Propositional symbols and logical connectives are the building blocks of formal logic. They allow us to represent complex statements and arguments using simple symbols, making it easier to analyze their structure and validity.

These tools help us break down complicated ideas into manageable pieces. By using symbols and connectives, we can focus on the logical relationships between statements, rather than getting caught up in the specific content of each claim.

Propositional Symbols and Logical Connectives

Propositional symbols in statements

• Represent statements in propositional logic using lowercase letters (p, q, r)
• Each symbol stands for a distinct statement that is either true or false
• Allow focusing on the logical structure of statements rather than their content
• Enable the abstraction and analysis of arguments based on their form
• Examples:
  • p could represent "The sky is blue"
  • q could represent "Grass is green"

Five basic logical connectives

• Negation (¬ or ~) represents the opposite truth value of a statement
  • If p is true, then ¬p is false, and vice versa
• Conjunction (∧ or &) combines two statements with a logical "and"
  • The resulting statement is true only if both component statements are true
• Disjunction (∨ or |) combines two statements with a logical "or"
  • The resulting statement is true if at least one component statement is true
• Conditional (→ or ⊃) represents an "if-then" relationship between two statements
  • The resulting statement is false only when the antecedent (first statement) is true and the consequent (second statement) is false
• Biconditional (↔ or ≡) represents an "if and only if" relationship between two statements
  • The resulting statement is true when both component statements have the same truth value (both true or both false)

Truth conditions of connectives

• Negation (¬p):
  • True when p is false
  • False when p is true
• Conjunction (p ∧ q):
  • True only when both p and q are true
  • False in all other cases
• Disjunction (p ∨ q):
  • False only when both p and q are false
  • True in all other cases
• Conditional (p → q):
  • False only when p is true and q is false
  • True in all other cases
• Biconditional (p ↔ q):
  • True when p and q have the same truth value (both true or both false)
  • False when p and q have different truth values

Combining symbols with connectives

• Create compound statements by combining propositional symbols with logical connectives
• Build complex statements using multiple symbols and connectives
• Examples:
  • If p represents "It is raining" and q represents "I have an umbrella":
    • p ∧ q means "It is raining and I have an umbrella"
    • p ∨ q means "It is raining or I have an umbrella"
    • p → q means "If it is raining, then I have an umbrella"
    • p ↔ q means "It is raining if and only if I have an umbrella"
• Complex statement example: $(p ∧ q) ∨ (¬p ∧ r)$
  • Represents "Either both p and q are true, or both not-p and r are true"
  • Could model a scenario like "Either I have both a car and a driver's license, or I don't have a car and I use public transportation"