Traditional or BooleanLogic treats purely binary logical states: a statement can only be either 100% true or 100% false.
BooleanLogic connectives are defined by truth tables where 1 is true and 0 is false, i.e.,
For "and" symbolized as "&" or "^" or "∧" (or "&&" in CeeLanguage/CeePlusPlus/JavaLanguage, etc.)
^ | A B
---+------
0 | 0 0
0 | 1 0
0 | 0 1
1 | 1 1
For "or" symbolized as "|" or "v" or "∨" (or "||" in the languages above)
v | A B
---+------
0 | 0 0
1 | 1 0
1 | 0 1
1 | 1 1
- "Not" is "~" (or "!" or "¬")
- | A ---+--- 1 | 0 0 | 1
and "implication" (see LogicalImplication) or "If..then.." is "->" or "⇒" (conditional)
-> | A B
----+------
1 | 0 0
0 | 1 0
1 | 0 1
1 | 1 1
"Iff" or "If and only if" is "<->" or "⇔" (biconditional)
is the same as (A -> B) ^ (B -> A)
There are many axioms of BooleanLogic, including DeMorgansLaws, and some deduction rule, such as
A^(A->B) -> B
This means, when doing proofs in logic or mathematics, in any step if A (which can be a complex expression) is true (or an assumption) and there is a definition, assumption, axiom or previously proved theorem of the form A->B, then B can be introduced into the proof chain.
For more truth tables, see
It should be noted that implication in BooleanLogic does not model the way the human mind does implication; it's a rather forced fit. This continues to cause theoretical and pragmatic issues that motivate a continuing search for a more suitable model.
See also FuzzyLogic, FirstOrderLogic, SymbolicLogic, TheoremProving, ThreeValuedLogic, TetralemmicLogic, MultiValuedLogic