# Sum

*By Robert Laing*

In 1863 William Stanley Jevons wrote to George Boole that surely Boole’s operation of addition should be replaced by the more natural ‘inclusive or’ (or ‘union’), leading to the law X+X=X. Boole completely rejected this suggestion (it would have destroyed his system based on ordinary algebra) and broke off the correspondence.— The Algebra of Logic Tradition

p | q | p + q |
---|---|---|

1 | 1 | 1 |

1 | 0 | 1 |

0 | 1 | 1 |

0 | 0 | 0 |

As can be seen from the above quote, that `1 + 1 = 1`

in logic arithmetic caused rancour between the field’s
founding fathers, and does to this day.
Jevons used a rotated ÷ symbol, ⋅∣⋅, for or instead of + to sidestep the
problem.

Moving from binary to any number of propositions, the existential quantification symbol ∃(p) tends to be used, as in

∃(p) = p_{1} + p_{2} + … + p_{n}

### Laws Analogous to Arithmetic

- The commutative law for OR: (p + q) ≡ (q + p)
- The associative law for OR: (p + (q + r)) ≡ ((p + q) + r)
- The distributive law of AND over OR: p(q + r) ≡ (pq + pr)
- 0 (FALSE) is the identity for OR: p OR 0 ≡ p

### Ways in Which AND and OR Differ from Plus and Times

- 1 is the annihilator for OR: (1 OR p) ≡ 1
- Idempotence of OR: p + p ≡ p