Logic: Definitions and Importance

Logic: Definitions and Importance

• Logic is the study of reasoning and argumentation.
• It structures valid arguments and differentiates correct from incorrect reasoning.
• It's essential in mathematics, computer science, philosophy, and everyday decision-making.

Statements and Propositions

• A statement (proposition) is either true or false, but not both.
• It's a declarative sentence.
• Examples: "2 + 2 = 4" (True) and "The sun is blue" (False).
• Non-statements include questions, commands, and opinions (e.g., "How are you?").

Logical Connectives

• Logical connectives combine or modify statements.

Types of Logical Connectives

• Negation (¬p): The opposite of a statement. If p is true, ¬p is false.
• Conjunction (p ∧ q): "And" statement. True only when both p and q are true.
• Disjunction (p ∨ q): "Or" statement. True if at least one of p or q is true.
• Implication (p → q): "If p, then q." False only if p is true and q is false.
• Biconditional (p ↔ q): "p if and only if q." True when both p and q have the same truth value.
• Exclusive Or (p ⊕ q): True if and only if p and q have different truth values.
• NAND (p | q): True unless both p and q are true.
• NOR (p ↓ q): True only if both p and q are false.
• Conditional (p => q): False only when p is true and q is false, True if p is false.

Truth Tables and Reasoning

• Truth tables determine the truth value of compound statements.
• Example for p → q:
  • p | q | p → q
  • T | T | T
  • T | F | F
  • F | T | T
  • F | F | T

Types of Reasoning

• Deductive Reasoning: Moves from general to specific; conclusions logically follow premises.
• Example: "All men are mortal; Socrates is a man; therefore, Socrates is mortal."
• Inductive Reasoning: Moves from specific to general; conclusions based on patterns or observations.
• Example: "The sun has risen every day, so it will rise tomorrow."

Logical Equivalence and Proofs

• Logical Equivalence: Two statements always have the same truth value.

• Denoted as p ≡ q ("p is logically equivalent to q").

• Statements are equivalent if their truth tables match.

• Methods to Determine Logical Equivalence:

  • Construct truth tables for both statements.
  • Compare columns. If they match, the statements are equivalent.

• Types of Statements:

  • Tautology: Always true.
  • Contradiction: Always false.
  • Contingency: Neither always true nor always false.

• Proof Techniques:

  • Converse: q → p
  • Inverse: ~p → ~q
  • Contraposition: ~q → ~p
  • Conditional Proof: Deduce the consequent from an additional premise.
  • Indirect Proof: Reduce to absurdity by negating the conclusion.

Fallacies

• Fallacies are errors in reasoning, leading to invalid arguments.

• Types of Fallacies:

  • Affirming the Consequent: p → q, q therefore p
  • Denying the Antecedent: p → q, ~p therefore ~q

Arguments

• An argument is a series of statements that persuade.
• Deduce conclusions to establish validity.