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2.2.2 Negations

~A

Alternative Forms  Other syntactic forms of negation are

• A, ~ A, -A, !A;
• non-A;
• "not A", "never A", "in no case A";
• not(A).

The last line denotes the input syntax of the Logic Evaluator.

Negation of infix atomic formulas is often expressed by crossing the predicate symbol, e.g.

x not < y

A	~A
false	true
true	false

In other words, ~A is true if and only if A is false. 

Operational Interpretation  A logical formula can be given an operational interpretation by a program that computes its truth value. For instance, in the Logic Evaluator, a negation is represented by an object of Java type

public final class Not implements Formula
{
  private Formula formula;

  public Not(Formula _formula)
  {
    formula = _formula;
  }

  public boolean eval() throws EvalException
  {
    if (formula.eval())
      return false;
    else
      return true;
  }
}

The Java expression (new Not(A)).eval() thus computes the truth value of the negation of A. The result is true only if the truth value of A is false.

We conclude the discussion of negation by stating a simple law and proving its correctness.

~~A iff A.

A	~A	~~A
false	true	false
true	false	true