Peirce Examples

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Some of the following examples are from Don D. Roberts’ Buch The Existential Graphs of Charles S. Peirce.

Derivation of Q from P(P(Q)) (modus ponens)

1. P(P(Q))		Premiss
2. P((Q))		1, by R4
3. ((Q))		2, by R1
4. Q			3, by R5

Derivation of (P(P(Q))(Q)) (modus ponens as a theorem)

1. (())			R5
2. (()(Q))		1, by R2
3. (P()(Q))		2, by R2
4. (P(P)(Q))		3, by R3
5. (P(P(Q))(Q))		4, by R3

Derivation of ((P((Q(R))))((((P(R)))(P(Q))))) (Selbstdistributivität der material implication)

1. (())					R5
2. ((P((Q(R))))())			1, by R2
3. ((P((Q(R))))((P((Q(R))))))		2, by R3
4. ((P((Q(R))))((PQ(R))))		3, by R5
5. ((P((Q(R))))((P(R)((Q)))))		4, by R5
6. ((P((Q(R))))((P(R)(P(Q)))))		5, by R3
7. ((P((Q(R))))((((P(R)))(P(Q)))))	6, by R5

Derivation of (((P(Q)))((Q(P)))), i.e. (P → Q) ∨ (Q → P) (paradox of the material implication)

1. (())			R5
2. (P(Q)())		1, by R2
3. (P(Q)(P))		2, by R3
4. (((P(Q)))(P))	3, by R5
5. (((P(Q)))(((P))))	4, by R5
6. (((P(Q)))((Q(P))))	5, by R2

Derivation of (P((Q(P)))), i.e. P → (Q → P) (syntactic variant of verum ex quodlibet)

1. (())			R5
2. (P())		1, by R2
3. (P(P))		2, by R3
4. (P(((P))))		3, by R5
5. (P((Q(P))))		4, by R2

Derivation of (P(Q)((Q)((P)))), i.e. (P → Q) → (¬Q → ¬P)

1. (())			R5
2. (P(Q)())		1, by R2
3. (P(Q)(P))		2, by R3
4. (P(Q)(P(Q)))		3, by R3
5. (P(Q)((Q)((P))))	4, by R5

Derivation of ((P)((P))), i.e. P ∨ ¬ P

1. (())                 R5
2. (()P)                1, by R2
3. ((P)P)               2, by R3
4. ((P)((P)))           3, by R5

Derivation of (P(P)), i.e. ¬(P ∧ ¬ P)

1. (())                 R5
2. (P())                1, by R2
3. (P(P))               2, by R3