MyWikiBiz, Author Your Legacy — Wednesday May 08, 2024
Jump to navigationJump to search
48 bytes added
, 13:56, 13 March 2009
Line 11: |
Line 11: |
| Here is Peirce's own statement and proof of the law: | | Here is Peirce's own statement and proof of the law: |
| | | |
− | <blockquote>
| + | {| align="center" cellpadding="8" width="90%" |
| + | | |
| <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> |
| | | |
Line 19: |
Line 20: |
| | | |
| <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 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 3.384).</p> |
− | </blockquote>
| + | |} |
| | | |
| Peirce goes on to point out an immediate application of the law: | | Peirce goes on to point out an immediate application of the law: |
| | | |
− | <blockquote>
| + | {| align="center" cellpadding="8" width="90%" |
| + | | |
| <p>From the formula just given, we at once get:</p> | | <p>From the formula just given, we at once get:</p> |
| | | |
Line 31: |
Line 33: |
| | | |
| <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 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 3.384).</p> |
− | </blockquote>
| + | |} |
| | | |
| '''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. |