Peirce's law

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Peirce's law is a formula in propositional calculus that is commonly expressed in the following form:

Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem.

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

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

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

Graphical proof
Under the existential interpretation of Peirce's logical graphs, Peirce's law is represented by means of the following formal equivalence or logical equation.

Proof. Using the axiom set given in the entry for logical graphs, Peirce's law may be proved in the following manner.

The following animation replays the steps of the proof.

Equational form
A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible:

Proof 1
Given what precedes, it remains to show that:

But this is immediate, since $$p \Rightarrow (r \Rightarrow p)$$ for any proposition $$r.\!$$

Proof 2
Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation:

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:

The following animation replays the steps of the proof.

Focal nodes

 * Inquiry Live
 * Logic Live

Peer nodes

 * Peirce's Law @ InterSciWiki
 * Peirce's Law @ Subject Wikis
 * Peirce's Law @ Wikiversity
 * Peirce's Law @ Wikiversity Beta

Logical operators

 * Exclusive disjunction
 * Logical conjunction
 * Logical disjunction
 * Logical equality


 * Logical implication
 * Logical NAND
 * Logical NNOR
 * Negation

Related topics

 * Ampheck
 * Boolean domain
 * Boolean function
 * Boolean-valued function
 * Differential logic


 * Logical graph
 * Minimal negation operator
 * Multigrade operator
 * Parametric operator
 * Peirce's law


 * Propositional calculus
 * Sole sufficient operator
 * Truth table
 * Universe of discourse
 * Zeroth order logic

Relational concepts

 * Continuous predicate
 * Hypostatic abstraction
 * Logic of relatives
 * Logical matrix


 * Relation
 * Relation composition
 * Relation construction
 * Relation reduction


 * Relation theory
 * Relative term
 * Sign relation
 * Triadic relation

Information, Inquiry

 * Inquiry
 * Dynamics of inquiry


 * Semeiotic
 * Logic of information


 * Descriptive science
 * Normative science


 * Pragmatic maxim
 * Truth theory

Related articles

 * Cactus Language
 * Futures Of Logical Graphs
 * Propositional Equation Reasoning Systems


 * Differential Logic : Introduction
 * Differential Propositional Calculus
 * Differential Logic and Dynamic Systems


 * Prospects for Inquiry Driven Systems
 * Introduction to Inquiry Driven Systems
 * Inquiry Driven Systems : Inquiry Into Inquiry

Document history
Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.


 * Peirce's Law, InterSciWiki
 * Peirce's Law, PlanetMath
 * Peirce's Law, Wikinfo
 * Peirce's Law, Wikiversity
 * Peirce's Law, Wikiversity Beta
 * Peirce's Law, Wikipedia