Propositional Logic

Propositional logic provides a formal language for representing and reasoning about logical relationships between propositions.

Propositional Logic:

Propositional logic, also known as sentential logic, is a branch of symbolic logic that deals with propositions or statements that are either true or false. It provides a formal framework for representing and reasoning about logical relationships between propositions.

Formal Language:

The formal language of propositional logic consists of:

• Alphabet: A set of symbols used to construct propositions and logical expressions.

• Syntax: Rules governing the construction of well-formed formulas (WFFs) from the alphabet.

• Semantics: Assigning truth values to propositions and evaluating the truth value of compound propositions.

Syntax:

In propositional logic, the syntax defines the rules for constructing well-formed formulas (WFFs) from atomic propositions and logical connectives. The syntax typically includes:

• Atomic Propositions: Basic statements or variables representing propositions (e.g., ( P, Q, R )).

• Logical Connectives: Symbols representing logical operations:

  • Negation ($¬$)

  • Conjunction (∧)

  • Disjunction ($∨$)

  • Implication ($→$)

  • Biconditional ($↔$)

Semantics:

The semantics of propositional logic involves interpreting the meaning of logical expressions and determining their truth values under different interpretations. It includes:

• Interpretation: Assigning truth values (true or false) to atomic propositions.

• Evaluation Function: A function that evaluates the truth value of compound propositions based on the truth values of their components.

• Properties of WFFs: Characteristics of well-formed formulas, such as validity, satisfiability, and contradiction.

Obtaining New Knowledge:

In propositional logic, new knowledge can be obtained through:

• Conceptualization: Abstracting real-world knowledge into propositional form.

• Representation: Expressing concepts and relationships using propositional logic symbols.

• Model Theory: Determining whether a given interpretation satisfies a set of propositions (knowledge base).

• Proof Theory: Deriving new propositions from existing ones using inference rules.

Inference Rules:

are formal principles or patterns of reasoning that allow one to derive new propositions or conclusions from existing propositions or premises.

In propositional logic, inference rules govern the derivation of new propositions from existing ones. Common inference rules include:

For example, in propositional logic, common inference rules include:

1. Modus Ponens: If $P$ implies $Q$, and $P$ is true, then $Q$ is true.

2. Modus Tollens: If $P$ implies $Q$, and $Q$ is false, then $P$ is false.

3. Conjunction Introduction: If both $P$ and $Q$ are true, then $P ∧ Q$ is true.

4. Disjunction Elimination: If $P∨Q$ is true, then either $P$ is true or $Q$ is true.