Some of the following examples are from Don D. Roberts' book
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)))))`

(self-distributive law of 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

© Christian Gottschall / christian.gottschall@univie.ac.at / 2012-03-31 01:19:53