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Chaining Conditional Statements

26 questions

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Question 1Question →

According to a corporation's restructuring plan, if the board of directors approves the merger, then the marketing division will be downsized. Any downsizing of the marketing division requires that the public relations team be outsourced. Furthermore, the company will undergo a comprehensive financial audit only if the public relations team is not outsourced.

Based on the corporation's restructuring plan, which one of the following must be true?

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Answer: If the board of directors approves the merger, the company will not undergo a comprehensive financial audit.

Answer

If the board of directors approves the merger, the company will not undergo a comprehensive financial audit.
The stimulus provides three conditional statements: (1) Merger Approved -> Marketing Downsized, (2) Marketing Downsized -> PR Outsourced, and (3) Financial Audit -> PR NOT Outsourced. By taking the contrapositive of the third statement, we get: PR Outsourced -> NO Financial Audit. Linking all these together creates a continuous chain: Merger Approved -> Marketing Downsized -> PR Outsourced -> NO Financial Audit. Therefore, if the merger is approved, the company absolutely cannot undergo a financial audit.

Step-by-Step Solution

1
Translate the first two premises into formal conditional logic.
Merger Approved -> Marketing Downsized; Marketing Downsized -> PR Outsourced.
To establish the initial chain of sufficient and necessary conditions.
2
Translate the third premise into formal conditional logic and find its contrapositive.
Audit -> PR NOT Outsourced. The contrapositive is: PR Outsourced -> NO Audit.
The phrase 'only if' introduces a necessary condition. Finding the contrapositive allows this premise to connect to the existing chain.
3
Chain all the conditional statements together.
Merger Approved -> Marketing Downsized -> PR Outsourced -> NO Audit.
The transitive property of conditional logic allows us to link shared necessary and sufficient conditions.
4
Evaluate the deduced chain to find a valid inference that matches an answer choice.
The chain directly supports the statement: If Merger Approved, then NO Audit.
In Must Be True questions, the correct answer is a direct, logically valid deduction from the provided premises.

Key Concept

Chaining Conditional Statements and Contrapositives
Estimated Time:1m 30s
Question 2Question →

A city planner is reviewing new zoning regulations for a commercial district, which state the following:

* A building will not be granted a Class A permit unless it features a subterranean parking garage.
* Any building featuring a subterranean parking garage must be evaluated by a structural engineer.
* Buildings are exempt from the deep excavation tax only if they are not evaluated by a structural engineer.
* No property is subject to the deep excavation tax unless it requires specialized foundation reinforcements.

Arrange the following logical derivations in the correct sequential order to form a valid deductive chain proving the conclusion: 'If a building is granted a Class A permit, it must require specialized foundation reinforcements.'

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Answer

The correct logical sequence begins by translating the initial 'unless' condition for a Class A permit, directly linking it to the structural engineer evaluation, contraposing the 'only if' statement to link to the deep excavation tax, contraposing the final 'unless' statement for foundation reinforcements, and finally chaining them all together.
The correct deductive sequence must build a valid transitive chain (A→BA \rightarrow B, B→CB \rightarrow C, C→DC \rightarrow D, D→ED \rightarrow E). We begin with the sufficient condition of the desired conclusion (Class A permit) and correctly translate each rule—using contrapositives for 'unless' and 'only if' statements—to link the necessary conditions sequentially until we arrive at the final necessary condition (specialized foundation reinforcements).

Step-by-Step Solution

1
Identify the initial trigger and necessary condition from the first premise.
CA→SPGCA \rightarrow SPG
The first premise uses 'unless'. 'Not A unless B' translates to ∼B→∼A\sim B \rightarrow \sim A, which contraposes to A→BA \rightarrow B. Thus, a Class A Permit (CACA) requires a Subterranean Parking Garage (SPGSPG).
2
Link the first necessary condition to the second premise.
SPG→SESPG \rightarrow SE
The second premise is a direct conditional ('Any A must B'), translating directly to SPG→SESPG \rightarrow SE (Subterranean Parking Garage requires a Structural Engineer).
3
Translate and contrapose the 'only if' statement to continue the chain.
SE→DETSE \rightarrow DET
'A only if B' means A→BA \rightarrow B. 'Exempt only if not evaluated' translates to Exempt→∼SEExempt \rightarrow \sim SE. The contrapositive is SE→∼ExemptSE \rightarrow \sim Exempt, meaning evaluated buildings are subject to the Deep Excavation Tax (DETDET).
4
Translate and contrapose the final 'unless' statement.
DET→SFRDET \rightarrow SFR
'No A unless B' translates to ∼B→∼A\sim B \rightarrow \sim A. The contrapositive is A→BA \rightarrow B. Thus, being subject to the Deep Excavation Tax (DETDET) requires Specialized Foundation Reinforcements (SFRSFR).
5
Transitively chain all conditional statements together.
CA→SFRCA \rightarrow SFR
By linking CA→SPGCA \rightarrow SPG, SPG→SESPG \rightarrow SE, SE→DETSE \rightarrow DET, and DET→SFRDET \rightarrow SFR, we prove the transitive conclusion CA→SFRCA \rightarrow SFR.

Key Concept

Chaining Conditional Statements
Question 3Question →

According to the new municipal regulations, the city council will authorize a tax abatement for a new project only if the project's commercial development phase proceeds. However, the commercial development phase will not proceed unless a zoning variance is granted. Furthermore, the commercial phase proceeds only if an environmental impact study is fully approved. The environmental impact study will be fully approved only when the local conservation group does not file a petition against the project. It is well known that the local conservation group will file a petition against any project unless the local wildlife habitat is strictly preserved and public access to the shoreline is maintained.

Based on the municipal regulations described above, which one of the following must be true?

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Answer: If a project does not strictly preserve the local wildlife habitat, the city council will not authorize a tax abatement for it.

Answer

If a project does not strictly preserve the local wildlife habitat, the city council will not authorize a tax abatement for it.
The correct answer accurately traces the contrapositive of the entire conditional chain. The stimulus establishes that a tax abatement requires the commercial phase to proceed, which requires the environmental study to be approved, which requires no petition to be filed. A petition is avoided ONLY IF the habitat is strictly preserved AND public access is maintained. Therefore, if the habitat is not strictly preserved, the compound necessary condition fails. This triggers the petition, which prevents study approval, stopping the commercial phase, and ultimately ensuring the city council will not authorize the tax abatement.

Step-by-Step Solution

1
Translate the first three conditional statements establishing the requirements for the tax abatement and commercial development.
Tax Abatement (TA) -> Commercial Development proceeds (CD). CD -> Zoning Variance Granted (ZVG). CD -> Environmental Impact Approved (EIA).
Identifying the baseline sufficient-necessary relationships, noting that 'only if' and 'unless' introduce necessary conditions.
2
Translate the final two statements regarding the environmental study and the conservation group.
EIA -> No Petition filed (~CGP). Not (Preserved AND Maintained) -> Petition filed (CGP).
The 'unless' clause modifies a compound condition. 'Unless X and Y' means 'If not (X and Y)'.
3
Form the contrapositive of the final compound condition.
~CGP -> (Local Wildlife Preserved AND Public Access Maintained).
This puts the statement in a format that can link directly to the end of the previous conditional chain.
4
Chain all the conditional statements together to see the full logical sequence.
TA -> CD -> EIA -> ~CGP -> (Preserved AND Maintained). Note that CD -> ZVG is also true but branches off.
Connecting the necessary condition of one statement to the sufficient condition of the next reveals the overarching logical architecture.
5
Form the contrapositive of the entire chain to find the correct valid inference.
Not (Preserved AND Maintained) -> CGP -> ~EIA -> ~CD -> ~TA.
Since 'Not (Preserved AND Maintained)' means 'Not Preserved OR Not Maintained', failing to meet EITHER condition triggers the entire negative chain, guaranteeing the tax abatement is denied.

Key Concept

Chaining Conditional Statements with Compound 'Unless' Clauses
Estimated Time:2m 0s
Question 4Question →

If a building is designated as a historical landmark, it cannot be demolished. Furthermore, any building that cannot be demolished is exempt from property taxes.

Which one of the following statements must be true based on the information provided?

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Answer: If a building is designated as a historical landmark, it is exempt from property taxes.

Answer

If a building is designated as a historical landmark, it is exempt from property taxes.
The correct answer accurately reflects the transitive property of conditional logic. The stimulus gives us A -> B (Historical Landmark -> Cannot be Demolished) and B -> C (Cannot be Demolished -> Exempt from Taxes). By linking these together, we logically deduce A -> C: if a building is a historical landmark, it is exempt from property taxes.

Step-by-Step Solution

1
Identify and diagram the first conditional premise.
Historical Landmark -> Cannot be Demolished
The text states that being a historical landmark is a sufficient condition for a building being protected from demolition.
2
Identify and diagram the second conditional premise.
Cannot be Demolished -> Exempt from Property Taxes
The text states that not being able to be demolished is a sufficient condition for being tax-exempt.
3
Link the two conditional premises to form a transitive chain.
Historical Landmark -> Cannot be Demolished -> Exempt from Property Taxes
The necessary condition of the first premise matches the sufficient condition of the second premise, allowing them to be chained together.
4
Derive the valid inference from the chain.
Historical Landmark -> Exempt from Property Taxes
By following the chain from the initial sufficient condition to the final necessary condition, we logically conclude that historical landmarks are exempt from property taxes.

Key Concept

Chaining Conditional Statements
Question 5Question →

Match each set of conditional premises with its logically valid chained conclusion or its valid contrapositive.

Click a left item, then click its matching right item

Items

If the meeting is held, the training is conducted. If the training is conducted, the project is approved.
If the training is conducted, the review is completed. If the review is completed, the meeting is held.
If the project is approved, the meeting is held. If the meeting is held, the review is completed.
If the review is completed, the project is approved. If the project is approved, the training is conducted.

Matches

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Answer

The logically valid pairs map each premise set to its direct transitive chain or the contrapositive of that chain.
Each valid pairing is derived by connecting two conditional statements to form a chain (A -> B and B -> C yields A -> C). For two of the pairs, the correct matching item provides the contrapositive of this chain (since A -> C implies ~C -> ~A).

Step-by-Step Solution

1
Analyze the first set of premises.
Chain: Meeting -> Training -> Project. Direct inference: Meeting -> Project. This matches the item stating 'If the meeting is held, the project is approved'.
Transitive logic allows us to bypass the intermediate necessary condition.
2
Analyze the second set of premises.
Chain: Training -> Review -> Meeting. Direct inference: Training -> Meeting. The contrapositive of this inference is 'Not Meeting -> Not Training'.
If the final necessary condition fails, the initial sufficient condition could not have occurred.
3
Analyze the third set of premises.
Chain: Project -> Meeting -> Review. Direct inference: Project -> Review. This matches the item stating 'If the project is approved, the review is completed'.
Linking the conditions yields a direct forward inference.
4
Analyze the fourth set of premises.
Chain: Review -> Project -> Training. Direct inference: Review -> Training. The contrapositive is 'Not Training -> Not Review'.
Reversing and negating the valid direct chain produces a valid contrapositive.

Key Concept

Chaining conditional statements and forming contrapositives
Question 6Question →

A software company has established the following protocol for deploying new applications:

- The code is merged into the main branch only if a beta testing phase is scheduled.
- If the software passes the final security audit, the code is merged into the main branch.
- A beta testing phase is scheduled only if the marketing team has been notified.

Arrange the following events in the correct logical sequence, starting with the initial sufficient condition (the trigger) and ending with the final necessary condition.

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Answer

The correct logical sequence is: The software passes the final security audit, followed by the code being merged, followed by a beta testing phase being scheduled, and finally the marketing team being notified.
By translating the conditional statements accurately (Audit -> Merged; Merged -> Beta; Beta -> Marketing), we can form a valid transitive chain from the initial trigger to the final requirement: Audit -> Merged -> Beta -> Marketing.

Step-by-Step Solution

1
Identify the initial sufficient condition that does not depend on any other event occurring first.
The rule 'If the software passes the final security audit...' makes passing the audit the starting trigger.
The word 'if' introduces a sufficient condition.
2
Determine the necessary condition for the first event.
Passing the audit requires that 'the code is merged into the main branch'.
The result of the 'if' clause forms the necessary condition.
3
Link the second event to its own necessary condition.
The code is merged 'only if a beta testing phase is scheduled', linking the code merge to beta testing.
The phrase 'only if' introduces a necessary condition.
4
Link the third event to its necessary condition.
A beta testing phase is scheduled 'only if the marketing team has been notified'.
This establishes the final link in the transitive logic chain.

Key Concept

Chaining Conditional Statements
Question 7Question →

Any painting restored by Dr. Vance will be included in the upcoming exhibition. If a painting is included in the exhibition, the museum must insure it for over a million dollars. Furthermore, the museum will not insure any painting for over a million dollars unless it has been appraised by an independent panel. Any painting belonging to the Renaissance period has been appraised by an independent panel. Finally, no painting featuring religious motifs was restored by Dr. Vance.

If the statements above are true, which one of the following must also be true?

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Answer: If a painting has not been appraised by an independent panel, it will not be included in the upcoming exhibition.

Answer

If a painting has not been appraised by an independent panel, it will not be included in the upcoming exhibition.
The correct answer properly chains two conditional statements and takes the contrapositive. The stimulus states: 'Included in exhibition -> Insured >1M' and 'Insured >1M -> Appraised by independent panel'. Chaining these gives: 'Included in exhibition -> Appraised by independent panel'. The logical contrapositive of this new statement is: 'NOT Appraised by independent panel -> NOT Included in exhibition', which perfectly matches this option.

Step-by-Step Solution

1
Translate the statements into formal conditional logic.
1: Restored by Vance -> Included in exhibition. 2: Included in exhibition -> Insured >1M. 3: Insured >1M -> Appraised by independent panel. 4: Renaissance -> Appraised by independent panel. 5: Religious motifs -> NOT Restored by Vance.
Mapping the logical relationships is necessary to see how the rules connect.
2
Identify connections between the conditional statements to form a chain.
Rules 1, 2, and 3 link together perfectly: Restored by Vance -> Included in exhibition -> Insured >1M -> Appraised by independent panel.
Chaining conditionals reveals deductive inferences that are not explicitly stated.
3
Derive the valid contrapositives of the logical chain.
The chain 'Included in exhibition -> Appraised by independent panel' yields the contrapositive: 'NOT Appraised by independent panel -> NOT Included in exhibition'.
Correct answers in Must Be True questions frequently rely on the contrapositive of a chained conditional statement.

Key Concept

Chaining Conditional Statements
Question 8Question →

Pair each set of operational spacecraft protocols with the logically valid deduction that can be correctly chained from its premises.

Click a left item, then click its matching right item

Items

If the probe enters the atmosphere, the heat shield deploys. If the heat shield deploys, telemetry data is transmitted.
The probe enters the atmosphere only if the solar panels retract. Telemetry data is transmitted only if the solar panels do not retract.
Unless the primary engine fires, the orbit will not stabilize. Telemetry data is transmitted only if the orbit stabilizes.
If the backup battery engages, the primary engine fires. The orbit stabilizes only if the primary engine does not fire.

Matches

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Answer

Each premise set must be translated into formal if-then logic to locate the shared term. By applying contrapositives where necessary, you can form a continuous transitive chain (A -> B -> C) to reach the required conclusion.
The correct pairings successfully utilize formal logic rules, particularly the contrapositive, to link the sufficient condition of one premise to the necessary condition of the second, resulting in an unassailable deduction.

Step-by-Step Solution

1
Translate each statement into formal conditional logic, paying close attention to indicator words like 'if', 'only if', and 'unless'.
You produce a set of logical relationships, such as A -> B and C -> Not B.
Standardizing the language allows you to see the logical flow without getting distracted by the narrative context.
2
Identify the shared variable between the two conditional statements.
You isolate the term that will act as the bridge in the deductive chain.
A chain can only be formed if the necessary condition of one statement matches the sufficient condition of another.
3
Apply the contrapositive to one of the statements if the shared variable has a mismatched polarity (e.g., B in one statement, Not B in the other).
The statements align cleanly (e.g., C -> Not B, and Not B -> Not A).
The contrapositive preserves the original truth value while rearranging the terms to enable transitive linking.
4
Follow the transitive property from the first sufficient condition to the final necessary condition.
You formulate the final valid deduction (e.g., C -> Not A).
This guarantees the final inference must logically be true based on the initial premises.

Key Concept

Chaining Conditional Statements
Question 9Question →

A municipal legal analyst is constructing a formal proof to demonstrate that if the city's Annual Arts Festival takes place this year, then the Mayor must not have signed the controversial Riverfront Zoning Act. The analyst relies on the following established municipal facts:

- The Mayor signs the Riverfront Zoning Act only if the Riverside Corporation initiates the condo project.
- The Riverside Corporation initiates the condo project only if they receive a property tax exemption.
- Any property tax exemption granted to developers requires the city to reduce its discretionary spending.
- Whenever the city reduces its discretionary spending, the Parks Maintenance Budget is eliminated.
- The Annual Arts Festival takes place only if the Parks Maintenance Budget is not eliminated.

Arrange the following intermediate deductions in the logically valid sequence that bridges the given premise 'The Annual Arts Festival is taking place' to the final conclusion 'The Mayor did not sign the Riverfront Zoning Act'.

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Answer

The logically valid sequence begins with the occurrence of the festival and proceeds backwards through a chain of contrapositives: deducing the budget is intact, leading to no reduction in spending, meaning no tax exemption, resulting in no condo project, and finally proving the Mayor did not sign the Act.
The correct sequence forms a continuous chain of valid deductive inferences using the contrapositive of each conditional premise. Starting with the given fact (Festival), we apply Festival -> ~Eliminated, then ~Eliminated -> ~Reduced, then ~Reduced -> ~Exemption, then ~Exemption -> ~Project, and finally ~Project -> ~Mayor. Each step correctly identifies the logical relationship (sufficient vs. necessary) required to bridge the previous deduction to the next.

Step-by-Step Solution

1
Identify the initial given fact and apply the first relevant conditional statement.
Deduce that the Parks Maintenance Budget remains intact.
The festival is taking place, and it taking place requires the budget to not be eliminated. Symbolically: Festival -> ~Eliminated.
2
Apply the derived fact (~Eliminated) to the next conditional statement in the chain.
Deduce that the city did not reduce its discretionary spending.
Reducing spending is sufficient for eliminating the budget (Reduced -> Eliminated). By the contrapositive (~Eliminated -> ~Reduced), spending was not reduced.
3
Continue the contrapositive chain using the absence of reduced spending.
Deduce that no property tax exemption was granted.
Exemptions require reduced spending (Exemption -> Reduced). By the contrapositive (~Reduced -> ~Exemption), no exemption was granted.
4
Apply the absence of the tax exemption to the condo project condition.
Deduce that the condo project is not initiated.
The project requires an exemption (Project -> Exemption). By the contrapositive (~Exemption -> ~Project), it is not initiated.
5
Apply the absence of the condo project to the final conditional statement.
Conclude that the Mayor did not sign the Act.
The Mayor signing requires the project to be initiated (Mayor -> Project). By the contrapositive (~Project -> ~Mayor), the Mayor did not sign the Act.

Key Concept

Chaining Conditional Statements via Contrapositives
Question 10Question →

According to the newly implemented botanical garden regulations, if a botanical specimen is classified as a delicate cultivar, it must be kept in the restricted conservatory. A specimen can be kept in the restricted conservatory only if it undergoes weekly pathogen screening. Furthermore, any specimen that undergoes weekly pathogen screening is exempt from the standard public exhibition rotation.

Based on the regulations above, which of the following must be true?

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Answer: If a botanical specimen is part of the standard public exhibition rotation, it is not classified as a delicate cultivar.

Answer

If a botanical specimen is part of the standard public exhibition rotation, it is not classified as a delicate cultivar.
The premises create a continuous logical chain: being a delicate cultivar guarantees placement in the restricted conservatory, which guarantees pathogen screening, which guarantees exemption from the public exhibition rotation (Delicate Cultivar -> Not in Exhibition). The correct answer is the logical contrapositive of this final chain: if a specimen IS in the public exhibition rotation, it cannot be a delicate cultivar (In Exhibition -> Not a Delicate Cultivar).

Step-by-Step Solution

1
Translate the rules into conditional statements.
1: Delicate Cultivar (DC) -> Restricted Conservatory (RC). 2: Restricted Conservatory (RC) -> Weekly Pathogen Screening (WPS). 3: Weekly Pathogen Screening (WPS) -> Not in Public Exhibition Rotation (~PER).
Formalizing the logic makes it easier to track the relationships, noting that 'only if' introduces a necessary condition.
2
Chain the conditional statements together.
DC -> RC -> WPS -> ~PER.
Since the necessary condition of each rule matches the sufficient condition of the next, they can be linked into a continuous chain.
3
Derive valid inferences and their contrapositives.
Valid chain inference: DC -> ~PER. Contrapositive: PER -> ~DC.
The correct answer to a Must Be True question involving formal logic is almost always the chained conclusion or its direct contrapositive.

Key Concept

Chaining Conditional Statements
Question 11Question →

A municipal planning board adheres to the following regulations for commercial developments:
- If a development is located in the waterfront zone, it must conduct an ecological impact study.
- Any development that conducts an ecological impact study is mandated to construct shoreline defenses.
- A development is granted a reduced property tax rate only if it is not mandated to construct shoreline defenses.
- Any development that receives a mayoral endorsement is granted a reduced property tax rate.

To validly deduce the conclusion 'If a development is located in the waterfront zone, it does not receive a mayoral endorsement,' arrange the following conditional deductions in the correct logical sequence of the transitive chain. Start with the initial trigger and end with the final deductive step.

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Answer

The correct sequence connects the waterfront zone to the lack of mayoral endorsement through a transitive chain involving ecological studies, shoreline defenses, and property tax rates by applying two contrapositives.
The correct sequence builds a valid transitive logic chain. It begins with the initial premise (Waterfront -> Ecological Study), continues to the second premise (Ecological Study -> Shoreline Defenses), utilizes the contrapositive of the third premise (Reduced Tax Rate -> NO Shoreline Defenses becomes Shoreline Defenses -> NO Reduced Tax Rate), and finally utilizes the contrapositive of the fourth premise (Mayoral Endorsement -> Reduced Tax Rate becomes NO Reduced Tax Rate -> NO Mayoral Endorsement).

Step-by-Step Solution

1
Identify the initial trigger from the required conclusion.
The chain must start with 'Waterfront zone'.
The conclusion we need to prove is 'Waterfront zone -> NO mayoral endorsement'.
2
Apply the first premise.
Waterfront zone -> Ecological impact study.
The rules explicitly state waterfront developments must conduct the study.
3
Chain the second premise.
Ecological impact study -> Shoreline defenses.
The rules explicitly state any development with a study must construct defenses.
4
Derive the contrapositive of the third premise.
Shoreline defenses -> NO reduced property tax rate.
The rule 'Reduced tax rate only if NO shoreline defenses' translates to (Reduced Tax Rate -> NO Shoreline Defenses). The contrapositive is (Shoreline Defenses -> NO Reduced Tax Rate).
5
Derive the contrapositive of the fourth premise.
NO reduced property tax rate -> NO mayoral endorsement.
The rule 'Mayoral endorsement -> Reduced tax rate' translates to (Mayoral Endorsement -> Reduced Tax Rate). The contrapositive is (NO Reduced Tax Rate -> NO Mayoral Endorsement).

Key Concept

Chaining Conditional Statements
Estimated Time:1m 30s
Question 12Question →

A deep-sea research station operates under strict automated environmental control protocols. Match each set of protocol conditions (Left) with the logically valid deduction that MUST be true based on those conditions (Right).

Click a left item, then click its matching right item

Items

Protocol 1: Ventilation Unit A's activation is sufficient to isolate Sector B. Sector G's activation is a necessary condition for Sector D to pressurize. However, Sector G cannot activate if Sector B is isolated.
Protocol 2: Filter E is purged unless Valve Z is secured. Manifold H failing precludes Valve Z from being secured. Furthermore, Pump T activating requires that Filter E not be purged.
Protocol 3: Generator I synchronizes if, and only if, Turbine K delays. Diagnostic L verifying depends on Generator I synchronizing. Relay M going offline is a necessary consequence of Turbine K delaying.
Protocol 4: Module N is sealed when, and only when, Hatch X is unlatched. Override O being engaged is sufficient to seal Module N. Sensor P's activation guarantees that Hatch X is latched.

Matches

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Answer

Protocol 1 matches with the deduction about Sector D and Unit A. Protocol 2 matches with the deduction about Pump T and Manifold H. Protocol 3 matches with the deduction about Relay M and Diagnostic L. Protocol 4 matches with the deduction about Sensor P and Override O.
Each correct pairing represents a flawless deductive chain formed by translating advanced conditional indicators (sufficient, necessary, unless, requires, precludes) into formal logic, finding the linkages, and tracing the contrapositive path to an unavoidable conclusion.

Step-by-Step Solution

1
Translate Protocol 1 into formal logic and chain the conditions.
A -> B; D -> G; B -> ~G. Chaining yields D -> G -> ~B -> ~A.
This establishes that if D pressurizes, A must be inactive, confirming the match.
2
Translate Protocol 2 into formal logic and chain the conditions.
The 'unless' statement becomes ~Z -> E (or ~E -> Z). 'Precludes' means H -> ~Z. 'Requires' means T -> ~E. Chaining yields T -> ~E -> Z -> ~H.
This proves that activating T guarantees H has not failed.
3
Translate Protocol 3 into formal logic and chain the conditions.
I <-> K; L -> I; K -> M. Forward chain: L -> I -> K -> M. Contrapositive chain: ~M -> ~K -> ~I -> ~L.
This confirms that if Relay M fails to go offline (~M), Diagnostic L does not verify (~L).
4
Translate Protocol 4 into formal logic and chain the conditions.
N <-> X_unlatched; O -> N; P -> X_latched (which is ~X_unlatched). Chain: P -> ~X_unlatched -> ~N -> ~O.
This proves that Sensor P activating guarantees Override O is not engaged.

Key Concept

Chaining Conditional Statements through advanced linguistic triggers and contrapositives.
Question 13Question →

According to the new international guidelines for clinical pharmaceutical trials, a trial advances to Phase III only if it demonstrates long-term efficacy. Furthermore, unless a trial is monitored for secondary adverse effects, it cannot demonstrate long-term efficacy. Every trial that is monitored for secondary adverse effects must be subject to a rigorous independent ethics review. Finally, a newly developed treatment contains synthetic bio-markers only if it advances to Phase III.

Based on the guidelines above, which one of the following must be true?

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Answer: Any newly developed treatment that contains synthetic bio-markers is subject to a rigorous independent ethics review.

Answer

Any newly developed treatment that contains synthetic bio-markers is subject to a rigorous independent ethics review.
The correct answer accurately connects the beginning of the logical chain to the end. The stimulus establishes: (1) Synthetic bio-markers require Phase III; (2) Phase III requires long-term efficacy; (3) Long-term efficacy requires monitoring for secondary effects; and (4) Monitoring requires an independent ethics review. Linking these together yields: Synthetic bio-markers → Ethics review.

Step-by-Step Solution

1
Translate the conditional rules regarding Phase III and efficacy.
Rule 1: Advances to Phase III → Demonstrates long-term efficacy. Rule 2: 'Unless monitored, cannot demonstrate efficacy' translates to 'If it demonstrates efficacy → Monitored for secondary effects'.
Accurate translation of indicator words like 'only if' and 'unless' is required to establish the logical relationships.
2
Translate the remaining rules and combine all statements into a single transitive chain.
Rule 3: Monitored for secondary effects → Independent ethics review. Rule 4: Contains synthetic bio-markers → Advances to Phase III. Combined Chain: Contains synthetic bio-markers → Advances to Phase III → Demonstrates long-term efficacy → Monitored for secondary effects → Independent ethics review.
Creating a unified logical sequence makes it possible to see the direct relationship between the first sufficient condition and the final necessary condition.
3
Evaluate the options against the established logical chain.
The chain proves that if a treatment contains synthetic bio-markers, it must ultimately be subject to an independent ethics review.
This confirms that the correct answer is a valid deduction drawn from linking the first term to the last term.

Key Concept

Chaining Conditional Statements
Question 14Question →

Match each set of corporate policy premises with the logically valid deduction that can be derived by chaining them.

Click a left item, then click its matching right item

Items

If a project undergoes an External Audit, it must receive Budget Approval. A project receives Budget Approval only if it passes the Compliance Review.
The Director will Sign-off on a project whenever it passes the Compliance Review. Unless the Executive Board Reviews it, the Director will not Sign-off on the project.
Final Implementation occurs only if the Executive Board Reviews the project. If a project undergoes an External Audit, then Final Implementation will occur.
A project is Signed-off by the Director whenever it receives Budget Approval. Unless a project undergoes an External Audit, it will not be Signed-off by the Director.

Matches

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Answer

Set 1 pairs with the deduction about Compliance and Audit; Set 2 pairs with the deduction about Executive Board and Compliance; Set 3 pairs with the deduction about Audit and Executive Board; Set 4 pairs with the deduction about Audit and Budget.
Each premise set forms a transitive chain of conditionals (A -> B -> C), and the matching right item represents either the direct inference (A -> C) or its valid contrapositive (~C -> ~A).

Step-by-Step Solution

1
Translate the premises of the first set into formal logic.
External Audit -> Budget Approval; Budget Approval -> Compliance Review.
Identifying the sufficient and necessary conditions allows us to form a logical chain.
2
Chain the premises of the first set and find the contrapositive.
Chain: External Audit -> Compliance Review. Contrapositive: ~Compliance Review -> ~External Audit.
This matches the right item stating 'If a project does not pass the Compliance Review, it does not undergo an External Audit.'
3
Translate the premises of the second set into formal logic.
Compliance Review -> Director Sign-off; Director Sign-off -> Executive Board Review.
'Whenever' introduces a sufficient condition. 'Unless X, not Y' translates to Y -> X.
4
Chain the premises of the second set and find the contrapositive.
Chain: Compliance Review -> Executive Board Review. Contrapositive: ~Executive Board Review -> ~Compliance Review.
This matches the right item stating 'If the Executive Board does not Review a project, then it has not passed the Compliance Review.'
5
Translate and chain the third set.
Final Implementation -> Executive Board Review; External Audit -> Final Implementation. Chain: External Audit -> Executive Board Review.
This directly matches the right item 'Any project that undergoes an External Audit must be Reviewed by the Executive Board.'
6
Translate and chain the fourth set.
Budget Approval -> Director Sign-off; Director Sign-off -> External Audit. Chain: Budget Approval -> External Audit. Contrapositive: ~External Audit -> ~Budget Approval.
This matches the right item stating 'If a project does not undergo an External Audit, it does not receive Budget Approval.'

Key Concept

Chaining Conditional Statements
Estimated Time:2m 0s
Question 15Question →

If the proposed highway expansion is approved, the adjacent wetlands will be drained. The wetlands will be drained only if the local wildlife council's objections are overruled. Furthermore, any drainage of the wetlands will result in a decline in the migratory bird population, and a decline in this population always leads to a drop in eco-tourism revenue.

Which one of the following statements must be true based on the information above?

Show answer & explanation

Answer: If local eco-tourism revenue does not drop, then the proposed highway expansion was not approved.

Answer

If local eco-tourism revenue does not drop, then the proposed highway expansion was not approved.
The stimulus establishes a direct sequence of conditional statements: Highway Approved -> Wetlands Drained -> Bird Decline -> Revenue Drop. The valid contrapositive of this entire sequence is: No Revenue Drop -> Highway Not Approved. The correct answer precisely states this logically guaranteed outcome.

Step-by-Step Solution

1
Formalize the conditional statements presented in the stimulus.
1. Highway Approved -> Wetlands Drained
2. Wetlands Drained -> Objections Overruled
3. Wetlands Drained -> Bird Decline
4. Bird Decline -> Revenue Drop
Translating English statements into formal logical notation reveals their precise relationships.
2
Identify the primary conditional chain that connects the majority of the variables.
Highway Approved -> Wetlands Drained -> Bird Decline -> Revenue Drop
Chaining allows us to see the ultimate necessary outcomes of the initial sufficient trigger.
3
Determine the valid contrapositive of the combined conditional chain.
No Revenue Drop -> No Bird Decline -> No Wetlands Drained -> Highway Not Approved
In 'Must Be True' formal logic questions, the correct answer is frequently the contrapositive of the main logical chain.

Key Concept

Chaining Conditional Statements
Question 16Question →

According to the newly ratified International Trade Arbitration framework, a penalized nation will have its economic sanctions lifted only if it submits to a binding independent audit. However, a penalized nation cannot submit to a binding independent audit unless it formally waives its right to appeal the tribunal's preliminary ruling. Furthermore, any nation that formally waives its right to appeal the tribunal's preliminary ruling will automatically be stripped of its emergency tariff protections. It is a strict rule of the framework that a penalized nation retains its emergency tariff protections if it maintains its original trade embargo.

If the statements above are true, which one of the following must also be true?

Show answer & explanation

Answer: Any penalized nation that maintains its original trade embargo will not have its economic sanctions lifted.

Answer

Any penalized nation that maintains its original trade embargo will not have its economic sanctions lifted.
The correct answer accurately reflects the contrapositive chain of the stimulus. By linking the premises, we establish that if sanctions are lifted (SL), the nation must not maintain its embargo (~ME), represented as SL -> ~ME. The contrapositive of this relationship is ME -> ~SL, meaning if a nation maintains its embargo, its sanctions cannot be lifted.

Step-by-Step Solution

1
Diagram the first sentence.
Sanctions Lifted (SL) -> Independent Audit (IA)
The phrase 'only if' introduces a necessary condition.
2
Diagram the second sentence.
Independent Audit (IA) -> Waives Appeal (WA)
The phrase 'cannot... unless' translates to '~WA -> ~IA', which contraposes to 'IA -> WA'.
3
Diagram the third sentence.
Waives Appeal (WA) -> Stripped of Protections (~RP)
The phrase 'any... will automatically' introduces a sufficient condition. Note that 'stripped of protections' is the logical negation of 'retains protections' (RP).
4
Diagram the fourth sentence.
Maintains Embargo (ME) -> Retains Protections (RP)
The phrase 'retains... if it maintains' places the condition following 'if' as the sufficient condition.
5
Create a unified conditional chain.
SL -> IA -> WA -> ~RP. The contrapositive of the fourth premise is ~RP -> ~ME. Combining them yields: SL -> IA -> WA -> ~RP -> ~ME.
Linking the common terms allows us to see the full transitive relationship.
6
Derive the full contrapositive chain to find the correct answer.
ME -> RP -> ~WA -> ~IA -> ~SL
The correct answer often relies on reading the contrapositive chain from left to right. 'ME -> ~SL' perfectly matches the correct option.

Key Concept

Chaining Conditional Statements
Question 17Question →

Match each set of cybersecurity defense protocols (1-4) with the logically valid deduction that must follow from those protocols. Assume no other protocols exist.

Click a left item, then click its matching right item

Items

Protocol Set 1: The system audit is triggered only if the auxiliary backup is activated. The server core is isolated whenever the auxiliary backup is activated.
Protocol Set 2: The server core is isolated only if the diagnostic protocols do not run. The system audit is triggered whenever the diagnostic protocols do not run.
Protocol Set 3: The diagnostic protocols run whenever the system audit is triggered. The server core is isolated only if the diagnostic protocols do not run.
Protocol Set 4: The server core is isolated whenever the auxiliary backup is not activated. The auxiliary backup is activated only if the system audit is triggered.

Matches

Show answer & explanation

Answer

Protocol Set 1 pairs with 'If the server core is not isolated, then the system audit is not triggered.' Protocol Set 2 pairs with 'If the system audit is not triggered, then the server core is not isolated.' Protocol Set 3 pairs with 'If the system audit is triggered, then the server core is not isolated.' Protocol Set 4 pairs with 'If the system audit is not triggered, then the server core is isolated.'
Each protocol set forms a unique transitive chain connecting the system audit (AA) and the server core (CC). By accurately translating 'only if' (necessary) and 'whenever' (sufficient) statements into formal logic, and applying the contrapositive rule (if X→YX \rightarrow Y, then ∼Y→∼X\sim Y \rightarrow \sim X), each set yields exactly one mathematically valid deduction provided in the options.

Step-by-Step Solution

1
Translate Protocol Set 1 into formal logic and derive its valid inference.
Let AA = Audit triggered, BB = Backup activated, CC = Core isolated. Premises: A→BA \rightarrow B and B→CB \rightarrow C. Chain: A→CA \rightarrow C. Contrapositive: ∼C→∼A\sim C \rightarrow \sim A.
'Only if' indicates a necessary condition (placed after the arrow), and 'whenever' indicates a sufficient condition (placed before the arrow).
2
Translate Protocol Set 2 into formal logic and derive its valid inference.
Let DD = Diagnostics run. Premises: C→∼DC \rightarrow \sim D and ∼D→A\sim D \rightarrow A. Chain: C→AC \rightarrow A. Contrapositive: ∼A→∼C\sim A \rightarrow \sim C.
Linking the common intermediate term (∼D\sim D) directly yields a continuous deductive chain from CC to AA.
3
Translate Protocol Set 3 into formal logic, apply a contrapositive to link terms, and derive the inference.
Premises: A→DA \rightarrow D and C→∼DC \rightarrow \sim D. The contrapositive of the second premise is D→∼CD \rightarrow \sim C. Chain: A→D→∼CA \rightarrow D \rightarrow \sim C. Result: A→∼CA \rightarrow \sim C.
The premises cannot be chained directly. Taking the contrapositive of the second premise aligns the terms so they can form a valid transitive sequence.
4
Translate Protocol Set 4 into formal logic, apply a contrapositive to link terms, and derive the inference.
Premises: ∼B→C\sim B \rightarrow C and B→AB \rightarrow A. The contrapositive of the second premise is ∼A→∼B\sim A \rightarrow \sim B. Chain: ∼A→∼B→C\sim A \rightarrow \sim B \rightarrow C. Result: ∼A→C\sim A \rightarrow C.
Just as in step 3, rearranging the second premise via contrapositive creates the bridge needed to link the variables AA and CC.

Key Concept

Translating conditional language into formal logic, chaining transitive statements, and deriving logically equivalent contrapositives.
Question 18Question →

The symphony orchestra will perform Mahler's Symphony No. 8 this season only if they hire additional brass musicians. Furthermore, the orchestra can hire additional brass musicians only if they secure the City Arts Grant. The terms of the City Arts Grant dictate that any recipient must host a free community concert. However, if the main concert hall's scheduled renovations are delayed, the orchestra will be unable to host a free community concert this season.

Which one of the following conclusions can be properly drawn from the information above?

Show answer & explanation

Answer: If the main concert hall's scheduled renovations are delayed, the symphony orchestra will not perform Mahler's Symphony No. 8.

Answer

If the main concert hall's scheduled renovations are delayed, the symphony orchestra will not perform Mahler's Symphony No. 8.
The stimulus establishes a linear chain of conditional logic: Performing Mahler requires hiring brass, which requires the grant, which requires a free concert. Thus, performing Mahler fundamentally requires hosting a free concert. The final sentence provides a trigger: if renovations are delayed, the free concert cannot happen. By taking the contrapositive of the main chain, if the free concert cannot happen, then the orchestra cannot perform Mahler. Therefore, a renovation delay logically guarantees that Mahler will not be performed.

Step-by-Step Solution

1
Diagram the first two conditional premises.
Perform Mahler -> Hire Brass -> Secure Grant
The phrase 'only if' introduces a necessary condition. Performing Mahler requires hiring brass, and hiring brass requires securing the grant.
2
Diagram the final two conditional premises.
Secure Grant -> Host Free Concert. Renovations Delayed -> NOT Host Free Concert.
The grant dictates the concert, making it a necessary condition. The delay prevents the concert, making the delay a sufficient condition for not hosting the concert.
3
Chain all the conditional statements together.
Perform Mahler -> Hire Brass -> Secure Grant -> Host Free Concert
Connecting the shared terms creates a complete logical chain of necessary conditions for performing Mahler.
4
Take the contrapositive of the entire chain and apply the final premise.
NOT Host Free Concert -> NOT Secure Grant -> NOT Hire Brass -> NOT Perform Mahler. Since Renovations Delayed -> NOT Host Free Concert, then Renovations Delayed -> NOT Perform Mahler.
If the final necessary condition in a chain fails, the initial sufficient condition cannot occur. A renovation delay triggers this failure.

Key Concept

Chaining Conditional Statements and applying Contrapositives
Estimated Time:1m 30s
Question 19Question →

The Faculty Research Committee has instituted the following rigid guidelines for grant allocations:
- A grant is approved for full funding only if the proposed project promises immediate commercial viability.
- Projects utilizing human subjects are invariably classified as High-Risk endeavors.
- Any project that is cleared by the Central Ethics Board necessarily incorporates multi-year longitudinal phases.
- Promising immediate commercial viability is impossible for any project that incorporates multi-year longitudinal phases.
- A project is classified as a High-Risk endeavor only when it is cleared by the Central Ethics Board.

Dr. Vance's proposed research project was just approved for full funding. Arrange the following logical deductions in the strictly required sequential order to construct a valid chain of inferences proving that Dr. Vance's project does not utilize human subjects.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence starts by deducing commercial viability from the funding approval, then the absence of longitudinal phases, followed by the lack of board clearance, then the lack of High-Risk classification, and finally concluding that human subjects are not utilized.
A valid logical deduction must proceed step-by-step from known facts. The known starting point is that the project received full funding (FF). First, the rule 'full funding only if... promises immediate commercial viability' translates to FF -> CV, yielding CV (item_cv). Second, the rule 'immediate commercial viability is impossible for any project that incorporates multi-year longitudinal phases' translates to MYL -> ~CV, and its contrapositive is CV -> ~MYL, yielding ~MYL (item_myl). Third, the rule 'cleared by the Central Ethics Board necessarily incorporates multi-year longitudinal phases' translates to CEB -> MYL, with the contrapositive ~MYL -> ~CEB, yielding ~CEB (item_ceb). Fourth, the rule 'High-Risk endeavor only when it is cleared by the Central Ethics Board' translates to HR -> CEB, with the contrapositive ~CEB -> ~HR, yielding ~HR (item_hr). Finally, the rule 'utilizing human subjects are invariably classified as High-Risk' translates to HS -> HR, with the contrapositive ~HR -> ~HS, proving the project does not use human subjects (item_hs).

Step-by-Step Solution

1
Translate the five policy rules into conditional statements.
1. FF -> CV; 2. HS -> HR; 3. CEB -> MYL; 4. MYL -> ~CV; 5. HR -> CEB.
Formalizing the statements prevents misinterpretation of necessary and sufficient conditions.
2
Identify the known starting fact regarding Dr. Vance's project.
The project has been approved for full funding (FF).
A valid deductive chain must begin with an established premise.
3
Chain the conditionals using modus ponens and contrapositives to reach the target conclusion (~HS).
FF -> CV -> ~MYL -> ~CEB -> ~HR -> ~HS.
This maps the exact logical progression required to prove the project does not utilize human subjects.
4
Match the formalized logical chain to the descriptive textual steps provided.
The correct order is item_cv, item_myl, item_ceb, item_hr, item_hs.
Each descriptive step explicitly relies on the conclusion generated by the immediately preceding logical step.

Key Concept

Chaining Conditional Statements and Contrapositives
Question 20Question →

A compliance officer is mapping the consequences of a bank adopting the new Alpha cybersecurity protocol. The regulatory framework dictates the following:

1. If the Alpha protocol is implemented, the core server infrastructure must be upgraded.
2. Current vendor contracts can be maintained only if the core server infrastructure is not upgraded.
3. Unless the bank maintains its current vendor contracts, it will incur early termination penalties.
4. Any institution that incurs early termination penalties is subject to a flagged annual audit.

Arrange the logical steps in the correct deductive sequence to validly prove that implementing the Alpha protocol guarantees a flagged annual audit.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct logical sequence starts by assuming the Alpha protocol is implemented, followed by deducing the server upgrade, then recognizing the broken vendor contracts via a contrapositive, leading to early termination penalties, and finally concluding with a flagged annual audit.
The correct sequence mirrors the logically valid transitive chain: Alpha -> Upgrade -> ~Maintain Contracts -> Penalties -> Flagged Audit. It properly accounts for the contrapositive of the 'only if' statement and correctly translates the 'unless' statement.

Step-by-Step Solution

1
Translate all premises into formal logic.
1: Alpha -> Upgrade. 2: Maintain -> ~Upgrade. 3: ~Maintain -> Penalties. 4: Penalties -> Flagged.
Formalizing the language (especially 'only if' and 'unless') is required before linking the conditions.
2
Establish the initial trigger.
Alpha protocol is implemented (Alpha).
To prove the final conditional statement (Alpha -> Flagged), you must assume the sufficient condition.
3
Follow the chain of necessary conditions.
Alpha -> Upgrade.
Direct application of the first premise.
4
Form the contrapositive of the second premise to link the chain.
Upgrade -> ~Maintain.
The necessary condition of premise 1 (Upgrade) is the negation of the necessary condition in premise 2 (~Upgrade). Taking the contrapositive of premise 2 creates the link.
5
Complete the chain through the remaining premises.
~Maintain -> Penalties -> Flagged.
The deductions line up perfectly with the sufficient conditions of premise 3 and 4.

Key Concept

Chaining conditional statements, utilizing contrapositives, and translating 'only if' and 'unless' statements into standard formal logic.
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