Conditional Proof (CP) is a powerful tool in formal logic. It lets us prove "if-then" statements by assuming the "if" part and showing the "then" part follows. This technique is key to tackling complex logical arguments.

CP fits into the broader world of proof methods. It's especially useful when direct proofs aren't possible, allowing us to reason hypothetically and draw valid conclusions from assumptions.

Conditional Proof Technique

Understanding Conditional Proof

• Conditional Proof (CP) is a technique used to prove the truth of a conditional statement
• Involves hypothetical reasoning where we assume the antecedent of the conditional statement is true
• Proceed to derive the consequent using valid inference rules and previously established premises
• If successful, we can conclude that the conditional statement as a whole is true

Applying the Conditional Proof Technique

• Begin by making an assumption, which is the antecedent of the conditional statement we want to prove
• Perform a series of valid inferences within the scope of the assumption, known as a subproof
• Aim to derive the consequent of the conditional statement within the subproof
• If the consequent is successfully derived, we can discharge the assumption and conclude the conditional statement is true

Components of Conditional Statements

Structure of Conditional Statements

• A conditional statement consists of two parts: an antecedent and a consequent
• The antecedent is the proposition that follows the word "if" in a conditional statement (if P)
• The consequent is the proposition that follows the word "then" in a conditional statement (then Q)
• The conditional statement asserts that if the antecedent is true, then the consequent must also be true (if P, then Q)

Scope and Discharging Assumptions

• The scope of an assumption refers to the portion of the proof where the assumption is in effect
• Typically, the scope is denoted by a vertical line or indentation, creating a subproof within the main proof
• All inferences made within the scope of the assumption are dependent on the assumption being true
• Once the desired conclusion (the consequent) is derived within the scope, the assumption can be discharged
• Discharging an assumption means that the conclusion of the subproof no longer depends on the assumption
• The conditional statement as a whole is then considered proven, with the antecedent implying the consequent