# Product

*By Robert Laing*

## 2 Value Algebra

Since conjunction is closely associated with the word *and*, it jarred me a bit to discover
that it is logic’s equivalent of multiplication, while disjunction
— commonly thought of as *or* — is logic’s addition,
the arithmetic operator I associate with *and*.

Why conjunction equates to *multiplication* is best illustrated by its truth table:

p | q | p · q |
---|---|---|

1 | 1 | 1 |

1 | 0 | 0 |

0 | 1 | 0 |

0 | 0 | 0 |

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

∀(p) = p_{1} · p_{2} · … · p_{n}

### Laws Analogous to Arithmetic

- The commutative law for AND: pq ≡ qp
- The associative law for AND: p(qr) ≡ (pq)r
- The distributive law of AND over OR: p(q + r) ≡ (pq + pr)
- 1 (TRUE) is the identity for AND: (p AND 1) ≡ p
- 0 is the annihilator for AND: (p AND 0) ≡ 0

## How conjunction differs from multiplication

- The distributive law for OR over AND: (p + qr) ≡ ((p + q)(p + r))
- Idempotence of AND: pp ≡ p