Table 1 Logical matrices of 16 Boolean operations
Logical connective | Logical operator | Logical matrix for 2-node |
|---|---|---|
symbol | Boolean networks | |
True | ⊤ | \(\left (\begin {array}{cccc} 1&1&1&1 \\ 0&0&0&0 \end {array} \right)\) |
False | ⊥ | \(\left (\begin {array}{cccc} 0&0&0&0 \\ 1&1&1&1 \end {array} \right)\) |
Proposition x 1 | x 1 | \(\left (\begin {array}{cccc} 1&1&0&0 \\ 0&0&1&1 \end {array} \right)\) |
Proposition x 2 | x 2 | \(\left (\begin {array}{cccc} 1&0&1&0 \\ 0&1&0&1 \end {array} \right)\) |
Negation (inhibition) x 1 | ¬x 1 | \(\left (\begin {array}{cccc} 0&0&1&1 \\ 1&1&0&0 \end {array} \right)\) |
Negation (inhibition) x 2 | ¬x 2 | \(\left (\begin {array}{cccc} 0&1&0&1 \\ 1&0&1&0 \end {array} \right)\) |
Conjunction | ∧ | \(\left (\begin {array}{cccc} 1&0&0&0 \\ 0&1&1&1 \end {array} \right)\) |
Disjunction (union) | ∨ | \(\left (\begin {array}{cccc} 1&1&1&0 \\ 0&0&0&1 \end {array} \right)\) |
Converse implication | ← | \(\left (\begin {array}{cccc} 1&1&0&1 \\ 0&0&1&0 \end {array} \right)\) |
Material conditional | → | \(\left (\begin {array}{cccc} 1&0&1&1 \\ 0&1&0&0 \end {array} \right)\) |
Converse nonimplication | \(\nleftarrow \) | \(\left (\begin {array}{cccc} 0&0&1&0 \\ 1&1&0&1 \end {array} \right)\) |
Material nonimplication | \(\nrightarrow \) | \(\left (\begin {array}{cccc} 0&1&0&0 \\ 1&0&1&1 \end {array} \right)\) |
Biconditional | ⇔ | \(\left (\begin {array}{cccc} 1&0&0&1 \\ 0&1&1&0 \end {array} \right)\) |
Alternative denial | ↑ | \(\left (\begin {array}{cccc} 0&1&1&1 \\ 1&0&0&0 \end {array} \right)\) |
Joint denial | ↓ | \(\left (\begin {array}{cccc} 0&0&0&1 \\ 1&1&1&0 \end {array} \right)\) |
Exclusive disjunction | ⊕ | \(\left (\begin {array}{cccc} 0&1&1&0 \\ 1&0&0&1 \end {array} \right)\) |