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Two small words that put the arrow the other way round, and the form that puts both arrows in at once.
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 translate if, only if, if and only if, and a declared unless reading; preserve permission versus action; and test each required direction with a distinguishing record.
You can tell a necessary condition from a sufficient one. English has a short phrase for each, they look almost the same, and swapping them turns a rule into a promise. That is this lesson.
| Term | What it means |
|---|---|
| P if Q | Q -> P: Q is sufficient for P. |
| P only if Q | P -> Q: Q is necessary for P. |
| P if and only if Q | P <-> Q: both directions hold. |
| P unless Q | Under the stated minimal reading, ~Q -> P. |
| Permission | Being allowed to act, distinct from actually performing the act. |
The phrases if and only if have different jobs. 'A file is approved if it is complete' makes completeness sufficient for approval. With A for approved and C for complete, write C -> A. 'A file is approved only if it is complete' makes completeness necessary, so write A -> C. A complete but unapproved file violates the first claim and is permitted by the second.
'A file is approved if and only if it is complete' combines both requirements: A <-> C. It rules out both an approved incomplete file and a complete unapproved file. The formula is true when both conditions hold and when both fail. A biconditional connects their values; it does not by itself assert that any file is approved.
For unless, this course declares the minimal truth-functional reading 'if not'. 'The event proceeds unless the venue closes' becomes ~V -> E, with V for venue closes and E for event proceeds. It requires the event when the venue does not close. It does not on this reading require cancellation when the venue closes. A speaker may intend a stronger exception rule, but that interpretation must be made explicit rather than silently added.
Before entering a formula, give each atom a complete meaning. If the notice says a member may borrow, the target atom should concern permission to borrow, not actual borrowing. A member who chooses not to borrow has not violated a guarantee of permission. Formal notation is only as accurate as its key, so distinguish eligibility, permission, and action before deciding which arrow the sentence needs.
Another way: steps
Let M mean current member and B mean borrowing is permitted. The notice 'borrowing is permitted only if membership is current' becomes B -> M. It excludes permission for nonmembers. It leaves open whether a particular member is permitted to borrow: perhaps another rule concerns overdue items. Do not add that other rule as a fact unless the task states it; recognize only that the one-way notice does not rule it out.
The notice 'borrowing is permitted if membership is current' becomes M -> B. It guarantees permission to current members. It leaves open whether nonmembers receive permission through a guest arrangement. A permitted nonmember violates the first notice but not the second. A member denied permission violates the second but not the first. These two diagnostic records make the difference visible.
An actual nonmember taking a book is not automatically the same as a nonmember permitted to borrow. The person might act without permission. If the exercise studies policy eligibility, code the permission claim. If it studies actual transactions, code the transaction and state the assumption that transactions obey permissions if that connection is needed. Switching between these claims mid-argument creates an apparent inference that the fixed key never supported.
Word order does not determine direction. 'Only if membership is current is borrowing permitted' still means B -> M. 'If membership is current, borrowing is permitted' means M -> B. Instead of following the order of the nouns, locate which condition is described as required and which is described as sufficient. Then test a mixed record to verify the translation.
A biconditional can express a definition, a theorem, a stipulated process rule, or an empirical claim about matching conditions. Its logical form alone does not tell you which. 'The indicator is on exactly when the latch is closed' is a proposed relationship between two device states; it is not automatically a definition of either state. Both directions require support if the claim describes an actual device.
To audit that indicator claim, inspect both possible mismatches. An on indicator with an open latch refutes indicator-on implies latch-closed. A closed latch with an off indicator refutes latch-closed implies indicator-on. Testing only closed-latch cases might check the second direction while missing a false indicator signal in the first. Design the evidence search around both arrows.
The both-false row is compatible with the biconditional. An off indicator and open latch agree, so they do not violate the exactly-when claim. This can feel surprising if the formula is mistaken for a conjunction. A conjunction would require both positive states; a biconditional requires their values to match. Reading the complete final column prevents that confusion.
If one direction has been established and the other remains unchecked, report the established conditional. Do not promote it to a biconditional for convenience. If a later complete audit establishes the other direction within the same scope, the combined statement is then warranted. The reasoning should show where that additional support came from rather than presenting a two-way guarantee as free wording.
Consider 'a report is released only if it is checked and signed'. Let R mean released, C checked, and S signed. The necessary condition is the whole conjunction, so write R -> (C & S). A released report missing either requirement violates this formula. A checked and signed report that is withheld does not violate it because the notice did not make the two checks sufficient for release.
Now change and to or. 'A report is released only if it is checked or signed', with inclusive or declared, becomes R -> (C | S). A released checked-but-unsigned report now satisfies the condition. This is a different rule, not a stylistic variant. Parentheses display the compound required condition and make the changed policy easy to test.
Negation also needs its proper scope. 'A report is released only if it is not both incomplete and unsigned' becomes R -> ~(I & U). That consequent permits either incomplete alone or unsigned alone; it prohibits their conjunction. If the author intends to prohibit each defect separately, the consequent must be ~I & ~U. The logical review can identify the difference, but it should not silently repair the author's policy by choosing the stronger one.
For a notice containing several clauses, translate them separately before deciding the outer connection. If approval requires completion and rejection follows a failed check, use a conjunction of the two conditional requirements. A single arrow spanning the whole English sentence may accidentally turn one requirement into the antecedent of another. Local translations followed by an explicit outer conjunction avoid that mistake.
Under the minimal reading, 'P unless Q' means ~Q -> P, equivalently P | Q. This allows P and Q both true. If the intended notice says that Q prevents P, it needs an additional restriction. For an event rule, a closed venue might still permit the event at another location, so the minimal reading leaves that possibility open. A complete cancellation policy may close it explicitly.
Do not use tone or expectation as if it were an extra formal premise. A listener may expect an exception to prevent the result, but the exercise's declared reading controls the calculation. In an actual conversation, clarification is appropriate when that distinction matters. Ask which combination the author intends to permit, then represent that interpretation consistently.
The final translation audit should include a read-back. Replace letters by their full meanings, including permission versus action and the relevant time. Check that every necessary restriction is preserved and that no sufficient guarantee has been added. This is especially useful for short notices because a tiny phrase can change the practical promise while leaving most of the sentence unchanged.
Suppose approval is granted only if a file is complete, and a file is complete only if its signature is present. Write A -> C and C -> S. Together they imply A -> S: approval requires a signature through the intermediate completeness condition. They do not imply S -> A. A signed file could still be incomplete or awaiting approval. This is a useful final audit because the necessary condition can travel through a chain while the direction stays fixed. Write the intermediate requirement before stating the combined result, and check that each occurrence of complete refers to the same standard and time.
A fictional library records twelve applicants. Seven are current members permitted to borrow, two are current members denied permission because of a separate restriction, one is a nonmember with guest permission, and two are nonmembers without permission. Let M mean current member and B mean borrowing is permitted. The record is complete for this exercise and does not claim that every permitted person actually borrowed a book.
Notice A says 'borrowing is permitted only if membership is current', or B -> M. The guest-permission record violates this notice because B is true and M false. The two denied members do not violate it. Notice B says 'borrowing is permitted if membership is current', or M -> B. Now the two denied members violate the notice, while the permitted guest does not.
The combined notice 'members and only members are permitted to borrow' is M <-> B. It has three violating records: the two denied members plus the permitted guest. Seven member-permission records and two nonmember-denial records agree with the biconditional, giving nine nonviolations out of twelve.
These counts do not tell the library which policy to adopt. They show that the notices make different commitments and cannot be interchanged without changing which records conflict with the wording. A careful policy review should identify the intended permission rule, preserve any additional restrictions explicitly, and then compare the record with that exact rule.
A biconditional is not nearly always a definition simply because it uses if and only if. It asserts two directions, whose justification must be examined in context. Also avoid saying an only-if rule says nothing useful: it can rule out permission when a necessary credential is absent. What it cannot do by itself is guarantee permission merely from that credential's presence. Finally, unless follows the expressly declared reading here; ordinary usage can call for clarification.
Declare the exact meanings.
B: borrowing permitted; M: current member
Permission differs from actual borrowing.
Read the only-if clause.
Borrowing permitted only if membership current
Membership is necessary.
Write the arrow.
B -> M
The required condition follows the arrow.
Construct a violation.
B=T,M=F
A permitted nonmember breaks this notice.
Check a permitted record.
B=F,M=T
A denied member does not violate this one-way requirement.
Write the two commitments.
M -> B; B -> M
Each direction excludes a different mismatch.
Combine their formulas.
M <-> B
The biconditional requires matching values.
Count member denials.
2
These violate M -> B.
Count nonmember permissions.
1
These violate B -> M.
Combine the disjoint violations.
2+1=3
The two mismatch categories do not overlap.
Check the remainder.
12-3=9 nonviolations
Matching true-true and false-false records satisfy the biconditional.
Read the whole notice.
Released only if checked and signed
Both requirements belong in the consequent.
Declare three claims.
R: released; C: checked; S: signed
Each has a fixed meaning.
Build the necessary package.
C & S
The notice requires both checks.
Attach it to release.
R -> (C & S)
Release is the antecedent.
Test a released unsigned record.
R=T,C=T,S=F
The full consequent is false.
Record the violation.
T -> F=F
Release without both checks breaks the notice.
Test a withheld complete record.
R=F,C=T,S=T gives T
The notice did not promise release from the checks alone.
State the declared convention.
P unless Q means if not Q then P
This exercise does not add exclusivity.
Write the conditional.
~Q -> P
The exception's denial is the antecedent.
Test both positive claims.
A means approved and C means complete. Translate: approved only if complete.
Answer:
For P <-> Q, count satisfying and violating assignments among TT,TF,FT,FF.
List matching pairs.
TT,FF: satisfying assignments
Both directions hold when values agree.
List mixed pairs.
TF,FT: violating assignments
Each has one failed arrow.
Read back the rule.
Same value on both sides
The biconditional does not assert both sides independently.
Complete P <-> Q over the displayed PQ rows.
| P | Q | P <-> Q |
|---|---|---|
E means event proceeds; V means venue closes. Translate E unless V, explicitly using the minimal if-not reading.
Answer:
A complete list has 8 permitted members, 3 denied members, 2 permitted nonmembers and 7 denied nonmembers. Write violations of B -> M, M -> B, and M <-> B, with B permission and M membership.
Only-if violations: b0
If violations: b1
Biconditional violations: b2
R means released, I incomplete, U unsigned. Translate: released only if not both incomplete and unsigned.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A means approved, C complete, R reviewed. Translate both clauses: approval only if complete, and approval if reviewed. Neither other direction is asserted.
Answer:
You can translate a notice with its scope intact and explain which record would violate each direction. Two valid inference patterns from conditionals come next; validity remains separate from premise truth.
15. Read unless minimally, step 3
P=T,Q=T satisfies it
Both can hold under the minimal reading.