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'''Peirce's law''' is a proposition in [[propositional calculus]] that is commonly expressed in the following form:

'''Peirce's law''' is a proposition in [[propositional calculus]] that is commonly expressed in the following form:

<p><center><math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math></center></p>

<center>

<p><math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math></p>

</center>

Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus.  The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem.

Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus.  The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem.

<p>A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it.  One of the simplest formulae of this kind is:</p>

<p>A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it.  One of the simplest formulae of this kind is:</p>

<p><center><math>\{ (x \prec y) \prec x \} \prec x.</math></center></p>

<center>

<p><math>\{ (x \prec y) \prec x \} \prec x.</math></p>

</center>

<p>This is hardly axiomatical.  That it is true appears as follows.  It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \prec y) \prec x</math> is true.  If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \prec y</math> is false.  But in the last case the antecedent of <math>x \prec y,</math> that is <math>x,\!</math> must be true.  (Peirce, CP&nbsp;3.384).</p>

<p>This is hardly axiomatical.  That it is true appears as follows.  It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \prec y) \prec x</math> is true.  If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \prec y</math> is false.  But in the last case the antecedent of <math>x \prec y,</math> that is <math>x,\!</math> must be true.  (Peirce, CP&nbsp;3.384).</p>

<p>From the formula just given, we at once get:</p>

<p>From the formula just given, we at once get:</p>

<p><center><math>\{ (x \prec y) \prec a \} \prec x,</math></center></p>

<center>

<p><math>\{ (x \prec y) \prec a \} \prec x,</math></p>

</center>

<p>where the <math>a\!</math> is used in such a sense that <math>(x \prec y) \prec a</math> means that from <math>(x \prec y)</math> every proposition follows.  With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math>  (Peirce, CP&nbsp;3.384).</p>

<p>where the <math>a\!</math> is used in such a sense that <math>(x \prec y) \prec a</math> means that from <math>(x \prec y)</math> every proposition follows.  With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math>  (Peirce, CP&nbsp;3.384).</p>