Addition

If P is true, then P or Q is true (for any Q)

P ⊢ P∨Q
Understanding Addition

Addition (also called Disjunction Introduction) is the rule that allows you to expand a true statement by adding alternatives. If you know P is true, then 'P or Q' must also be true, regardless of what Q is. This might seem counterintuitive at first, but it's logically sound.

Why Addition Works:

Since disjunction (OR) is true when at least one disjunct is true, and we know P is true, then P∨Q is automatically true regardless of Q's truth value.

Strategic Use:

While this rule might seem to add unnecessary information, it's strategically important in proofs where you need to create disjunctions for later use with other inference rules.

Symbolic Logic Examples
P ⊢ P ∨ Q (from any P, we can derive P or Q)

Basic Addition

formal
Given:
P
Therefore:
P ∨ Q

Real World Example

demonstration
Fact:
It is raining
Therefore:
It is raining OR it is sunny

Complex Propositions

advanced
Given:
A ∧ B
Addition:
(A ∧ B) ∨ (C → D)

Key Point: Addition allows us to introduce disjunctions - we can always add alternatives to true statements.

Examples & Applications

Example 1: Basic Addition(Simple expansion)

beginner
Known fact:
The meeting is confirmed
Addition result:
Either the meeting is confirmed OR the backup plan is ready

Explanation: Addition allows us to expand our conclusions by introducing new possibilities, which is useful for creating fallback options.

Example 2: System Reliability(Fault tolerance)

intermediate
Primary system:
The main server is operational
Expanded reliability:
Either the main server is operational OR the backup server can handle requests

Explanation: In system design, addition helps establish multiple paths to success, improving overall reliability.

Example 3: Proof Strategy(Logical reasoning)

advanced
Established premise:
P (from previous steps)
Strategic addition:
P ∨ Q (preparing for case analysis)

Explanation: Addition is strategically used in proofs to set up for elimination rules or case analysis by introducing relevant alternatives.

Key Insights
Logical Strengthening: Addition creates logically weaker statements (easier to satisfy) from stronger ones.
Proof Strategy: Often used in proof by cases - if we need to prove P∨Q, we can prove P and apply Addition.
Proof Strategy
Information Preservation: While it adds alternatives, the original true statement remains true and its truth is preserved.
Disjunction Introduction: This is the primary way to introduce new disjunctions into logical reasoning chains.
Foundation for Choices: Essential for modeling scenarios where multiple options lead to acceptable outcomes.
Relationship to Other Rules

Addition (Expanding)

P ⊢ P∨Q

"From a specific fact, create a broader alternative"

Used for introducing disjunctions and expanding possibilities.

Disjunctive Syllogism (Eliminating)

P∨Q, ¬P ⊢ Q

"From alternatives, eliminate one to find the other"

Used for narrowing down from disjunctions to specific conclusions.

When to Use Addition

Proof Strategy

  • • Setting up for Disjunctive Syllogism
  • • Creating proof by cases
  • • Establishing disjunctive premises
  • • Constructive dilemma setup

Practical Reasoning

  • • Creating broader categories
  • • Option enumeration
  • • Fallback planning
  • • Risk assessment

Academic Contexts

  • • Classification tasks
  • • Multiple choice construction
  • • Hypothesis formation
  • • Research methodology

Programming Logic

  • • Boolean OR operations
  • • Condition expansion
  • • Error handling paths
  • • State machine design

Related Concepts

Understanding this concept connects to these important logical concepts: