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Binary logic, also referred to as two-value or Boolean logic, is a set of rules for dealing with propositions that must be either true or false. Its primary applications are in computer programming and mathematics, although there are also recreational games and puzzles based upon more formal logic. The alternative to binary logic is “fuzzy” logic, which allows for statements that are neither true nor false and/or statements with degrees of truthfulness.
Propositions are the core operands of binary logic in the same way that numbers are the core operands of arithmetic. Generally symbolically denoted by a single letter, a proposition is a statement that must be either true or false, such as “Bill is over six feet tall,” or “Two plus two equals five.” Subjective statements such as “Suzi is pretty,” generally cannot be treated as propositions, as their truth depends on perspective. Propositions should also avoid pronouns, as a change in the pronoun’s referent changes the nature of the proposition.
There are three operations common to all binary logic systems, AND, OR, and NOT. In addition, many logic systems add the operations IF . . . THEN, IF AND ONLY IF, and EOR. Notations vary greatly, so it is important to remain consistent in how one writes out binary logic.
The negation operation, NOT, is a unitary operation that is applied to a single proposition. For a given proposition A, NOT-A is false if A is true and NOT-A is true if A is false. The AND operation creates a new compound proposition from two simpler propositions, such as “Bill is over six feet tall and two plus two equals five.” This new proposition is true if both of the propositions that make it up are true; otherwise it is false. The OR operation also creates a new proposition from two simpler propositions, such as “Bill is over six feet tall or two plus two equals five.” A OR B is a true proposition if A is true, if B is true, or if both are true. It is only false if both A and B are false.
The other operations are not included in all binary logic systems. The conditional operation, IF A THEN B, is only false when A is true and B is false and true otherwise, so it can also be expressed as NOT-A OR B. The IF AND ONLY IF operation, also called the biconditional operation, is true if A and B are both true and false if A and B have differing truth values. The EOR operation is a strict alternative, either A or B but not both. It is the opposite of the biconditional, true if A and B have differing values and false if they have matching truth values.
The advantage of binary logic is that it provides a set of formal rules that can be used to test propositions for contradictions. For this reason, the logic has many applications in theoretical math and computer science. The disadvantage is that those rules only work with statements that are absolutely true or absolutely false, and can provide unreliable results when used with vaguer statements.