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2.2.5 Implications

(A => B)

Alternative Forms  An implication A => B is often also expressed as

• "A implies B" ("A impliziert B");
• "if A, then B" ("wenn A, dann B");
• "B, (only) if A" ("B (nur) dann, wenn A");
• "B follows from A" ("aus A folgt B");
• "A is sufficient for B" ("A ist hinreichend für B");
• "B is necessary for A" ("B ist notwendig für A");
• implies(A, B).

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

A	B	A => B
false	false	true
false	true	true
true	false	false
true	true	true

In other words, A => B is false if and only if A is true and B is false. 

Please note that an implication is always true if its premise is false. This may appear strange at first glance, because it makes sentences like "if 2 is odd, then 3 is even" true. However, by an implication we actually want to express the fact "if the premise holds, then also the conclusion holds", i.e.,

• either the premise does not hold,
• or (the premise holds and) the conclusion holds.

This behavior is also expressed by the first of the following laws.

A => B	iff	~A \/ B
A => B	iff	~B => ~A
~(A => B)	iff	A /\  ~B.

Because of the first law, the implication A => B is frequently defined just as a syntactic abbreviation for ~A \/ B.

Operational Interpretation  In the Logic Evaluator, an implication is represented by an object of the Java type

public final class Implies implements Formula
{
  private Formula formula0;
  private Formula formula1;

  public Implies(Formula _formula0, Formula _formula1)
  {
    formula0 = _formula0;
    formula1 = _formula1;
  }

  public boolean eval() throws EvalException
  {
    if (formula0.eval())
    {
      if (formula1.eval())
        return true;
      else
        return false;
    }
    else
      return true;
  }
}