Analytical Reasoning and Logical Reasoning

1. Learning path: analytical reasoning and syllogisms#

Analytical reasoning means breaking given information into precise conditions and checking what follows. Logical reasoning is the wider process of drawing justified conclusions. This first chapter focuses on categorical syllogisms: statements about groups such as students, readers and players. The same discipline supports later chapters on arrangements, directions, relations and puzzles.

Study in this order: four statement types → Venn diagrams → the six requested combinations → analytical method → additional patterns → possibility and either-or → worked questions → independent practice.

A statement is a premise supplied by the question. A conclusion is a claim to test. A conclusion follows only if it is true in every arrangement satisfying all premises. Finding one picture that supports it is insufficient; finding one valid counterexample disproves a claim of necessity.

Treat the supplied statements as true even when they conflict with everyday knowledge. If the question says “All books are birds”, reason from that sentence; do not reject it because real books cannot fly. Do not add facts, reverse arrows or assume different “some” groups are the same objects.

2. The four basic statements: A, E, I and O#

The statement codes A/E/I/O are traditional labels; group names A/B/C are separate placeholders. “Some” means at least one, possibly all. “All” does not mean “only”. “No” excludes every shared member. “Some A are not B” says nothing definite about whether other A objects belong to B.

Existence convention used here: All and No describe relationships but do not by themselves assert that a class has members. Some and Some-not explicitly assert existence. Some competitive-exam instructions additionally assume that named classes are nonempty; under that extra assumption, All A are B gives Some B are A. Read the paper's stated convention. This chapter labels conclusions requiring existence instead of silently changing conventions.

Example: All dragons are animals does not prove that dragons exist. If Some dragons exist is additionally given, then Some animals are dragons follows.

3. How to read a Venn diagram#

A circle represents a class; its size does not represent the number of members. A contained circle shows inclusion, separate circles show exclusion, and an × marks an existing witness. Overlap drawn without an × indicates a permitted region, not proof that an object exists there.

basics

The first two pictures show relationships without asserting existence. In the third picture the × is in A ∩ B. In the fourth it is in A but outside B; the drawing leaves other A–B relationships open.

Draw the minimum forced relationship. All A are B permits A and B to be equal; do not conclude that some B lie outside A merely because the outer circle looks larger. For unknown relations, test more than one picture. A neat drawing is a working model, not a substitute for the words.

Three-circle technique: consider all eight regions: A only, B only, C only, AB only, AC only, BC only, ABC, and outside all three. In a full Venn method, mark forbidden regions empty and put × only where existence is required. If a witness could occupy either of two regions, keep its location undecided; do not invent two witnesses or fix it in the convenient region.

4. All + All: follow the arrow direction#

Chain: All A are B. All B are C. Therefore All A are C.

all-all

Example: All poets are readers. All readers are learners. All poets are learners follows. All learners are poets does not follow. Some learners are poets needs the additional fact that poets exist under this chapter's convention.

Common container: All A are B; All C are B. Nothing definite follows between A and C. They may overlap, be separate or coincide inside B. Example: all roses and all lilies are flowers does not prove all roses are lilies.

Common contained class: All B are A; All B are C. All B belong to A ∩ C. If B exists, Some A are C follows; without that existence condition, it does not. The identical words “All + All” therefore do not determine the answer; the position of the common term matters.

5. Some + Some: the two witnesses may differ#

Some A are B. Some B are C. No definite A–C relationship follows. The first witness may be x and the second may be y. They need not be the same member of B.

some-some

Counterexample: A = {x}, B = {x, y}, C = {y}. Both premises are true but Some A are C is false. Another valid model is A = B = C = {x}; there Some A are C is true. Since both types of model are possible, neither Some A are C nor No A is C is individually forced.

Example: Some artists are teachers. Some teachers are runners. Some artists are runners is possible but not definite. No artists are runners is also not definite. Existing conversions still hold: Some teachers are artists and Some runners are teachers.

The rule “two particular premises give no necessary relation between the end terms” is useful for ordinary three-term syllogisms. It does not mean that the original premises or their valid conversions stop being true.

6. No + No: a shared exclusion is not a connection#

No A is B. No B is C. No definite relation follows between A and C. Being separate from the same class does not make two other classes separate from each other.

no-no

Countermodels: A = C = {x}, B = {y} satisfies both premises and allows A–C overlap. A = {x}, B = {y}, C = {z} also satisfies both and makes A and C separate. Thus “No A is C”, “Some A are C” and “All A are C” are not individually guaranteed.

Example: No cats are birds. No birds are dogs. Using only these statements, you cannot infer No cats are dogs. Real-world knowledge about cats and dogs is outside the question.

Safe conversions still apply: No B is A and No C is B. Two negative premises do not establish a necessary connection between the end terms in this pattern.

7. All + Some: locate the existing subgroup#

Pattern 1: All A are B; Some B are C. Some A are C does not follow. The B–C witness may lie outside A.

all-some

Counterexample: A = {x}, B = {x, y}, C = {y}. Both statements hold, yet A and C do not intersect.

Pattern 2: All B are A; Some B are C. Some A are C follows: take the B–C witness; because every B is A, that same witness belongs to A and C. Some C are A also follows.

Pattern 3: All A are B; Some C are A. Some C are B follows. The witness begins inside the smaller class A and travels along A → B.

Equivalent order: Some A are B; All B are C gives Some A are C. Swapping the written order of premises does not alter their logical effect. Swapping the subject and predicate of an All statement does.

8. All + No: exclusion travels to a contained class#

Pattern 1: All A are B; No B is C. Therefore No A is C, and equivalently No C is A.

all-no

Example: All squares are shapes. No shapes are sounds. No squares are sounds follows. “Some squares are not sounds” additionally requires that squares exist under the declared convention.

Pattern 2: No A is B; All C are B. Then No A is C follows. C is inside the class from which A is excluded.

Invalid direction: All A are B; No A is C does not give No B is C. The portion of B outside A may overlap C. Counterexample: A = {x}, B = {x, y}, C = {y}.

Think “exclusion of a whole applies to its parts”; exclusion of a part does not automatically apply to the whole. This explains the valid patterns without memorising a slogan that ignores term positions.

9. Some + No: preserve the witness#

Some A are B. No B is C. Therefore Some A are not C.

some-no

Take the known A–B witness x. Because no B can be C, x is outside C. The premises do not exclude every A from C: other A members may be C. Therefore No A is C does not follow. Nor does Some C are not A follow, because it reverses an O-type statement and may even invent C's existence.

Example: Some musicians are teachers. No teachers are pilots. Some musicians are not pilots follows. No musicians are pilots does not follow.

A reordered valid pattern is No A is B; Some B are C → Some C are not A. Convert the Some statement or trace the same B–C witness. Always preserve which class definitely contains the witness and which class definitely excludes it.

10. Analytical method: arrows, exclusions and witnesses#

Use A → B for All A are B; A ⟂ B for No A is B; x ∈ A,B for Some A are B; x ∈ A and x ∉ B for Some A are not B.

1. Identify the end terms in the proposed conclusion and the middle term connecting the premises.
2. Write every All arrow in its actual direction. Join forward chains only.
3. Write No as a symmetric exclusion: A ⟂ B also means B ⟂ A.
4. Give each Some/Some-not statement its own witness x, y, etc. Do not merge them without a reason.
5. Propagate a witness forward through All arrows and out of classes excluded by No.
6. Test each conclusion separately. If it is not forced, try to construct a countermodel.

Worked chain: Some A are B; All B are C; No C is D. Let x be A and B. B → C puts x in C; C ⟂ D puts x outside D. Thus Some A are C and Some A are not D follow. No A is D does not follow because the argument tracked only a particular A witness.

Negation test: add the opposite of the proposed conclusion to the premises. If no consistent model is possible, the conclusion is necessary. To test All A are C, try to place one A outside C. If every such placement violates a premise, All A are C follows. First check that the original premises are themselves consistent; ordinary exam questions are expected to be consistent.

11. Distribution method: an additional analytical check#

A term is distributed when the statement speaks about every member of that class.

In an ordinary three-term categorical syllogism, the middle term must be distributed at least once. A term distributed in the conclusion must also be distributed in its corresponding premise. Two negative premises cannot establish an end-term conclusion; a negative premise requires a negative conclusion in a valid standard form. Two particular premises cannot establish the usual necessary end-term conclusion.

Example: All A are B; All C are B → All A are C fails because B is undistributed in both premises. Example: All A are B; No A is C → No B is C fails because B is distributed in the conclusion but not in its premise.

These checks diagnose familiar standard forms. Do not use them blindly for nonstandard wording, extra premises, possibility questions or contradictory premises. The all-model/witness test remains the final check, especially where a particular conclusion would require an unstated existence assumption.

12. Some-not patterns and missing combinations#

For the first row, a witness outside B must also lie outside C because C is wholly inside B. For the second, the A witness moves into C while staying outside B. In the fourth row, the B witness could be outside A; never send it backward through A → B.

Some A are not B does not imply Some B are not A. Counterexample: A = {x, y}, B = {y}. The first statement is true, but every B is A. Some-not is the most frequently misconverted statement.

13. Only, only a few, not all and at least#

Only A are B means All B are A: only readers are poets → every poet is a reader. A are only B, when used as “A are exclusively B”, means All A are B. Read the grammatical direction carefully; do not reverse every sentence merely because it contains “only”.

Only a few A are B, in the usual aptitude-test convention, combines Some A are B with Some A are not B. It does not prove Some B are not A. Some A are B alone does not mean “only a few”; all A could be B.

Not all A are B means Some A are not B. It is not equivalent to No A is B. “At least some” asserts one or more. All A are not B is ambiguous in ordinary English; a well-written question should use either “No A is B” or “Not all A are B”. Follow an explicit definition in the paper rather than guessing.

If a question says only a few, use its supplied convention if different. If it says “only A are B” and “Some B exist”, those B witnesses are A; the word “only” alone does not create a witness.

14. Definite, possible and impossible conclusions#

A definite conclusion is true in every valid model. A possible conclusion is true in at least one valid model. An impossible conclusion is false in every valid model. A conclusion that does not follow can be either possible or impossible; does not follow is not the same as false.

From All A are B; Some B are C, Some A are C is possible but not necessary. It is also possible for A and C to be separate. From All A are B; No B is C, Some A are C is impossible.

To test “All A can be C”, try placing all A inside C without violating a premise. Under strict logic an empty A may satisfy a universal; aptitude possibility questions often intend nonempty classes. State or follow the question's convention. With Some A exist and No A is C, All A can be C is impossible. Without an existence assumption, treating that universal possibility as automatically impossible is unsound.

Some exam keys use “possibility” to mean possible but not already definite. Logical possibility in this chapter includes definite truths, because a truth in every valid model is also true in at least one when the premises are consistent. Read the test instructions if its answer categories use a narrower convention.

15. Either-or conclusions and contradictory pairs#

First test conclusions I and II individually. An either-or answer is appropriate when neither is individually forced but exactly one must hold in every valid model. Merely finding two uncertain conclusions is insufficient.

The clean contradictory pairs for the same ordered terms are:

Example: Some A are B; Some B are C. Conclusions: I. Some A are C. II. No A is C. Neither is individually forced, but they are exact opposites; exactly one holds. Therefore either I or II follows under this answer convention.

Beware: Some A are C and Some A are not C may both be true, so they are not an exclusive either-or pair. All A are C and No A is C are not generally exhaustive: with a partly overlapping nonempty A, both are false. If A is empty, both universal claims can be true under strict logic.

Use valid conversion to align term order where needed, especially E and I statements. Do not reverse an O statement to manufacture a contradictory pair. If one conclusion already follows individually, select that definite option rather than replacing it with an unnecessary either-or answer.

16. Solved examination examples#

Example 1 — forward chain. All pens are tools; All tools are objects. I: All pens are objects. II: All objects are pens. Only I follows. A smaller class travels forward through containment; reversal is not justified.

Example 2 — witness chain. Some students are singers; All singers are performers. I: Some students are performers. II: Some performers are students. Both follow. The same existing student is a performer; an I statement converts.

Example 3 — restricted witness. Some workers are readers; No readers are swimmers. I: Some workers are not swimmers. II: No workers are swimmers. Only I follows. There may be other workers who swim.

Example 4 — common container. All roses are flowers; All lilies are flowers. I: Some roses are lilies. II: No roses are lilies. Either I or II, under the exclusive complementary-pair convention. Neither is individually forced, but one must hold. Do not select “both follow”.

Example 5 — only a few. Only a few artists are singers; All singers are performers. I: Some artists are performers. II: Some artists are not performers. Only I follows. The artist outside singers may still be a performer; outside a subset is not necessarily outside its superset.

Example 6 — reverse containment and exclusion. Some A are not B; All C are B. I: Some A are not C. II: Some C are not A. Only I follows. The known A outside B cannot enter its subset C; no C witness outside A is established.

Example 7 — three premises. All A are B; Some C are A; No B is D. I: Some C are not D. II: No A is D. Both follow. Trace the C–A witness for I and the full A → B exclusion chain for II.

Example 8 — existence trap. All A are B; No B is C. I: No A is C. II: Some A are not C. Only I under the declared no-unspoken-existence convention. With an additional premise that A exists, both follow. A good answer identifies the convention before using a shortcut.

17. Practice: twelve questions with explanations#

For questions 1–10, choose: (a) only I; (b) only II; (c) both; (d) neither individually and no exclusive either-or; (e) either I or II, but neither individually. Use the existence convention stated earlier.

1. All A are B; All B are C. I: All A are C. II: All C are A.
2. Some A are B; Some B are C. I: All A are C. II: Some C are not A.
3. No A is B; No B is C. I: No A is C. II: All A are C.
4. All B are A; Some B are C. I: Some A are C. II: Some C are A.
5. All A are B; No B is C. I: No C is A. II: Some C are not A.
6. Some A are B; No B is C. I: Some A are not C. II: No A is C.
7. Some A are not B; All A are C. I: Some C are not B. II: Some B are not C.
8. Only A are B; Some B are C. I: Some A are C. II: All C are A.
9. Only a few A are B; All B are C. I: Some A are C. II: Some A are not C.
10. Some A are B; Some B are C. I: Some A are C. II: No A is C.
11. Some A are B; No B is C. Which is impossible? (a) Some A are C; (b) Some A are not C; (c) All A are C; (d) Some C are not A.
12. All A are B; Some B are C. Which model disproves “Some A are C”? (a) A={x}, B={x,y}, C={y}; (b) A=B=C={x}; (c) A={x}, B={y}, C={x}; (d) A={x}, B={x}, C=empty.

Answers and reasoning#

Try each before reading its answer. For every “does not follow”, produce a model where the premises stay true and that conclusion becomes false. This is more reliable than recognising a familiar-looking option.

18. One-page revision and a student study plan#

Seven mistakes to avoid: reversing All; reversing Some-not; reading Some as “some but not all”; merging different witnesses; enlarging a particular conclusion into a universal; treating one convenient picture as proof; ignoring existence or possibility conventions.

Study plan: Session 1—translate 20 sentences into A/E/I/O and draw the four basic diagrams. Session 2—solve the six combinations with at least one counterexample for every invalid inference. Session 3—use arrows and distribution to explain the same answers. Session 4—complete the practice set without the key and classify errors by cause. Reattempt only missed patterns until you can justify each answer in one sentence.

Before selecting an option, ask: Did I use only the premises? Did I preserve arrow direction? Is this the same witness? Does the conclusion hold in every valid model? Did I test I and II separately before checking either-or? Accuracy comes before speed.

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