Rules for Proofs

Two types of rules can be used to justify steps in formal proofs: rules of inference and rules of replacement. In order to use these properly, you should understand the differences between them.

The main difference is that rules of inference are forms of valid arguments (that’s why they have a therefore ∴  symbol), but rules of replacement are forms of equivalent propositions (which is why they have the equivalence sign  ≡  between the two parts).  This fundamental distinction is the cause of all other differences in how they are applied in proofs.

  1.  Rules of inference only work in one direction, but rules of replacement work in either direction. For example, you may not start with A ⊃ (A • C) in step 4, then in step 5 conclude A ⊃ C by 4 Abs., because Absorption is a rule of inference. But you may start with ~M ⊃ ~L in step 4, then conclude L ⊃ M in step 5 by 4 Trans. because Transposition is a rule of replacement, and the equivalence works both directions.
  2. Rules of inference may not be used within a larger compound proposition, but rules of replacement may be applied wherever they occur, even inside compound propositions. For example, you may not start with ~(E • G) in step 3, then conclude ~E in step 5 by 3 Simp. You may not simplify inside a larger proposition, because Simplification is a rule of inference. But you may start with P ⊃ (~P v Q) in step 6, and then conclude P ⊃ (P ⊃ Q) in step 7 by 6 Impl., because Material Implication is a rule of replacement, and can be applied within the parentheses.
  3. Rules of inference can have two premises, and thus the justification may have two numbers before the abbreviated rule name. But rules of replacement state that one proposition is equivalent to another, and thus the justification will always only have one number in front of the abbreviated rule name. Thus, while you will sometimes have 5, 3 H.S. as a justification, you cannot have 5, 3 Equiv. as a justification.

Learn these rules!

Rules of Inference and Replacement

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