Propositional Logic

Propositional Logic is a branch of logic that deals with propositions and their relationships. A proposition is a declarative statement that can be either true or false, but not both. Propositional logic serves as the foundation for more complex logical systems, including First Order Logic (FOL).

2.1 Mathematical Logic and Inference

• Mathematical Logic: This field studies formal logical systems and their applications in mathematics and computer science. It provides the tools for reasoning about mathematical statements, helping to understand how conclusions follow from premises.

• Inference: Inference is the process of deriving logical conclusions from premises. In mathematical logic, inference rules dictate how one can validly move from one statement to another, forming the basis of logical proofs.

2.2 First Order Logic: Syntax and Semantics, Inference in FOL

• First Order Logic (FOL): FOL extends propositional logic by allowing the use of quantifiers and predicates, enabling more expressive statements about objects and their properties.

• Syntax: Syntax in FOL refers to the formal structure of expressions. It consists of:

  • Predicates: Functions that return true or false (e.g., P(x) means "x is a property").
  • Terms: Variables, constants, or functions representing objects in the domain.
  • Quantifiers: Existential (∃) and universal (∀) quantifiers allow for statements about some or all objects.

• Semantics: Semantics gives meaning to the syntactical expressions. It defines how to interpret the symbols and what truth values they hold based on the structure of the domain.

• Inference in FOL: Inference in FOL involves drawing conclusions from premises using rules and axioms. The conclusions are valid if they hold under all interpretations of the premises.

2.3 Forward Chaining, Backward Chaining

• Forward Chaining: This is a data-driven inference technique. It starts with known facts and applies inference rules to derive new facts until a goal is reached. It’s commonly used in expert systems and databases.

• Backward Chaining: This is a goal-driven inference method. It begins with a goal and works backward to determine if there are sufficient facts to support the goal. If the goal can be reduced to known facts, the conclusion is reached.

2.4 Language, Semantics, and Reasoning

• Language: In logic, language consists of symbols, expressions, and rules for forming statements. The language of logic allows for the formulation of arguments and reasoning processes.

• Semantics: It provides the meanings of the statements within the logical language, detailing how truth values are assigned based on interpretations.

• Reasoning: Reasoning involves using logic to draw conclusions, make decisions, and solve problems. It can be deductive (where conclusions necessarily follow from premises) or inductive (where conclusions are probable based on evidence).

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2.5 Syntax and Truth Values, Valid Arguments and Proof Systems

• Syntax and Truth Values: The syntax defines the structure of logical statements, while truth values (true or false) determine their validity. A statement's syntax must be correctly formed for it to be evaluated semantically.

• Valid Arguments: An argument is valid if, whenever the premises are true, the conclusion must also be true. Validity is a crucial concept in constructing logical proofs.

• Proof Systems: Proof systems are formal methods for deriving conclusions from premises. These systems use rules of inference to establish the validity of arguments.

2.6 Rules of Inference and Natural Deduction

• Rules of Inference: These are logical rules that justify the steps taken in logical proofs. Common rules include:

  • Modus Ponens: If "P implies Q" and "P" is true, then "Q" must be true.
  • Modus Tollens: If "P implies Q" and "Q" is false, then "P" must also be false.
  • Disjunctive Syllogism: If "P or Q" is true and "P" is false, then "Q" must be true.

• Natural Deduction: A method for deriving conclusions through a structured sequence of statements. It allows for introducing and eliminating logical operators systematically.

2.7 Axiomatic Systems and Hilbert Style Proofs

• Axiomatic Systems: These are logical frameworks consisting of a set of axioms (self-evident truths) and rules of inference. Axiomatic systems provide a foundation for proving theorems.

• Hilbert Style Proofs: Named after David Hilbert, this proof system is based on a formal structure using axioms and rules of inference to derive theorems. It emphasizes the importance of formalization in mathematical logic.

2.8 The Tableau Method

• Tableau Method: This is a decision procedure for propositional and first-order logic. It involves breaking down complex formulas into simpler components and constructing a tree-like structure (tableau) to test for satisfiability. The tableau method helps visualize logical relationships and determine if a formula is consistent.

2.9 The Resolution Refutation Method

• Resolution Refutation: This is a powerful inference rule used primarily in propositional logic and first-order logic. It operates on clauses in conjunctive normal form (CNF). The basic idea is to refute a proposition by deriving a contradiction. If the negation of a statement leads to a contradiction, the original statement is considered valid.