DWG NO. 02.3 — Lesson 3 of 5

Deductive Reasoning and the Laws of Logic

Unit 2: Reasoning and Proof · ~15–30 min

Objective

Apply the Law of Detachment and the Law of Syllogism to draw logically valid conclusions from true conditional statements.

Deductive vs. inductive reasoning

Deductive reasoning uses accepted facts, definitions, and postulates to reach a conclusion that must be true. That's different from the inductive reasoning of Lesson 2.1, which only spots a pattern that's likely true. Deductive reasoning is how conjectures get turned into proven theorems — and it runs on two core laws.

The Law of Detachment

If a conditional p → q is true, and its hypothesis p is also true, then the conclusion q must be true. In other words: once you know the “if” part happened, the “then” part is guaranteed.

The Law of Syllogism

If p → q is true and q → r is true, then p → r is true. Two conditionals chain together into a new one, as long as the conclusion of the first matches the hypothesis of the second.

Two true conditionals chain into a third: p→q and q→r combine into p→r.

Worked Example 1 · Law of Detachment

ProblemGiven: “If a figure is a square, then it is a rectangle” (true), and “Figure ABCD is a square” (true).

1.The hypothesis of the conditional (“a figure is a square”) matches the given fact about ABCD.

2.By the Law of Detachment, the conclusion must follow: ABCD is a rectangle.

Conclusion: ABCD is a rectangle

Worked Example 2 · Law of Syllogism

ProblemGiven: “If an angle is acute, then it measures less than 90°” and “If an angle measures less than 90°, then it is not a right angle.”

1.The conclusion of the first statement (“measures less than 90°”) matches the hypothesis of the second.

2.By the Law of Syllogism, chain them into one new conditional.

If an angle is acute, then it is not a right angle.

Guided practice

1.Given “If a triangle is equilateral, then it is isosceles” (true) and “Triangle XYZ is equilateral” (true), what can you conclude?
Show answer
By the Law of Detachment: Triangle XYZ is isosceles.
2.Combine using the Law of Syllogism: “If a number ends in 0, then it is divisible by 5” and “If a number is divisible by 5, then it is not prime (unless it's 5 itself).”
Show answer
If a number ends in 0, then it is not prime (unless it's 5 itself).
3.Given “If a shape is a square, then it has four right angles” (true) and “Shape J has four right angles,” is it valid to conclude Shape J is a square? Explain.
Show answer
No. This affirms the conclusion instead of the hypothesis — that's not a valid use of the Law of Detachment. A rectangle also has four right angles without being a square, so the conclusion doesn't have to follow.
4.Combine three true conditionals via syllogism: “If p, then q,” “If q, then r,” and “If r, then s.”
Show answer
If p, then s. (Chain p→q with q→r to get p→r, then chain that with r→s to get p→s.)
5.In one sentence each, explain the difference between inductive and deductive reasoning.
Show answer
Inductive reasoning finds a likely pattern from specific examples (a conjecture); deductive reasoning uses true statements and logical laws to reach a conclusion that is guaranteed to be true.

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