1.1.5 Comprehending and critiquing arguments

What you'll learn

How to identify the assumptions, logical steps and conclusion in a mathematical argument.
How to distinguish a valid proof from evidence based only on examples.
How to find counterexamples, hidden assumptions and gaps in reasoning.
How to assess mathematical arguments used in real-world applications.

What is a mathematical argument?

A mathematical argument is a sequence of statements intended to establish a conclusion. Each statement should follow logically from information already given, an accepted definition, or a known result.

Arguments appear in proofs, derivations of formulae, solutions to problems and mathematical models. To understand an argument, you should be able to explain not only what has been done, but why each step is allowed.

Definition

Premise and conclusion

A premise is a fact or assumption from which an argument begins. The conclusion is the statement that the argument claims follows from its premises.

For example, consider:

Every multiple of 4 is even. The integer $n$ is a multiple of 4. Therefore, $n$ is even.

The first two statements are premises. The final statement is the conclusion.

Key Idea

Follow the logical chain

A complete argument has the structure: premises → justified steps → conclusion. When critiquing it, check every link rather than judging only whether the conclusion looks plausible.

Example

Unpacking an argument about odd integers

Consider the claim: “The square of every odd integer is odd.”

Express an arbitrary odd integer as $n = 2 k + 1$ , where $k$ is an integer. This uses the definition of an odd integer.
Square the expression and simplify:
$n2=(2k+1)2=4k2+4k+1=2(2k2+2k)+1.\begin{aligned} n^2&=(2k+1)^2\\ &=4k^2+4k+1\\ &=2(2k^2+2k)+1. \end{aligned}$
Since $2k^2+2k$ is an integer, $n^2$ has the form $2 m + 1$ for an integer $m$ . Therefore, $n^2$ is odd.

Understanding the scope of a claim

The scope of a statement tells you which values or objects it concerns. Words such as “all”, “every”, “some” and “there exists” are mathematically important.

A universal claim says that a statement is true for every permitted value. For example, “Every prime number is odd” is a universal claim, although it is false because 2 is prime and even.

An existential claim says that at least one value with a stated property exists. For example, “There exists an even prime number” is true because 2 is such a number.

Definition

Counterexample

A counterexample is one permitted example for which a universal claim is false. A single counterexample disproves the whole universal claim.

Example

Disproving a claim about squares

A student claims that $n^2>n$ for every integer $n$ .

Because the claim concerns every integer, test values at which the behaviour might change, particularly zero, one and negative integers.
Taking $n = 1$ gives $n^2=1$ , so $n^2=n$ , not $n^2>n$ .
Therefore, $n = 1$ is a counterexample and the original universal claim is false. In fact, $n^2>n$ holds for integers satisfying $n < 0$ or $n > 1$ .

Common Mistake

Treating examples as proof

Checking many examples can suggest that a claim is true, but it does not prove a universal claim. However, one valid counterexample is enough to disprove it.

Valid arguments and true conclusions

An argument is valid if its conclusion must follow whenever its premises are true. Validity concerns the logical structure of the reasoning.

A sound argument is valid and also has true premises. A valid argument can begin from a false premise, so validity alone does not guarantee that its conclusion describes reality.

Consider:

All square numbers are even. The number 9 is square. Therefore, 9 is even.

The structure is valid: if both premises were true, the conclusion would follow. However, the first premise is false, so the argument is not sound.

Key Idea

Check logic and starting information

Ask two separate questions: “Are the premises true or justified?” and “Does the conclusion actually follow from them?”

Necessary and sufficient conditions

A condition is sufficient if it guarantees a result. A condition is necessary if the result cannot occur without it.

Being divisible by 4 is sufficient for an integer to be even, but it is not necessary: 6 is even without being divisible by 4. Being divisible by 2 is both necessary and sufficient for an integer to be even.

This distinction matters when an argument incorrectly reverses an implication.

Definition

Implication and converse

The implication $P⇒QP\Rightarrow Q$ means that whenever $P$ is true, $Q$ must be true. Its converse is $Q⇒PQ\Rightarrow P$ , which is a different statement and may be false.

Example

Critiquing a reversed implication

A student argues: “If an integer is divisible by 6, then it is divisible by 3. Since 12 is divisible by 3, it must be divisible by 6.”

The original implication is: divisible by 6 $⇒\Rightarrow$ divisible by 3.
The student uses the converse: divisible by 3 $⇒\Rightarrow$ divisible by 6. The original implication does not justify this reversal.
The converse is false; for example, 9 is divisible by 3 but not by 6. Although 12 happens to be divisible by 6, the student’s method does not establish that conclusion.

Common Mistake

Accepting a true conclusion from invalid reasoning

A conclusion may happen to be true even when the argument is invalid. You must assess the reasoning used, not just the final statement.

Checking algebraic and logical steps

When reading a proof or derivation, check that each operation is permitted. Common gaps include:

dividing by an expression that might equal zero;
taking a square root without considering both signs;
squaring both sides and introducing additional solutions;
applying a result outside its stated domain;
assuming the statement that is supposed to be proved.

An argument that assumes its own conclusion is called circular reasoning. It gives no independent reason for accepting the conclusion.

Example

Finding an illegal division

The following argument claims to prove that $1 = 2$ . It begins by assuming $a = b$ , where both values are non-zero.

From $a = b$ , multiplying by $a$ gives $a^2=ab$ . Subtracting $b^2$ gives
$a^2-b^2=ab-b^2.$
Factorising both sides gives
$(a - b) (a + b) = b (a - b) .$
The argument then divides by $a - b$ to obtain $a + b = b$ . However, the premise $a = b$ means that $a - b = 0$ . This step is division by zero, which is undefined.
Every later conclusion depends on that invalid step, so the argument does not prove that $1 = 2$ .

Common Mistake

Equivalent steps

An algebraic step should preserve the relevant solutions. Squaring an equation may create extra solutions, while dividing by an expression may remove values for which that expression is zero.

Critiquing justifications of methods and formulae

A derivation obtains a result or formula from known facts through justified steps. When reading one, identify:

the definitions or earlier results being used;
the assumptions under which the formula holds;
whether the algebra is correct;
whether the conclusion has the same scope as the reasoning.

For example, the quadratic formula is derived for equations $ax^2+bx+c=0$ with $a≠0a\neq0$ . The restriction matters because dividing by $a$ is part of the derivation. If $a = 0$ , the equation is not quadratic.

Example

Checking a claimed formula for an arithmetic sequence

A student claims that the sum of the first $n$ terms of an arithmetic sequence is $Sn=n2(a+l)S_n=\frac{n}{2}(a+l)$ , where $a$ is the first term and $l$ is the final term.

Write the sum forwards and backwards:
$Sn=a+(a+d)+⋯+l,Sn=l+(l−d)+⋯+a.\begin{aligned} S_n&=a+(a+d)+\cdots+l,\\ S_n&=l+(l-d)+\cdots+a. \end{aligned}$
Adding corresponding terms gives $a + l$ each time. Since there are $n$ pairs,
$2S_n=n(a+l).$
Dividing by 2 gives
$Sn=n2(a+l).S_n=\frac{n}{2}(a+l).$
The derivation is justified because an arithmetic sequence has a constant difference, making every paired sum equal.

Arguments in applications and modelling

A mathematical model is a simplified mathematical representation of a real situation. An argument within a model may be mathematically valid but still give an unreliable real-world conclusion if its assumptions are unsuitable.

For example, a population model might assume a constant growth rate, unlimited resources and no migration. Calculations based on those assumptions could be correct, while long-term predictions remain unrealistic.

When critiquing an applied argument, check:

whether relevant quantities have been included;
whether units and numerical accuracy are consistent;
whether the assumptions suit the context;
whether the data support the claimed conclusion;
whether the final answer is interpreted sensibly.

Example

Critiquing a constant-speed model

A journey of 120 km takes 2 hours. A student concludes that the vehicle travelled at exactly 60 km h⁻¹ throughout.

The calculation $1202=60\frac{120}{2}=60$ correctly gives the average speed over the whole journey.
Average speed does not determine the speed at each instant. The vehicle could have stopped, accelerated or travelled at several different speeds.
The conclusion “average speed was 60 km h⁻¹” is justified. The stronger conclusion “speed was exactly 60 km h⁻¹ throughout” requires the additional assumption that the speed remained constant.

Tip

Strength of conclusions

Use the weakest conclusion fully supported by the evidence. Do not replace “may”, “on average” or “under these assumptions” with an unjustified definite claim.

Writing a clear critique

A strong critique identifies the exact issue and explains its effect. Avoid vague comments such as “the proof is wrong”.

A useful structure is:

State the step or assumption being challenged.
Explain why it is invalid, incomplete or unjustified.
Give a counterexample or corrected condition where possible.
State whether the conclusion is false, or merely not proved by that argument.

Exam technique

In the exam

Identify the premises, the conclusion and the scope of the claim before checking individual steps.
Test universal claims using boundary values, zero, negative values and other special cases where permitted.
Name the precise flaw, such as reversing an implication, dividing by zero or using an unsuitable modelling assumption.
Distinguish “the conclusion is false” from “the argument does not prove the conclusion”.
Check that any formula or applied result is used within its stated conditions and domain.

Self review

Check yourself

Why can one counterexample disprove a universal claim, while many supporting examples cannot prove it?
How would you distinguish a valid argument from a sound argument?
What assumptions would you check before accepting a conclusion produced by a mathematical model?

What you'll learn

How to identify the assumptions, logical steps and conclusion in a mathematical argument.
How to distinguish a valid proof from evidence based only on examples.
How to find counterexamples, hidden assumptions and gaps in reasoning.
How to assess mathematical arguments used in real-world applications.

What is a mathematical argument?

Definition

Premise and conclusion

A premise is a fact or assumption from which an argument begins. The conclusion is the statement that the argument claims follows from its premises.

For example, consider:

Every multiple of 4 is even. The integer $n$ is a multiple of 4. Therefore, $n$ is even.

The first two statements are premises. The final statement is the conclusion.

Key Idea

Follow the logical chain

A complete argument has the structure: premises → justified steps → conclusion. When critiquing it, check every link rather than judging only whether the conclusion looks plausible.

Example

Unpacking an argument about odd integers

Consider the claim: “The square of every odd integer is odd.”

Express an arbitrary odd integer as $n = 2 k + 1$ , where $k$ is an integer. This uses the definition of an odd integer.
Square the expression and simplify:
$n2=(2k+1)2=4k2+4k+1=2(2k2+2k)+1.\begin{aligned} n^2&=(2k+1)^2\\ &=4k^2+4k+1\\ &=2(2k^2+2k)+1. \end{aligned}$
Since $2k^2+2k$ is an integer, $n^2$ has the form $2 m + 1$ for an integer $m$ . Therefore, $n^2$ is odd.

Understanding the scope of a claim

The scope of a statement tells you which values or objects it concerns. Words such as “all”, “every”, “some” and “there exists” are mathematically important.

A universal claim says that a statement is true for every permitted value. For example, “Every prime number is odd” is a universal claim, although it is false because 2 is prime and even.

An existential claim says that at least one value with a stated property exists. For example, “There exists an even prime number” is true because 2 is such a number.

Definition

Counterexample

A counterexample is one permitted example for which a universal claim is false. A single counterexample disproves the whole universal claim.

Example

Disproving a claim about squares

A student claims that $n^2>n$ for every integer $n$ .

Because the claim concerns every integer, test values at which the behaviour might change, particularly zero, one and negative integers.
Taking $n = 1$ gives $n^2=1$ , so $n^2=n$ , not $n^2>n$ .
Therefore, $n = 1$ is a counterexample and the original universal claim is false. In fact, $n^2>n$ holds for integers satisfying $n < 0$ or $n > 1$ .

Common Mistake

Treating examples as proof

Checking many examples can suggest that a claim is true, but it does not prove a universal claim. However, one valid counterexample is enough to disprove it.

Valid arguments and true conclusions

An argument is valid if its conclusion must follow whenever its premises are true. Validity concerns the logical structure of the reasoning.

A sound argument is valid and also has true premises. A valid argument can begin from a false premise, so validity alone does not guarantee that its conclusion describes reality.

Consider:

All square numbers are even. The number 9 is square. Therefore, 9 is even.

The structure is valid: if both premises were true, the conclusion would follow. However, the first premise is false, so the argument is not sound.

Key Idea

Check logic and starting information

Ask two separate questions: “Are the premises true or justified?” and “Does the conclusion actually follow from them?”

Necessary and sufficient conditions

A condition is sufficient if it guarantees a result. A condition is necessary if the result cannot occur without it.

This distinction matters when an argument incorrectly reverses an implication.

Definition

Implication and converse

The implication $P⇒QP\Rightarrow Q$ means that whenever $P$ is true, $Q$ must be true. Its converse is $Q⇒PQ\Rightarrow P$ , which is a different statement and may be false.

Example

Critiquing a reversed implication

A student argues: “If an integer is divisible by 6, then it is divisible by 3. Since 12 is divisible by 3, it must be divisible by 6.”

The original implication is: divisible by 6 $⇒\Rightarrow$ divisible by 3.
The student uses the converse: divisible by 3 $⇒\Rightarrow$ divisible by 6. The original implication does not justify this reversal.
The converse is false; for example, 9 is divisible by 3 but not by 6. Although 12 happens to be divisible by 6, the student’s method does not establish that conclusion.

Common Mistake

Accepting a true conclusion from invalid reasoning

A conclusion may happen to be true even when the argument is invalid. You must assess the reasoning used, not just the final statement.

Checking algebraic and logical steps

When reading a proof or derivation, check that each operation is permitted. Common gaps include:

dividing by an expression that might equal zero;
taking a square root without considering both signs;
squaring both sides and introducing additional solutions;
applying a result outside its stated domain;
assuming the statement that is supposed to be proved.

An argument that assumes its own conclusion is called circular reasoning. It gives no independent reason for accepting the conclusion.

Example

Finding an illegal division

The following argument claims to prove that $1 = 2$ . It begins by assuming $a = b$ , where both values are non-zero.

From $a = b$ , multiplying by $a$ gives $a^2=ab$ . Subtracting $b^2$ gives
$a^2-b^2=ab-b^2.$
Factorising both sides gives
$(a - b) (a + b) = b (a - b) .$
The argument then divides by $a - b$ to obtain $a + b = b$ . However, the premise $a = b$ means that $a - b = 0$ . This step is division by zero, which is undefined.
Every later conclusion depends on that invalid step, so the argument does not prove that $1 = 2$ .

Common Mistake

Equivalent steps

An algebraic step should preserve the relevant solutions. Squaring an equation may create extra solutions, while dividing by an expression may remove values for which that expression is zero.

Critiquing justifications of methods and formulae

A derivation obtains a result or formula from known facts through justified steps. When reading one, identify:

the definitions or earlier results being used;
the assumptions under which the formula holds;
whether the algebra is correct;
whether the conclusion has the same scope as the reasoning.

Example

Checking a claimed formula for an arithmetic sequence

A student claims that the sum of the first $n$ terms of an arithmetic sequence is $Sn=n2(a+l)S_n=\frac{n}{2}(a+l)$ , where $a$ is the first term and $l$ is the final term.

Write the sum forwards and backwards:
$Sn=a+(a+d)+⋯+l,Sn=l+(l−d)+⋯+a.\begin{aligned} S_n&=a+(a+d)+\cdots+l,\\ S_n&=l+(l-d)+\cdots+a. \end{aligned}$
Adding corresponding terms gives $a + l$ each time. Since there are $n$ pairs,
$2S_n=n(a+l).$
Dividing by 2 gives
$Sn=n2(a+l).S_n=\frac{n}{2}(a+l).$
The derivation is justified because an arithmetic sequence has a constant difference, making every paired sum equal.

Arguments in applications and modelling

When critiquing an applied argument, check:

whether relevant quantities have been included;
whether units and numerical accuracy are consistent;
whether the assumptions suit the context;
whether the data support the claimed conclusion;
whether the final answer is interpreted sensibly.

Example

Critiquing a constant-speed model

A journey of 120 km takes 2 hours. A student concludes that the vehicle travelled at exactly 60 km h⁻¹ throughout.

The calculation $1202=60\frac{120}{2}=60$ correctly gives the average speed over the whole journey.
Average speed does not determine the speed at each instant. The vehicle could have stopped, accelerated or travelled at several different speeds.
The conclusion “average speed was 60 km h⁻¹” is justified. The stronger conclusion “speed was exactly 60 km h⁻¹ throughout” requires the additional assumption that the speed remained constant.

Tip

Strength of conclusions

Use the weakest conclusion fully supported by the evidence. Do not replace “may”, “on average” or “under these assumptions” with an unjustified definite claim.

Writing a clear critique

A strong critique identifies the exact issue and explains its effect. Avoid vague comments such as “the proof is wrong”.

A useful structure is:

State the step or assumption being challenged.
Explain why it is invalid, incomplete or unjustified.
Give a counterexample or corrected condition where possible.
State whether the conclusion is false, or merely not proved by that argument.

Exam technique

In the exam

Identify the premises, the conclusion and the scope of the claim before checking individual steps.
Test universal claims using boundary values, zero, negative values and other special cases where permitted.
Name the precise flaw, such as reversing an implication, dividing by zero or using an unsuitable modelling assumption.
Distinguish “the conclusion is false” from “the argument does not prove the conclusion”.
Check that any formula or applied result is used within its stated conditions and domain.

Self review

Check yourself

Why can one counterexample disprove a universal claim, while many supporting examples cannot prove it?
How would you distinguish a valid argument from a sound argument?
What assumptions would you check before accepting a conclusion produced by a mathematical model?

What you'll learn

What is a mathematical argument?

Understanding the scope of a claim

Valid arguments and true conclusions

Necessary and sufficient conditions

Checking algebraic and logical steps

Critiquing justifications of methods and formulae

Arguments in applications and modelling

Writing a clear critique

Review 1.1.5 Comprehending and critiquing arguments by teaching Genie

1.1.5 Comprehending and critiquing arguments

What you'll learn

What is a mathematical argument?

Understanding the scope of a claim

Valid arguments and true conclusions

Necessary and sufficient conditions

Checking algebraic and logical steps

Critiquing justifications of methods and formulae

Arguments in applications and modelling

Writing a clear critique

Review 1.1.5 Comprehending and critiquing arguments by teaching Genie

1.1.5 Comprehending and critiquing arguments

What you'll learn

What is a mathematical argument?

Understanding the scope of a claim

Valid arguments and true conclusions

Necessary and sufficient conditions

Checking algebraic and logical steps

Critiquing justifications of methods and formulae

Arguments in applications and modelling

Writing a clear critique

Review 1.1.5 Comprehending and critiquing arguments by teaching Genie

1.1.5 Comprehending and critiquing arguments Revision Guide

1.1.5 Comprehending and critiquing arguments

What you'll learn

What is a mathematical argument?

Understanding the scope of a claim

Valid arguments and true conclusions

Necessary and sufficient conditions

Checking algebraic and logical steps

Critiquing justifications of methods and formulae

Arguments in applications and modelling

Writing a clear critique

Review 1.1.5 Comprehending and critiquing arguments by teaching Genie

1.1.5 Comprehending and critiquing arguments Revision Guide