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Deny both halves of an if-then and swap them, and you have the same claim read from the far end.
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.
You will construct the contrapositive of a conditional, preserve the scope of compound sides, verify equivalence, and distinguish rewriting a rule from establishing the negative observation needed to apply it.
You can build the column for P → Q and for its converse, and you have seen them disagree. Now there are two more sentences that can be made from the same two halves, and one of them is worth a great deal.
| Term | What it means |
|---|---|
| Contrapositive | From A -> B, form ~B -> ~A. |
| Inverse | From A -> B, form ~A -> ~B. |
| Equivalent | The formulas agree on every assignment. |
| Observed denial | Evidence that the complete consequent fails in the relevant case. |
Four sentences out of two letters, and one table settles all of them.
| P | Q | P → Q | Q → P | ~P → ~Q | ~Q → ~P |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | T | F |
| F | T | T | F | F | T |
| F | F | T | T | T | T |
Read the columns. The first and the last are identical. The middle two are identical to each other and different from the first.
So:
This is a rewrite you are allowed to make without adding a premise, and it is genuinely useful: it lets you run a rule backwards. A rule says what follows when something happens; the contrapositive tells you what must not have happened when the consequence is missing.
Another way: picture
Small circle inside a big one again: rainy days inside wet-pitch days. Standing outside the big circle — the pitch is dry — you are certainly outside the small one too, because the small one is entirely inside the big one. That picture is the contrapositive, and it is the same picture as before, seen from outside.
Another way: steps
To take the contrapositive of a claim:
The contrapositive of P -> Q is ~Q -> ~P. The two formulas have the same truth conditions. Rewriting the rule therefore requires no new factual premise. But applying that rewritten rule to a particular case requires evidence that ~Q holds. The transformation alone does not announce that the consequent is absent or that the antecedent is false.
Suppose a fictional archive rule says that every uploaded file appears in the completed index. Let U mean uploaded and I mean appears in that index. Its contrapositive is ~I -> ~U. To conclude that a particular file was not uploaded, an investigator needs the further premise ~I. A failed search screen is not automatically that premise. The index might not have loaded, the search term might be misspelled, or the search might cover only part of the archive.
If the index is explicitly complete and current for the relevant time, and the file is confirmed absent, the denial has the right form. Together with the original guarantee it supports ~U. If either the guarantee or the observation is uncertain, the practical conclusion inherits that uncertainty. Formal equivalence preserves the content of a rule; it does not improve the evidence for it.
This distinction corrects the misleading slogan that backwards reasoning is free evidence. The equivalent conditional is available without an extra logical assumption. A conclusion about an actual file still depends on a trustworthy rule and an actual denied consequence. Write all those inputs down so a reader can see which are logical transformations and which come from inspection.
For (P & Q) -> R, the contrapositive is ~R -> ~(P & Q). The denial applies to the entire conjunction. It says the two conditions cannot both hold if R fails. It does not say both fail individually. P true and Q false is compatible with the denied conjunction. Writing ~R -> (~P & ~Q) would add an unwarranted restriction.
For P -> ~Q, negate the consequent ~Q to obtain ~~Q and negate the antecedent P to obtain ~P. Exchange them to get ~~Q -> ~P, then simplify the double negation to Q -> ~P in classical logic. The visible result has only one denial because the original consequent was already negative. Counting negation signs without tracking their scope is therefore an unreliable method.
If the antecedent is already a denial, the same careful process works. The contrapositive of ~P -> Q is ~Q -> ~~P, equivalently ~Q -> P. Again, the double denial arises from negating a whole side, not from a special exception to the rule. Keep the unsimplified intermediate formula in your working so someone can check how the final expression was obtained.
For an antecedent that is a disjunction, (P | Q) -> R becomes ~R -> ~(P | Q). Here De Morgan's rule would allow ~R -> (~P & ~Q). That stronger-looking pair of denials is justified because the original antecedent needed only one disjunct. Compare it with the conjunction case rather than assuming every compound antecedent behaves identically.
After transforming, test a diagnostic row. If you suspect you produced the inverse by mistake, choose P false and Q true for a simple P -> Q. The original and contrapositive are true there, while the converse and inverse are false. A single row can expose the error; a complete table or valid equivalence reasoning establishes the correct all-row relationship.
Imagine a workshop process that guarantees an accepted batch receives a permanent acceptance mark. If the mark is confirmed absent, the guarantee excludes acceptance. It does not identify why the batch was not accepted. It might never have been submitted, still await inspection, or have failed a test. The denied antecedent is a limited conclusion, not a full causal diagnosis.
If a mark can later fall off, the time and wording of the original rule matter. 'Accepted batches were marked at acceptance' does not imply that every accepted batch has a mark now. The contrapositive of the historical rule concerns not having been marked at that earlier event, not a current missing tag. Replacing the historical consequent with a present observation silently changes the premise.
Conditional rules about tendencies need similar care. 'Most accepted batches have visible marks' permits exceptions, so a missing mark does not deductively rule out acceptance. It may still be evidence, depending on rates and alternative explanations. The material conditional exercises intentionally use an exceptionless stated guarantee within a defined scope. An everyday investigation must check whether its own claim really has that form.
In a written audit, separate three lines: the original guarantee, its equivalent contrapositive, and the independently established denial used to apply it. Then state the bounded conclusion and any remaining unanswered question. This format exposes whether disagreement concerns the rule's accuracy, the observation, the time reference, or the logical transformation.
Original and contrapositive form one equivalent pair. Converse and inverse form another. The pairs generally differ from each other because they exclude different mixed assignments. A converse can nevertheless be true for independent reasons; if both directions hold, a biconditional is appropriate. Avoid describing every converse as false or every transformed conditional as a new piece of evidence.
The practical advantage of contraposition is that a rule may be easier to apply through a clearly absent result than through an unobservable starting condition. That advantage depends on the available evidence. If neither the antecedent nor the consequent can be checked reliably, changing the formula's wording does not solve the measurement problem. Logical equivalence and investigative convenience should both be stated, but they should not be confused.
Before comparing a rule with its contrapositive, verify that both mention the same object, observation time, and complete condition.
A fictional archive states that every uploaded file is listed in the final index for that day's upload batch. The index is declared complete after processing finishes. Today there are twelve submitted files. Nine appear in the final index and three do not. Let U mean a file was successfully uploaded in today's batch and I mean it appears in that batch's final index. The guarantee is U -> I.
The contrapositive ~I -> ~U allows the three confirmed absences to rule out successful upload for those files, assuming the guarantee and complete-index observations are correct. The nine listed files cannot be called successfully uploaded solely by reversing the original rule. The system might list waited in line files too; a separate I -> U rule would be needed to eliminate that possibility.
Now suppose the three absences came from an early screen before indexing finished. The observation no longer establishes ~I for the final index named in the key. Rerun the check after completion rather than declaring failed uploads from an incomplete view. The logical rule has not changed; the evidence no longer matches its consequent.
The report should therefore record the processing status, the three confirmed final-index absences, and the guarantee used. It should not diagnose a network failure without additional evidence. Absence of successful upload leaves several possible causes. This example shows both the useful exclusion supplied by contraposition and the causal questions it leaves open.
Taking the inverse for the contrapositive. If it is not raining, the pitch is not wet denies both halves and stops. That is the inverse, its column matches the converse, and it does not follow.
Denying only one half. ~P → Q, or P → ~Q. These are different transformations and are not generally equivalent to the original conditional.
Thinking the contrapositive is new information. It is the same claim. If it tells you something you did not know, what you did not know was what the original claim already said.
Forgetting that the rule has to be true. Reading a rule backwards is exactly as reliable as the rule. A shaky rule read backwards gives a shaky conclusion, stated with more confidence.
Declare the rule's meaning.
U: uploaded; I: in final index
The same batch and time apply to both atoms.
Write the original guarantee.
U -> I
Upload requires index inclusion.
Negate and exchange the sides.
~I -> ~U
This is the equivalent contrapositive.
Add the verified observation.
~I
The complete final index has been checked.
State the bounded conclusion.
~U
The guarantee and denial exclude upload without identifying its cause.
Identify the original formula.
P -> ~Q
The consequent is already a denial.
Negate the whole consequent.
~~Q
The new negation applies outside the existing one.
Negate the antecedent.
~P
The original antecedent is P.
Exchange the denied sides.
~~Q -> ~P
Negated consequent becomes antecedent.
Simplify the double denial.
Q -> ~P
Classical double negation preserves truth values.
Check a diagnostic assignment.
P=T,Q=T makes both original and rewrite F
Both formulas exclude the same combination.
State the original rule.
(P & Q) -> R
Both P and Q form its antecedent.
Identify the whole consequent.
R
Its denial will begin the transformed formula.
Negate the whole antecedent.
~(P & Q)
The conjunction must remain within the denial.
Write the contrapositive.
~R -> ~(P & Q)
This preserves the original truth conditions.
Use the observed absence.
R=F; therefore ~(P & Q)
The guarantee excludes the complete sufficient condition.
Test a remaining possibility.
P=T,Q=F
One conjunct may still hold.
Reject an overstrong conclusion.
~P & ~Q does not follow
The missing result does not identify which part failed.
State the guaranteed consequence.
Uploaded -> listed in final complete index
The time and completeness are part of the premise.
Inspect the supplied observation.
An early search found no entry
This may not describe the final index.
State the evidence gap.
Write the contrapositive of P -> ~Q, simplifying double negation.
Answer:
Compare P -> Q and ~Q -> ~P across TT,TF,FT,FF.
Calculate both columns.
TFTT; TFTT
Both conditionals fail only when P is true and Q false.
Count matching positions.
matches matches
Every assignment produces equal values.
Count discrepancies.
differences differences
The formulas are equivalent.
Complete ~Q -> ~P in PQ row order to compare with P -> Q.
| P | Q | ~Q -> ~P |
|---|---|---|
Write the contrapositive of (P | Q) -> R without distributing the denial.
Answer:
A complete final-index audit has 15 files: 11 listed and 4 absent. Given upload -> listed, write confirmed absences, uploads ruled out by those absences, and listed files whose upload is not established by reversal alone.
Absences: b0
Ruled out: b1
Unsettled listed files: b2
Do (P & Q) -> R and ~R entail ~P? Construct a countermodel if not.
(P & Q) -> R
~R
∴ ~P
valid invalid — countermodel:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Write the contrapositive of (P & Q) -> ~R, simplifying the double negation only.
Answer:
You can write the contrapositive of a claim, show with a table that it says the same thing, and use it to rule something out. Tell someone why the inverse and the converse are the same mistake twice. Next: necessary and sufficient conditions, which is this pair of ideas given the names people argue with.
15. Separate a search failure from absence, step 3
~I is not yet established
Contraposition cannot supply a missing factual premise.