Changes

Here is Peirce's own statement and proof of the law:

Here is Peirce's own statement and proof of the law:

<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>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>

Peirce goes on to point out an immediate application of the law:

Peirce goes on to point out an immediate application of the law:

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

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

<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>

'''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.'''  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.