Propositional Logic · step 3 of 13

Logical equivalence & the standard laws

Two formulas are logically equivalent, written A ≡ B, when they have the same truth value under every valuation — equivalently, when A ↔ B is a tautology. Exams almost always want the algebraic route rather than a table: you rewrite one side step by step using named laws until it becomes the other. The core toolkit is De Morgan, distributivity, the implication law p → q ≡ ¬p ∨ q, contraposition, absorption and double negation.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Eliminate → and ↔ firstRewrite p → q as ¬p ∨ q and p ↔ q as (p → q) ∧ (q → p). Almost every equivalence proof starts here, because the remaining laws are stated for ¬, ∧, ∨.
Push negations inwardApply De Morgan repeatedly: ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q, cancelling double negations as they appear.
Restructure with distributivity and absorptionDistribute to line the two sides up structurally; use p ∨ (p ∧ q) ≡ p and p ∧ (p ∨ q) ≡ p to delete redundant clauses.
Name every law you useWrite the justification beside each line. The marks are for the named steps, not the final formula.

Worked example

Show that ¬(p → q) ≡ p ∧ ¬q.

¬(p → q) ≡ ¬(¬p ∨ q) by the implication law.
≡ ¬¬p ∧ ¬q by De Morgan.
≡ p ∧ ¬q by double negation.

Answer. ¬(p → q) ≡ p ∧ ¬q — the negation of an implication is not another implication, it is a conjunction.

Where marks get dropped

These are the specific errors that cost credit on logical equivalence & the standard laws questions — QED's rubric penalises each of them separately.

Negating an implication as ¬p → ¬q. That is the inverse, which is not equivalent; the correct negation is p ∧ ¬q.
Applying De Morgan without flipping the connective. ¬(p ∧ q) becomes ¬p ∨ ¬q — the ∧ must turn into ∨.
Chaining steps with = or → instead of ≡. Equivalence is a relation between formulas, not a connective inside one, and mixing them is treated as a genuine error.

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Logical equivalence & the standard laws — frequently asked questions

Is A ≡ B the same as A ↔ B?

They are closely related but not the same object. A ↔ B is a formula that may be true or false; A ≡ B is a claim about all valuations. The link is that A ≡ B holds exactly when A ↔ B is a tautology.

Can I just use a truth table instead?

For two or three variables a table is a valid proof and QED accepts it. But many exam questions explicitly say "using the standard laws", and then a table earns no method marks.

Which laws am I allowed to assume?

The standard set: commutativity, associativity, distributivity, identity, domination, idempotence, double negation, De Morgan, absorption, implication and contraposition. Anything else should be derived.

The rest of Propositional Logic

Connectives, truth tables, equivalences and normal forms. Each subtopic below has its own method, worked example and mark-losing traps.

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