Traditional or BooleanLogic treats purely binary logical states: a statement can only be either 100% true or 100% false.

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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
* http://whyslopes.com/volume1a/ch23_Truth_Tables_From_Occurence_Tables.html

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.

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See also FuzzyLogic, FirstOrderLogic, SymbolicLogic, TheoremProving, ThreeValuedLogic, TetralemmicLogic, MultiValuedLogic
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CategoryLogic