# Peirce Example Derivations

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@posteo.de / 2012-03-31 01:19:53