Essential Terms

Fundamental logical terminology and concepts

Understanding basic logic vocabulary

What are Essential Terms?

Essential terms form the vocabulary of logic. Just as you need to understand basic mathematical terms like "addition" and "equation" to do math, you need to understand logical terms like "proposition," "conjunction," and "validity" to work with logic effectively.

Core Categories:

• Propositions: Statements that are true or false
• Operators: Symbols that connect propositions (∧, ∨, →, etc.)
• Truth Values: True and False
• Argument Structure: Premises, conclusions, validity

Why Learn These?

Understanding these terms is like learning the alphabet before reading. They provide the foundation for everything else in logic, from simple truth tables to complex proofs and reasoning systems.

Core Logical Terms

Proposition

A statement that is either true or false, but not both. This is the basic unit of logical reasoning.

Examples: "It is raining," "2 + 2 = 4," "The door is open"

Non-examples: Questions ("Is it raining?"), commands ("Close the door"), expressions ("Ouch!")

Premise

A proposition that serves as the starting point or assumption in an argument. Premises provide the foundation for reasoning.

Example: In "All humans are mortal, and Socrates is human," both statements are premises.

Conclusion

A proposition that follows logically from the premises. The conclusion is what we aim to establish or prove.

Example: "Therefore, Socrates is mortal" is the conclusion that follows from the premises above.

Inference Rule

A logical principle that allows us to derive new propositions from existing ones. These rules preserve truth.

Example: Modus Ponens - from "If P then Q" and "P", we can infer "Q".

Argument

A structured sequence of propositions where premises are offered as support for a conclusion.

Structure: Premise₁, Premise₂, ..., Therefore Conclusion

Validity

An argument is valid if the conclusion must be true whenever all premises are true. This is about logical structure.

Key point: Validity is about the connection between premises and conclusion, not about whether premises are actually true.

Soundness

An argument is sound if it is both valid AND all its premises are actually true. This is the gold standard for arguments.

Formula: Sound = Valid + True Premises

Tautology

A proposition that is true under every possible assignment of truth values. No matter what, a tautology is always true.

Example: P ∨ ¬P ("It is raining or it is not raining") is always true regardless of the weather.

Contradiction

A proposition that is false under every possible assignment of truth values. No matter what, a contradiction is always false.

Example: P ∧ ¬P ("It is raining and it is not raining") can never be true.

Logical Operators (Preview)

These symbols combine propositions to create more complex statements. Each will be covered in detail later.

Negation

"not P"

∧

Conjunction

"P and Q"

∨

Disjunction

"P or Q"

→

Implication

"if P then Q"

↔

Biconditional

"P if and only if Q"

⊢

Entailment

"therefore"

⊤

Tautology

"always true"

⊥

Contradiction

"always false"

Putting It All Together

Understanding how these terms work together is crucial for analyzing any logical argument or reasoning process.

The Structure of Logical Reasoning:

Premises are the propositions we start with (assumptions)

Inference Rules are the methods we use to reason

Conclusions are what we derive (the result)

🎯 Practice Tip:

When analyzing any argument, first identify: What are the premises? What inference rules are being used? What conclusion is being drawn? Is the reasoning valid?

Examples & Applications

Example 1: Propositions(Understanding statements)

beginner

Proposition:

"It is raining outside"

Non-Proposition:

"Please close the door" (a command)

Explanation: Propositions must be statements that can be definitively true or false. Commands, questions, and exclamations are not propositions.

Example 2: Logical Operators(Connecting statements)

intermediate

Individual Propositions:

P: "It is sunny" | Q: "It is warm"

Connected with AND:

P ∧ Q: "It is sunny AND it is warm"

Explanation: Logical operators like ∧ (AND), ∨ (OR), and → (IF-THEN) allow us to build complex statements from simple ones.

Example 3: Valid Arguments(Structure vs. content)

advanced

Valid Structure:

If P then Q; P; therefore Q

Example Instance:

If it rains, the ground gets wet; It is raining; Therefore, the ground is wet

Explanation: Validity depends on logical structure, not content. The same valid pattern works regardless of what specific propositions you substitute.

Key Insights

Foundation First: Master these essential terms before diving into complex logical operations. They are the building blocks of all logical reasoning.

Precision Matters: Logic requires precise terminology. The difference between "valid" and "sound," or "proposition" and "sentence," is crucial for clear thinking.

Accuracy

Universal Language: These terms form a universal language for logical discussion, allowing clear communication about reasoning across different fields.

Pattern Recognition: Understanding these terms helps you recognize logical patterns in everyday language, academic arguments, and formal proofs.

Related Concepts

Understanding this concept connects to these important logical concepts: