1. What is a Proposition in Logic? Complete Guide

In this comprehensive guide, you will learn what a proposition in Logic is and everything you need to know about this fundamental concept in mathematical logic.

You will discover what a logical proposition is, how to identify one, which expressions do NOT qualify as propositions, how to represent them symbolically, and you will test your knowledge with practical exercises.

What is a Proposition in Logic?

A logical proposition (or simply proposition) is a declarative sentence that has a single truth value: it can be either true (T) or false (F), but never both at the same time.

Fundamental Characteristics

1. It is a declarative sentence: It affirms or denies something about the world.
2. It has a defined truth value: It is always T or F, without exceptions.
3. It is objective: Its truth or falsity does not depend on personal opinions.

Examples of Propositions

Proposition	Truth Value
Washington D.C. is the capital of the United States	T
\( 2 + 3 = 7 \)	F
Water boils at 100°C at sea level	T
The Moon is larger than the Earth	F
\( \sqrt{16} = 4 \)	T
Brazil is in Europe	F

Notice that each statement can be clearly classified as true or false. That is the essence of a proposition.

What is NOT a Proposition?

Not all sentences are propositions. For a statement to be considered a logical proposition, it must be possible to assign it a truth value. The following expressions are NOT propositions:

Questions (Interrogative Sentences)

Questions do not assert anything; they only request information.

• How old are you?
• What time is it?
• Will you come to class tomorrow?

Commands and Imperatives

Commands express desires for someone to do something; they do not assert facts.

• Close the door!
• Study for the exam.
• Please, sit down.

Exclamations

Exclamations express emotions, not verifiable facts.

• What a beautiful day!
• Incredible!
• Happy birthday!

Wishes and Pleas

Wishes express hopes, not assertions.

• I hope I pass the course.
• I hope it doesn’t rain tomorrow.
• I wish I could travel.

Open Sentences (With Undefined Variables)

An expression with undefined variables has no truth value until a specific value is assigned.

• \( x + 5 = 10 \) (What is the value of \( x \)?)
• He is an engineering student (Who is “he”?)
• \( y > 3 \) (depends on the value of \( y \))

Important Note: If we assign a value to the variable, the open sentence becomes a proposition. For example, if \( x = 5 \), then “\( x + 5 = 10 \)” becomes “\( 5 + 5 = 10 \)”, which is a true proposition.

Terminology: Open sentences are also known as open statements or propositional functions. This last term is widely used in first-order logic (or predicate logic), a topic we will cover in future posts.

Paradoxes

Paradoxes generate logical contradictions and cannot have a consistent truth value.

• “This sentence is a lie”
• “I always lie”

If the sentence “This sentence is a lie” were true, then it would be lying, which would make it false. But if it were false, then it wouldn’t be lying, which would make it true. This contradiction prevents assigning a truth value.

Truth Value

The truth value is the quality that indicates whether a proposition is true or false.

Notation

The truth value is represented in different equivalent ways:

True	False
T	F
1	0

Examples of Determining Truth Value

Determine the truth value of the following propositions:

1. “The number 7 is prime”
   • Analyzing: Prime numbers are only divisible by 1 and themselves. 7 meets this condition.
   • Truth value: T
2. “\( 3^2 = 6 \)”
   • Analyzing: \( 3^2 = 3 \times 3 = 9 \neq 6 \)
   • Truth value: F
3. “Mount Everest is the tallest mountain in the world”
   • Analyzing: Mount Everest stands at 8,848 meters, making it the tallest mountain on Earth when measured from sea level.
   • Truth value: T
4. “\( \pi \approx 3.1415 \)”
   • Analyzing: \( \pi \approx 3.14159265… \) is an irrational number. The value 3.1415 is only an approximation, not the exact value.
   • Truth value: T (if we’re talking about the approximate value)
   • Note: If \( \pi = 3.1415 \), then it’s false because \( 3.1415 \neq 3.14159265… \)

Fundamental Logical Principles

Classical logic is based on three principles that every proposition must satisfy:

Principle of Identity

Every proposition is identical to itself.

In symbols: \( p \equiv p \) (read: “p is equivalent to p”)

If we affirm that “a circle is a circle,” it’s because it cannot be a square or a triangle, as that would be illogical.

Principle of Non-Contradiction

A proposition cannot be true and false at the same time.

In symbols: \( \neg(p \land \neg p) \) (read: “it is not the case that p and not-p”)

It is impossible for “\( 2 + 2 = 4 \)” to be true and false simultaneously.

Principle of the Excluded Middle

Every proposition is either true or false; there is no third possibility.

In symbols: \( p \lor \neg p \) (read: “p or not-p”)

A proposition like “Mars has water” is either true or false. There is no intermediate state.

Note: The symbols \( \neg \), \( \land \), and \( \lor \) are logical connectives that we will study in detail in the next post.

Are These Principles Axioms?

An axiom is a proposition that is accepted as true without the need for proof. It serves as a starting point for building a logical or mathematical system.

In classical logic, these three principles function as axioms or fundamental truths that are accepted without proof. They are the foundation of deductive reasoning and can be verified as tautologies (always true formulas) using truth tables.

However, there are non-classical logics where some of these principles are not accepted:

• Intuitionistic Logic: Rejects the Excluded Middle (does not accept \( p \lor \neg p \) as an axiom)
• Paraconsistent Logic: Allows controlled contradictions
• Fuzzy Logic: Admits intermediate truth values between 0 and 1

These alternative systems have applications in artificial intelligence, quantum computing, and philosophy. We will study them in future posts.

Propositional Variables

To work more efficiently with propositions, we use propositional variables (also called sentential variables or propositional letters): lowercase letters that represent complete propositions.

Standard Notation

The letters used are: \( p, q, r, s, t, u, v, w… \)

Subscripts can also be used when more variables are needed: \( p_1, p_2, p_3… \)

Examples of Assignment with Truth Value

Variable	Proposition	Truth Value
\( p \)	“The triangle has three sides”	T
\( q \)	“Paris is in Germany”	F
\( r \)	“\( 5 > 3 \)”	T

By assigning letters to propositions, we can refer to them in abbreviated form. This will be especially useful when working with compound propositions and truth tables.

Advantages of Using Propositional Variables

1. Simplicity: It’s easier to write \( p \) than “The triangle has three sides”
2. Generalization: Allows working with logical formulas regardless of specific content
3. Algebraic manipulation: Facilitates applying logical rules
4. Abstraction: Allows studying logical structure independent of content

Classification of Propositions

Propositions are classified into two types according to their structure:

Simple Propositions (Atomic)

These are propositions that do not contain logical connectives and express a single idea. They cannot be broken down into smaller propositions.

Examples:

• “The sun rises in the east”
• “Mary studies medicine”
• “\( 5 > 3 \)”
• “Water is liquid at room temperature”

Compound Propositions (Molecular)

These are propositions that contain one or more logical connectives and are formed by two or more simple propositions.

Examples:

• “It rains and it’s cold” (two propositions joined by “and”)
• “I study or I work” (two propositions joined by “or”)
• “If I pass the exam, then I will celebrate” (conditional structure)
• “It is not true that the Earth is flat” (negation)

Introduction to Logical Connectives

Logical connectives (also called logical operators) are symbols that allow combining simple propositions to form compound propositions. Below is a brief introduction to each one:

Negation (\( \neg \))

Negation reverses the truth value of a proposition. If \( p \) is true, then \( \neg p \) is false, and vice versa.

Structure: \( \neg p \)

Conjunction (\( \land \))

Conjunction joins two propositions with “and.” It is true only when both propositions are true.

Structure: \( p \land q \)

Disjunction (\( \lor \))

Disjunction joins two propositions with “or.” It is true when at least one of the propositions is true.

Structure: \( p \lor q \)

Conditional (\( \rightarrow \))

The conditional establishes an implication relationship: “if… then…” It is false only when the antecedent is true and the consequent is false.

Structure: \( p \rightarrow q \), where \( p \) is the antecedent and \( q \) is the consequent.

Biconditional (\( \leftrightarrow \))

The biconditional indicates equivalence: “if and only if.” It is true when both propositions have the same truth value.

Structure: \( p \leftrightarrow q \)

Note: This is just an introduction to logical connectives. In the next post, we will study each connective in depth, including their complete truth tables, properties, and advanced examples.

Why Study Logical Propositions?

Now that you understand the theory, let’s see how this applies in practice:

Programming and Computer Science

Propositions are the foundation of Boolean logic, the heart of all programming. Every if-else condition is a proposition:

if (age >= 18 AND hasDocument == true) {
    allowAccess();
}

Here: \( p \): “age >= 18”, \( q \): “hasDocument == true” → The condition is: \( p \land q \)

What does this mean? If you have no programming experience, let me explain:

• if-else means “if… then… otherwise…” It’s how computers make decisions.
• AND is the same as the logical connective conjunction (\( \land \)) we saw earlier.
• The code says: “If age is greater than or equal to 18 and has a document, then allow access.”

In other words: only if both propositions are true (\( p \land q = T \)), the action is executed. It’s exactly logical conjunction in action!

Artificial Intelligence

AI systems use propositional logic for expert systems, medical diagnosis, and software verification.

Digital Circuits

Logic gates (AND, OR, NOT) work exactly like logical connectives. Your computer processes information using these principles.

Mathematics

Fundamental for building rigorous proofs, understanding theorems, and working with sets.

Critical Thinking

Helps you identify fallacies, construct valid arguments, and detect false information.

Databases (SQL)

Queries directly use propositional logic:

SELECT * FROM students WHERE average > 14 AND attendance >= 80;

What does this mean? SQL is the language for querying databases. This code says:

• SELECT * FROM students → “Select all data from the students table”
• WHERE → “Where the following conditions are met”
• average > 14 AND attendance >= 80 → \( p = \text{average} > 14 \), \( q = \text{attendance} \geq 80 \): this is the same as \( p \land q \)

In plain English: “Give me all students whose average is greater than 14 and whose attendance is at least 80%.” Only students where both conditions are true appear.

In summary: Propositions are not just theory. They are tools we use in technology, science, law, and everyday life.

Limitations of Propositional Logic

Although propositional logic is fundamental, it has important limitations you should know:

What Propositional Logic CANNOT Do:

Limitation	Example	Hidden Detail	Solution
Does not analyze internal structure	“Socrates is mortal” is just “p”	Subject: Socrates, Predicate: mortal → Mortal(Socrates)	Predicate Logic
Cannot express quantifiers	“All humans are mortal”	Quantifier: ∀ (for all) → ∀x: Human(x) → Mortal(x)	Predicate Logic (∀, ∃)
Cannot represent relations	“John is taller than Peter”	Relation: Taller(John, Peter) — connects two individuals	Predicate Logic
Cannot handle uncertainty	“It will probably rain tomorrow”	Degree of truth: 0.7 (70% probable), not just T or F	Fuzzy Logic
Cannot reason about time	“Tomorrow it will be true”	Temporal operator: F (future) → Fp (“p will be true in the future”)	Temporal Logic
Cannot express possibility	“It’s possible that it rains”	Modal operator: ◇ (possibility) → ◇p (“it’s possible that p”)	Modal Logic
Does not require relation between propositions	“If 2+2=4, then she is beautiful”	Material implication: Only evaluates T or F, not meaning or causality between p and q	Relevant Logic

When to Use Each Type of Logic?

Need	Type of Logic
Basic connectives (and, or, not, if-then)	Propositional
Quantifiers (all, some, none)	Predicate
Degrees of truth (0.7, 0.3)	Fuzzy
Possibility and necessity	Modal
Logical relevance between premises and conclusion	Relevant

Note: We will study predicate logic (or first-order logic) in future posts, where you will learn to overcome these limitations.

Practical Exercises

Test your understanding with the following exercises.

Section A: Is It or Isn’t It a Proposition?

Indicate whether each statement is a proposition (P) or not a proposition (NP). Briefly justify.

1. The number 15 is divisible by 3.
2. What is your favorite color?
3. \( x^2 – 4 = 0 \)
4. Congratulations on your achievement!
5. The sum of the internal angles of a triangle is 180°.
6. I hope tomorrow is a good day.
7. Please close the window.
8. \( 2 + 2 = 5 \)
9. “This sentence has five words”
10. London is in England.

Section B: Determine the Truth Value

For each proposition, determine if it is true (T) or false (F).

1. \( \sqrt{25} = 5 \)
2. The square of a negative number is negative.
3. Peru has a coast on the Pacific Ocean.
4. \( 7 \times 8 = 54 \)
5. The number 1 is prime.
6. All mammals are vertebrates.
7. \( (-3)^2 = -9 \)
8. An equilateral triangle has all its sides equal.

Section C: Identify the Propositional Variables

Assign propositional variables (\( p \), \( q \), \( r \)…) to the simple propositions in the following text:

“If you study mathematics and practice exercises, then you will pass the exam. Additionally, if you pass the exam, you can enroll in the next course.”

Section D: Classify as Simple or Compound

Indicate whether each proposition is Simple (S) or Compound (C). If compound, identify the simple propositions that compose it.

1. Water is a chemical compound.
2. It rains or it’s sunny.
3. If I have money, then I will travel.
4. The Moon orbits the Earth.
5. It is not true that 2 + 2 = 5.
6. I study and work at the same time.

Answers to the Exercises

Section A

1. P – It is an assertion that can be T or F (in this case, T)
2. NP – It is a question
3. NP – It is an open sentence (depends on the value of \( x \))
4. NP – It is an exclamation
5. P – It is a verifiable assertion (T)
6. NP – It is a wish
7. NP – It is a command
8. P – It is an assertion that can be T or F (in this case, F)
9. P – It is true: “This sentence has five words” actually has 5 words
10. P – It is a verifiable assertion (T)

Section B

1. T – \( \sqrt{25} = 5 \) is correct
2. F – The square of any real number is non-negative. \( (-3)^2 = 9 > 0 \)
3. T – Peru has a coast on the Pacific
4. F – \( 7 \times 8 = 56 \), not 54
5. F – The number 1 is not considered prime by definition
6. T – All mammals are vertebrates
7. F – \( (-3)^2 = (-3) \times (-3) = 9 \), not -9
8. T – By definition, an equilateral triangle has all three sides equal

Section C

• \( p \): “You study mathematics”
• \( q \): “You practice exercises”
• \( r \): “You will pass the exam”
• \( s \): “You can enroll in the next course”

The structure would be: \( (p \land q) \rightarrow r \) and \( r \rightarrow s \)

Section D

1. S – It is a single assertion
2. C – Composed of: “It rains” (p) and “it’s sunny” (q), joined by “or”
3. C – Composed of: “I have money” (p) and “I will travel” (q), with conditional structure
4. S – It is a single assertion
5. C – It is the negation of “2 + 2 = 5”
6. C – Composed of: “I study” (p) and “I work at the same time” (q), joined by “and”

Summary

In this post, we have learned the fundamental concepts about logical propositions:

Concept	Description
Proposition	Declarative statement with truth value T or F
Truth value	Quality of being true (T/1) or false (F/0)
Propositional variables	Letters (\( p, q, r… \)) that represent propositions
Simple proposition	Does not contain logical connectives
Compound proposition	Contains one or more logical connectives

The Three Fundamental Principles

1. Identity: \( p \equiv p \)
2. Non-contradiction: \( \neg(p \land \neg p) \)
3. Excluded middle: \( p \lor \neg p \)

These principles function as axioms in classical logic.

The five logical connectives (introduction)

Connective	Symbol	Example
Negation	\( \neg \)	“It is not raining”
Conjunction	\( \land \)	“It is raining and it is cold”
Disjunction	\( \lor \)	“It is raining or it is sunny”
Conditional	\( \rightarrow \)	“If it rains, then I get wet”
Biconditional	\( \leftrightarrow \)	“I pass if and only if I study”

Limitations of Propositional Logic

Propositional logic cannot express quantifiers (“all,” “some”), relationships between objects, uncertainty, or time. For that, there exist predicate logic, fuzzy logic, modal logic, and temporal logic.

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What’s Next?

In the next post about logical connectives you will learn in detail: negation, conjunction, inclusive disjunction, conditional, biconditional, and exclusive disjunction. We will learn to construct truth tables and evaluate compound propositions.

Did you find this post useful? Leave me a comment with your questions or suggestions! And don’t forget to check the next installment of this series on mathematical logic.