Precision in Mathematical Language
A free Logic lesson from the “Foundations of Logical Thinking” unit, with a worked example and practice problems including step-by-step solutions.
Mathematical reasoning depends on precise language. Words like always, sometimes, at least, exactly, and if must be read carefully because small wording changes can change the meaning of a claim. Learning objective: Rewrite vague claims so they can be checked. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: 'All squares are rectangles' is a claim; 'Draw a square' is a command. Example 2: In 'All squares are rectangles, and this is a square, so this is a rectangle,' the first two sentences are premises and the last sentence is the conclusion. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.
What you'll learn
- Rewrite vague claims so they can be checked
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Precision in Mathematical Language): Classify the sentence "The number 18 is divisible by 3."
- It makes a claim that can be checked.
- Logic starts by deciding whether a sentence makes a true-or-false claim.
- The best classification is Statement.
Answer: Statement
Practice problems
1. Practice case A (Precision in Mathematical Language): Classify the sentence "The number 18 is divisible by 3."
Choices: Statement · Question · Command · Fragment
Show solution
- It makes a claim that can be checked.
- Logic starts by deciding whether a sentence makes a true-or-false claim.
- The best classification is Statement.
Answer: Statement
2. Practice case B (Precision in Mathematical Language): In the argument "All multiples of 6 are even. 24 is a multiple of 6. Therefore, 24 is even," which sentence is the conclusion?
Choices: 24 is even. · All multiples of 6 are even. · 24 is a multiple of 6. · 6 is even.
Show solution
- The conclusion is what the premises support.
- The word therefore signals the conclusion.
- So the conclusion is 24 is even.
Answer: 24 is even.
3. Practice case C (Precision in Mathematical Language): Which wording is most precise for a math claim?
Choices: Every integer greater than 2 is positive. · Numbers are usually big. · It works most of the time. · That answer feels right.
Show solution
- Precise claims say exactly which objects are included.
- Every integer greater than 2 is positive can be checked.
- The other choices use vague language.
Answer: Every integer greater than 2 is positive.
4. Practice case D (Precision in Mathematical Language): A valid argument is best described as one where:
Choices: the conclusion must follow if the premises are true · the premises are always short · the conclusion is probably popular · the words sound mathematical
Show solution
- Validity is about structure.
- Assume the premises are true and ask whether the conclusion is forced.
- That is the definition of validity.
Answer: the conclusion must follow if the premises are true
5. Practice case E (Precision in Mathematical Language): Which sentence is a premise in "If a figure is a square, then it has four sides. This figure is a square. So it has four sides."?
Choices: This figure is a square. · So it has four sides. · Four is even. · The answer is 4.
Show solution
- A premise is a reason used to support the conclusion.
- This figure is a square is one of the reasons.
- The conclusion is that it has four sides.
Answer: This figure is a square.
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