Evaluate conditional reasoning
Use necessary and sufficient conditions correctly and expose invalid reversals with counterexamples.
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Before you begin
- Read an English passage closely
- Distinguish a statement from its supporting reason
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before evaluating support, what should you identify?
What you will learn
- Recognise invalid deduction and confused conditions
- Explain the impact of an invalid inference
Write the implication in one direction
‘If the detector is working and detects smoke, the alarm sounds’ gives a conditional rule. An alarm may also sound during a test, so sounding does not prove smoke detection. If P implies Q, accepting P establishes Q; rejecting Q excludes P. Accepting Q does not establish P, and rejecting P does not exclude Q.
The last two mistakes are affirming the consequent and denying the antecedent. The terminology matters less than explaining why the conclusion is not required by the rule. Construct a case with the reason true and conclusion false to demonstrate the gap.
Distinguish necessary from sufficient
A necessary condition must hold for an outcome, but may not guarantee it. A sufficient condition guarantees the outcome when it holds, but may not be the only route. Passing a safety check may be necessary for entry without guaranteeing an available place; a gold pass may guarantee entry without being required for every visitor.
State whether the flaw undermines the claimed certainty or only weakens a probable inference. Do not conclude that the outcome is impossible merely because the supplied argument has not proved it.
Sources for this lesson
Pause and explain
From P implies Q and not-Q, what follows?
Worked example
‘All gold passes admit entry. Lin entered, so Lin had a gold pass.’ Evaluate.
Show the worked solution
- Gold implies entry.
- The rule does not say only gold admits entry.
- A standard pass could be another route.
Answer The conclusion does not follow. The argument affirms the consequent by treating a sufficient route as necessary; Lin could have entered through another permitted route.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Guided: Give a counterexample to the gold-pass inference.
Give me a hint
Keep the admission rule true.
Compare my reasoning
- All gold passes can still admit entry.
- Standard passes can also admit entry.
- Lin uses a standard pass.
Lin could use a standard pass while every gold pass still admits entry. The original premise remains true and the conclusion about Lin’s gold pass is false, proving the deduction invalid.
Look for these in your work
- Preserves the premise
- Makes the conclusion false
- Explains the counterexample
Independent: ‘Everyone entering must register. Sam registered, therefore Sam entered.’ Evaluate.
Give me a hint
Registration may occur without a visit.
Compare my reasoning
- Entry implies registration.
- Registration is necessary, not sufficient.
- Sam might register and then stay home.
The inference is invalid. Registration is a necessary entry condition, but someone may register without attending. The facts permit Sam to stay home, so registration does not prove entry.
Look for these in your work
- Identifies the direction
- Distinguishes conditions
- Provides a counterexample
Application: ‘If the network is down, uploads fail. The network is not down, so uploads cannot fail.’ Evaluate.
Give me a hint
Other causes can produce failure.
Compare my reasoning
- The rule gives one sufficient cause of failure.
- An incorrect password could also cause failure.
- Not having that cause does not guarantee success.
This denies the antecedent. Uploads may fail for other reasons even when the network is working; the stated rule does not establish that network failure is the only cause.
Look for these in your work
- Identifies the invalid step
- Gives another permitted cause
- Limits the conclusion
Common mistakes
- Reversing a one-way rule
- Treating a necessary step as a guarantee
- Inferring the opposite when proof fails
What can you explain now?
From P implies Q and not-Q, what follows?
Compare with the explanation
Not-P
If P were true, Q would have to be true, contradicting not-Q.
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Make it stick
From P implies Q and not-Q, what follows?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.