2.1 Propositional logic

Propositional logic is the foundation of mathematical reasoning and formal logic. It provides a systematic way to analyze and evaluate the truth values of complex statements, enabling mathematicians to construct valid arguments and proofs.

This topic covers the basic elements of propositional logic, including propositions, logical connectives, and truth tables. It also explores logical operators, laws, and properties, as well as argument forms, proof techniques, and practical applications in fields like circuit design and computer programming.

Basic elements of propositional logic

• Propositional logic forms the foundation for mathematical reasoning and formal logic
• Provides a systematic way to analyze and evaluate the truth values of complex statements
• Enables mathematicians to construct valid arguments and proofs

Propositions and statements

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• Declarative sentences that can be either true or false
• Atomic propositions represent simple, indivisible statements
• Compound propositions combine multiple atomic propositions using logical connectives
• Denoted by lowercase letters (p, q, r) in formal notation

Logical connectives

• Symbols or words used to join simple propositions into compound propositions
• Include AND (∧), OR (∨), NOT (¬), IF-THEN (→), and IF AND ONLY IF (↔)
• Allow for the creation of more complex logical expressions
• Determine the truth value of compound propositions based on their components

Truth tables

• Visual representations of all possible truth values for a logical expression
• Rows represent different combinations of truth values for atomic propositions
• Columns show the resulting truth values for each logical connective
• Used to evaluate the truth of complex propositions and determine logical equivalence

Logical equivalence

• Two propositions are logically equivalent if they have the same truth values for all possible inputs
• Denoted by the symbol ≡ (three horizontal lines)
• Determined by comparing truth tables or using logical laws and properties
• Allows for simplification and transformation of logical expressions

Logical operators

• Fundamental tools for constructing and manipulating logical expressions in propositional logic
• Enable the creation of complex statements from simpler ones
• Form the basis for more advanced logical reasoning and proof techniques

Negation and conjunction

• Negation (¬) reverses the truth value of a proposition
  • ¬p is true when p is false, and false when p is true
  • Used to express the opposite of a given statement
• Conjunction (∧) represents the logical AND operation
  • p ∧ q is true only when both p and q are true
  • Used to combine two or more propositions that must all be true simultaneously

Disjunction and exclusive or

• Disjunction (∨) represents the logical OR operation
  • p ∨ q is true when at least one of p or q is true
  • Used to express alternatives or possibilities
• Exclusive OR (⊕) is true when exactly one of its operands is true
  • p ⊕ q is true when p and q have different truth values
  • Useful in situations where only one option can be chosen

Conditional and biconditional

• Conditional (→) represents implication or "if-then" statements
  • p → q is false only when p is true and q is false
  • Used to express logical consequences or dependencies
• Biconditional (↔) represents "if and only if" statements
  • p ↔ q is true when p and q have the same truth value
  • Used to express necessary and sufficient conditions

Logical laws and properties

• Fundamental principles that govern the behavior of logical expressions
• Allow for simplification, transformation, and manipulation of logical statements
• Essential for constructing valid arguments and proofs in mathematics

Commutative and associative laws

• Commutative law states that the order of operands does not affect the result
  • p ∧ q ≡ q ∧ p and p ∨ q ≡ q ∨ p
  • Applies to conjunction and disjunction operations
• Associative law allows regrouping of operands without changing the result
  • (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) and (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
  • Enables flexible manipulation of complex expressions

Distributive and De Morgan's laws

• Distributive law relates conjunction and disjunction operations
  • p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) and p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
  • Allows for expansion or factoring of logical expressions
• De Morgan's laws describe the relationship between negation and conjunction/disjunction
  • ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q
  • Used to simplify negations of complex statements

Law of excluded middle

• States that for any proposition p, either p or its negation (¬p) must be true
• Expressed as p ∨ ¬p, which is always true (a tautology)
• Forms the basis for proof by contradiction and other logical reasoning techniques
• Controversial in some branches of mathematics (intuitionistic logic)

Argument forms and validity

• Logical structures used to analyze and evaluate the validity of arguments
• Essential for constructing sound mathematical proofs and reasoning
• Help identify common patterns of valid and invalid arguments

Modus ponens and modus tollens

• Modus ponens (affirming the antecedent)
  • If p → q and p is true, then q must be true
  • Structure: ((p → q) ∧ p) → q
  • Used in direct proofs and logical deductions
• Modus tollens (denying the consequent)
  • If p → q and q is false, then p must be false
  • Structure: ((p → q) ∧ ¬q) → ¬p
  • Often used in proofs by contradiction

Hypothetical syllogism

• Combines two conditional statements to form a new conditional
• If p → q and q → r, then p → r
• Structure: ((p → q) ∧ (q → r)) → (p → r)
• Allows for chaining of logical implications in complex arguments

Disjunctive syllogism

• Based on the principle of elimination
• If p ∨ q is true and p is false, then q must be true
• Structure: ((p ∨ q) ∧ ¬p) → q
• Used in proofs by cases and logical deductions

Proof techniques

• Systematic methods for demonstrating the truth of mathematical statements
• Essential skills for advanced mathematical reasoning and problem-solving
• Rely on the principles and rules of propositional logic

Direct proof

• Assumes the hypothesis and uses logical deduction to reach the conclusion
• Follows a step-by-step reasoning process based on known facts and definitions
• Often employs modus ponens and other valid argument forms
• Effective for proving straightforward implications and equalities

Proof by contradiction

• Assumes the negation of the statement to be proved
• Derives a logical contradiction from this assumption
• Concludes that the original statement must be true
• Utilizes the law of excluded middle and modus tollens
• Powerful technique for proving statements indirectly

Proof by cases

• Breaks down the problem into exhaustive, mutually exclusive cases
• Proves the statement for each case separately
• Combines the results to conclude the statement holds for all possibilities
• Often used when dealing with conditional statements or disjunctions
• Employs the principle of disjunctive syllogism

Applications of propositional logic

• Propositional logic extends beyond pure mathematics into various practical domains
• Provides a foundation for designing and analyzing digital systems and algorithms
• Enables formal verification and reasoning in computer science and engineering

Boolean algebra

• Algebraic structure that represents the behavior of logical operations
• Isomorphic to propositional logic, with operations corresponding to logical connectives
• Used in circuit design, database queries, and set theory
• Allows for simplification and optimization of logical expressions

Circuit design

• Digital circuits implement Boolean functions using logic gates
• AND, OR, and NOT gates correspond directly to logical connectives
• Complex circuits can be designed and analyzed using propositional logic
• Minimization of logical expressions leads to more efficient circuit designs

Computer programming

• Conditional statements in programming languages based on propositional logic
• Boolean expressions used for control flow and decision-making in algorithms
• Logical operators in programming languages (&&, ||, !) derived from propositional connectives
• Program verification and correctness proofs rely on propositional logic principles

Limitations of propositional logic

• While powerful, propositional logic has certain constraints and limitations
• Understanding these limitations helps in choosing appropriate logical systems for different problems
• Motivates the development of more expressive logical frameworks

Expressiveness vs predicate logic

• Propositional logic cannot express relationships between objects or quantification
• Unable to represent statements like "For all x, P(x)" or "There exists an x such that P(x)"
• Predicate logic extends propositional logic to handle these more complex statements
• Many mathematical concepts require the additional expressiveness of predicate logic

Paradoxes in propositional logic

• Certain statements lead to logical paradoxes within propositional logic
• The Liar's Paradox ("This sentence is false") cannot be consistently represented
• Self-referential statements pose challenges for truth value assignment
• Highlights the need for more sophisticated logical systems in some contexts

Advanced topics

• Build upon the foundations of propositional logic to address more complex problems
• Provide tools for automated reasoning and theorem proving
• Connect propositional logic to broader areas of mathematical logic and computer science

Normal forms

• Standard ways of representing logical formulas to simplify analysis and manipulation
• Conjunctive Normal Form (CNF) expresses formulas as conjunctions of disjunctions
• Disjunctive Normal Form (DNF) expresses formulas as disjunctions of conjunctions
• Useful for algorithmic processing of logical expressions and satisfiability checking

Resolution principle

• Powerful inference rule used in automated theorem proving
• Based on the idea of resolving two clauses to produce a new clause
• Allows for the systematic derivation of new logical consequences
• Forms the basis for many automated reasoning systems and SAT solvers

Automated theorem proving

• Computational methods for proving mathematical theorems automatically
• Utilizes resolution, normal forms, and other advanced propositional logic techniques
• Applications in formal verification of hardware and software systems
• Challenges include managing the search space and handling complex mathematical concepts