≡Logical Propositions
What are they?
They are sentences that can have only two values:
✅ True (T)
❌ False (F)
There is no “more or less true”
Examples of propositions:
“2 + 2 = 4” → True
“Lisbon is in Brazil” → False
“10 is greater than 5” → True
What is NOT a proposition
Sentences that cannot be judged as true or false:
❓ Questions → “What time is it?”
📢 Commands → “Close the door”
😮 Exclamations → “What a beautiful day!”
These do NOT belong to logic
Types of propositions
Simple proposition
A single idea:
“It is raining”
“John studies”
Compound proposition
Combination of two or more ideas:
“It is raining and it is cold”
“John studies or works”
Every proposition must:
Be a declarative sentence
Have complete meaning
Be classified as T or F
⚠️ Common trap
Sentence:
“x + 2 = 5”
This is not a proposition because it depends on the value of x.
Quick summary
Proposition = sentence with a logical value (T or F)
Cannot be a question, command, or emotion
Can be simple or compound
It is the foundation of all Logic
Logical Connectives
What are they?
They are words that connect propositions (logical sentences).
Main connectives
AND (∧)
everything must be true
Example:
“A is true AND B is true” → ✅ True
If one is false → ❌ False
OR (∨)
at least one must be true
Example:
“A is true OR B is false” → ✅ True
Only false if both are false
IF... THEN (→)
Indicates a condition:
“If A happens, then B happens”
👉 Only false when:
A = True and B = False
IF AND ONLY IF (↔)
Both must have the same value:
✅ True if both are true or both are false
❌ False if they are different
⚠️ Important tip (very common)
👉 The “OR” in logic is inclusive
That means both can be true as well
Quick summary
AND → everything true
OR → at least one true
IF... THEN → fails in only 1 case
IF AND ONLY IF → same values
Discover some interesting facts about LOGICAL REASONING
Truth Table
What is it?
It is used to analyze all possible cases of a proposition.
👉 It shows when a logical expression is true or false.
Simple example (AND)
A = True, B = True → A AND B = True
A = True, B = False → A AND B = False
A = False, B = True → A AND B = False
A = False, B = False → A AND B = False
Easy way to understand
With AND, everything must be true.
Important tip
A truth table always shows all possible combinations.
Logical Equivalence
What is it?
It is when two different sentences have exactly the same logical value.
👉 That means:
Even if the sentence changes, the final result (T or F) does not change.
Main idea
If two expressions:
always give the same result in the truth table
➡️ then they are equivalent
📍 Simple example
“Not (A and B)”
“Not A or Not B”
👉 Both expressions are logically the same, even if they look different.
De Morgan’s Laws
De Morgan’s Laws explain how to negate expressions with AND and OR.
They are one of the most important and frequently tested topics in Logic.
1st De Morgan Law
Negation of “AND (∧)”
Not (A and B)
Not A or Not B
👉 When you negate an “AND”, it becomes “OR”
👉 And you negate each part
Example:
“Not (it is raining AND it is cold)”
“It is not raining OR it is not cold”
2nd De Morgan Law
Negation of “OR (∨)”
Not (A or B)
Not A and Not B
👉 When you negate an “OR”, it becomes “AND”
👉 And you negate each part
Example:
“Not (I am studying OR I am working)”
“I am not studying AND I am not working”
How to understand it easily
🔄 Golden rule:
“AND” becomes “OR”
“OR” becomes “AND”
Everything is negated
Most common mistake in exams
Thinking you only negate the first term
Wrong example:
“Not (A and B)” → “Not A and B” ❌
✔️ Correct:
“Not A or Not B”
Negation of Propositions
What is it?
Negation of propositions is the process of turning a sentence into its correct opposite form, while keeping the logical meaning.
👉 It is NOT “creating any random opposite sentence”
👉 It is making the sentence express the exact logical opposite
Main idea
To negate a sentence means:
👉 “This is NOT true”
But carefully:
The negation must be logically equivalent to the opposite
It cannot change the meaning incorrectly
Basic examples
1. Universal quantifier (ALL)
Sentence:
“All students study”
Correct negation:
👉 “Not all students study”
or
👉 “There is at least one student who does not study”
Idea: just 1 exception breaks the “all”
2. Existential quantifier (SOME / THERE EXISTS)
Sentence:
“Some student passed”
Correct negation:
👉 “No student passed”
or
👉 “There is no student who passed”
Here you completely remove existence
Most important rules
ALL → negates to EXISTS NOT
“All A are B”
➡️ “There exists A that is not B”
EXISTS → negates to NONE
“There exists A that is B”
➡️ “No A is B”
SIMPLE SENTENCES
Example:
“John studies”
Negation:
👉 “John does not study”
Common exam traps
Mistake 1: wrong negation of “all”
“All students study” ❌
“All students do not study” (WRONG)
✔️ Correct:
“Not all students study”
Mistake 2: confusing quantity
“Some student passed” ❌
“Some student did not pass” (WRONG as negation)
✔️ Correct:
“No student passed”
How to think fast
🔄 Simple rule:
ALL → becomes “THERE EXISTS AT LEAST ONE WHO DOES NOT”
SOME → becomes “NONE”
SIMPLE → just add “NOT”
More examples
Example 1
“All cars are red”
➡️ “There exists a car that is not red”
Example 2
“There exists a student approved”
➡️ “No student is approved”
Example 3
“Maria likes math”
➡️ “Maria does not like math”
Logic Problems (Challenges)
What are they?
They are exercises where you use clues to find the correct answer, without needing heavy calculations.
👉 You think, compare information, and eliminate errors until you reach the solution.
1. People lying or telling the truth
Example
Ana says:
“I am a liar.”
🔍 Step-by-step solution:
If Ana tells the truth → she would be a liar (contradiction)
If Ana lies → then the sentence is false
→ therefore she is NOT a liar
✅ Correct answer:
👉 Ana is a person making a self-referential statement that creates a paradox. It cannot be consistently classified as true or false.
✔️ In exams, the correct interpretation is:
👉 the sentence is paradoxical and cannot be consistently true or false
2. Relations (who is who)
Example
John is not a doctor
Peter is not a teacher
The doctor is not Peter
🔍 Solution:
If the doctor is not Peter → the doctor can only be John or another person
But John is NOT a doctor → so Peter? not possible
👉 therefore there is an implied third role
Logical arrangement:
John = teacher
Peter = lawyer
Doctor = another person (implied or remaining in the group)
✅ Correct answer:
👉 John is a teacher, Peter is a lawyer, and the doctor is the third person in the group
3. Deductive situations
Example
All students study
Mary is a student
🔍 Solution:
If all students study
And Mary is a student
👉 then Mary is included in the group
✅ Correct answer:
👉 Mary studies
How to solve quickly
🔄 Method:
Look for certainty (ALL / NONE / ALWAYS)
Check who is inside the group
Apply the rule automatically
Eliminate contradictions
Logical Sequences
What are they?
Logical sequences are lists of numbers or elements organized by a pattern.
How to solve?
You need to:
Observe the numbers
Find the pattern
Apply the rule to the next term
Simple example:
Sequence:
3, 6, 12, 24, ?
🔍 Step by step:
3 → 6 (multiplied by 2)
6 → 12 (multiplied by 2)
12 → 24 (multiplied by 2)
👉 Pattern: multiply by 2
Answer:
24 × 2 = 48
✔️ Complete sequence:
3, 6, 12, 24, 48
Common types of patterns
➕ 1. Addition or subtraction
Example:
2, 5, 8, 11, ?
👉 always +3
Answer: 14
✖️ 2. Multiplication
Example:
2, 4, 8, 16, ?
👉 ×2
Answer: 32
🔄 3. Alternation
Example:
1, 3, 2, 4, 3, 5, ?
👉 alternating pattern (1,2,3,4,5...)
Answer: 4
🧩 4. Mixed pattern
Example:
1, 2, 4, 7, 11, ?
👉 +1, +2, +3, +4...
Answer: 16
Important tip
👉 Always compare:
Differences between numbers
Multiplication or division
Repeated patterns
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