Classical logic truth table magma algebra
Truth table expressed as an algebraic magma with operations ¬, ∧, ∨, →, ←, ↔ or alternatively !, &, |, -->, <--, <-->.
Based on classical logic with TruthValues{N} and TruthTable{N,M}, the @truthtable macro assigns named variable propositional projections as generators of the truth.
When using DeMorgan together with PrettyTables, the output is automatically formatted.
julia> using DeMorgan, PrettyTables
julia> @truthtable p q
julia> (p-->q)<-->(¬q-->¬p)
┌───┬───┬───────────┬──────┬──────┬───────────────────┐
│ p │ q │ p→q │ ¬(q) │ ¬(p) │ ⊤ │
│ │ │ ¬(q)→¬(p) │ │ │ (p→q)↔(¬(q)→¬(p)) │
├───┼───┼───────────┼──────┼──────┼───────────────────┤
│ 1 │ 1 │ 1 │ 0 │ 0 │ 1 │
│ 1 │ 0 │ 0 │ 1 │ 0 │ 1 │
│ 0 │ 1 │ 1 │ 0 │ 1 │ 1 │
│ 0 │ 0 │ 1 │ 1 │ 1 │ 1 │
└───┴───┴───────────┴──────┴──────┴───────────────────┘
julia> @truthtable p q r
julia> ((p-->q)∧(q-->r))-->(p-->r)
┌───┬───┬───┬─────┬─────┬─────────────┬─────┬─────────────────────┐
│ p │ q │ r │ p→q │ q→r │ (p→q)∧(q→r) │ p→r │ ⊤ │
│ │ │ │ │ │ │ │ ((p→q)∧(q→r))→(p→r) │
├───┼───┼───┼─────┼─────┼─────────────┼─────┼─────────────────────┤
│ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │
│ 1 │ 1 │ 0 │ 1 │ 0 │ 0 │ 0 │ 1 │
│ 1 │ 0 │ 1 │ 0 │ 1 │ 0 │ 1 │ 1 │
│ 1 │ 0 │ 0 │ 0 │ 1 │ 0 │ 0 │ 1 │
│ 0 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │
│ 0 │ 1 │ 0 │ 1 │ 0 │ 0 │ 1 │ 1 │
│ 0 │ 0 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │
│ 0 │ 0 │ 0 │ 1 │ 1 │ 1 │ 1 │ 1 │
└───┴───┴───┴─────┴─────┴─────────────┴─────┴─────────────────────┘With finitely generated truth basis, equivalence classes of truth expressions are formed and kept track of, including contradiction ⊥ and tautology ⊤ class expressions.