Pr
        (
        P
        ∣
        Q
        )
        =
        
          
            
              Pr
              (
              Q
              ∣
              P
              )
              
              a
              (
              P
              )
            
            
              Pr
              (
              Q
              ∣
              P
              )
              
              a
              (
              P
              )
              +
              Pr
              (
              Q
              ∣
              ¬
              P
              )
              
              a
              (
              ¬
              P
              )
            
          
        
        
        
        
      
    
    {\displaystyle \Pr(P\mid Q)={\frac {\Pr(Q\mid P)\,a(P)}{\Pr(Q\mid P)\,a(P)+\Pr(Q\mid \lnot P)\,a(\lnot P)}}\;\;\;}
  
 and  
  
    
      
        Pr
        (
        P
        ∣
        ¬
        Q
        )
        =
        
          
            
              Pr
              (
              ¬
              Q
              ∣
              P
              )
              
              a
              (
              P
              )
            
            
              Pr
              (
              ¬
              Q
              ∣
              P
              )
              
              a
              (
              P
              )
              +
              Pr
              (
              ¬
              Q
              ∣
              ¬
              P
              )
              
              a
              (
              ¬
              P
              )
            
          
        
        .
      
    
    {\displaystyle \Pr(P\mid \lnot Q)={\frac {\Pr(\lnot Q\mid P)\,a(P)}{\Pr(\lnot Q\mid P)\,a(P)+\Pr(\lnot Q\mid \lnot P)\,a(\lnot P)}}.}

</p><p>In the equations above 
  
    
      
        Pr
        (
        Q
        )
      
    
    {\displaystyle \Pr(Q)}
  
 denotes the probability of 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
, and 
  
    
      
        a
        (
        P
        )
      
    
    {\displaystyle a(P)}
  
 denotes the <a href="/facts/Base_rate/HnRytNkF">base rate</a> (aka. <a href="/facts/Prior_probability/JQKAD4o0">prior probability</a>) of 
  
    
      
        P
      
    
    {\displaystyle P}
  
. The <a href="/facts/Conditional_probability/QcN2UERV">conditional probability</a> 
  
    
      
        Pr
        (
        Q
        ∣
        P
        )
      
    
    {\displaystyle \Pr(Q\mid P)}
  
 generalizes the logical statement 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
, i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. Assume that 
  
    
      
        Pr
        (
        Q
        )
        =
        1
      
    
    {\displaystyle \Pr(Q)=1}
  
 is equivalent to 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 being TRUE, and that 
  
    
      
        Pr
        (
        Q
        )
        =
        0
      
    
    {\displaystyle \Pr(Q)=0}
  
 is equivalent to 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 being FALSE.  It is then easy to see that 
  
    
      
        Pr
        (
        P
        )
        =
        0
      
    
    {\displaystyle \Pr(P)=0}
  
 when 
  
    
      
        Pr
        (
        Q
        ∣
        P
        )
        =
        1
      
    
    {\displaystyle \Pr(Q\mid P)=1}
  
 and 
  
    
      
        Pr
        (
        Q
        )
        =
        0
      
    
    {\displaystyle \Pr(Q)=0}
  
. This is because 
  
    
      
        Pr
        (
        ¬
        Q
        ∣
        P
        )
        =
        1
        −
        Pr
        (
        Q
        ∣
        P
        )
        =
        0
      
    
    {\displaystyle \Pr(\lnot Q\mid P)=1-\Pr(Q\mid P)=0}
  
 so that 
  
    
      
        Pr
        (
        P
        ∣
        ¬
        Q
        )
        =
        0
      
    
    {\displaystyle \Pr(P\mid \lnot Q)=0}
  
 in the last equation. Therefore, the product terms in the first equation always have a zero factor so that 
  
    
      
        Pr
        (
        P
        )
        =
        0
      
    
    {\displaystyle \Pr(P)=0}
  
 which is equivalent to 
  
    
      
        P
      
    
    {\displaystyle P}
  
 being FALSE. Hence, the <a href="/facts/Law_of_total_probability/rq6NTRn1">law of total probability</a> combined with <a href="/facts/Bayes%27_theorem/wQqFB2Dt">Bayes' theorem</a> represents a generalization of <i>modus tollens</i>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h3>Subjective logic</h3>
<p><i>Modus tollens</i> represents an instance of the abduction operator in <a href="/facts/Subjective_logic/X1njTWYj">subjective logic</a> expressed as:
</p><p>
  
    
      
        
          ω
          
            P
            
              
                
                  ‖
                  ~
                
              
            
            Q
          
          
            A
          
        
        =
        (
        
          ω
          
            Q
            
              |
            
            P
          
          
            A
          
        
        ,
        
          ω
          
            Q
            
              |
            
            ¬
            P
          
          
            A
          
        
        )
        
          
            
              ⊚
              ~
            
          
        
        (
        
          a
          
            P
          
        
        ,
        
        
          ω
          
            Q
          
          
            A
          
        
        )
        
        ,
      
    
    {\displaystyle \omega _{P{\tilde {\|}}Q}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A}){\widetilde {\circledcirc }}(a_{P},\,\omega _{Q}^{A})\,,}

</p><p>where 
  
    
      
        
          ω
          
            Q
          
          
            A
          
        
      
    
    {\displaystyle \omega _{Q}^{A}}
  
 denotes the subjective opinion about 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
, and 
  
    
      
        (
        
          ω
          
            Q
            
              |
            
            P
          
          
            A
          
        
        ,
        
          ω
          
            Q
            
              |
            
            ¬
            P
          
          
            A
          
        
        )
      
    
    {\displaystyle (\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})}
  
 denotes a pair of binomial conditional opinions, as expressed by source 
  
    
      
        A
      
    
    {\displaystyle A}
  
. The parameter 
  
    
      
        
          a
          
            P
          
        
      
    
    {\displaystyle a_{P}}
  
 denotes the <a href="/facts/Base_rate/HnRytNkF">base rate</a> (aka. the <a href="/facts/Prior_probability/JQKAD4o0">prior probability</a>) of 
  
    
      
        P
      
    
    {\displaystyle P}
  
. The abduced marginal opinion on 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is denoted 
  
    
      
        
          ω
          
            P
            
              
                
                  ‖
                  ~
                
              
            
            Q
          
          
            A
          
        
      
    
    {\displaystyle \omega _{P{\tilde {\|}}Q}^{A}}
  
. The conditional opinion 
  
    
      
        
          ω
          
            Q
            
              |
            
            P
          
          
            A
          
        
      
    
    {\displaystyle \omega _{Q|P}^{A}}
  
 generalizes the logical statement 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
, i.e. in addition to assigning TRUE or FALSE the source 
  
    
      
        A
      
    
    {\displaystyle A}
  
 can assign any subjective opinion to the statement. The case where 
  
    
      
        
          ω
          
            Q
          
          
            A
          
        
      
    
    {\displaystyle \omega _{Q}^{A}}
  
 is an absolute TRUE opinion is equivalent to source 
  
    
      
        A
      
    
    {\displaystyle A}
  
 saying that 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 is TRUE, and the case where 
  
    
      
        
          ω
          
            Q
          
          
            A
          
        
      
    
    {\displaystyle \omega _{Q}^{A}}
  
 is an absolute FALSE opinion is equivalent to source 
  
    
      
        A
      
    
    {\displaystyle A}
  
 saying that 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 is FALSE.  The abduction operator 
  
    
      
        
          
            
              ⊚
              ~
            
          
        
      
    
    {\displaystyle {\widetilde {\circledcirc }}}
  
 of <a href="/facts/Subjective_logic/X1njTWYj">subjective logic</a> produces an absolute FALSE abduced opinion 
  
    
      
        
          ω
          
            P
            
              
                
                  ‖
                  ~
                
              
            
            Q
          
          
            A
          
        
      
    
    {\displaystyle \omega _{P{\widetilde {\|}}Q}^{A}}
  
 when the conditional opinion 
  
    
      
        
          ω
          
            Q
            
              |
            
            P
          
          
            A
          
        
      
    
    {\displaystyle \omega _{Q|P}^{A}}
  
 is absolute TRUE and the consequent opinion 
  
    
      
        
          ω
          
            Q
          
          
            A
          
        
      
    
    {\displaystyle \omega _{Q}^{A}}
  
 is absolute FALSE. Hence, subjective logic abduction represents a generalization of both <i>modus tollens</i> and of the <a href="/facts/Law_of_total_probability/rq6NTRn1">Law of total probability</a> combined with <a href="/facts/Bayes%27_theorem/wQqFB2Dt">Bayes' theorem</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Evidence_of_absence/Cs2BbgsB">Evidence of absence</a> – Relevance fallacy</li>
<li><a href="/facts/Latin_phrases/AEtMoFf9">Latin phrases</a></li>
<li><a href="/facts/Modus_operandi/eIfQAbr9"><i>Modus operandi</i></a> – Habits of working</li>
<li><a href="/facts/Modus_ponens/Wq4NcmrG"><i>Modus ponens</i></a> – Rule of logical inference</li>
<li><a href="/facts/Modus_vivendi/w2RAXAfQ"><i>Modus vivendi</i></a> – Arrangement that allows conflicting parties to coexist in peace</li>
<li><a href="/facts/Non_sequitur_(logic)/1c3P1BEH"><i>Non sequitur</i></a> – Faulty deductive reasoning due to a logical flawPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Performative_contradiction/rRegL6jh">Performative contradiction</a> – Concept in logic</li>
<li><a href="/facts/Proof_by_contradiction/vHtFzFQ5">Proof by contradiction</a> – Proving that the negation is impossible</li>
<li><a href="/facts/Proof_by_contrapositive/ProBe0FP">Proof by contrapositive</a> – Mathematical logic conceptPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Stoic_logic/lBc4QlpH">Stoic logic</a> – System of propositional logic developed by the Stoic philosophers</li>
<li><a href="/facts/Law_of_excluded_middle/tDhbugCN">Law of excluded middle</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="sources">Sources</h2>
<ul><li>Audun Jøsang, 2016, <i><a href="https://books.google.com/books?id=nqRlDQAAQBAJ&q=%22Modus+tollens%22">Subjective Logic; A formalism for Reasoning Under Uncertainty</a></i> Springer, Cham, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-42337-1</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><i><a href="http://mathworld.wolfram.com/ModusTollens.html">Modus Tollens</a></i> at Wolfram MathWorld</li></ul>

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

<ol>
<li id="fn:1"><p>Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London: Routledge. p. 60. ISBN 978-0-415-91775-9. <a href="978-0-415-91775-9" target="_blank">978-0-415-91775-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Sanford, David Hawley (2003). If P, Then Q: Conditionals and the Foundations of Reasoning (2nd ed.). London: Routledge. p. 39. ISBN 978-0-415-28368-7. [Modus] tollens is always an abbreviation for modus tollendo tollens, the mood that by denying denies. <a href="978-0-415-28368-7" target="_blank">978-0-415-28368-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Susanne Bobzien (2002). "The Development of Modus Ponens in Antiquity", Phronesis 47. <a href="/wiki/Susanne_Bobzien" target="_blank">/wiki/Susanne_Bobzien</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"Ancient Logic: Forerunners of Modus Ponens and Modus Tollens". Stanford Encyclopedia of Philosophy. <a href="http://plato.stanford.edu/entries/logic-ancient/#StoSyl" target="_blank">http://plato.stanford.edu/entries/logic-ancient/#StoSyl</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Audun Jøsang 2016:p.2 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Audun Jøsang 2016:p.92 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Modus tollens open-in-new

Modus tollens