The converse is a different claim

Exchange a conditional's complete sides, and test the reversed claim with its own evidence.

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 construct a converse, compare its truth conditions with the original, and identify the evidence needed for the reverse direction without assuming it is either automatically true or automatically false.

2. What you already have

You can read P → Q as the one combination it forbids, and you can build its column. This lesson swaps the two halves round and builds the column again, which is the shortest proof there is that the swap changes the claim.

3. Different directions

4. Swap the halves and you have a new claim

Term	What it means
Converse	The result of exchanging a conditional's complete antecedent and consequent.
Independent evidence	Support for a claim beyond merely repeating a different conditional.
Biconditional	A claim asserting both directions together.
Counterexample to a converse	A case with the original consequent true and original antecedent false.

If A then B and if B then A are two different claims. One can be true while the other is false, and swapping them is the commonest mistake anybody makes with an if. The test is always the same: find one case where the first half fails and the second half holds anyway.

They disagree in two rows out of four. Nothing could be a cleaner demonstration that these are different sentences: each one is false in a row where the other is true.

P	Q	P → Q	Q → P
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	T

In words: if it is raining, the pitch is wet is broken by rain on a dry pitch. If the pitch is wet, it is raining is broken by a wet pitch under a clear sky — a groundsman with a hose, a burst pipe, dew. The first claim is modest and probably true. The second makes a different requirement: wetness requires rain, excluding other routes to wetness.

That is the pattern to watch for. Neither direction is logically stronger in general: each excludes a row the other permits. Adding the converse to the original creates a stronger combined claim.

Another way: picture

Draw a small circle inside a big one. The small circle is the rainy days, the big one the wet-pitch days. If raining then wet says the small circle sits inside the big one. The converse says the big circle sits inside the small one — which, if both directions hold, would mean they are the same circle, and that is a far bigger thing to claim.

Another way: steps

To test a converse:

Write the original with its two halves underlined.
Swap them and read the new sentence aloud. It should sound different; if it does not, read it again more slowly.
Hunt for a case where the new if-part holds and the new then-part fails — a wet pitch with no rain.
One such case settles it. No such case means you have not shown the converse false, which is not the same as showing it true.

5. Compare like claims before calling one a converse

'Every registered volunteer has an identification card' has the form V -> C for a particular person. Its converse is C -> V: everyone with such a card is a registered volunteer. Cards issued to visitors would refute the converse without refuting the original. This is a clean reversal because the same two complete claims exchange roles without any additional change.

'Successful students often read' and 'reading causes success' are not a clean conditional-converse pair. The second also introduces causation and changes a frequency claim. Such a move may be unjustified, but diagnosing it accurately requires naming the extra changes. Likewise, changing 'possible' into 'certain' is a change in strength, not merely a reversed arrow. A careful analysis reconstructs what actually changed before selecting a familiar logical pattern.

Use fictional screening rules when the exercise supplies exceptionless guarantees. If every marked component triggers a scanner, the converse says every scanner trigger identifies a marked component. A trigger caused by an unmarked component separates the two rules. For an actual scanner, the rates and operating conditions need evidence; the formal exercise does not declare any real test perfect.

6. Ask which group each direction covers

A conditional and its converse select different antecedent groups. The original V -> C concerns volunteers and asks whether they have cards. The converse C -> V concerns cardholders and asks whether they are volunteers. Even if the same people appear in the records, the denominators for these two audit questions can differ. A perfect result in one group does not automatically imply a perfect result in the other.

Imagine six volunteers, all with cards, and four visitors who also have cards. The original rule has no violation among the six volunteers. The converse has four violations among ten cardholders. These numbers explain the informational gap without needing an abstract label. Card possession is compatible with volunteering, but the record does not make it exclusive to volunteers.

Now imagine a different club in which cards are issued exactly to volunteers. Both groups contain the same six people. The two directions hold in that complete stipulated record. This does not make reversal a generally valid operation. It shows that extra facts can establish the converse in a particular setting. Distinguish the rule of inference from the truth of this specific reversed claim.

When the record is incomplete, a missing counterexample is not a proof of either direction. If you inspect only volunteers, you might never encounter the visitor cards that refute the converse. An evidence search designed around the original antecedent group can therefore be badly designed for the reversed question. State the new target population before treating an earlier audit as support.

7. Test reversals with unchanged meanings

To construct the converse, exchange the complete sides of the arrow without adding or removing negation. The converse of P -> ~Q is ~Q -> P. The consequent was already a denial; it moves intact into the antecedent position. Writing Q -> ~P instead would negate and exchange the original sides, which is a different transformation.

Compound conditions move intact as well. The converse of (P & Q) -> R is R -> (P & Q). It requires both P and Q whenever R holds. Reversing only one conjunct to get R -> P produces a different claim from the complete converse. It might be a consequence of that converse, but it does not state the same requirement.

For a concrete witness, give enough information to evaluate both formulas. If P is 'this item was ordered' and Q is 'this item arrived', an unordered arrival gives P false and Q true. The original P -> Q survives that record; the converse Q -> P fails. The story could involve a gift or a mistaken delivery. You do not need to show which story actually happened to demonstrate that the original statement alone permits the assignment.

However, if a prompt includes an additional rule that no unordered item can arrive, that new premise closes this possibility. You must then test the enlarged argument. A counterexample valid against one premise alone may fail against a richer premise set. Preserve all stated evidence, and label any proposed repair as an addition rather than silently importing it into the original claim.

8. Use the failed reversal to improve a question

A converse error can reveal what evidence an investigation actually needs. If a manager wants to infer registration from card possession, ask how cards are issued, whether visitors receive them, and whether cards remain valid after registration ends. These are specific ways the original rule could hold while its converse fails. The counterexample directs inquiry toward the missing restriction.

Do not respond by rejecting every use of indicators. An indicator can provide probabilistic evidence even when it is not a sufficient logical guarantee. A card might make volunteering more likely without establishing it with certainty. The task here asks whether an exceptionless arrow follows, so a single permitted alternative defeats that deductive claim. A different task about probability needs counts and comparison rates.

In a final explanation, write the original, write the converse, identify their different violating cases, and say what additional information would settle the converse. That is more informative than announcing 'converse fallacy' without showing the missing step. It also allows someone to defend the reverse claim with genuine independent evidence instead of being told that a converse can never be true.

9. Record why a direction was accepted

When both directions are supported, document the evidence for each rather than erasing the distinction. One direction may follow from a definition while the other depends on a complete record or an additional operating rule. The final biconditional should make those separate warrants visible.

10. Inspect the right records at a club desk

A club keeps a complete fictional record of sixteen people. Six are registered volunteers and have cards. Four are visitors with cards. The other six are visitors without cards. No volunteer lacks a card. Let V mean registered volunteer and C mean cardholder. The rule V -> C is satisfied for all sixteen people because the volunteer-without-card box is empty.

The converse C -> V fails for the four visitors with cards. There are ten cardholders altogether, six plus four, and only six are volunteers. In this declared record, sixty percent of cardholders are volunteers. By contrast, one hundred percent of volunteers have cards. These are different conditional percentages because they use different groups as denominators.

Suppose the desk wants a card to be sufficient evidence of volunteer status. The existing record shows that its current card system does not support that use. Staff could check a separate registration list or redesign the card system to distinguish visitors. The logic alone does not choose between those practical responses; it identifies the current inference that is unsupported.

If a revised record later shows no visitor cards, retest the new situation. A changed policy and complete new evidence might establish both directions. Do not carry a past counterexample into a different time without checking whether its conditions still apply. Conversely, do not erase the earlier failure merely because the system is now repaired.

11. Do not replace one error with another

A conditional does not generally entail its converse. That is different from saying a converse is always false. Both directions can hold because of an independently established definition or theorem, or because the stated model makes them hold. For arbitrary P and Q, the mixed rows show why reversal is not a valid general rule.

Neither arrow is automatically stronger than the other. P true and Q false violates the original but satisfies the converse; P false and Q true does the reverse. The conjunction of both arrows is stronger than either alone. Also keep the inverse separate: ~P -> ~Q negates both sides without exchanging them. The contrapositive, taught next, negates and exchanges them.

12. Reverse a whole conditional

13. Compare two evidence denominators

14. Move a compound antecedent intact

15. Keep a denial when exchanging sides

16. Guided practice

17. Guided practice

Compare P -> Q with Q -> P over TT,TF,FT,FF. Complete the count of matching and differing entries.

18. Guided practice

19. Practice

20. Practice

A complete club list has 9 volunteers with cards, 0 volunteers without cards, 3 visitors with cards and 8 visitors without. Write original V -> C violations, converse C -> V violations, and cardholders.

21. Somewhere new

Given the original ~(P & Q) -> R, write its converse without simplifying either side.

22. Lesson test

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

23. Test question

The original is (P | Q) -> ~R. Write its converse with complete sides preserved.

24. What you can do now

You can write a converse, build its column, and find the case that refutes it. Tell someone why a wet pitch means rain is a far bigger claim than rain makes the pitch wet. Next: the one transformation of an if-then that you are allowed to make.