Chapter 3: Logical Reasoning

3.1 Statements

• A statement is a sentence whose truth value can be determined as either true or false, but not both. (p.56)
• Questions, exclamations and commands are not statements because their truth values cannot be determined. (p.56)
• Not all mathematical statements are true; their truth values can be determined. A false statement is disproved with a counter-example. (p.57)
• x ∈ R means x is an element of the real numbers (any rational or irrational number). (p.57)
• Statements using “all” or “some” can be true or false; e.g. “All polygons have diagonals” is false because a triangle has no diagonals. (p.58)

3.1 Negation and Compound Statements

• Negation: use the word “no” or “not” to negate a statement p; the negation is written ~p (read “tilde p”). (p.59)
• The truth value changes (true to false or false to true) through negation. (p.59)
• A compound statement combines two or more statements using “and” or “or”. (p.60)
• “and” means both; “or” means one of them or both. (p.61)

Truth table for compound statements (p.61)

p	q	p and q	p or q
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	False

• “p and q” is true only when both p and q are true; “p or q” is false only when both are false. (p.61)

3.1 Implications

• Implication “if p, then q”: p is the antecedent and q is the consequent. (p.63)
• Implication “p if and only if q” consists of two implications: “if p, then q” and “if q, then p”. (p.64)

3.1 Converse, Inverse and Contrapositive

• Statement: If p, then q. Converse: If q, then p. Inverse: If ~p, then ~q. Contrapositive: If ~q, then ~p. (p.66)

Truth values of an implication and its converse, inverse and contrapositive (p.68)

p	q	If p, then q	Converse	Inverse	Contrapositive
True	True	True	True	True	True
True	False	False	True	True	False
False	True	True	False	False	True
False	False	True	True	True	True

• An implication “if p, then q” is always true except when p is true and q is false. If the antecedent is false, the implication is always true. (p.68)
• The contrapositive has the same truth value as the implication; the converse and inverse also have the same truth value as each other. (p.68)
• Counter-example: for a false statement, at least one counter-example negates its truth (e.g. a triangle has no diagonal). (p.69)
• When a converse/inverse is false, give a specific counter-example, e.g. 4 > 3 but 4 > 5 is false. (p.70)

3.2 Arguments

• An argument is the process of making a conclusion based on statements; it consists of premises and one conclusion. A simple argument has at least two premises. (p.71)
• A premise is a statement that gives information before a conclusion is made; a conclusion is an outcome or decision. (p.71)
• Specific statements refer to a particular case; general statements describe a concept. (p.72)
• Deductive argument: making a specific conclusion based on general premises. (p.73)
• Inductive argument: making a general conclusion based on specific premises. (p.73)

3.2 Validity and Soundness (Deductive)

• Form I: Premise 1: All A are B. Premise 2: C is A. Conclusion: C is B. (p.75)
• Form II: Premise 1: If p, then q. Premise 2: p is true. Conclusion: q is true. (p.75)
• Form III: Premise 1: If p, then q. Premise 2: Not q is true. Conclusion: Not p is true. (p.75)
• Validity is decided by the form of the argument, not by the truth of the premises or conclusion. True premises do not guarantee validity. (p.75)
• A deductive argument is sound if (i) it is valid and (ii) all the premises and the conclusion are true. (p.76)
• Not all valid arguments are sound: a valid argument with a false premise is not sound. (p.76)
• To form a valid argument, write a premise/conclusion so that it complies with Form I, II or III. (p.78)

3.2 Strength and Cogency (Inductive)

• Deductive arguments emphasise validity; inductive arguments emphasise strength. (p.81)
• Strength depends on the probability that the conclusion is true, assuming all premises are true. (p.81)
• A strong argument is cogent only if all its premises are true; a weak argument is never cogent. (p.82)
• The number of premises does not guarantee a strong argument; strength depends on the truth of the conclusion. (p.82)
• Forming an inductive argument: observe examples, observe common features, then make a general conclusion (e.g. a formula for a number sequence). (p.83)
• Examples: 1/1, 1/2, 1/3, … → n⁻¹; 1, 3, 5, 7, … → 2n + 1 (n = 0, 1, 2, …); 0, 9, 24, 45, … → 3n² − 3. (p.83)

3.2 Problem Solving

• Deduction: substitute a given value into a general formula, e.g. P(t) = 2^t → P(8) = 256 new cells on day 8; solve 2^t = 2 048 → t = 11. (p.84)
• Induction: observe the pattern in data to build a general formula, e.g. toy cars produced = 250n + 20. (p.85)