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# Propositional Logic Basics

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### What is the main importance of the concept of completeness?

• To ensure that every tautology can be proved within the system (correct)
• To guarantee the soundness of a logical calculus
• To ensure that all false propositions are considered
• To determine the relevance of a logical calculus
• ### What is the condition for a logical calculus to be considered complete?

• If every formula can be proved within the system
• If every clause contains every literal only once
• If every tautology can be proved within the system (correct)
• If every proposition can be derived from the axioms
• ### Why does the resolution rule terminate for propositional logic?

• Because the algorithm is not guaranteed to terminate
• Because the resolution rule is incomplete
• Because the number of clauses is finite (correct)
• Because the number of clauses is infinite
• ### What is the satisfiability problem (SAT) in propositional logic?

<p>The problem of determining whether a proposition is satisfiable</p> Signup and view all the answers

### What is the Hilbert calculus used for in propositional logic?

<p>To derive new formulas from axioms and laws</p> Signup and view all the answers

### What is the significance of the P = NP-problem?

<p>It is a famous open problem in mathematics and computer science</p> Signup and view all the answers

### What is the main characteristic of NP-complete problems?

<p>They have no known efficient algorithms to solve them</p> Signup and view all the answers

### What is the relationship between a proposition being a tautology and being non-satisfiable?

<p>A proposition is a tautology if its negation is non-satisfiable</p> Signup and view all the answers

### What is the significance of completeness in propositional logic?

<p>It ensures that all tautologies can be proved within the system</p> Signup and view all the answers

### What is the significance of soundness in a logical calculus?

<p>It ensures that all formulas derived from the rules are valid</p> Signup and view all the answers

### What is true about a logical calculus that is semantically complete?

<p>It can prove every tautology</p> Signup and view all the answers

### What is the resolution rule used for in propositional logic?

<p>To combine clauses to form a resolvent</p> Signup and view all the answers

### What is the limitation of the theorem on completeness of logical calculus for propositional logic?

<p>It only applies to propositional logic</p> Signup and view all the answers

### What is the characteristic of the runtime of algorithms for NP-complete problems?

<p>It grows exponentially with the number of input variables</p> Signup and view all the answers

### What is the relationship between the number of propositional variables and the number of possible clauses in the resolution rule?

<p>The number of possible clauses is 2^n, where n is the number of propositional variables</p> Signup and view all the answers

### What is the significance of propositional logic in programming?

<p>It is used to formulate and evaluate conditions</p> Signup and view all the answers

### What is the property of soundness in a logical calculus?

<p>It means that every formula derived is valid under all interpretations</p> Signup and view all the answers

### What is the main difference between the resolution rule and the Hilbert calculus?

<p>The resolution rule uses clauses to derive new formulas, while the Hilbert calculus uses axioms and deduction laws</p> Signup and view all the answers

### What is the significance of the number of propositional variables in the resolution rule?

<p>It determines the number of possible clauses</p> Signup and view all the answers

### What is the relationship between the completeness of a logical calculus and the soundness of a logical calculus?

<p>Completeness implies soundness, but not vice versa</p> Signup and view all the answers

## Study Notes

### Propositional Logic

• Propositional logic is a subfield of mathematical logic that studies 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.
• Propositional logic investigates the truth values of propositions, which are defined as phrases to which a truth value "true" or "false" can be attached.

### Basic Concepts

• Propositional variables are statements that can be assigned a truth value, such as "it is raining" or "London is the capital of the United Kingdom."
• Propositional logic studies how to reason about the truth of propositions, including assessing how the truth of one proposition relates to the truth of another.

### Boolean Operators

• Boolean operators are used to combine propositional variables, including:
• NOT (¬): inverts the truth value of a given propositional variable
• AND (∧): corresponds to "and" in colloquial language, connecting two propositional variables
• OR (∨): corresponds to "or" in everyday language, connecting two propositional variables
• IF-THEN (→): represents a logical implication, connecting two propositional variables
• IF-AND-ONLY-IF (): represents an equivalence or biconditional, connecting two propositional variables
• Truth tables are used to define the behavior of these operators.

### Complete Basis

• A set of Boolean operators is a complete basis if every other Boolean operator can be expressed with elements of the set.
• OR and NOT form a complete basis, as do NAND and other operators.

### Propositional Logic as a Language

• Propositional logic can be understood as a formal language, following a set of rules defined by its syntax and semantics.
• The language is defined via a set of rules, including the use of propositional variables, Boolean operators, and parentheses.

### Calculation Rules and Normal Forms

• Laws in propositional logic, such as the idempotent law, commutative law, associative law, absorption law, distributive law, and De Morgan's law, allow for the transformation of formulas.
• These laws can be used to simplify and manipulate propositional logic formulas.

### Applications

• Propositional logic plays a key role in computer science and appears in various situations, including:
• Formulation of conditions and constraints in programming (e.g., if-then-else statements)
• Formulation of complex formulas in spreadsheets
• Setting up logic circuits in hardware design
• Artificial intelligence
• Logic programming (e.g., in Prolog)
• Program verification, formal specifications, and formal semantics### Propositional Logic
• De Morgan's Laws: allow to transform a negated bracket or to switch between AND and OR connectives.
• Normal Forms: standardized ways of describing a fact in propositional logic.
• Disjunctive Normal Form (DNF): a proposition is represented as a disjunction of conjunctions.
• Conjunctive Normal Form (CNF): a proposition is represented as a conjunction of disjunctions.
• Reduced Normal Forms: require that every literal appears only once in a conjunction of a DNF, or in every disjunction of a CNF.

### Proofs

• Deduction Laws: used to deduce new formulas from existing ones.
• Axioms: formulas that are considered true and are used to derive new formulas.
• Formal System: an abstract structure used to infer or prove formulas from axioms according to a set of rules.
• Hilbert Calculus: a logical calculus that consists of axioms and transformation laws.
• Truth Tables: can be used to prove or disprove whether a proposition is a tautology.

### Interpretation and Satisfiability

• Interpretation: a function that assigns a truth value to each propositional variable.
• Model: a formula that is true under a given interpretation.
• Satisfiability: a formula is satisfiable if at least one interpretation exists that makes the formula true.
• Tautology: a formula that is true for every possible interpretation.
• Falsifiable: a formula that is false for at least one interpretation.
• Non-satisfiable: a formula that is false for every possible interpretation.

### Three-Valued Logic

• Undefined or Unknown Values: an extension of two-valued logic to include a third truth value.
• Short Evaluation: a conjunction or disjunction is evaluated as soon as the outcome is determined.
• Strict Interpretation: the conjunction and disjunction are evaluated in the classical manner.

### Proof by Contradiction and Resolution

• Proof by Contradiction: a method of proof that assumes the opposite of what is to be proven, and then derives a contradiction.
• Resolution: a method of proof that is based on the complement rule and can be automated.
• Clause: a propositional formula that is a disjunction of literals.
• Resolution Rule: a rule for combining two clauses to form a new one.

### Soundness and Completeness

• Soundness: a property of a logical system that ensures that every formula that can be deduced is a tautology.
• Completeness: a property of a logical system that ensures that it is always sufficient to prove or disprove whether a proposition is a tautology.
• Theorem: the logical systems of propositional logic discussed are sound, including the calculus of truth tables, the Hilbert calculus, and the resolution rule.### Completeness of a Logical Calculus
• A logical calculus is (semantically) complete if every tautology can be proved within the system.
• A calculus is complete if ⊨ φ always implies ⊢ φ.
• In propositional logic, the following logical calculi are complete:
• The calculus of truth tables
• The Hilbert calculus with axioms and laws of propositional logic
• The resolution rule described in the previous algorithm

### Propostional Logic

• Propostional logic studies the truth values of propositions.
• A proposition is a statement that can be assigned a truth value (true or false).
• Propostional variables are the simplest well-formed formulas.
• Boolean operators are used to combine propositional variables:
• NOT (¬): inverts the truth value of a propositional variable
• AND (∧): true if and only if both propositional variables are true
• OR (∨): true if at least one propositional variable is true
• IF-THEN (→): true unless the antecedent is true and the consequent is false
• IF-AND-ONLY-IF (): true if and only if both propositional variables have the same truth value

### Boolean Operators

• The list of Boolean operators includes:
• XOR (exclusive OR)
• NAND (not-and)
• NOR (not-or)
• A set of Boolean operators is a complete basis if every other Boolean operator can be expressed with elements of that set.
• Theorem: OR and NOT form a complete basis, and NAND forms a complete basis.

### Propositional Logic as a Language

• Propositional logic can be understood as a formal language.
• A language is defined via its syntax (which expressions are allowed?) and its semantics (what do certain expressions mean in the language?).
• The language of propositional logic follows a specific set of rules, making it a context-free language (CFL).
• In programming languages, Boolean operators are available for use in logical expressions.

• Fuzzy logic: a framework for working with imprecise or unclear terms.
• Temporal logic: deals with propositional variables whose truth values are time-dependent.
• Predicate logic (first-order logic): deals with statements that contain variables that must be interpreted.

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## Description

Learn the fundamentals of propositional logic, a crucial subfield of mathematical logic that deals with deriving new facts from known propositions, with applications in programming.

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