Statements and connectives
A statement (proposition) is a sentence that is either true (T) or false (F), not both. 'Delhi is in Asia' is a statement. 'Close the door!' and 'x + 2 = 5' (without knowing x) are not.
We join statements with connectives:
- NOT P (¬P): the opposite truth value.
- P AND Q (P ∧ Q): true only if both are true.
- P OR Q (P ∨ Q): true if at least one is true (inclusive or).
- IF P THEN Q (P → Q): false only when P is true and Q is false.
- P IF AND ONLY IF Q (P ↔ Q): true when both have the same truth value.
A truth table lists every combination. With 2 statements there are 2² = 4 rows; with 3, there are 8.
Negating: NOT (P AND Q) = (NOT P) OR (NOT Q); NOT (P OR Q) = (NOT P) AND (NOT Q). The negation of 'all students passed' is 'at least one student did not pass', not 'no student passed'.
If-then: converse, inverse, contrapositive
For 'If P then Q':
- Converse: if Q then P.
- Inverse: if not P then not Q.
- Contrapositive: if not Q then not P.
Only the contrapositive always has the same truth value as the original. Example: 'If a number ends in 0, it is divisible by 5' is true. Contrapositive: 'If it is not divisible by 5, it does not end in 0' (true). Converse: 'If divisible by 5, it ends in 0' (false: 15).
'P is sufficient for Q' and 'Q is necessary for P' both mean P → Q.
Deductive reasoning: valid argument forms and syllogisms
In deduction, the conclusion follows necessarily. An argument is valid if true premises cannot give a false conclusion; it is sound if it is valid and its premises really are true.
Rules of inference
- Modus ponens: P → Q; P; so Q.
- Modus tollens: P → Q; not Q; so not P.
- Hypothetical syllogism: P → Q; Q → R; so P → R.
- Disjunctive syllogism: P or Q; not P; so Q.
- Constructive dilemma: (P → Q) and (R → S); P or R; so Q or S.
Invalid look-alikes: affirming the consequent (P → Q; Q; so P) and denying the antecedent (P → Q; not P; so not Q).
Categorical propositions
Four forms: A 'All S are P', E 'No S are P', I 'Some S are P', O 'Some S are not P'. A categorical syllogism has two premises and a conclusion with three terms. Test it with a Venn diagram: draw three overlapping circles, shade empty regions for 'all/no', put an × for 'some', then check whether the conclusion is already shown.
Induction, analogy, proofs and fallacies
Non-deductive reasoning
- Induction: from many observed cases to a general rule ('every swan I saw is white, so all swans are white'). Stronger with more, varied and unbiased cases; one counterexample can break it.
- Analogy: two things are alike in some ways, so probably alike in another. Stronger when the similarities are relevant.
Direct and indirect justification
Direct proof: start from known facts and go step by step to the result. Example: the sum of two even numbers 2a + 2b = 2(a + b) is even.
Indirect proof (by contradiction): assume the statement is false and show this leads to something impossible. Example: assume there is a largest whole number N; but N + 1 is bigger, a contradiction, so no largest whole number exists. A counterexample disproves an 'all' claim.
Fallacies and paradoxes
A fallacy is a mistake in reasoning: attacking the person (ad hominem), appeal to the crowd (ad populum), appeal to false authority, false dilemma (only two options), circular reasoning, equivocation (one word, two meanings), hasty generalisation. A paradox seems to prove something impossible, like 'This sentence is false'.
Puzzles with conditions
Seating and order puzzles use the same skills: write each clue as a statement, place the sure facts first, then test cases and drop those that lead to a contradiction.
Try it yourself: catch the faulty argument
- Read ads or posts for one day. Write down two claims of the form 'If you use X, you will Y'.
- For each, write the converse and the contrapositive. Which one does the ad want you to believe?
- Look for a fallacy (crowd, famous person, only-two-choices).
- Marble test: put 9 white and 1 coloured button in a bag. Take out 5. Does 'all are white' feel true? Keep going until it breaks.
In the 3D, use the last step to test every connective with P and Q.
Key formulas and definitions
- ¬P; P ∧ Q (and); P ∨ Q (or); P → Q (if-then, false only for T → F); P ↔ Q (same value)
- Truth table rows = 2ⁿ for n statements
- Contrapositive ¬Q → ¬P ≡ P → Q; converse and inverse are not equivalent
- Modus ponens: P → Q, P ⊢ Q; Modus tollens: P → Q, ¬Q ⊢ ¬P
- De Morgan: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q; ¬(P ∨ Q) ≡ ¬P ∧ ¬Q
- Categorical forms: A (all), E (no), I (some), O (some … not)
Worked examples
1. P: 'It is Sunday' (true). Q: 'The school is open' (false). Find P ∧ Q, P ∨ Q, P → Q and ¬Q.
P ∧ Q = T and F = F. P ∨ Q = T or F = T. P → Q = T → F = F (the only false case). ¬Q = not F = T.
2. Is this valid? 'If a shape is a square, it has 4 sides. This shape has 4 sides. So it is a square.'
Not valid. It affirms the consequent (P → Q; Q; so P). A rectangle or a kite also has 4 sides. Valid would be modus ponens: 'it is a square, so it has 4 sides'.
3. Test with a Venn diagram: 'All cats are animals. Some animals are pets. So some cats are pets.'
Draw circles C (cats), A (animals), P (pets). 'All cats are animals': shade the part of C outside A. 'Some animals are pets': put × in A ∩ P, but it could be in the part outside C. The diagram does not force an × in C ∩ P, so the syllogism is invalid (even though the conclusion happens to be true).
4. Prove by contradiction: if n² is even, then n is even (n a whole number).
Assume n² is even but n is odd. Then n = 2k + 1, so n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd. This contradicts 'n² is even'. So n must be even.
5. Write the converse, inverse and contrapositive of 'If it rains, the match is cancelled.' Which is equivalent to the original?
Converse: if the match is cancelled, it rained. Inverse: if it does not rain, the match is not cancelled. Contrapositive: if the match is not cancelled, it did not rain. Only the contrapositive is equivalent.
6. Five friends A, B, C, D, E sit in a row. C is in the middle. A is at the left end. B is next to C on the right. E is not next to A. Find the order.
Positions 1–5: A is 1, C is 3, B is 4. Left: 2 and 5 for D and E. E is not next to A, so E is not 2; E = 5, D = 2. Order: A, D, C, B, E.
Common mistakes
- Thinking 'if P then Q' also means 'if Q then P'. The converse can be false.
- Negating 'all are' as 'none are'. The correct negation is 'at least one is not'.
- Calling an argument valid because its conclusion is true. Validity is about the form, not the truth of the conclusion.
- Treating induction as proof. Many examples make a rule likely; one counterexample disproves it.