Propositional Logic Basics

Introduction to Propositional Logic

• Propositional logic is a subfield of mathematical logic that investigates how to utilize the structure of known propositions to derive new and valid facts and propositions.
• It provides the basis for the formulation of conditions that are often used in programming.

Basic Concepts

• Propositional logic studies how to reason about the truth of propositions.
• It includes assessing how the truth of one proposition relates to the truth of another.
• A proposition is a statement that can be assigned a truth value (true or false).

Propositional Variables

• A propositional variable is a statement that can be assigned a truth value.
• Examples of propositional variables: "London is the capital of the United Kingdom", "It is raining", etc.
• A sentence that cannot be assigned a truth value is not a propositional variable (e.g., questions, commands).

Boolean Operators

• NOT (¬): inverts the truth value of a proposition
• AND (∧): connects two propositions, evaluates to true if both propositions are true
• OR (∨): connects two propositions, evaluates to true if at least one proposition is true
• IF-THEN (): implies that one proposition is true if another is true
• IF-AND-ONLY-IF (): indicates that two propositions have the same truth value

Truth Tables

• A truth table lists all possible combinations of truth values of propositional variables and the corresponding truth values of the entire proposition.
• Used to define and illustrate the behavior of Boolean operators.

Complete Basis

• A set of Boolean operators is a complete basis if every other Boolean operator can be expressed using elements of the set.
• OR and NOT form a complete basis, as do NAND and NOR.

Propositional Logic as a Language

• Propositional logic can be understood as a formal language with its own syntax and semantics.
• The language includes propositional variables, Boolean operators, and logical constants (true and false).

Calculation Rules and Normal Forms

• Propositional logic has various transformation rules, such as the idempotent law, commutative law, associative law, absorption law, and distributive law.
• De Morgan's laws allow for the resolution of negated brackets and the transformation of AND-connectives into OR-connectives.
• Normal forms, such as disjunctive normal form (DNF) and conjunctive normal form (CNF), provide standardized ways of describing propositions.- Propositional Logic*

Idempotent Law

• Can be applied to remove duplicates of a literal from a proposition without changing its meaning
• If a literal appears more than once with the same sign, it can be simplified

DNF (Disjunctive Normal Form) and CNF (Conjunctive Normal Form)

• Every proposition can be transformed into DNF or CNF using transformation laws
• Truth tables can be used to transform arbitrary propositions into normal form
• Example of a proposition in normal form is shown in Table 7

Proofs

• Transformation laws can be used to deduce new formulas from existing ones
• A formal system consists of axioms and rules to infer new formulas
• Hilbert calculus is a logical calculus that consists of axioms and transformation laws
• Truth tables can be used to prove formulas
• Symbol ⊢ is used to indicate that a formula can be derived in a given logical calculus

Interpretation and Satisfiability

• An interpretation assigns a truth value to each propositional variable
• A formula is true under an interpretation if and only if it is satisfied by that interpretation
• A formula is satisfiable if it is true under at least one interpretation
• A formula is falsifiable if it is false under at least one interpretation
• A formula is tautology if it is true under all interpretations
• A formula is non-satisfiable if it is false under all interpretations

Three-Valued Logic

• In three-valued logic, a proposition can have three truth values: true, false, and undefined
• In Java, there are two variants of conjunction and disjunction: bitwise and Boolean
• Short evaluation versus strict interpretation in three-valued logic

Proof by Contradiction and Resolution

• Proof by contradiction is a proof method that assumes the negation of a proposition and derives a contradiction
• Resolution is a method of automated theorem proving
• Resolution rule can be automated and is used in programming languages like Prolog
• Resolution can be used to check whether a propositional formula is a tautology

Soundness and Completeness

• Soundness of a logical calculus means that every formula that can be deduced in the system is a tautology

• Completeness of a logical calculus means that every tautology can be proved within the system

• The calculi of truth tables, Hilbert calculus, and resolution rule are sound and complete for propositional logic### Propositional Logic and Satisfiability

• Satisfiability and tautology are distinct attributes, but closely related concepts.

• A proposition is a tautology if its negation is non-satisfiable.

Complexity Theory and NP-Complete Problems

• The SAT problem is a famous NP-complete problem in complexity theory.
• Algorithms are classified based on their runtime growth in relation to input variables.
• No efficient algorithms are known for NP-complete problems, meaning their runtime cannot be bounded by a polynomial in the number of input variables.
• The existence of efficient algorithms for NP-complete problems is related to the "P = NP" problem, a famous open problem in mathematics and computer science.

Propositional Logic

• Propositional logic is a subfield of mathematical logic that studies the structure of statements to derive new and valid statements.
• It forms the basis for formulating and evaluating conditions important in programming.

Logical Deduction Systems

• There are several formal deduction systems, known as logical calculus, to evaluate propositions, including:
  • Truth tables
  • The Hilbert calculus with axioms and deduction laws
  • The resolution rule
• These logical systems are sound, meaning all derived formulas are valid under all interpretations.
• The three logical calculi are also complete, meaning every tautology can be derived within the system.

Introduction to Propositional Logic

• Propositional logic is a subfield of mathematical logic that investigates how to utilize the structure of known propositions to derive new and valid facts and propositions.
• It provides the basis for the formulation of conditions that are often used in programming.

Basic Concepts

• Propositional logic studies how to reason about the truth of propositions.
• It includes assessing how the truth of one proposition relates to the truth of another.
• A proposition is a statement that can be assigned a truth value (true or false).

Propositional Variables

• A propositional variable is a statement that can be assigned a truth value.
• Examples of propositional variables: "London is the capital of the United Kingdom", "It is raining", etc.
• A sentence that cannot be assigned a truth value is not a propositional variable (e.g., questions, commands).

Boolean Operators

• NOT (¬): inverts the truth value of a proposition
• AND (∧): connects two propositions, evaluates to true if both propositions are true
• OR (∨): connects two propositions, evaluates to true if at least one proposition is true
• IF-THEN (): implies that one proposition is true if another is true
• IF-AND-ONLY-IF (): indicates that two propositions have the same truth value

Truth Tables

• A truth table lists all possible combinations of truth values of propositional variables and the corresponding truth values of the entire proposition.
• Used to define and illustrate the behavior of Boolean operators.

Complete Basis

• A set of Boolean operators is a complete basis if every other Boolean operator can be expressed using elements of the set.
• OR and NOT form a complete basis, as do NAND and NOR.

Propositional Logic as a Language

• Propositional logic can be understood as a formal language with its own syntax and semantics.
• The language includes propositional variables, Boolean operators, and logical constants (true and false).

Calculation Rules and Normal Forms

• Propositional logic has various transformation rules, such as the idempotent law, commutative law, associative law, absorption law, and distributive law.
• De Morgan's laws allow for the resolution of negated brackets and the transformation of AND-connectives into OR-connectives.
• Normal forms, such as disjunctive normal form (DNF) and conjunctive normal form (CNF), provide standardized ways of describing propositions.- Propositional Logic*

Idempotent Law

• Can be applied to remove duplicates of a literal from a proposition without changing its meaning
• If a literal appears more than once with the same sign, it can be simplified

DNF (Disjunctive Normal Form) and CNF (Conjunctive Normal Form)

• Every proposition can be transformed into DNF or CNF using transformation laws
• Truth tables can be used to transform arbitrary propositions into normal form
• Example of a proposition in normal form is shown in Table 7

Proofs

• Transformation laws can be used to deduce new formulas from existing ones
• A formal system consists of axioms and rules to infer new formulas
• Hilbert calculus is a logical calculus that consists of axioms and transformation laws
• Truth tables can be used to prove formulas
• Symbol ⊢ is used to indicate that a formula can be derived in a given logical calculus

Interpretation and Satisfiability

• An interpretation assigns a truth value to each propositional variable
• A formula is true under an interpretation if and only if it is satisfied by that interpretation
• A formula is satisfiable if it is true under at least one interpretation
• A formula is falsifiable if it is false under at least one interpretation
• A formula is tautology if it is true under all interpretations
• A formula is non-satisfiable if it is false under all interpretations

Three-Valued Logic

• In three-valued logic, a proposition can have three truth values: true, false, and undefined
• In Java, there are two variants of conjunction and disjunction: bitwise and Boolean
• Short evaluation versus strict interpretation in three-valued logic

Proof by Contradiction and Resolution

• Proof by contradiction is a proof method that assumes the negation of a proposition and derives a contradiction
• Resolution is a method of automated theorem proving
• Resolution rule can be automated and is used in programming languages like Prolog
• Resolution can be used to check whether a propositional formula is a tautology

Soundness and Completeness

• Soundness of a logical calculus means that every formula that can be deduced in the system is a tautology

• Completeness of a logical calculus means that every tautology can be proved within the system

• The calculi of truth tables, Hilbert calculus, and resolution rule are sound and complete for propositional logic### Propositional Logic and Satisfiability

• Satisfiability and tautology are distinct attributes, but closely related concepts.

• A proposition is a tautology if its negation is non-satisfiable.

Complexity Theory and NP-Complete Problems

• The SAT problem is a famous NP-complete problem in complexity theory.
• Algorithms are classified based on their runtime growth in relation to input variables.
• No efficient algorithms are known for NP-complete problems, meaning their runtime cannot be bounded by a polynomial in the number of input variables.
• The existence of efficient algorithms for NP-complete problems is related to the "P = NP" problem, a famous open problem in mathematics and computer science.

Propositional Logic

• Propositional logic is a subfield of mathematical logic that studies the structure of statements to derive new and valid statements.
• It forms the basis for formulating and evaluating conditions important in programming.

Logical Deduction Systems

• There are several formal deduction systems, known as logical calculus, to evaluate propositions, including:
  • Truth tables
  • The Hilbert calculus with axioms and deduction laws
  • The resolution rule
• These logical systems are sound, meaning all derived formulas are valid under all interpretations.
• The three logical calculi are also complete, meaning every tautology can be derived within the system.