Implications

• How conditional statements (if…then) express implications and their converse, inverse, and contrapositive.
• To recognise which forms are logically equivalent and safe to swap in proofs and puzzles.
• To chain implications and use syllogistic patterns in verbal reasoning.
• To avoid the classic affirming the consequent and denying the antecedent fallacies.

Level 1 — Foundations

Verbal: "If P then Q" (P→Q) means P is sufficient for Q; Q is necessary for P. False only when P true and Q false.

Related forms (statement P→Q):

Examples: P = "studies daily", Q = "passes exam".

• P→Q: If studies daily, passes.
• Contrapositive: If did not pass, did not study daily (valid rephrase).
• Converse: If passes, studies daily (invalid — might pass by luck).

Chain: P→Q and Q→R implies P→R (hypothetical syllogism).

Level 2 — Exam depth

Biconditional: P↔Q when both P→Q and Q→P — "if and only if".

Necessary vs sufficient language:

• "Only if" introduces necessary condition: "Pass only if study" = P→study.
• "If" often introduces sufficient: "If study, pass" = study→pass.

Multiple implications puzzle: Map as arrows between nodes; follow chains; use contrapositive on any edge.

Exam phrasing: "Unless it rains, we play" → ¬rain→play, or play∨rain.

Counterexample method: To disprove P→Q, find one case P=T, Q=F.

Identify valid reformulation

Statement: "If a number is divisible by 6, it is divisible by 3."
Contrapositive: "If a number is not divisible by 3, it is not divisible by 6." **Valid.**
Converse: "If divisible by 3, divisible by 6." **Invalid** (counter: 9).
Inverse: "If not divisible by 6, not divisible by 3." **Invalid** (counter: 9).

Chain two implications

Clues: (1) If A then B. (2) If B then C. (3) A is true.
Modus ponens: A→B + A ⇒ B. B→C + B ⇒ C. **Therefore C is true.**

• Rewrite "You may enter only with a pass" as an implication.
• Give counterexample to converse of "If square then rectangle."
• From P→Q and ¬Q, what follows?
• Stretch: Translate "All cats are mammals" into conditional form.

Revision tip: Revisit adjacent topics in Logical Deduction before mixed practice on Implications.

Open the Practice tab for graded questions on Implications.

Underline only if, if and only if, and unless in verbal statements before symbolising. Practice translating ten daily sentences into P→Q form — speed here wins deduction sections. On multiple-choice items, test the contrapositive against options when the original conditional appears. Never select the converse unless the statement is biconditional.

Implication logic appears in constraint puzzles phrased as "if seat A then not seat B" — translate immediately. Syllogisms are categorical implications — contrapositive reasoning resolves many Venn items faster than drawing three circles. Debate rebuttal often attacks a hidden implication ("Your plan assumes costs stay flat"). When reading verbal passages, arrow diagrams for conditionals prevent reversing causal claims.