Logical Deductions notes

MDCAT Analytical and Logical Reasoning

Logical deductions require deciding what must be true from given statements, without adding information from personal knowledge. This chapter covers categorical statements, syllogisms, assumptions, conditional reasoning, comparisons, directions, and fact-based conclusions.

Basic Rules of Logical Deduction

A deduction is a conclusion obtained from the information given in the statements. Treat every statement as true, even if it sounds unusual in real life. Do not use outside knowledge or common assumptions unless the statement supports them.

A conclusion follows only when it is definitely supported. If a conclusion may be true but is not certain, it does not follow. The words all, some, no, only, unless, older, younger, east and west must be read carefully.

All means every member of one group belongs to another group. Example: All pencils are erasers.
Some means at least one, but not necessarily all. Example: Some chocolates are sweets.
No means zero members overlap. Example: No phone is a computer.
A conclusion that is merely possible is not a definite conclusion.
Do not reverse a statement automatically. All roses are flowers does not mean all flowers are roses.
Do not connect two separate some statements unless the statements prove that they refer to the same object.
The conclusion must be based only on the stated information.

Categorical Statements and Set Relationships

Most logical deduction questions use groups or classes. Each group can be imagined as a circle. An all statement places one group inside another. A no statement keeps two groups separate. A some statement shows that at least one member belongs to both groups.

For example, All fruits are juicy places fruits inside juicy things. Some juicy things are sour does not prove that any fruit is sour, because the juicy things mentioned may be different from the fruits.

All A are B means A is a subset of B.
No A is B means A and B have no common member.
Some A are B means at least one object is both A and B.
Some A are not B means at least one A lies outside B.
All A are B and some A exist together prove that some B exist.
All A are B does not prove that some A exist unless existence is stated.
Some A are B can also be read as some B are A.
All A are B and all B are C prove all A are C.

Combining Statements and Drawing Syllogistic Conclusions

To solve a syllogism, link the groups through a common term. Then check whether the conclusion is forced. The direction of the links matters. If all pencils are erasers and some pens are pencils, the same pens must be erasers.

A negative statement can also be carried through a definite chain. If no plastic is biodegradable and some bags are plastic, those particular bags are not biodegradable. However, a statement about one group does not automatically apply to every related group.

Some pens are pencils plus all pencils are erasers gives some pens are erasers.
Some pens are pencils also gives some erasers are pens, because the same objects are involved.
No metal object is fragile plus this key is metal gives this key is not fragile.
No plastic is biodegradable plus some bags are plastic gives some bags are not biodegradable.
All students are intelligent plus some intelligent people are scholars does not prove that some students are scholars.
All students are intelligent does not prove all scholars are intelligent.
All fruits are juicy plus some juicy things are sour does not prove that some fruits are sour.
All toys are plastic plus some plastics are harmful does not prove that some toys are harmful.

Checking Conclusions and Common Errors

A conclusion may fail because the statements do not establish the required connection. In Some candies are chocolates and Some chocolates are sweets, the chocolates in the two statements may be different. Therefore, neither some candies are sweets nor some sweets are not candies is definite.

When a conclusion has the opposite direction, test whether reversal is valid. If some cameras are phones, then some phones are cameras, because the same objects are being described. But All dresses are cloth does not mean all cloth is dresses.

Some cameras are phones proves some phones are cameras.
No phone is a computer and some cameras are phones proves that those cameras are not computers, so no camera is a computer follows for the stated cameras.
All dresses are cloth and no cloth is plastic proves no dress is plastic.
All roses are flowers and some flowers fade quickly does not prove that some roses fade quickly.
All cats are animals and some animals are wild does not prove that some cats are wild.
Some watches are digital and all digital objects are gadgets proves some watches are gadgets, but not all watches are digital.
All metals are solids and some solids are liquids does not prove that some metals are liquids.
If all metals are solids and some solids are liquids, no definite statement about whether the metals are liquids follows.

Negative, Universal, and Existential Conclusions

Words such as no, not, all and some determine the strength of a conclusion. A universal statement about all members cannot normally be obtained from a statement about only some members. Similarly, the existence of a group should not be assumed unless it is stated or forced by another statement.

Some conclusions can be obtained in two forms. If some pens are pencils, then some pens are erasers follows when all pencils are erasers. The same objects are also some erasers that are pens.

Some is weaker than all. Some intelligent people are scholars does not mean all intelligent people are scholars.
All A are B does not mean some A are B unless at least one A is known to exist.
No A is B is equivalent to no B is A.
Some A are B is equivalent to some B are A.
Some A are not B does not mean all A are not B.
If all A are B and some A exist, then some B exist.
A conclusion saying all scholars are intelligent cannot be proved from some intelligent people being scholars.
If two groups are not shown to overlap, do not infer that they overlap.

Assumptions in Statements

An assumption is an unstated idea that must be accepted for a statement, request, warning or argument to make sense. It is not simply a sentence that may be true. Ask whether the speaker must be relying on that idea.

For the message, Please do not wait for me, I may be late, start taking lunch as soon as the guests arrive, the speaker assumes that the managers can start lunch without the speaker. The message does not necessarily assume that the speaker will definitely be late or that the guests will arrive early.

An implicit assumption is necessary for the statement to be meaningful or workable.
A possibility is not automatically an assumption.
The message about starting lunch assumes that lunch may begin in the director's absence.
The same message does not prove that the director will certainly be late.
In a statement saying reform is necessary to remove corruption and prejudice, the relevant assumption may be that the existing administrative system contributes to the problem.
Do not select an assumption merely because it repeats words from the statement.
Test each proposed assumption by asking whether the statement still makes sense without it.

Conditional Statements and Unless

Conditional reasoning connects a condition with a result. In the statement If P, then Q, P is sufficient for Q, while Q is necessary for P. The valid reverse form is not generally available unless it is logically forced by the wording.

Unless means if not. Unless you submit the form, your application will be rejected means that failure to submit causes rejection. If the application was not rejected, the condition that would cause rejection did not occur, so the form was submitted.

If P, then Q means P leads to Q.
P is a sufficient condition for Q.
Q is a necessary condition for P.
If P then Q, and P is true, Q must be true.
If P then Q, and Q is false, P must be false.
If P then Q, and Q is true, P is not necessarily true.
Unless P, Q can usually be read as if not P, then Q.
Unless you submit the form, your application will be rejected plus no rejection proves that you submitted the form.

Comparisons, Order, and Transitive Reasoning

Comparison questions use a chain of relations. If Ali is taller than Bilal and Bilal is taller than Hamza, Ali must be taller than Hamza. The same method applies to age, position, quantity and birth order.

Read more than, less than, older than and younger than in their exact direction. A relationship involving the same three people may be rewritten in an easier order before reaching a conclusion.

If A is taller than B and B is taller than C, A is taller than C.
If A is younger than B, then B is older than A.
Joe is younger than Kathy means Kathy is older than Joe.
Mark was born after Joe means Mark is older than Joe.
Joe younger than Kathy and Mark born after Joe do not alone establish whether Mark is older or younger than Kathy.
If Kathy is older than Mark, then the relationship is consistent with Joe being younger than Kathy and Mark being older than Joe.
A basket with more apples than lemons and more lemons than oranges must have more apples than oranges.
Transitivity works when the comparison direction is consistent.

Directions, Distance, and Position

For direction questions, draw a simple path or use horizontal and vertical displacement. North and south affect the vertical position. East and west affect the horizontal position. Opposite movements on the same line are subtracted.

In the example, Kamran walks 10 km north and then 6 km south. His net vertical movement is 4 km north. He then walks 3 km east. The shortest distance from the starting point is 5 km, obtained from a right triangle with sides 4 km and 3 km. His direction is north-east.

North and south are opposite directions.
East and west are opposite directions.
Opposite movements are subtracted, while movements in the same direction are added.
After 10 km north and 6 km south, the net position is 4 km north.
A 4 km north displacement and a 3 km east displacement give a 5 km resultant distance.
The direction with both north and east displacement is north-east.
For two blocks east of a drugstore, the hotel lies east of the drugstore.
If the market is one block west of the hotel, it may still be east of the drugstore when the hotel is two blocks east of it. Therefore, the drugstore is west of the market.

Fact-Based Inference and True, False, or Uncertain Statements

Some questions provide facts and ask whether a third statement is true, false or uncertain. Use only the first statements. A third statement is true when it must follow, false when it contradicts the facts, and uncertain when both possibilities remain.

A fact should not be strengthened. If pictures can tell a story and all story books have a picture, it does not follow that pictures tell stories better than words. If some story books have words, that exact statement remains a fact.

More apples than lemons and more lemons than oranges proves more apples than oranges.
Joe younger than Kathy, Mark born after Joe, and Kathy older than Mark are mutually consistent, so the stated age order can be true.
If a statement is not proved and is not disproved, it is uncertain.
Pictures can tell a story does not prove that pictures tell a story better than words.
All story books have a picture does not prove that all stories are simple.
Some story books have words remains a fact when it is directly stated.
A conclusion must not add words such as better, always, only or all unless the facts support them.
Use the exact strength of the original fact when judging the result.

Key terms

Deduction: A conclusion obtained logically from given statements.
Statement: A sentence accepted as true for the purpose of reasoning.
Conclusion: A claim tested to see whether it follows from the statements.
Universal statement: A statement about every member of a group, usually using all or no.
Particular statement: A statement about at least one member, usually using some.
Syllogism: A form of reasoning in which statements are combined to produce a conclusion.
Subset: A group whose every member belongs to another larger group.
Implicit assumption: An unstated idea that must be accepted for a statement to make sense.
Conditional statement: A statement expressing that one condition leads to another result.
Sufficient condition: A condition that is enough to guarantee a result.
Necessary condition: A condition that must be present for another statement to occur.
Transitivity: The rule that A greater than B and B greater than C gives A greater than C.
Existential statement: A statement claiming that at least one object exists in a group or relationship.
Contradiction: A situation in which two statements cannot both be true.
Net displacement: The final change in position after opposite movements are cancelled or subtracted.

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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