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, 14:38, 26 October 2008
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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 formula in [[propositional calculus]] that is commonly expressed in the following form: |
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| <center> | | <center> |
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| ==Equational form== | | ==Equational form== |
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| + | A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible: |
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| + | <center> |
| + | <p><math>((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p</math></p> |
| + | </center> |
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| + | ===Proof 1=== |
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| + | Given what precedes, it remains to show that: |
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| + | <center> |
| + | <p><math>p \Rightarrow ((p \Rightarrow q) \Rightarrow p)</math></p> |
| + | </center> |
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| + | But this is immediate, since <math>p \Rightarrow (r \Rightarrow p)</math>, for any proposition <math>r.\!</math> |
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| + | ===Proof 2=== |
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| + | Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation: |
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| + | {| align="center" border="0" cellpadding="10" cellspacing="0" |
| + | | [[Image:Peirce's_Law_Figure_3.jpg|500px]] || (3) |
| + | |} |
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| + | Using the axioms and theorems listed in the article on [[logical graphs]], the equational form of Peirce's law may be proved in the following manner: |
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| + | {| align="center" border="0" cellpadding="10" cellspacing="0" |
| + | | [[Image:Peirce's_Law_Figure_4.jpg|500px]] || (4) |
| + | |} |
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| ==Bibliography== | | ==Bibliography== |