Examkin

Quantifier Logic (Some, Most, All)

16 soru

Tüm türler16Çoktan seçmeli8Doğru / Yanlış5Eşleştirme3
Soru 1Soru →

Consider the following premises regarding the state judiciary:

Every judge appointed to the appellate court has over a decade of trial experience. Most judges appointed to the appellate court have published legal scholarship. None of the judges who have published legal scholarship sit on the disciplinary board.

Based on these premises, the following deduction must logically be true:
Most judges with over a decade of trial experience do not sit on the disciplinary board.

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Cevap: False

Cevap

The statement is False because it commits a quantifier overreach error, attempting to apply a characteristic of a specific subset (appellate judges) to the majority of a broader superset (judges with trial experience).
The deduction provided in the stem is invalid. It correctly chains the premises to realize that most appellate judges do not sit on the board. However, it illegally extends that conclusion to state that most judges with a decade of trial experience do not sit on the board. We only know that *some* (at least the appellate judges) do not. We do not have information about the rest of the judges with trial experience.

Adım Adım Çözüm

1
Map the formal logic of the premises.
Premise 1: All Appellate Judges (AJ) -> Trial Experience (TE). Premise 2: Most AJ -> Published (P). Premise 3: No P -> Disciplinary Board (DB), which means All P -> ~DB.
Translating statements into formal notation clarifies the boundaries of the quantifier relationships.
2
Chain the 'Most' and 'None/All' premises to form a valid intermediate deduction.
Combine Premise 2 (Most AJ -> P) with Premise 3 (All P -> ~DB) to deduce: Most AJ -> ~DB (Most appellate judges do not sit on the disciplinary board).
In quantifier logic, combining a 'Most' statement with an 'All/None' statement yields a valid 'Most' statement if the 'Most' group points to the sufficient condition of the 'All' group.
3
Evaluate the relationship between the intermediate deduction and the target group.
We know 'Most AJ -> ~DB' and 'All AJ -> TE'. The deduction asks if 'Most TE -> ~DB' must be true.
We must test if the target conclusion strictly follows from the relationship between AJ and TE.
4
Determine the validity of the final deduction.
Invalid. Because AJ is a subset of TE, we can only conclude that 'Some TE -> ~DB'. We cannot conclude 'Most TE' share any characteristic because the broader group of TE judges could be infinitely larger than the AJ subset.
Moving from a subset to a superset only guarantees a 'Some' relationship, never a 'Most' relationship without additional numerical data.

Anahtar Kavram

Quantifier Logic (Some, Most, All)
Soru 2Soru →

Consider the following premises about an ecosystem:

All migratory birds that nest in the Alpine Ridge rely on the silver-leaf willow for food.
Most migratory birds that nest in the Alpine Ridge have developed a resistance to the local avian flu.
Every bird that relies on the silver-leaf willow for food has a beak specially adapted for cracking seeds.
Most birds that have developed a resistance to the local avian flu carry a specific genetic marker, GM-42.

Based on these premises, it must be true that at least some birds with beaks specially adapted for cracking seeds carry the GM-42 genetic marker.

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Cevap: False

Cevap

The statement could be false, so it does not logically have to be true.
The statement is logically invalid because mathematically connecting 'Some A are B' with 'Most B are C' does not guarantee 'Some A are C'. The 'most' group of B could entirely exclude the 'some' group of A.

Adım Adım Çözüm

1
Deduce the relationship between Alpine Ridge birds and beak adaptations.
All Alpine Ridge birds rely on the willow, and all willow-reliant birds have adapted beaks. Therefore, all Alpine Ridge birds have adapted beaks (A → B, B → C means A → C).
This establishes a formal necessary condition for all Alpine Ridge birds, simplifying the first part of the stimulus.
2
Integrate the 'most' premise regarding Alpine Ridge birds and resistance.
Since most Alpine Ridge birds are resistant, and all Alpine Ridge birds have adapted beaks, we can validly deduce that at least some birds with adapted beaks are resistant (Some A are B).
This maps the relationship between the two target variables presented in the conclusion: adapted beaks and flu resistance.
3
Evaluate the final link between flu resistance and the GM-42 genetic marker.
The premises yield two isolated relationships: 'Some birds with adapted beaks are resistant' and 'Most resistant birds have the GM-42 marker'.
Isolating these formal relationships allows us to test the proposed conclusion strictly against the rules of quantifier logic.
4
Test the validity of combining 'Some X are Y' with 'Most Y are Z'.
There is no mathematical guarantee that the 'some' overlap in the first relationship shares any members with the 'most' majority in the second relationship. Thus, no valid deduction can be made.
This proves the target statement is a logical overreach and could be false.

Anahtar Kavram

Formal deductions from overlapping quantifiers (the invalidity of chaining 'Some' and 'Most')
Soru 3Soru →

Evaluate the logical validity of the deduction based on the provided premises.

Premises:
1. All ceremonial masks recovered from the archaeological site are adorned with lapis lazuli.
2. Most of the artifacts adorned with lapis lazuli recovered from the site date back to the Second Dynasty.

Deduction: Based on these premises, it must be true that most of the ceremonial masks recovered from the site date back to the Second Dynasty.

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Cevap: False

Cevap

The deduction is false. We cannot logically conclude that most of the ceremonial masks date back to the Second Dynasty.
The correct evaluation is false because the logical chain 'All A are B' and 'Most B are C' does not yield 'Most A are C'. The ceremonial masks are a subset of the lapis lazuli artifacts. It is possible that the lapis lazuli artifacts from the Second Dynasty consist entirely of non-mask objects, meaning zero masks might actually date to the Second Dynasty.

Adım Adım Çözüm

1
Translate the premises into formal quantifier logic structures.
Let A = ceremonial masks, B = artifacts adorned with lapis lazuli, and C = artifacts dating to the Second Dynasty. Premise 1: All A are B. Premise 2: Most B are C.
Translating natural language into categorical relationships prevents relying on outside assumptions.
2
Determine the necessary overlap between the specific subgroup (A) and the characteristic (C).
Because 'A' is merely a subset of 'B', and only 'most' (not all) of 'B' are 'C', the subset 'A' could entirely exist within the minority portion of 'B' that is NOT 'C'.
To test if the 'most' quantifier logically guarantees a trait for every contained subset.
3
Evaluate the proposed deduction.
The deduction claims that 'Most A are C' must be true. Since it is entirely possible that zero 'A' are 'C', the deduction is invalid.
In formal logic, a deduction is only valid if it MUST be true under all scenarios permitted by the premises.

Anahtar Kavram

Quantifier Logic: Invalid chaining of 'All' and 'Most'
Soru 4Soru →

In the city's botanical garden, all luminescent plants require high humidity. Furthermore, some luminescent plants in the botanical garden bloom at night.

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

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Cevap: Some plants in the botanical garden that bloom at night require high humidity.

Cevap

Some plants in the botanical garden that bloom at night require high humidity.
The correct answer must be true because if 'some' (at least one) luminescent plants bloom at night, and 'all' luminescent plants require high humidity, then those specific night-blooming luminescent plants must also require high humidity. Therefore, there is at least one plant (some) that both blooms at night and requires high humidity.

Adım Adım Çözüm

1
Analyze the first premise.
All luminescent plants require high humidity (Luminescent -> High Humidity).
This establishes a universal requirement for a specific group of plants.
2
Analyze the second premise.
Some luminescent plants bloom at night (Luminescent-some-Night Blooming).
This establishes that at least one plant belongs to both the 'luminescent' category and the 'night blooming' category.
3
Combine the premises to find the overlap.
Since the plants that are both luminescent and night-blooming are luminescent, they must also require high humidity.
Applying the universal rule from step 1 to the specific subset identified in step 2 proves that at least some night-blooming plants require high humidity.

Anahtar Kavram

Quantifier Logic Overlaps (All + Some)
Soru 5Soru →

All of the medieval manuscripts currently housed in the monastery's east wing were written by monks from the local order. Most of the manuscripts housed in the east wing contain intricate gold illuminations. Furthermore, none of the manuscripts written by monks from the local order are bound in leather.

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

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Cevap: At least some manuscripts that contain intricate gold illuminations are not bound in leather.

Cevap

At least some manuscripts that contain intricate gold illuminations are not bound in leather.
The stimulus establishes that every manuscript in the east wing was written by local monks, and no manuscripts written by local monks are bound in leather. By chaining these statements, we deduce that no manuscripts in the east wing are bound in leather. Because most of the manuscripts in the east wing feature gold illuminations, there is a guaranteed overlapping group of manuscripts that both have gold illuminations and are strictly not bound in leather. Consequently, it must be logically true that at least some manuscripts containing intricate gold illuminations are not bound in leather.

Adım Adım Çözüm

1
Identify the formal logic relationships given in the stimulus.
1. East Wing -> Local Monks; 2. East Wing -most-> Gold Illuminations; 3. Local Monks -> NOT Leather Bound.
Mapping the premises allows for accurate logical chaining and prevents unwarranted assumptions.
2
Chain the universal and absolute negative statements.
Since East Wing -> Local Monks, and Local Monks -> NOT Leather Bound, we can deduce: East Wing -> NOT Leather Bound.
Transitive property of conditional logic establishes that no manuscripts in the east wing are bound in leather.
3
Overlap the deduced absolute statement with the quantifier statement.
We know most East Wing manuscripts have gold illuminations. We also know that zero East Wing manuscripts are leather-bound.
This establishes the traits of the specific overlapping subgroup of manuscripts.
4
Draw the final inference based on the overlap.
Therefore, there must be some manuscripts that contain gold illuminations that are not bound in leather.
If most of a group (East Wing) has a trait (Gold), and all of that group lacks another trait (Leather), then at least some things with the first trait must lack the second trait.

Anahtar Kavram

Quantifier Logic (Some, Most, All)
Soru 6Soru →

Consider the following premises: Most corporate lawyers at the firm hold advanced degrees in finance, and most corporate lawyers at the firm have handled international mergers. Based on these premises, it must be logically true that at least some individuals who have handled international mergers hold advanced degrees in finance.

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Cevap: True

Cevap

True
The statement presents a valid logical deduction. When two 'most' statements share the same sufficient condition (the same subject group, which is corporate lawyers), their necessary conditions must overlap. If out of a group of 100100 lawyers, at least 5151 have finance degrees and at least 5151 have handled international mergers, there must be at least 22 lawyers with both traits. Thus, it must be true that at least some people who handled international mergers have finance degrees.

Adım Adım Çözüm

1
Identify the base group and the associated quantifiers in the premises.
Base group: Corporate lawyers at the firm. Quantifiers: 'Most' (meaning strictly greater than 50%50\%).
To formalize the logical structure of the premises.
2
Translate the premises into formal logic.
Premise 1: Most A (corporate lawyers) are B (hold finance degrees). Premise 2: Most A (corporate lawyers) are C (have handled international mergers).
To clearly see the relationships between the groups.
3
Evaluate the necessary overlap between the two subsets.
Since both subsets (B and C) comprise more than 50%50\% of the total group A, they must overlap by at least one member. Thus, at least some A are both B and C.
To determine what mathematically must be true when two majorities are taken from the exact same finite set.
4
Analyze the conclusion given in the statement.
The conclusion states that 'some C are B'. Since we proved that there is an overlap (some A are both B and C), it logically follows that some C are B.
To compare the logically required overlap with the statement's conclusion.

Anahtar Kavram

Quantifier Overlap (Two 'Most' statements about the same subject)
Tahmini Süre:1m 0s
Soru 7Soru →

Analyze the four formal logic setups involving professional and social categories. Match each premise setup to the precise logical deduction or scope limit that uniquely follows from it.

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Öğeler

All judges are lawyers. Most judges are politicians. Most politicians are wealthy.
Most judges are lawyers. Most judges are politicians. All politicians are wealthy.
All judges are politicians. Most politicians are lawyers. Most politicians are wealthy.
Most judges are politicians. All politicians are lawyers. All lawyers are wealthy.

Eşleşmeler

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Cevap

Setup 1 matches the deduction that some lawyers are politicians but none are necessarily wealthy. Setup 2 matches the deduction that most judges are wealthy but not all politicians are lawyers. Setup 3 matches the deduction that some lawyers are wealthy but no judges are necessarily lawyers. Setup 4 matches the deduction that all politicians are lawyers and wealthy, but most lawyers are not necessarily politicians.
Each setup uniquely allows for precise deductions based on formal logic rules: universal chaining (All A are B + All B are C -> All A are C), majority overlaps within the same group (Most A are B + Most A are C -> Some B are C), and universal/majority combinations (Most A are B + All B are C -> Most A are C).

Adım Adım Çözüm

1
Analyze Setup 1: All judges are lawyers. Most judges are politicians. Most politicians are wealthy.
Deduce that since judges are a subset of lawyers, and most judges are politicians, there is an overlap guaranteeing some lawyers are politicians. Note that 'Most politicians are wealthy' cannot be chained back.
To identify that while 'some lawyers are politicians' must be true, we lack evidence to prove any lawyers are wealthy.
2
Analyze Setup 2: Most judges are lawyers. Most judges are politicians. All politicians are wealthy.
Deduce that combining 'Most judges are politicians' and 'All politicians are wealthy' yields 'Most judges are wealthy'. We cannot prove all politicians are lawyers.
To find the unique matching deduction regarding what must be true (most judges are wealthy) versus what is unsupported.
3
Analyze Setup 3: All judges are politicians. Most politicians are lawyers. Most politicians are wealthy.
Deduce that two majorities of the same group (politicians) must intersect, proving some lawyers are wealthy. However, the subset 'judges' might not intersect with the 'lawyers' majority.
To confirm that while some lawyers must be wealthy, we cannot guarantee any judges are lawyers.
4
Analyze Setup 4: Most judges are politicians. All politicians are lawyers. All lawyers are wealthy.
Deduce through universal chaining that all politicians are wealthy (and lawyers). However, 'All politicians are lawyers' does not mean 'Most lawyers are politicians'.
To verify that all politicians are both lawyers and wealthy, but it is not necessarily true that most lawyers are politicians.

Anahtar Kavram

Quantifier Logic (Some, Most, All) Valid Inferences and Scope Limitations
Tahmini Süre:3m 0s
Soru 8Soru →

Marine biologists studying the newly discovered Marianas deep-sea caldera have cataloged various unique species. According to their findings, all organisms that inhabit the caldera's core depend entirely on chemosynthesis for survival. Furthermore, most of the organisms inhabiting the caldera's core lack photobiological receptors. Interestingly, any organism that migrates to the epipelagic zone possesses photobiological receptors. Finally, researchers noted that some organisms dependent on chemosynthesis are active filter feeders.

Which one of the following logically follows from the researchers' findings?

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Cevap: At least some organisms that depend entirely on chemosynthesis do not migrate to the epipelagic zone.

Cevap

At least some organisms that depend entirely on chemosynthesis do not migrate to the epipelagic zone.
The correct answer properly chains the provided formal logic. Since any organism that migrates possesses photobiological receptors (Migrate →\rightarrow Receptors), it logically follows via contrapositive that any organism lacking these receptors does not migrate (No Receptors →\rightarrow No Migrate). We are told that most caldera core inhabitants lack these receptors; therefore, most caldera core inhabitants do not migrate. Finally, because all caldera core inhabitants depend entirely on chemosynthesis, this specific non-migrating group forms a part of the chemosynthesis-dependent population. Thus, it must be true that at least some organisms dependent on chemosynthesis do not migrate.

Adım Adım Çözüm

1
Diagram the formal relationships provided in the stimulus.
1. Core →\rightarrow Chemosynthesis. 2. Core (most) →\rightarrow No Receptors. 3. Migrate →\rightarrow Receptors. 4. Chemosynthesis (some) →\rightarrow Filter Feeder.
Mapping the exact logical quantifiers prevents relying on outside assumptions and highlights formal connections.
2
Find the contrapositive of the strict conditional regarding migration.
If Migrate →\rightarrow Receptors, then No Receptors →\rightarrow No Migrate.
Contrapositives allow us to connect negative premises to strict conditional rules.
3
Chain the 'most' statement with the contrapositive.
Core (most) →\rightarrow No Receptors →\rightarrow No Migrate. Therefore, most Core inhabitants do not migrate.
A 'most' statement can be validly chained into a strict sufficient/necessary conditional to yield a new 'most' deduction.
4
Combine the deduction with the first premise to find the overlapping subset.
All Core inhabitants are Chemosynthesis-dependent. Since most Core inhabitants do not migrate, at least some Chemosynthesis-dependent organisms do not migrate.
If a subset (Core) has a certain property (No Migrate), and that entire subset belongs to a larger group (Chemosynthesis), then at least some members of the larger group must have that property.

Anahtar Kavram

Chaining 'most' statements with conditional contrapositives and inferring 'some' from subset relationships.
Tahmini Süre:2m 0s
Soru 9Soru →

At the city's architectural preservation site, most of the historic buildings were constructed before the year 1900. Furthermore, most of the historic buildings at the site feature masonry facades. All buildings constructed before 1900 at the site are protected by the heritage registry.

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

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Cevap: At least some buildings protected by the heritage registry feature masonry facades.

Cevap

The correct answer is the statement that at least some buildings protected by the heritage registry feature masonry facades.
The premises establish that 'most' historic buildings are pre-1900 and 'most' feature masonry facades. Because both statements apply to the majority of the exact same group, there must be a mathematical overlap: at least some historic buildings are both pre-1900 and feature masonry. The stimulus then states that all pre-1900 buildings are protected by the heritage registry. Since we already proved that some of those pre-1900 buildings have masonry facades, those specific overlapping buildings must also be protected. Therefore, it is definitively true that at least some protected buildings feature masonry facades.

Adım Adım Çözüm

1
Identify and map the formal quantifier relationships provided in the premises.
1. Most Historic -> Pre-1900
2. Most Historic -> Masonry
3. All Pre-1900 -> Protected
Deconstructing the stimulus into formal logical notation is the first step to making rigorous deductions.
2
Combine the two 'most' statements concerning the same subject group (Historic buildings).
Since 'most' indicates more than 50%, the two groups (Pre-1900 and Masonry) must overlap. Therefore, at least some Pre-1900 buildings feature Masonry.
In quantifier logic, any two 'most' statements applied to the exact same population logically guarantee an intersection (a 'some' statement).
3
Chain the resulting 'some' statement with the 'all' statement.
Since 'Some Pre-1900 -> Masonry' and 'All Pre-1900 -> Protected', it logically follows that 'Some Protected -> Masonry'.
A 'some' statement can be chained with an 'all' statement if the 'all' condition applies to the shared term, transferring the trait to the new group.

Anahtar Kavram

Deducing overlaps from multiple 'most' statements and chaining them with 'all' statements.
Soru 10Soru →

Consider the following premises regarding staff at the Central Research Institute:

All research fellows at the institute possess a doctoral degree.
Most research fellows at the institute are currently directing a grant-funded project.
Most research fellows at the institute have served on the ethics board.

Based on these premises, the following deduction must be true: At least some individuals who are directing a grant-funded project and have served on the ethics board possess a doctoral degree.

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Cevap: True

Cevap

True. The formal logic premises guarantee that at least one individual satisfies all three conditions.
The conclusion logically follows from the premises. The intersection of two 'most' statements concerning the same base group (research fellows) proves that 'some' research fellows have both characteristics. Applying the universal 'all' condition to these specific fellows proves they must also hold a doctoral degree.

Adım Adım Çözüm

1
Analyze the relationship between the two 'most' premises applied to the same group.
Identify that because 'most' means strictly more than half, the group of research fellows directing projects and the group serving on the board must overlap by at least one person.
Two 'most' statements applied to the exact same finite group mathematically require an intersection (yielding a 'some' statement).
2
Apply the 'all' premise to the overlapping group identified in Step 1.
Conclude that since all research fellows possess a doctoral degree, the specific research fellow(s) who both direct a project and serve on the board must inherently possess a doctoral degree.
A universal 'all' statement applies to every single member of the group, which inherently includes any and all subsets.
3
Evaluate the target statement against the deductions.
The target statement claims 'some individuals' (meaning at least one) have all three characteristics, which perfectly matches the deduction.
To confirm if the statement logically must be true based strictly on the formal logic of the provided premises.

Anahtar Kavram

Quantifier Overlap (Most + Most = Some)
Soru 11Soru →

In formal logic, specific quantifier combinations yield strictly valid deductions. Analyze the four multi-premise setups below, each from a different domain. Match each setup to the abstract description of its logically valid deduction.

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Öğeler

Ecology: All wetland ecosystems support amphibian populations, and most wetland ecosystems are highly sensitive to pollution.
Astronomy: Most exoplanets in the catalog orbit red dwarf stars, and every exoplanet that orbits a red dwarf star is tidally locked.
Archaeology: All artifacts found in the lower stratum predate the Bronze Age, yet some artifacts found in the lower stratum are not made of stone.
Policy: Most of the university's board members voted for the expansion, and most of the university's board members are alumni.

Eşleşmeler

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Cevap

Ecology matches the 'some members of the broader category...' deduction; Astronomy matches the 'majority of the initial group...' deduction; Archaeology matches the 'some members of the enclosing category must lack...' deduction; Policy matches the 'two traits must overlap...' deduction.
To solve quantifier logic setups, you must track the formal relationships without being distracted by the subject matter. 1. Ecology (All A are B + Most A are C) proves that the trait 'C' transfers to the broader category 'B'. 2. Astronomy (Most A are B + All B are C) proves that the majority of 'A' is enclosed in 'C'. 3. Archaeology (All A are B + Some A are not C) proves that the 'not C' trait transfers to the enclosing category 'B'. 4. Policy (Most A are B + Most A are C) proves that traits 'B' and 'C' must overlap.

Adım Adım Çözüm

1
Diagram the Ecology setup.
All W -> A (W is a subset of A). Most W -> S. The W's that are S are also A's, yielding the deduction: Some A are S.
To identify the abstract logical result, we must map the specific categories (W, A, S) to their roles: subset, broader category, and trait.
2
Diagram the Astronomy setup.
Most E -> R. All R -> T. Because the 'Most E' group is entirely contained within R, and R is entirely within T, that 'Most E' group must be within T. Deduction: Most E are T.
This establishes a transitive 'Most' relationship passing through an 'All' relationship.
3
Diagram the Archaeology setup.
All L -> P (L is a subset of P). Some L -> ~S. Those specific L's that are not S must still be inside P. Deduction: Some P are ~S.
This confirms how a negative 'Some' statement interacts with an enclosing 'All' statement.
4
Diagram the Policy setup.
Most B -> V. Most B -> A. Because both represent more than 50% of the exact same parent group (B), they must mathematically intersect. Deduction: Some V are A.
This tests the classic 'Most + Most = Some' overlap rule.

Anahtar Kavram

Formal deduction using combinations of 'All', 'Most', and 'Some' quantifiers, specifically recognizing transitive properties and mathematical overlaps of majorities.
Soru 12Soru →

Most urban planners employed by the city hold degrees in architecture. Furthermore, most urban planners employed by the city have worked in the public sector for at least five years. According to municipal guidelines, all individuals who have worked in the public sector for at least five years are eligible for the municipal leadership grant.

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

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Cevap: At least one person holding a degree in architecture is eligible for the municipal leadership grant.

Cevap

At least one person holding a degree in architecture is eligible for the municipal leadership grant.
The correct answer properly identifies the guaranteed overlap between two 'most' statements applied to the same group. Since most urban planners have architecture degrees and most have five years of public sector experience, at least one person must have both. Furthermore, because the stimulus states that ALL individuals with five years of public sector experience are eligible for the grant, that specific overlapping person (who is an architect) must be eligible for the grant.

Adım Adım Çözüm

1
Analyze the first two premises regarding 'most' relationships within a single group.
Identify that 'most' urban planners are architects, and 'most' urban planners have five years of public sector experience.
When two 'most' statements apply to the exact same population (urban planners), the two subgroups must logically overlap by at least one member.
2
Form a deduction from the overlapping 'most' statements.
Conclude that there is at least one urban planner who is BOTH an architect AND has five years of public sector experience.
If group A is >50% of the total and group B is >50% of the total, their combined sum exceeds 100%, meaning they must share at least one individual.
3
Apply the conditional 'all' rule to the deduced overlap.
Because the overlapping individual has five years of public sector experience, and ALL such individuals are eligible for the grant, this individual must be eligible.
Applying a universal rule ('all') to a specific subject confirms a property of that subject.
4
Synthesize the final deduction.
Since the overlapping individual is an architect and is eligible for the grant, it must be true that at least one person holding an architecture degree is eligible for the grant.
This directly matches the logically proven facts.

Anahtar Kavram

Deducing overlap from two 'most' statements about the same subject group, and chaining that overlap with an 'all' statement.
Soru 13Soru →

A recent audit of the city's public transit network revealed several operational details. Most of the transit routes in the northern sector currently operate at a financial loss. Furthermore, all transit routes that operate at a financial loss are heavily subsidized by the municipal government. Interestingly, the audit also showed that most of the transit routes in the northern sector are serviced exclusively by hybrid vehicles.

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

Cevabı ve açıklamayı göster

Cevap: At least some transit routes serviced exclusively by hybrid vehicles are heavily subsidized by the municipal government.

Cevap

At least some transit routes serviced exclusively by hybrid vehicles are heavily subsidized by the municipal government.
The correct answer is derived by combining three premises. First, because most northern routes operate at a loss and all loss-making routes are subsidized, we can definitively deduce that most northern routes are subsidized. Second, we are explicitly told that most northern routes are serviced by hybrid vehicles. Because more than half of the northern routes are subsidized and more than half are serviced by hybrid vehicles, there must be an overlap between these two groups. Therefore, it is absolutely true that at least some routes with hybrid vehicles are subsidized.

Adım Adım Çözüm

1
Chain the first two premises to form a new deduction.
Deduction: Most of the transit routes in the northern sector are heavily subsidized.
If most northern routes operate at a loss, and every route operating at a loss is subsidized, then logically, most northern routes must be subsidized.
2
Analyze the final premise in relation to the new deduction.
We now have two facts about the northern sector: Most of its routes are subsidized, and most of its routes are serviced exclusively by hybrid vehicles.
Identifying multiple 'most' statements about the exact same subject group (northern sector routes) is the key to unlocking the final inference.
3
Apply the overlapping quantifiers rule.
Conclusion: At least some routes serviced by hybrid vehicles are subsidized.
If a majority of a defined group possesses trait X, and a majority of that same group possesses trait Y, there is a mathematical necessity that at least one member of the group possesses both trait X and trait Y.

Anahtar Kavram

Formal logic deductions using overlapping quantifiers, specifically deriving a 'some' statement from two intersecting 'most' statements applied to the same group.
Soru 14Soru →

Match each set of formal premises concerning ancient linguistics and archaeology to the logically valid deduction that must follow from it.

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Öğeler

All translated scrolls from the Qumari archive discuss astronomical phenomena. Furthermore, most of the translated scrolls from the Qumari archive were authored by temple priests.
Most scholars specializing in Qumari syntax are proficient in early Veldtian dialects. Additionally, most scholars specializing in Qumari syntax have published peer-reviewed research.
Every known Veldtian dialect employs a base-eight counting system. However, some known Veldtian dialects lack dedicated future-tense verb forms.
Most excavated Veldtian trade outposts contained mercantile ledgers. Every excavated Veldtian trade outpost that contained mercantile ledgers was situated along a major coastal route.

Eşleşmeler

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Cevap

Setup 1 matches with the inference about works authored by temple priests. Setup 2 matches with the inference about published individuals being proficient in Veldtian dialects. Setup 3 matches with the inference about base-eight dialects lacking future tense. Setup 4 matches with the inference about the majority of outposts being on coastal routes.
Each correct pairing strictly follows formal quantifier logic rules without introducing outside assumptions. The overlaps of 'All/Most', 'Most/Most', and 'All/Some' establish mathematically guaranteed intersections, while the 'Most/All' chain allows a valid transitive 'Most' deduction.

Adım Adım Çözüm

1
Analyze the first set of premises: All Q are A, and Most Q are P.
Deduce that because more than half of Q is P, and 100% of Q is A, there must be an overlap. Therefore, Some P are A.
Combining an 'All' and a 'Most' statement about a shared sufficient condition always yields a 'Some' conclusion.
2
Analyze the second set of premises: Most S are V, and Most S are R.
Deduce that two majorities of the same group must intersect. Therefore, Some R are V.
If >50% of a group has trait X, and >50% has trait Y, at least one member must have both traits.
3
Analyze the third set of premises: All V are B, and Some V are not F.
Deduce that the specific 'V's that are not 'F' must still be 'B'. Therefore, Some B are not F.
A 'Some' statement introduces an existing subset, which inherits all universal ('All') traits of the parent group.
4
Analyze the fourth set of premises: Most O are L, and All L within O are C.
Deduce that because the subset L makes up most of O, and all L are C, C must also make up most of O. Therefore, Most O are C.
This is a restricted transitive property where a universal condition applied to a majority subset transfers the majority status to the conclusion.

Anahtar Kavram

Logical deduction using combinations of 'All', 'Most', and 'Some' quantifiers to determine guaranteed overlaps.
Soru 15Soru →

A comprehensive survey of flora in the Veridia Valley documented various characteristics of the local plant species. Every plant species in the valley that produces nectar is pollinated by indigenous insects. Most of the valley's plant species that are pollinated by indigenous insects flower during the early spring. In addition, some of the plant species in the valley that produce nectar are classified as toxic to mammals.

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

Cevabı ve açıklamayı göster

Cevap: At least some of the plant species in the valley that are toxic to mammals are pollinated by indigenous insects.

Cevap

At least some of the plant species in the valley that are toxic to mammals are pollinated by indigenous insects.
The correct answer follows strictly from the first and third premises. The third premise establishes that there is an intersection between nectar-producing plants and toxic plants (i.e., there are plants that are both). The first premise establishes a universal rule: every single nectar-producing plant is pollinated by insects. Applying this universal rule to the intersection, those specific toxic, nectar-producing plants must also be pollinated by insects. Therefore, it is absolutely true that some toxic plants are pollinated by insects.

Adım Adım Çözüm

1
Translate the statements into formal logical relationships.
Statement 1: All Nectar -> Pollinated by Insects. Statement 2: Most Pollinated by Insects -> Flower in Spring. Statement 3: Some Nectar -> Toxic to Mammals.
Mapping the quantifiers allows us to see how the groups interact without being distracted by the subject matter.
2
Identify connections between the 'Some' statement and the 'All' statement.
We know that at least one plant is both Nectar-producing and Toxic (from Statement 3). We also know that every single Nectar-producing plant is Pollinated by Insects (from Statement 1).
The most reliable valid deductions in quantifier logic come from combining 'Some' statements with 'All' statements.
3
Derive the logically guaranteed conclusion.
Since the toxic, nectar-producing plants mentioned in Statement 3 must follow the rule established in Statement 1, they must be pollinated by insects. Therefore, at least some toxic plants are pollinated by insects.
If an entity possesses trait X (Nectar), and ALL things with trait X have trait Y (Pollinated by Insects), then that specific entity must also have trait Y.

Anahtar Kavram

Quantifier Logic (Some, Most, All)
Soru 16Soru →

Botanical surveys of the lower river valley reveal several consistent patterns. All native shrubs in the valley require alkaline soil to thrive. Furthermore, most of these native shrubs bloom in late spring. Finally, none of the plants that bloom in late spring are frost-resistant. Which one of the following conclusions must logically follow from the survey results described above?

Cevabı ve açıklamayı göster

Cevap: At least some plants that require alkaline soil are not frost-resistant.

Cevap

At least some plants that require alkaline soil are not frost-resistant.
The correct answer accurately synthesizes the overlapping sets presented in the premises. Since most native shrubs bloom in late spring, and no late-blooming plants are frost-resistant, it must be true that most native shrubs are not frost-resistant. Since every native shrub requires alkaline soil, those specific non-frost-resistant shrubs must also have alkaline soil, proving that at least some plants requiring alkaline soil are not frost-resistant.

Adım Adım Çözüm

1
Diagram the first premise to establish a universal relationship.
Native Shrubs (NS) -> Alkaline Soil (AS)
This establishes that anything classified as a native shrub possesses the trait of requiring alkaline soil.
2
Diagram the remaining premises to identify characteristics applying to the majority.
Native Shrubs (NS) -most- Bloom Late Spring (BLS). Bloom Late Spring (BLS) -> NOT Frost-Resistant (~FR). Therefore, Native Shrubs (NS) -most- NOT Frost-Resistant (~FR).
Chaining a 'most' statement into a universal 'none/not' statement yields a valid 'most' deduction.
3
Synthesize the overlapping traits to find a guaranteed deduction.
Since Native Shrubs have the universal trait of needing alkaline soil, and most of them are not frost-resistant, the groups 'Alkaline Soil' and 'NOT Frost-Resistant' must overlap.
Whenever a single group (Native Shrubs) entirely belongs to Group X and mostly belongs to Group Y, Groups X and Y must share at least some members.

Anahtar Kavram

Quantifier overlapping and valid deduction from combinations of 'all', 'most', and 'none'.
Quantifier Logic (Some, Most, All) Alıştırma Soruları — LSAT | Examkin