Back to the on-screen lesson · Print

Valid is not the same as true

Whether the conclusion follows and whether the premises are true are two questions, and only one box out of four settles anything.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

You will take any argument and ask its two questions in order: granting every premise, could the conclusion still be false, and are the premises in fact true. You will give a countermodel when the form fails, say which of the four combinations you are looking at, and name which half of the argument your objection is aimed at — which is what turns a disagreement into a question somebody can answer.

2. What you already have

You can judge a move from an if-then and produce a countermodel when it fails. Every one of those judgments was about form. This last lesson says what form does and does not buy you, and it is the sentence the whole course has been heading for.

3. Keep the two audits distinct

4. Two properties that require different evidence

Valid is a fact about the shape of an argument: if the premises were true, the conclusion would have to be. True is a fact about the world. An argument can be valid with a false premise and therefore unsound, and it can be invalid with every premise true and a true conclusion into the bargain. Two questions, asked separately, always.

Here are all four combinations, with an example of each.

Only the valid, all-true-premise combination guarantees the conclusion through this argument. Other evidence may still establish the conclusion independently.

Two consequences are worth saying out loud.

A valid argument can have a false conclusion. With a false premise, validity no longer guarantees the conclusion's truth. The conclusion may be true or false; valid inference does not automatically transmit falsity.

An invalid argument can have a true conclusion. People arrive at right answers by bad routes all the time. The conclusion being true alone does not establish validity — and this is the one that is hardest to hold on to, because agreeing with a conclusion makes an argument feel good.

5. Why the form question comes first

The form question can be answered by somebody who knows nothing about the subject. Every glorp is a flurb; this is a glorp; so this is a flurb is valid, and you have no idea what any of those words mean. That is not a trick — it is the reason validity is worth having as a separate idea, and the reason a grader can settle it exactly while it cannot settle whether a premise is true.

It also makes arguments easier to have. If the form fails, this proposed deductive route fails; the factual disagreement may still need investigation. If the form holds, inspect the premises and the accuracy of the translation to identify what remains disputed.

That is the practical payoff of the whole course: not winning arguments, but finding out what the argument is actually about.

6. What this course settles, and what it does not

This course checks one thing at a time and says which: whether a conclusion follows from the reading the item states, what number the counts give, and which group a piece of evidence covers. It does not report whether you reason well in general, and no question here is about what you believe.

So the fourteen lessons check exactly these things: which group a piece of evidence covers, what a count out of a sample permits, what the four boxes of a table say, what an if-then forbids, which of four moves is valid, and whether a conclusion follows from stated premises. Each has an exact answer, and that is why each is asked.

What is left out is left out on purpose. Whether a premise is true is usually a question for a different subject — history, biology, the evening news — and this course hands it over rather than pretending to answer it. Whether you should believe the conclusion is yours. The most a grader can honestly say is this follows and that does not, and it is more useful than it sounds.

7. False premises can lead to either kind of conclusion

Consider the valid form P -> Q, P; therefore Q. In an actual assignment with P false and Q true, the conditional is true, P is false, and the conclusion Q is true. The argument is unsound because one premise is false, yet its conclusion is true. This is a direct counterexample to the slogan that a valid argument always carries falsehood from input to output.

Now choose actual P true and Q false. The conditional premise becomes false while the atomic premise is true, and the conclusion is false. The form remains valid because no assignment makes both premises true and Q false. These two actual situations demonstrate the correct limit: false premises remove the guarantee, leaving either conclusion value possible.

An invalid form is also compatible with a true conclusion. For P -> Q,Q; therefore P, the actual assignment P true,Q true makes every displayed statement true. But P false,Q true remains a countermodel to the form. Actual truth on one assignment does not erase a possible true-premise false-conclusion assignment elsewhere. Keep the actual row and the countermodel row clearly labeled.

If the actual premises are known true and the actual conclusion false, that actual situation itself witnesses invalidity. This is a special combination with a decisive consequence. If at least one premise is false, the actual row alone cannot diagnose validity. You must return to the all-assignments test rather than infer the form from the observed conclusion.

8. An audit can leave a question open without marking it false

Suppose an argument has a valid form and its first premise is supported by a current record, but its second premise is unchecked. Soundness has not been established. That verdict differs from proven unsoundness: the unchecked premise might turn out true. An honest report marks the missing evidence instead of pretending that uncertainty is a false truth value.

The same applies to the translation. A phrase such as approved might refer to approval in principle, approval for payment, or final release. If different occurrences use different meanings, a neat symbolic pattern can hide an equivocation. Fix the complete atom meanings before testing the form, then check that the records concern those same claims. The notation cannot validate a shift in what a word denotes.

When more than one defect exists, you may report both, provided each is explained separately. A premise can be false and the inference can also be invalid. There is no requirement to suppress one justified objection. Give the countermodel for the formal defect and the relevant record for the factual defect, so neither is mistaken for evidence of the other.

Finally, distinguish the conclusion's truth from the quality of this particular argument for it. A failed route can motivate a search for better evidence rather than a reversal of belief. Saying 'this argument has not established the claim' does not assert the opposite claim. This restraint is the final application of the course's recurring rule: report exactly what the supplied evidence and inference warrant.

9. Preserve the audit record

Keep the actual assignment beside the factual verdict and the countermodel beside the inference verdict. Labeling these two rows prevents a reader from treating a hypothetical possibility as an observation.

10. Review a fictional dispatch report

A dispatch report argues: every dispatched box has a tracking record; box A has been dispatched; therefore box A has a tracking record. The form is valid. Suppose the actual audit establishes that box A is still waiting, but a tracking record was prepared in advance. Let D mean dispatched and K mean tracking record exists. The actual assignment is D false and K true.

The first premise D -> K is true on this assignment, and the second premise D is false. The conclusion K is true. This particular argument is unsound, but the actual tracking record establishes its conclusion independently. The correct review therefore identifies the false dispatch premise without claiming that the record does not exist.

A second report argues from D -> K and K to D. It uses true premises in this actual case and reaches the false conclusion D. The actual row D false,K true is a countermodel, so this reversed inference is invalid. The two reports concern the same box and same record but have different logical defects because their premise lists and targets differ.

For a third box, suppose dispatch is verified but the record system's guarantee has not been checked. MP still supplies the conditional formal result, but soundness is not established from the evidence available. An audit should request the missing support for the general premise, rather than treating the familiar pattern as proof that the database is complete. The form check and the record check work together.

11. Do not overread either failure

A false conclusion alone does not prove invalidity; an actually true premise set with that false conclusion does. A true conclusion alone does not prove validity. False premises do not force false conclusions, even under valid inference. Finally, uncertainty about a premise leaves soundness unestablished rather than automatically disproved. You may raise both factual and formal objections when both are justified, but explain them with their separate evidence.

12. A valid route with a true conclusion and false premise

1. State the formal route.

   D -> K,D; therefore K

   It has MP form.

2. Check the guarantee of the form.

   D=T,K=F would falsify D -> K

   No countermodel preserves both premises.

3. Record the actual facts.

   D=F,K=T

   The box is waiting but has a prepared record.

4. Evaluate the actual premises.

   D -> K=T; D=F

   One premise is false.

5. Report the two results.

   Valid but unsound; K actually true

   False premises remove a guarantee without forcing false conclusions.

13. Use an actual failure as a countermodel

1. State the reversed route.

   D -> K,K; therefore D

   The inference affirms the consequent.

2. Use the actual assignment.

   D=F,K=T

   The prepared record exists before dispatch.

3. Check the first premise.

   D -> K=T

   The antecedent is false.

4. Check the second premise.

   K=T

   The record exists as asserted.

5. Check the conclusion.

   D=F

   Dispatch has not happened.

6. State the inference defect.

   Invalid and unsound

   The actual facts preserve all premises and falsify the target.

14. Keep an unchecked premise open

1. State the proposed argument.

   P -> Q,P; therefore Q

   The formal pattern is MP.

2. Audit the form.

   Valid

   A true antecedent and true conditional guarantee the consequent.

3. Record the first evidence status.

   P verified true

   The individual event has been checked.

4. Record the second evidence status.

   P -> Q not yet verified

   The claimed general guarantee needs support.

5. Avoid assigning falsehood from ignorance.

   Unchecked does not mean false

   The evidence gap has not settled the premise's truth.

6. State the current soundness status.

   Not established

   One required factual component remains unsupported.

7. Identify the next evidence task.

   Check the guarantee within its stated scope

   That investigation can address the open premise without retesting the already valid form.

15. Correct a garbage-in slogan

1. Use a valid form.

   P -> Q,P; therefore Q

   The form preserves truth when both premises are true.

2. Choose an actual assignment.

   P=F,Q=T

   One premise is false while the conclusion is true.

3. Your turn: work this step out. Its working is at the end of the packet.

   State the corrected principle.

16. Guided practice

P -> Q,P; therefore Q. Actual P=F,Q=T. Count countermodels to the form across all four P,Q assignments, count actually false premises, and write T (true) or F (false) for the actual conclusion. A countermodel makes every premise true and the conclusion false; zero countermodels establishes validity.

Countermodels: _b0

False premises: _b1

Conclusion: _b2

17. Guided practice

For P -> Q,Q; therefore P, let actual P=T,Q=T. Count actually false premises and countermodels over all assignments.

1. Evaluate the actual premise values.

   T,T; _false false premises

   The supplied actual assignment satisfies the arrow and Q.

2. Search the other assignments.

   FT; _{countermodels} countermodel

   There the premises hold while P fails.

3. Keep the judgments separate.

   Actual true statements; invalid argument

   Truth at the actual row does not establish an all-row guarantee.

18. Guided practice

Assess only the form P -> Q,~P; therefore ~Q. Give a countermodel if invalid.

P -> Q
~P
∴ ~Q

valid invalid — countermodel:

19. Practice

P -> Q,Q; therefore P. Actual P=T,Q=T. Count countermodels to the form across all four P,Q assignments, count actually false premises, and write T (true) or F (false) for the actual conclusion. A countermodel makes every premise true and the conclusion false; zero countermodels establishes validity.

Countermodels: _b0

False premises: _b1

Conclusion: _b2

20. Practice

A report uses D -> K,D; therefore K. The actual dispatch audit shows D false and K true. Write the number of false premises and the number of true conclusions in this one argument. Then count how many of these two soundness requirements fail: (1) the form is valid; (2) every actual premise is true.

False premises: _b0

True conclusions: _b1

Failed soundness requirements: _b2

21. Somewhere new

P -> ~Q,P; therefore ~Q. Actual P=T,Q=T. Count countermodels to the form across all four P,Q assignments, count actually false premises, and write T (true) or F (false) for the actual conclusion. A countermodel makes every premise true and the conclusion false; zero countermodels establishes validity.

Countermodels: _b0

False premises: _b1

Conclusion: _b2

22. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

23. Test question

P -> Q,Q; therefore P. Actual P=F,Q=T. Count countermodels to the form across all four P,Q assignments, count actually false premises, and write T (true) or F (false) for the actual conclusion. A countermodel makes every premise true and the conclusion false; zero countermodels establishes validity.

Countermodels: _b0

False premises: _b1

Conclusion: _b2

24. What you can do now

You can separate validity from truth, give a countermodel, and say what an argument has and has not settled. Tell someone why a valid argument with a false premise does not guarantee its conclusion, and why arriving at the right answer is not the same as having shown it.

Working for the steps left to you

15. Correct a garbage-in slogan, step 3