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) = p1 · p2 · … · pn
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