Changes

<center>

<center>

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

<p><math>\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.</math></p>

</center>

</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 \,-\!\!\!< y) \,-\!\!\!< 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 \,-\!\!\!< y</math> is false.  But in the last case the antecedent of <math>x \,-\!\!\!< y,</math> that is <math>x,\!</math> must be true.  (Peirce, CP&nbsp;3.384).</p>

|}

|}

<center>

<center>

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

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

</center>

</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 \,-\!\!\!< y) \prec a</math> means that from <math>(x \,-\!\!\!< 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>

|}

|}

'''Note.'''  The above transcription uses the "precedes sign" (<math>\prec</math>) for the "sign of illation" that Peirce customarily wrote as a cursive symbol somewhat like a gamma (<math>\gamma\!</math>) turned on its side or else typed as a bigram consisting of a dash and a "less than" sign.

'''Note.'''  Peirce uses the ''sign of illation'' “<math>-\!\!\!<</math>” for implication.  In one place he explains “<math>-\!\!\!<</math>” as a variant of the sign “<math>\le</math>” for ''less than or equal to'';  in another place he suggests that <math>A \,-\!\!\!< B</math> is an iconic way of representing a state of affairs where <math>A,\!</math> in every way that it can be, is <math>B.\!</math>

==Graphical proof==

==Graphical proof==