The Subject of Extending Logical Minds

Combine some logic with some bomb defusing; now how about understanding this to make your life easier?

• The original Logic manual can be found here.
• This contains ONLY specific logical identities and certain logic gate equivalents for this module and other “Logic” modules.
• The experts are allowed to use this for a better reference.

“A” and “B” are boolean variables in the reference sheet provided.

An “!” character represents a negate operator, E.G. !A represents NOT A.

Some Logic Identities
Involution	!!A = A
Dominance (OR)	TRUE ∨ A = TRUE
Dominance (AND)	FALSE ∧ A = FALSE
Complementartity (OR)	!A ∨ A = TRUE
Complementartity (AND)	!A ∧ A = FALSE
DeMorgan’s (OR)	A ∨ B = !A | !B
DeMorgan’s (AND)	A ∧ B = !A ↓ !B
Absorption	A ∧ (A ∨ B) = A
Idempotence	A ∧ A = A ∨ A = A

Other Identities
A ↔ A = TRUE	A ⊻ A = FALSE

Logical Gate Equivalents
A ↔ B = !A ⊻ B = A ⊻ !B = !(A ⊻ B)	A ⊻ B = !A ↔ B = A ↔ !B = !(A ↔ B)
A → B = !A ∨ B	A ← B = A ∨ !B
A ∨ B = A ← !B = !A → B	A ↓ B = !(A ∨ B)
A | B = !(A ∧ B)

Table ALT: Logical connective symbol list

Logical Connective	Symbol	Logic Gate Equivalent	Interpterion
Conjunction	∧	AND	Both TRUE returns TRUE. Otherwise at least 1 FALSE returns FALSE.
Disjunction	∨	OR	At least 1 TRUE returns TRUE. Otherwise both FALSE returns FALSE.
Exclusive Disjunction	⊻	XOR	Matching inputs returns FALSE. NOT matching inputs returns TRUE.
Alternative Denial	|	NAND	At least 1 FALSE returns TRUE. Otherwise, both TRUE returns FALSE.
Joint Denial	↓	NOR	Both FALSE returns TRUE. Otherwise, at least 1 TRUE returns FALSE.
Biconditional	↔	XNOR	Matching inputs returns TRUE. NOT matching inputs returns FALSE.
Implication (Left)	→	-	Left TRUE and right FALSE returns FALSE. Otherwise TRUE.
Implication (Right)	←	-	Left FALSE and right TRUE returns FALSE. Otherwise TRUE.