Hypothetical Syllogism
A free Logic lesson from the “Argument Validity” unit, with a worked example and practice problems including step-by-step solutions.
If p leads to q and q leads to r, then p leads to r. This lets students connect multi-step reasoning. Learning objective: Chain conditional statements together. 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: 'If p then q; p; therefore q' is valid modus ponens. Example 2: 'If p then q; q; therefore p' is affirming the consequent and is invalid. 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
- Chain conditional statements together
- 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 (Hypothetical Syllogism): Worked example: If a student completes the practice, then they unlock the quiz. If they unlock the quiz, then they can submit it. What conclusion follows?
- Worked Example: First identify exactly what the question is asking: Example case A (Hypothetical Syllogism): Worked example: If a student completes the practice, then they unlock the quiz. If they unlock the quiz, then they can submit it. What conclusion follows?
- For a syllogism, follow how one premise rules in, rules out, or chains to the next claim.
- Hypothetical syllogism chains conditionals.
- p leads to q, and q leads to r.
Answer: If the student completes the practice, then they can submit the quiz.
Practice problems
1. Practice case A (Hypothetical Syllogism): Practice: If a student completes the practice, then they unlock the quiz. If they unlock the quiz, then they can submit it. What conclusion follows?
Choices: If the student completes the practice, then they can submit the quiz. · If the student submits the quiz, then they completed the practice. · The student cannot unlock the quiz. · No conditional can be formed.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case A (Hypothetical Syllogism): Practice: If a student completes the practice, then they unlock the quiz. If they unlock the quiz, then they can submit it. What conclusion follows?
- For a syllogism, follow how one premise rules in, rules out, or chains to the next claim.
- Hypothetical syllogism chains conditionals.
- p leads to q, and q leads to r.
- Therefore p leads to r.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: If the student completes the practice, then they can submit the quiz.
2. Practice case B (Hypothetical Syllogism): Practice: Which argument form does this use? "If it freezes, then the pipes contract. If the pipes contract, then they may crack."
Choices: Hypothetical Syllogism · Modus Ponens · Modus Tollens · Disjunctive Syllogism
Show solution
- Hypothetical Syllogism has the pattern p -> q; q -> r; therefore p -> r.
- Match the premises and conclusion to the pattern.
- This form is valid.
Answer: Hypothetical Syllogism
3. Practice case C (Hypothetical Syllogism): Practice: Which symbolic pattern names Hypothetical Syllogism?
Choices: p -> q; q -> r; therefore p -> r · p -> q; p; therefore q · p -> q; ¬q; therefore ¬p · p or q; ¬p; therefore q
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case C (Hypothetical Syllogism): Practice: Which symbolic pattern names Hypothetical Syllogism?
- For a syllogism, follow how one premise rules in, rules out, or chains to the next claim.
- Hypothetical Syllogism has a specific shape.
- p -> q; q -> r; therefore p -> r
- This form is valid.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: p -> q; q -> r; therefore p -> r
4. Practice case D (Hypothetical Syllogism): Practice: Is Hypothetical Syllogism a valid argument form?
Choices: Valid: the conclusion must follow from the premises. · Invalid: the conclusion does not have to follow. · It depends only on whether the conclusion sounds realistic. · It is valid only when the sentences use variables.
Show solution
- Hypothetical Syllogism preserves logical force.
- Hypothetical syllogism chains conditionals.
- p leads to q, and q leads to r.
Answer: Valid: the conclusion must follow from the premises.
5. Practice case E (Hypothetical Syllogism): Practice: Which sentence best describes Hypothetical Syllogism?
Choices: Chain p → q and q → r into p → r. · From p → q and p, conclude q. · It picks the conclusion that sounds most familiar. · It works only when the sentences are short.
Show solution
- Hypothetical Syllogism has a precise shape.
- Chain p → q and q → r into p → r.
- The distractors describe a different form or no real rule.
Answer: Chain p → q and q → r into p → r.
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