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Recipe

Build a chain of equivalences

Use when: you must show two sentences are logically equivalent step by step, with a law named at each step, or simplify a sentence as far as possible

Goal: a numbered list of sentences from the start to the target, each equivalent to the one before, each step labeled with its law

Taught in Chains of equivalences

Quick recipe

  1. Remove the arrows
  2. Push each ¬ down to the letters
  3. Reorder and regroup
  4. Distribute or factor
  5. Simplify
  1. 1

    Replace each \leftrightarrow with (A \to B) \wedge (B \to A) or (A \wedge B) \vee (\neg A \wedge \neg B), then each \to with material implication, working from the innermost arrow out.

    Example

    (P \to R) \wedge (Q \to R) becomes (\neg P \vee R) \wedge (\neg Q \vee R), one arrow per step.

    Why: every other law is stated for \neg, \wedge and \vee, so arrows block most patterns from matching.

  2. 2

    Use De Morgan on a \neg in front of a bracket, \neg(A \to B) \equiv A \wedge \neg B if an arrow is still there, and double negation on every \neg\neg.

    Why: the distributive and simplification laws need negations on single letters to see the pattern.

  3. 3

    Use the commutative and associative laws so that a shared part sits in the same position in each group.

    Why: factoring needs the shared part in matching positions, and a law's pattern must match symbol for symbol.

  4. 4

    Expand with a distributive law, or factor a shared part out.

    • aiming for a short sentence — factor, which usually exposes a tautology or contradiction.
    • aiming for a list shape (next section) — expand.

    Why: distribution is the only law that changes which connective is in charge.

  5. 5

    Remove repeats (idempotent), replace A \wedge \neg A or A \vee \neg A (negation laws), and drop or absorb always-true and always-false parts (identity, domination).

    ⚡ Shortcut

    If the two ends are both known, simplify each one separately and look for a common middle sentence. Reading your second chain backward joins the two.

    Why: these are the only laws that make a sentence shorter.

Common mistakes

  • Applying a law to a part that only looks like its pattern, such as De Morgan on \neg P \wedge Q.
  • Changing two things in one step without naming both laws.
  • Regrouping across two different connectives as if the associative law covered them.
  • Concluding "not equivalent" because no chain was found. Only a disagreeing row shows that.