George Boole is known for introducing a logical calculus for propositions in the mid-19th century. Although others before him such as Leibniz worked on logical calculi, Boole developed the first systematic one. Later C. S. Peirce and Gottlob Frege developed calculi that took into account the difference between universal and existential propositions. Since then many logical calculi have been developed, such as the Calculus of Indications previously noted here.
However, these calculi are not necessarily easy to calculate with. For that it is best to use something close to the familiar arithmetic and algebra. Below are two examples of Boolean logic calculation methods. Note that every Boolean expression has a dual expression by interchanging the OR and AND operations and the 0 and 1 elements of the expression.
MIN-MAX LOGIC
The Boolean operations are negation (NOT, ¬, ~), conjunction (AND, ∧), and disjunction (OR, ∨), with the constants 0 (contradiction) and 1 (tautology). These correspond to the set operations complement (^{c}, ´ ), intersection (∩), and union (∪) with constants ∅ (null set) and Ω (universal set).
Boolean logic may be represented as 0 (contradiction, null set) and 1 (tautology, universal set) with the following minimum and maximum operations:
¬a = a^{c} = 1 – a | a ∧ b = a ∩ b = min(a, b) | a ∨ b = a ∪ b = max(a, b) |
Other operations may be defined from these such as material implication, a → b = ¬a ∨ b, which corresponds to the subset proposition a ⊆ b, and is represented by max(1 – a, b).
FINITE FIELD LOGIC
Propositional logic may be represented by any functionally complete binary calculus such as the finite (Galois) field of order 2. The constants are 0 and 1 with 1 + 1 = 0. Since ordinary arithmetic is a field, this representation is mostly familiar:
¬a = a + 1 | a ∧ b = a · b = ab | a ∨ b = ab + a + b |
Then a → b is represented by ab + a + 1.
See also here.