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The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived.  But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.

The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived.  But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.

In discussing this Example, it is convenient to observe that the implication relation that is ordinarily indicated by the propositional form ''x'' ⇒ ''y'' is equivalent to an order relation ''x'' ≤ ''y'' on boolean values, where 0 is taken to be less than 1.

In discussing this Example, it is convenient to observe that the implication relation ordinarily indicated by the propositional form <math>x \Rightarrow y</math> is equivalent to an order relation <math>x \le y</math> on the boolean values <math>0, 1 \in \mathbb{B},</math> where <math>0\!</math> is taken to be less than <math>1.\!</math>

* Example 2.  Transitivity

* Example 2.  Transitivity