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