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2.2.2 Negations

If A is a formula, then
is a formula.

Alternative Forms  Other syntactic forms of negation are

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
represents ~(x < y).

Definition 5 (Semantics of Negation) For every formula A, the meaning of ~A is denoted by the following  truth table (Wahrheitstabelle) that lists the possible truth values for A in the first column and the truth value of its negation in the second column:
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;
      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.

Proposition 2 (Inversion of Negation) For every formula A, we have
~~A iff A.

Proof  Let A be an arbitrary formula. We have to show that the meaning of ~~A is the same as the meaning of A, i.e., that they have the same truth values. From Definition Semantics of Negation, we can construct the following truth table.
A ~A ~~A
false true false
true false true
Since the last column coincides with the first column, we are done. 

Author: Wolfgang Schreiner
Last Modification: October 4, 1999

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