Natural deduction in propositional logic lets us prove complex statements step-by-step. We start with premises and use inference rules to reach a conclusion. It's like building a logical argument brick by brick.

Constructing simple proofs is all about connecting the dots between what we know and what we want to prove. We'll learn how to structure our reasoning, justify each step, and develop strategies for tackling different types of proofs.

Proof Components

Essential Elements of a Proof

• Premise
  • A statement or proposition assumed to be true for the purpose of the argument
  • Serves as the starting point or foundation for the logical reasoning process
  • Can be based on given information, axioms, or previously proven statements
• Conclusion
  • The statement or proposition that is derived or inferred from the premises
  • Represents the final outcome or result of the logical reasoning process
  • Must be logically supported by the premises and the line of proof

Structuring a Proof

• Line of Proof
  • A sequence of logical steps that connect the premises to the conclusion
  • Each step in the line of proof must be justified by a valid inference rule or logical principle
  • Ensures a clear and systematic progression from the given information to the desired conclusion
• Justification
  • An explanation or rationale provided for each step in the line of proof
  • Demonstrates the validity and correctness of the logical reasoning process
  • Commonly refers to specific inference rules, axioms, or previously proven statements

Logical Reasoning

Inference and Deduction

• Inference
  • The process of deriving a conclusion from one or more premises
  • Involves applying logical rules and principles to draw new information from given statements
  • Can be inductive (generalizing from specific instances) or deductive (deriving specific conclusions from general principles)
• Valid Deduction
  • A form of inference where the conclusion necessarily follows from the premises
  • If the premises are true, the conclusion must also be true
  • Ensures the preservation of truth from the premises to the conclusion (modus ponens, modus tollens)

Developing a Proof Strategy

• Proof Strategy
  • A systematic approach or plan for constructing a proof
  • Involves identifying the given information, the desired conclusion, and the logical steps needed to connect them
  • May involve breaking down the problem into smaller sub-goals or lemmas
  • Requires selecting appropriate inference rules and logical principles to justify each step in the proof
  • Common strategies include direct proof, proof by contradiction, and proof by induction