What an if-then claim says · printable packet

A conditional forbids one combination out of four and stays silent about the other three.

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 find the if-part and the then-part of a conditional wherever English puts them, write the claim as a formula with the arrow starting from the if-part, and complete its truth-table column. You will be able to name the single combination that would show the claim false, and say why a case in which the if-part does not hold leaves the claim standing however it turns out.

2. What you already have

From Grade 7 you can write a sentence as a formula and complete a truth-table column for up to three letters. Both are used straight away here. What is new is not the machinery but the question: which cases does this claim cover?

3. The parts of a conditional

4. A conditional forbids one combination and permits the rest

There are four ways the world can be, and the claim has an opinion about only one of them.

Term	What it means
Antecedent	The condition before the arrow.
Consequent	The result required when that condition holds.
Violating case	A true antecedent together with a false consequent.
Material conditional	A two-valued combination false only on that violating case.

Raining	Pitch wet	Is the claim broken?
yes	yes	no — this is what it said would happen
yes	no	yes — it promised a wet pitch and got a dry one
no	yes	no — somebody watered it; the claim said nothing about this
no	no	no — the claim was never tested

So P → Q is false in exactly one row and true in the other three. It is a promise about the cases where P holds, and it is completely silent about the cases where P does not.

The two bottom rows are where people object. Surely if it is not raining and the pitch is wet, something has gone wrong with the claim? No. The claim never said rain was the only thing that wets a pitch. It said rain is enough to wet one. Those are different sentences, and telling them apart is the next two lessons.

Another way: picture

Picture a promise: if you finish your homework, I will take you to the pool. You break that promise in one way only — homework finished, no pool. If the homework is not finished and you go to the pool anyway, you have been generous, not dishonest, and the promise stands.

Another way: steps

To read any conditional:

Underline the if-part and the then-part. The if-part is not always first: whenever, provided that and as long as all introduce it in the middle of a sentence.
Say the forbidden combination out loud: if-part true, then-part false.
Ask whether the case in front of you is that combination. If it is not, the claim survives it.

5. Read the role of the clause, not its position

Let R mean it rains and M mean the match is off. 'If it rains, the match is off', 'the match is off if it rains', and 'the match is off whenever it rains' all put rain in the antecedent: R -> M. Moving the if-clause to the end of a sentence does not reverse its logical role.

'It rains only if the match is off' also has the truth-functional reading R -> M. Here only if identifies the necessary consequent M. It sounds unusual as an everyday claim, but the word order does not change the arrow. By contrast, 'the match is off only if it rains' becomes M -> R. That new claim makes rain necessary for cancellation and rules out a cancellation on a dry day.

To avoid guessing, state a case the sentence forbids. The original conditional rejects rain with the match not called off. If your formula rejects a different combination, inspect which clause you treated as necessary. Reading a sentence back from its formula is a check on interpretation, not a claim that rain causes a cancellation in every real setting.

6. A rule about records and a claim about every record

A truth-functional conditional describes the allowed combination for a case. An everyday statement such as 'every returned kit includes its charger' additionally ranges over a group of cases. We use one pair of atoms at a time: R means this kit is returned, and C means this kit includes its charger. The conditional R -> C is evaluated for each kit. One returned kit without a charger violates the general rule, provided it belongs to the group and time the rule covers.

A kit that has not been returned cannot supply that particular violation. It might contain a charger or lack one; neither combination gives R true and C false. This does not show that the kit is satisfactory under every policy. A separate rule might require chargers to remain attached even during use. Testing one requirement does not settle all possible requirements at once.

Distinguish surviving a test from being established. If you inspect one returned kit and find its charger, that case agrees with the universal rule. It does not prove that every other returned kit complies. A small set of nonviolating observations may leave many uninspected cases. If the relevant group is finite and you actually inspect every member reliably, the complete audit gives stronger evidence about that specified group.

The table itself does not require a survey. Its four rows describe the possible truth combinations of the two keyed claims. A record audit supplies counts of cases in those combinations. The table tells you which box contains violations; the audit tells you how many observed records are in that box. Keep these contributions distinct when explaining why a rule failed or what further evidence is needed.

7. Make a negative observation strong enough for the test

To record a violating case, you need evidence for both parts: the antecedent holds and the consequent fails. 'We did not find the charger in our first glance' may not establish that the returned kit lacks a charger. Perhaps it is in a hidden compartment. The logical test is exact, while the observations used as its inputs can be incomplete or mistaken. Specify what was checked before treating an absence as a false consequent.

Time can also change the result. If the policy concerns the state at check-in, a charger added the next morning does not show compliance at check-in. Conversely, a charger removed after a compliant check-in does not by itself refute the earlier record. Include the relevant time in both atom meanings so you do not combine an antecedent from one moment with a consequent from another.

Some rules have compound antecedents. 'If the kit is returned and its inspection is complete, a receipt is issued' has (R & I) -> S as its declared reading. A returned but uninspected kit without a receipt does not violate that statement, because the whole antecedent is false. A returned, inspected kit without a receipt does violate it. Checking only the first conjunct would incorrectly broaden the policy's coverage.

A conditional can contain a denied consequent as well. 'If an item is damaged, it is not released' becomes D -> ~L. Its violating case has D true and L true: the item is damaged and released. Do not mechanically look for the letter after the arrow to be false; look for the complete consequent to be false. Here the complete consequent is the denial ~L, which fails when L is true.

8. Preserve the difference between a guarantee and a tendency

The material conditional used here represents a declared exceptionless relationship within its scope. A statement that returned kits usually include chargers is different: it permits some violating-looking records because it does not make an all-cases guarantee. Translating usually as an unrestricted arrow would make the evidence appear to refute a stronger claim than was actually asserted.

Likewise, a rule does not by itself explain why its consequent occurs. Charger inclusion could result from a checklist, a packaging design, or careful staff. A correct conditional record does not identify the cause. Causal investigation needs additional comparisons and assumptions, as the earlier evidence lessons explain.

When reporting an audit, name the exact rule, its intended group and time, and the observation that would violate it. Then report whether such a case was actually established. This lets someone disagree about the evidence or the scope without confusing that disagreement with the truth rule for a conditional.

9. Audit a kit-return promise

A fictional equipment desk promises: every kit checked in before closing includes its charger at check-in. The complete record for today's twenty kits contains eight checked in with chargers, two checked in without chargers, six not checked in but known to have chargers, and four not checked in without chargers. Let R mean checked in before closing and C mean charger present at check-in for a returned kit, or at the stated audit time otherwise.

The four boxes total twenty. The promise concerns the ten checked-in kits: eight plus two. Among those, the two lacking chargers violate R -> C. The six unreturned kits with chargers do not violate the rule; charger possession was never said to require check-in. The four unreturned kits without chargers also do not supply this particular violation, although they might matter under another policy.

Report two violations among ten covered returns. Do not divide by all twenty when describing the compliance rate among returns: that would include ten cases outside the antecedent group. Nor should a ninety-percent nonviolation rate across all twenty be described as ninety-percent compliance among returns. The relevant denominator follows the question asked.

Before acting on the result, confirm that the record is complete for today's declared scope and that charger absence was checked at the right time. If one of the two apparently missing chargers was actually present in a compartment, correct the observation and recount. The truth rule identifies the violating combination; it does not establish the accuracy of the inspection.

10. Where this goes wrong

Thinking the claim is broken when the if-part is false. No rain and a wet pitch feels like a counterexample and is not one. The claim covered the rainy cases; this is not one of them.

*Reading if as only if. If it rains the pitch is wet* does not say rain is the only route to a wet pitch. Most arguments that go wrong with conditionals go wrong here, which is why the next four lessons are about it.

*Reading if as because.* A conditional says what goes with what. It does not say the first thing caused the second, and an argument that quietly upgrades one into the other has added something nobody agreed to.

Treating an if-then as untestable. It is easy to test: find a case with the if-part true and check the then-part. What makes it feel slippery is that most cases do not test it at all.

11. Find the one violating combination

12. Count covered records and violations

13. Respect a compound antecedent

14. A denied consequent

15. Guided practice

16. Guided practice

Audit the count of violating and nonviolating logical assignments for P -> Q. The four input pairs are TT,TF,FT,FF.

17. Guided practice

R means receipt sent and C means check complete. Translate: a receipt is sent whenever the check is complete.

18. Practice

The claim is P -> ~Q. The record has P true and Q true. Write the antecedent value, consequent value, and conditional value as T or F.

19. Practice

A return audit has TT=12,TF=3,FT=4,FF=1 for R:returned and C:charger present. For rule R -> C, write the number of returned kits, violations, and unreturned kits.

20. Somewhere new

The rule is (P & Q) -> R. The record has P true, Q false, R false. Write the antecedent value, consequent value, and conditional value as T or F.

21. Lesson test

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

22. Test question

The rule is P -> (Q & R). The record has P true, Q true, R false. Write the antecedent value, consequent value, and conditional value as T or F.

23. What you can do now

You can read a conditional as the one combination it forbids, complete its column, and name what would refute it. Tell someone why a wet pitch on a dry day does not break the claim that rain wets the pitch. Next: the converse, which is the claim people hear instead of the one that was made.

Working for the steps left to you

A damaged item that is released violates the rule.