Logic

A logical statement \(X\) is a statement that is either true or false.

A statement \(X\) is necessary for a statement \(Y,\) \(X \Leftarrow Y,\) if \(X\) is required to be true for \(Y\) to be true. Neccessity does not guarantee that \(Y\) is true even if \(X\) is true. Neccessary conditions are expressed as “\(X\) if \(Y\)”.

A statement \(X\) is sufficient for a statement \(Y,\) \(X \Rightarrow Y,\) if \(X\) being true guarantees that \(Y\) is true. Sufficiency does not imply that \(X\) is necessary for \(Y.\) Sufficient conditions are expressed as “\(X\) only if \(Y\)”.

If a statement \(X\) is both necessary and sufficient for a statement \(Y,\) \(X \Leftrightarrow Y,\) then these statements are logically equivalent. Equivalent conditions are expressed as “\(X\) if and only if \(Y\)”.

The negation statement “\(X\) is false” is true if and only if \(X\) is false; otherwise the statement is false.

The conjunction statement “\(X\) and \(Y\)” is true if and only if the statements \(X\) and \(Y\) are both true; otherwise the statement is false.

The disjunction statement “\(X\) or \(Y\)” is true if either statement \(X\) or \(Y\) is true or both are true; otherwise the statement is false.