Lesson 8: Arguments and validity (intermediate, 15 min)

State premises and a conclusion, and ask whether the conclusion follows. This intermediate lesson, about fifteen minutes, covers a valid argument, a plausible fallacy, the classic valid forms, and the case where inconsistent premises make anything follow.

Level: intermediate15 min6 claims in this lesson were confirmed by the exact checker at build time

State premises and a conclusion, check whether the conclusion really follows, and recognise valid forms, common fallacies, and vacuous validity.

In argument mode you give premises, claims you accept, and a conclusion. The argument is valid when no assignment makes every premise TRUE and the conclusion FALSE. The first step checks rain -> wet, rain, therefore wet: valid, modus ponens. The second tries rain -> wet, wet, therefore rain: invalid, affirming the consequent, with the counterexample rain = FALSE, wet = TRUE, a street wet for another reason. The third lists more valid forms, modus tollens and the chain rule rain -> wet, wet -> slippery, therefore rain -> slippery. The fourth is the vacuous case: from rain and NOT rain, snow follows, and the engine labels the validity vacuous.

Each of the four steps carries a verified claim about validity or invalidity. The exercises are four arguments to judge: modus tollens with an alarm, the matching fallacy of denying the antecedent, a disjunctive syllogism with card or cash, and an argument with premises that contradict each other.

The worked examples section has a page per argument form with the engine's verdict and witness, and the argument validity checker in the tools accepts any premises and conclusion. The counterexamples lesson that follows makes the refuting row the central object.

Do this lesson interactively in the app

Steps

1. When does a conclusion follow?

In argument mode you state premises, claims you accept, and a conclusion. The argument is valid when no assignment makes every premise TRUE and the conclusion FALSE.

Classic example (modus ponens): from rain -> wet and rain, conclude wet. Tangibool proves validity by searching for a counterexample and finding none.

p1 — rain -> wet · premisep2 — rain · premiseconclusion — wet · conclusion

rain: it is raining; wet: the street is wet;

When does a conclusion follow?
#	rain	wet	p1	p2	conclusion
0	FALSE	FALSE	TRUE	FALSE	FALSE
1	FALSE	TRUE	TRUE	FALSE	TRUE
2	TRUE	FALSE	FALSE	TRUE	FALSE
3	TRUE	TRUE	TRUE	TRUE	TRUE

Run “Does the conclusion follow?” and read the result.

valid · holds

2. Plausible, but invalid

From rain -> wet and wet, can you conclude rain? No: the street could be wet for another reason. The counterexample rain=FALSE, wet=TRUE makes both premises TRUE and the conclusion FALSE.

This fallacy is called affirming the consequent: it silently assumes the converse, wet -> rain.

p1 — rain -> wet · premisep2 — wet · premiseconclusion — rain · conclusion

rain: it is raining; wet: the street is wet;

Plausible, but invalid
#	rain	wet	p1	p2	conclusion
0	FALSE	FALSE	TRUE	FALSE	FALSE
1	FALSE	TRUE	TRUE	TRUE	FALSE
2	TRUE	FALSE	FALSE	FALSE	TRUE
3	TRUE	TRUE	TRUE	TRUE	TRUE

Check validity and replay the counterexample the checker returns.

invalid · holds

3. More valid forms

Some forms are valid whatever the inputs mean:

- modus tollens: from rain -> wet and NOT wet, conclude NOT rain; - chain rule: from rain -> wet and wet -> slippery, conclude rain -> slippery; - disjunctive syllogism: from card OR cash and NOT card, conclude cash.

The workspace shows the chain rule.

p1 — rain -> wet · premisep2 — wet -> slippery · premiseconclusion — rain -> slippery · conclusion

rain: it is raining; wet: the street is wet; slippery: the street is slippery;

More valid forms
#	rain	wet	slippery	p1	p2	conclusion
0	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE
1	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE
2	FALSE	TRUE	FALSE	TRUE	FALSE	TRUE
3	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	FALSE	TRUE	FALSE
5	TRUE	FALSE	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE	FALSE	FALSE
7	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

Remove premise p2 and check again: the conclusion no longer follows.

valid · holds

4. Inconsistent premises: valid, but vacuous

If the premises can never all be TRUE together, no counterexample can exist, so any conclusion is formally valid, even snow from rain and NOT rain.

Tangibool still reports such an argument as valid, but always with a warning that it holds only vacuously. Treat that warning as a sign that the premises contradict each other.

p1 — rain · premisep2 — NOT rain · premiseconclusion — snow · conclusion

rain: it is raining; snow: it is snowing;

Inconsistent premises: valid, but vacuous
#	rain	snow	p1	p2	conclusion
0	FALSE	FALSE	FALSE	TRUE	FALSE
1	FALSE	TRUE	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE	FALSE	FALSE
3	TRUE	TRUE	TRUE	FALSE	TRUE

Ask “Can all premises be true together?”

vacuously-valid · holds

Exercises

Premises: alarm -> noise and NOT noise. Conclusion: NOT alarm. Valid or not? (argument-validity)
Premises: alarm -> noise and NOT alarm. Conclusion: NOT noise. Valid or not? If not, give a counterexample. (argument-validity)
Premises: card OR cash and NOT card. Conclusion: cash. Valid or not? (argument-validity)
Premises: member -> discount, member, and NOT discount. Conclusion: refund. Valid or not? (argument-validity)

Do this lesson interactively in the app

Questions

What is a sound argument?

A valid argument whose premises are actually true. Logic can check validity; whether the premises hold in the world is a separate question that no truth table answers.

Why does the engine call the vacuous case valid?

Because validity is defined by the absence of a counterexample, and contradictory premises leave no row to build one on. The engine says valid and adds the vacuous warning, which is the honest reading of both facts.

Guides

Tools

Argument validity checker: premises, conclusion, counterexample or proof

Practice

Lessons

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.