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Turning a sentence into a formula: which connective the joining word demands, which way the arrow points, and how far a negation reaches.
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.
By the end of this lesson you will be able to write a dictionary of statement letters, symbolize sentences that use and, or, if, only if, unless, neither and not both, decide which way a conditional points, fix the scope of a negation with brackets, and check a symbolization by asking which situation it rules out.
You can read a formula and find its main connective. Symbolizing runs the other way: you are given a sentence and have to produce the formula whose truth table is the sentence's own.
| Term | What it means |
|---|---|
| Symbolization key | A fixed mapping from letters to complete statements. |
| Necessary condition | A condition required by another: P only if Q is P -> Q. |
| Sufficient condition | A condition that guarantees another within the stated claim: P if Q is Q -> P. |
| Declared reading | The particular meaning an exercise asks the formula to preserve. |
Fix a dictionary first: each letter stands for a complete statement, written out. Then the joining word decides the connective, and only the joining word does. And, but, although, yet are all $\wedge$ — the contrast they add is rhetorical and has no truth table. Or is $\vee$ unless the sentence says otherwise. The conditional is where the work is: if $P$ then $Q$ and $Q$ if $P$ are both $P \to Q$, while $P$ only if $Q$ is also $P \to Q$, because only if marks $Q$ as necessary for $P$. $P$ unless $Q$ is $\neg Q \to P$. Negation needs a decision about scope: neither $P$ nor $Q$ is $\neg P \wedge \neg Q$, while not both $P$ and $Q$ is $\neg (P \wedge Q)$, and the two differ on every row where exactly one half holds. The test of a symbolization is never how similar it looks: it is whether the formula is false in exactly the situations the sentence rules out.
Another way: steps
Another way: example
The door opens only if the card is valid. $P$: the door opens. $Q$: the card is valid. Validity is necessary, so $P \to Q$. The ruled-out situation is an opening door with an invalid card, which is the one row the formula makes false.
A plausible-looking arrow is not enough. Suppose the claim is that a file is accepted only if it has a signature. Set acceptance true and signature false. This must violate the claim, so the correct formula must be false on that assignment. Set acceptance false and signature true. The original claim allows that case: perhaps the file is incomplete. A formula that rejects it has added a sufficient condition.
Two formulas can use different connectives and still express the same declared reading. P -> Q and ~P | Q agree on every row. The task therefore concerns truth conditions, not copying one preferred string. It remains essential to use the atom key supplied in the question. A formula cannot silently redefine P halfway through.
Ordinary English can leave genuine ambiguity. 'Either' does not always settle whether both alternatives are allowed; 'unless' may carry suggestions beyond its minimal truth-functional reading. In these exercises inclusive or and the reading of unless as 'if not' are stated conventions. In an actual discussion, ask the speaker which cases they intend to permit before you evaluate their argument.
A translation should preserve the situations an assertion permits and excludes. Merely finding one situation in which the English sentence and the formula agree is insufficient. Many different formulas are true when every atom is true. Test a difficult case: one that separates a required condition from a sufficient condition, or separates a denial of a conjunction from a conjunction of denials.
Take 'the cabinet opens only if the key is turned'. Let O mean the cabinet opens and K mean the key is turned. The claim excludes O true with K false, so O -> K is appropriate. It does not guarantee opening whenever the key turns; the mechanism could be jammed. K true with O false is permitted by the stated requirement. The reversed arrow would reject that permitted case and therefore add information.
Now consider 'the cabinet opens if the key is turned'. The clause introduced by if supplies the sufficient condition. Under this declared reading the formula is K -> O. The word order of the English sentence places the result first, but the arrow places its antecedent first. A useful intermediate sentence is 'if K, then O'. Rewrite the relationship in that clear form before entering the formula.
If the notice says 'the cabinet opens if and only if the key is turned', both arrows are required. The biconditional O <-> K excludes either mixed assignment while allowing both true and both false. This is a stronger claim than either one-way conditional by itself. Do not supply the stronger relation because it feels natural; translate what the notice explicitly says.
A good explanation of a translation names the excluded case and one revealing permitted case. This checks both undertranslation and overtranslation. A formula that fails to exclude opening without the required key has omitted a restriction. A formula that also excludes a turned key with a stuck cabinet has added a restriction to the only-if notice. Accuracy requires avoiding both changes.
The sentence 'the room is not both booked and occupied' denies a conjunction. Let B mean booked and O mean occupied. Write ~(B & O). This leaves three combinations available: booked but empty, unbooked but occupied, and neither. It rules out only the combination where both claims hold. Writing ~B & ~O would leave just the last combination, which is much stronger.
By contrast, 'the room is neither booked nor occupied' denies each alternative. Its formula can be ~B & ~O or, equivalently, ~(B | O). Both formulations exclude every assignment in which either positive claim holds. Reading the formula back in complete sentences is an effective way to check that a negation has not moved across a connective without a corresponding change in meaning.
The English word 'but' ordinarily contributes the conjunction of its clauses, while also suggesting a contrast. In a declared truth-functional reading, 'the room is small but bright' becomes S & L. The formula retains the joint assertion and leaves the conversational contrast unrepresented. State that limit rather than pretending conjunction captures every feature of ordinary speech.
For 'either P or Q, but not both', combine an inclusive disjunction with the denial of their conjunction: (P | Q) & ~(P & Q). The first part excludes neither being true; the second excludes both being true. Each part does a different job. Writing only ~(P & Q) would allow neither, while writing only P | Q would allow both. The two restrictions together preserve the declared exclusive reading.
When unless is explicitly defined as 'if not', 'P unless Q' becomes ~Q -> P. It excludes the case where Q fails and P also fails. It does not on this minimal reading say that Q prevents P; both may be true. Ordinary speakers sometimes intend additional contrast or exception conditions. A formal exercise can declare its convention, but an actual policy should clarify the intended cases before a formula is used as its specification.
Suppose an author states two conditions for a grant: a grant is paid only if the application is approved, and an application is approved if its review succeeds. With P for paid, A for approved, and R for review succeeds, the conjunction is (P -> A) & (R -> A). The second arrow does not run from approval to successful review. Other approval routes remain possible under the stated text.
It may be sensible to require a successful review for every approval, but adding A -> R would revise the policy. Translation and revision are different tasks. First represent the actual declared requirements. Then identify a missing restriction and label it as a proposed addition. This makes disagreements visible and avoids attributing a stronger claim to somebody who never made it.
For a long sentence, translate each clause locally before joining them. Fix the atom key, identify the connective within each clause, and decide how the clauses relate at the outer level. If two requirements must both hold, use conjunction between their formulas. Parenthesize compound clauses so the outer structure is visible. Then choose a diagnostic assignment for each requirement independently.
Two syntactically different formulas can still be acceptable translations when they agree on every assignment under the same key. For example, P -> Q and ~P | Q express the same truth condition. Matching a preferred spelling is therefore less important than preserving the complete conditions. But agreement on a single observed case does not establish equivalence. You need a truth-table comparison or a justified equivalence rule for that stronger conclusion.
Finally, retain a short note when the declared reading excludes part of the ordinary meaning. A temporal 'and then', a causal 'because', or a promise can convey more than a truth-functional connective represents. The notation is useful precisely when its intended scope is clear. A faithful limited model identifies what it preserves and gives a reader enough context to avoid treating the omitted meaning as formally established.
A community room notice says: a booking is confirmed only if payment has arrived, and a booking is canceled if the safety inspection fails. Let C mean confirmed, P mean payment arrived, X mean canceled, and S mean the inspection passes. Under the declared reading that a failed inspection is a non-passing inspection, the two clauses become C -> P and ~S -> X. Together they are (C -> P) & (~S -> X).
The first clause is tested by a confirmed unpaid booking. C is true and P false, so that clause fails. A paid unconfirmed booking does not fail it: the notice does not promise availability merely because money arrived. The second clause is tested by a failed inspection with no cancellation. S is false and X false, so ~S is true while X is false, and that conditional fails.
There are four atoms, so a complete table would contain sixteen assignments. We do not need all sixteen merely to reject the reversed translation P -> C: a paid but unconfirmed booking already distinguishes it from the original clause. Nor does either clause assert that confirmation and cancellation cannot occur together. If the booking system requires that extra restriction, write ~(C & X) as an additional requirement. Logical translation makes omissions visible. It does not repair a policy by silently inserting the conditions its author probably wanted. Keep the formula, the intended policy, and the actual booking record separate.
The commonest error is reading only if as if and pointing the arrow backwards; P only if Q is $P \to Q$, not $Q \to P$. The second is letting a negation spread further than the English does: not both denies the pair and allows either one alone, while neither denies each. The third is symbolizing a letter as something that is not a statement — $P$ has to be the card is valid, never the card.
Declare the complete atoms.
O: cabinet opens; K: key is turned
Letters retain these meanings throughout the translation.
Identify the requirement.
O only if K
K is necessary for O.
Write the conditional.
O -> K
It forbids opening without a turned key.
Test the prohibited case.
O=T,K=F makes O -> K false
The formula rejects exactly this violation of the requirement.
Test a permitted case.
O=F,K=T makes O -> K true
The notice does not guarantee opening whenever the key turns.
Declare the options.
P: paper ticket; D: digital ticket
Each atom names possession of one ticket type.
Require an available option.
P | D
At least one must hold.
Describe the prohibited overlap.
P & D
This says both types are present.
Exclude that overlap.
~(P & D)
The stated reading rules out both.
Join the requirements.
(P | D) & ~(P & D)
Both at-least-one and not-both restrictions apply.
Test the four cases.
TT:F; TF:T; FT:T; FF:F
Only the assignments with exactly one true atom survive.
Fix the key.
A: alarm sounds; B: backup runs; P: power available
All clauses use the same meanings.
Read the first declared clause.
Alarm sounds unless backup runs: ~B -> A
Unless is explicitly read as if not.
Read the second clause.
Backup runs only if power available: B -> P
Power is necessary for backup operation.
Join both asserted requirements.
(~B -> A) & (B -> P)
The policy asserts both clauses together.
Test absence of backup and alarm.
B=F,A=F makes first clause F
A missing backup requires the alarm on this reading.
Test backup without power.
B=T,P=F makes second clause F
That case violates the necessary condition.
Check the permitted overlap.
A=T,B=T,P=T satisfies both
The minimal unless clause does not prohibit the alarm when backup runs.
State the positive claims.
B: booked; O: occupied
Negation must apply to the claims in the fixed key.
Deny each alternative.
~B; ~O
Neither rules out each positive claim.
Assert both denials.
P means parcel arrives; N means notice is sent. The notice is sent if the parcel arrives. Translate this one-way claim.
Answer:
Complete the worked audit of exactly one of P and Q: (P | Q) & ~(P & Q).
List satisfying assignments.
TF and FT: allowed assignments
Each makes the disjunction true and the conjunction false.
Check the overlap assignment.
TT contributes overlap satisfying assignments
The not-both clause rejects TT.
Check neither alternative.
FF is rejected
The at-least-one clause is false.
B means booked and O means occupied. Translate: the room is not both booked and occupied.
Answer:
A means alarm sounds; B means backup runs. Translate 'the alarm sounds unless backup runs', with unless explicitly meaning if not.
Answer:
A booking is confirmed only if the room is available. C means confirmed and A means available. Availability alone does not guarantee confirmation. Formalize the notice.
Answer:
L means loan approved, I means identity checked, and D means documents complete. Translate both requirements: approval only if identity checked; approval if documents complete. No other restriction is stated.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
R means room released, C means cleaned, and I means inspected. Translate: the room is released only if it is both cleaned and inspected. Do not make those checks sufficient.
Answer:
You can symbolize a compound English sentence and check the result against the situations the sentence rules out. Say in your own words why 'P only if Q' and 'P if Q' point the arrow in opposite directions.
14. Neither alternative, step 3
~B & ~O
Both restrictions hold together.