Syllogisms notes

MDCAT Analytical and Logical Reasoning

Syllogisms are arguments in which one or more statements are used to derive a conclusion. In Analytical and Logical Reasoning, solve them by identifying set relationships, checking whether the conclusion must follow, and avoiding assumptions that are not given.

Basic Idea of a Syllogism

A syllogism contains statements called premises and a statement called the conclusion. The conclusion is valid only when it must be true whenever the given statements are true.

Use the statements exactly as written. Do not use general knowledge. For example, from all cats are mammals and all dogs are mammals, we cannot conclude that all cats are dogs because both groups may be different parts of the mammal group.

Premise: A given statement used as evidence.
Conclusion: A statement tested against the premises.
Valid conclusion: A conclusion that necessarily follows from the premises.
Invalid conclusion: A conclusion that may be true, but is not guaranteed by the premises.
Deductive reasoning moves from general rules to a specific result.
Example: All metals conduct electricity. Copper is a metal. Therefore, copper conducts electricity. This is valid deductive reasoning.

Four Main Forms of Statements

Most syllogism questions use four forms: all, no, some, and some are not. The words all and no describe a complete relationship between two groups. The word some shows that at least one member exists.

Translate each sentence into a set relationship before checking the conclusion. A group can be represented as a circle inside, outside, or partly overlapping another group.

All A are B means every member of A is included in B. A is a subset of B.
No A is B means A and B do not overlap at all.
Some A are B means at least one object belongs to both A and B.
Some A are not B means at least one object belongs to A but does not belong to B.
All A are B does not mean all B are A.
No A is B is equivalent to no B is A. For example, no fish is a crocodile and no crocodile is a fish have the same meaning.

The Chain Rule for All Statements

A chain is formed when the first group is included in a second group and the second group is included in a third group. The first group is then included in the third group.

This rule explains conclusions such as all roses are plants. If all roses are flowers and all flowers are plants, every rose must be a plant.

All A are B plus all B are C gives all A are C.
All fish are tortoises plus no tortoise is a crocodile gives no fish is a crocodile.
All medicines are drugs plus no drug is a food gives no medicine is a food.
All daffodils are plants plus all plants are living things gives all daffodils are living things.
All directors are film stars plus all film stars are playback singers gives all directors are playback singers.
All cats are mammals plus all dogs are mammals does not give all cats are dogs. Sharing a common group does not make two groups identical.

Using Some Statements Correctly

A statement containing some gives evidence about at least one actual member. This member can be carried through a valid all relationship.

For example, some birds are parrots and all parrots are green. The birds that are parrots must be green, so some birds are green.

Some A are B plus all B are C gives some A are C.
Some bags are pouches plus all pouches are cases gives some bags are cases.
Some careless people are drivers plus no student is careless gives some drivers are not students.
Some honest people are rich plus no politician is honest gives some rich people are not politicians.
Some students are hardworking plus all hardworking people succeed does not give all students succeed. Only the hardworking students are known to succeed.
Some fluids are gases plus all liquids are fluids does not give some liquids are gases. The fluids that are gases may not be liquids.

Rules for No Statements

A no statement means that two groups cannot share a member. It can be used with an all statement to exclude a smaller group from another group.

The direction can be reversed without changing the meaning. If no tortoise is a crocodile, then no crocodile is a tortoise. This allows both forms of the same conclusion in suitable questions.

All A are B plus no B are C gives no A are C.
All fish are tortoises plus no tortoise is a crocodile gives both no crocodile is a fish and no fish is a crocodile.
No politician is honest plus some honest people are rich gives some rich people are not politicians.
No student is careless plus some careless people are drivers gives some drivers are not students.
No cities are countries and no countries are villages do not show any relationship between cities and villages.
From no cities are countries, we cannot conclude that some countries are cities. The first statement actually denies overlap.

Existence and the Word Some

A universal statement such as all A are B describes a relationship but does not normally tell us that A exists. In many traditional aptitude syllogism answer keys, universal categories are treated as existing when a conclusion uses some. This convention explains answers such as all P are Q, all Q are R, therefore some R are P.

For examination questions, follow the stated convention in the answer choices. Still, do not create a new object without a stated or accepted existence basis. A conclusion must remain compatible with every premise.

Some means at least one, not all.
All A are B does not normally mean some A are B in strict modern logic.
In the supplied exam convention, all P are Q and all Q are R may support some R are P.
All squares are quadrilaterals may be treated as supporting some quadrilaterals are squares under the same convention.
Some quadrilaterals are not squares does not itself prove that some quadrilaterals are squares in strict logic, but the supplied answer treats the latter as valid.
A conclusion claiming existence is unsafe when no member of the relevant group is given and the examination convention does not require existence.

Necessary Conclusions and Common Traps

A conclusion may sound reasonable without being necessary. Test it by asking whether the premises allow a situation in which the conclusion is false. If such a situation is possible, the conclusion is invalid.

Do not reverse a relationship. The statement all watches are alarms is different from all alarms are watches. Also, do not confuse a group being inside another group with the two groups being equal.

All clocks are watches and some clocks are alarms gives some alarms are watches.
The same premises do not give all watches are alarms because watches may exist that are not alarms.
All cats are mammals and all dogs are mammals does not give all cats are dogs.
All surgeons are doctors and some doctors are researchers gives no definite conclusion about surgeons and researchers.
All A are B and no B are C makes some C are A invalid because A must lie inside B, while B cannot overlap C.
All A are B does not give some B are A unless existence of A is established or accepted by the stated convention.

Handling Several Conclusions

When more than one conclusion is listed, check each one independently. A conclusion can be true while another conclusion from the same premises is unsupported.

Some questions use choices such as only I, only II, both I and II, or neither. Select the combination that contains only conclusions that necessarily follow.

All fish are tortoises and no tortoise is a crocodile: both no crocodile is a fish and no fish is a crocodile follow.
All daffodils are plants, all plants are living things, and some living things are real: only all daffodils are living things follows.
Some bags are pouches, all pouches are cases, and no cases are purses: some bags are cases follows; some pouches are purses and no bags are purses do not follow.
No rabbit is a lion, some horses are lions, and all rabbits are tables: none of some tables are lions, some horses are rabbits, and no lion is a table is forced.
No cities are countries and no countries are villages: neither some countries are cities nor no villages are cities follows.
A conclusion must be supported by a complete chain or a direct overlap or exclusion.

Special Words and Direction of Meaning

Words such as only, can apply, and successfully applied change the logical direction of a statement. Read them carefully before drawing a conclusion.

Only graduates can apply means that being a graduate is necessary for applying. If Sana applied successfully, Sana must be a graduate.

Only A can be B means all B are A, not all A are B.
Only graduates can apply means all applicants are graduates.
Sana applied successfully plus only graduates can apply gives Sana is a graduate.
If all A are B, then every A has property B, but a B may or may not be an A.
No A is B and no B is A are equivalent statements.
A necessary condition must be present whenever the result occurs. A sufficient condition guarantees the result when it is present.
Do not change only graduates can apply into all graduates can apply. The latter is not stated.

A Reliable Method for Solving Questions

First, identify the important groups and mark the relationship between them. Then carry a known member through the relationships. Finally, reject conclusions that reverse a statement, introduce an unsupported overlap, or claim existence without a valid basis.

For diagram questions, place a smaller group inside a larger group for all, keep groups separate for no, and overlap groups for some. Check every conclusion against the diagram.

Step 1: Underline words such as all, no, some, not, and only.
Step 2: Convert each statement into an inclusion, exclusion, or overlap.
Step 3: Join chains such as A inside B inside C.
Step 4: Carry some members through an all relationship.
Step 5: Use no relationships to prove that a group is outside another group.
Step 6: Test each conclusion separately.
Step 7: Reject reverse conclusions and unrelated group connections.
Step 8: Choose only the conclusions that are definitely supported by the accepted logic convention.

Key terms

Syllogism: An argument containing premises from which a conclusion is derived.
Premise: A statement given as the basis for reasoning.
Conclusion: A statement that is tested to see whether it follows from the premises.
Valid conclusion: A conclusion that must be true if the premises are true.
Invalid conclusion: A conclusion that is not guaranteed by the premises.
Deductive reasoning: Reasoning that applies a general rule to reach a specific conclusion.
Universal statement: A statement about every member of a group, usually using all or no.
Particular statement: A statement about at least one member of a group, usually using some.
Subset: A group whose every member is included in another group.
Overlap: A relationship in which two groups share at least one member.
Disjoint groups: Groups that have no members in common.
Existence: The presence of at least one actual member of a group.
Necessary condition: A condition that must be present for an event or action to occur.
Sufficient condition: A condition that is enough to guarantee an event or result.
Existential conclusion: A conclusion that claims that at least one member of a group exists.
Chain rule: The rule that A inside B and B inside C places A inside C.

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Analytical and Logical Reasoning shortcuts

Choosing the stronger supporting argument

Choose the option that directly explains why the claim is true. Prefer relevant evidence with a clear link to the claim, not a statement that is merely related.

Identify the claim and ask, “Why would this make the claim true?”
Reject personal opinions, unrelated facts and extreme statements.
Example: “Teachers should be paid more” is supported by “They shape future generations.”

This shortcut does not apply when the option gives direct statistical or experimental evidence that is stronger than a general reason.

Finding the pattern in letter and number series

Check letters and numbers separately. Look for equal increases, alternating changes, multiplication, or a repeating pattern.

Convert letters to positions if needed, such as A = 1 and D = 4.
Find the change in the letters and in the numbers independently.
Example: A2, D4, G6, J8 has letters increasing by 3 and numbers by 2, so the answer is M10.

Do not assume a simple addition pattern if alternating or multiplication patterns fit all the given terms.

15 more Analytical and Logical Reasoning shortcuts are in the MDCAT pack. Already have it? See all shortcuts