Derivation Rules in Logic

• A derivation rule, also known as an inference rule, is a logical form consisting of a function that takes premises, analyzes their syntax, and returns a conclusion.
• Derivation rules are syntactic transformations applicable for deriving new statements from existing ones.
• They are fundamental in logic and formal systems, enabling proof construction and theorem deduction.
• Each derivation rule specifies a valid method to infer a conclusion from a set of premises.
• A derivation rule's general format is: If premise 1, premise 2, ..., premise n are true, then the conclusion is true.
• Premises are statements assumed true, and the conclusion is derived from them.

Components of a Derivation Rule

• Premises: Initial statements or assumptions on which the rule operates.
• Conclusion: The statement inferred or derived from the premises.
• Logical Connectives: Symbols like AND, OR, NOT, IMPLIES, and EQUIVALENT express relationships between statements.
• Variables: Symbols represent arbitrary terms or formulas.
• Quantifiers: Symbols such as "for all" and "there exists" specify the quantity of objects for which a statement holds true.

Common Derivation Rules in Propositional Logic

• Modus Ponens: If P is true, and P implies Q is true, then Q is true.
• Modus Tollens: If P implies Q is true, and Q is false, then P is false.
• Hypothetical Syllogism: If P implies Q is true, and Q implies R is true, then P implies R is true.
• Disjunctive Syllogism: If P or Q is true, and P is false, then Q is true.
• Addition: If P is true, then P or Q is true.
• Simplification: If P and Q is true, then P is true.
• Conjunction: If P is true, and Q is true, then P and Q is true.
• Resolution: If P or Q is true, and NOT Q or R is true, then P or R is true.

Common Derivation Rules in Predicate Logic

• Universal Instantiation: If "for all x, P(x)" is true, then P(a) is true for any specific a.
• Universal Generalization: If P(a) is true for an arbitrary a, then "for all x, P(x)" is true.
• Existential Instantiation: If "there exists x such that P(x)" is true, then P(c) is true for some specific c.
• Existential Generalization: If P(a) is true for some specific a, then "there exists x such that P(x)" is true.

Application of Derivation Rules

• Derivation rules serve to construct formal proofs.
• A formal proof comprises a sequence of statements, each either an assumption or derived from prior statements using a derivation rule.
• The goal of a proof is demonstrating that a particular conclusion logically stems from a given set of premises.
• Derivation rules are employed in automated theorem provers.
• Automated theorem provers are computer programs attempting to find proofs of mathematical theorems.
• They are also employed in expert systems.
• Expert systems are computer programs using knowledge and inference rules to solve problems in a specific domain.

Example: Modus Ponens

• Rule: If P, and P → Q, then Q.
• P is a proposition.
• P → Q means "If P, then Q".
• Knowing that P is true, and that P → Q, allows the conclusion that Q is true.
• Example: "If it is raining, then the ground is wet."
• P = "It is raining."
• Q = "The ground is wet."
• P → Q = "If it is raining, then the ground is wet."
• If we know that "It is raining" is true, then we can conclude that "The ground is wet" is true.

Example: Modus Tollens

• Rule: If P → Q, and ¬Q, then ¬P.
• P → Q means "If P, then Q".
• ¬Q means "Not Q".
• Knowing that P → Q, and that ¬Q, allows the conclusion that ¬P.
• Example: "If it is raining, then the ground is wet."
• P = "It is raining."
• Q = "The ground is wet."
• P → Q = "If it is raining, then the ground is wet."
• If we know that "The ground is not wet" is true, then we can conclude that "It is not raining" is true.

Formal Systems

• Formal systems consist of sets of symbols and rules for manipulating those symbols.
• Derivation rules form a key component of formal systems.
• Logic: Used to derive conclusions from premises.
• Mathematics: Used to prove theorems.
• Computer science: Used in programming language semantics.

Properties of Derivation Rules

• Soundness: A derivation rule is sound if it always produces true conclusions when applied to true premises.
• Completeness: A set of derivation rules is complete if it can be used to prove all true statements in a given formal system.
• Consistency: A set of derivation rules is consistent if it does not allow for the derivation of contradictory statements.

Importance of Derivation Rules

• Derivation rules are essential for reasoning and making logical inferences.
• They provide a structured and formal way to derive new knowledge from existing knowledge.
• They are used in a wide variety of fields, including mathematics, computer science, philosophy, and artificial intelligence.
• Derivation rules allow for the automation of reasoning and the development of intelligent systems.
• Utilization of derivation rules ensures the correctness and reliability of deductions.

Explore derivation rules, the core of logical deduction. These rules, also known as inference rules, guide syntactic transformations to derive conclusions from premises. They're essential in constructing proofs and deducing theorems within formal systems.