7.1 Conditional Proof (CP) Technique

Conditional Proof Technique

Conditional Proof (CP) lets you prove "if-then" statements by temporarily assuming the "if" part and showing the "then" part follows. Instead of proving a conditional directly, you get to work from the inside out: assume the antecedent, derive the consequent, and then conclude the whole conditional is true.

This technique is especially useful when a direct proof isn't practical. Many arguments in formal logic have conditional conclusions, and CP gives you a structured way to handle them through hypothetical reasoning.

Understanding Conditional Proof

A Conditional Proof works on a simple but powerful idea: if assuming $P$ always leads you to $Q$, then $P \rightarrow Q$ must be true.

Here's how it works in practice:

1. Assume the antecedent. You temporarily suppose that $P$ is true. This opens a subproof, a section of the proof that operates under that assumption.
2. Derive the consequent. Within the subproof, use valid inference rules (Modus Ponens, Hypothetical Syllogism, etc.) along with any previously established premises to arrive at $Q$.
3. Discharge the assumption. Once you've derived $Q$ inside the subproof, you close it and write $P \rightarrow Q$ as a line in the main proof, justified by "CP."

The key point: you aren't claiming $P$ is actually true. You're showing that if it were true, $Q$ would follow. That's exactly what a conditional statement asserts.

Components of Conditional Statements

Structure

A conditional statement has two parts:

• The antecedent is the proposition after "if." In $P \rightarrow Q$, the antecedent is $P$.
• The consequent is the proposition after "then." In $P \rightarrow Q$, the consequent is $Q$.

The conditional $P \rightarrow Q$ claims that whenever $P$ is true, $Q$ must also be true. It does not claim that $P$ is true on its own.

Scope and Discharging Assumptions

Scope refers to the portion of the proof where your assumption is active. In most proof formats, the scope is shown by indentation or a vertical line running alongside the subproof. Every inference you make inside that indented block depends on the assumption.

A few rules to keep in mind about scope:

• You can use any premise from the main proof inside a subproof, but lines derived inside a subproof cannot be used outside it after the assumption is discharged.
• Discharging the assumption means closing the subproof. At that point, the individual lines inside the subproof are no longer available. What you get instead is the conditional $P \rightarrow Q$ as a new line in the main proof.
• The discharged conditional no longer depends on the assumption. It stands on its own, supported by whatever premises the main proof already established.

Applying CP: Step-by-Step Example

Suppose your premises are:

And you want to prove: $A \rightarrow C$

Here's the proof:

1. $A \rightarrow B$ (Premise)
2. $B \rightarrow C$ (Premise)
3.     $A$ (Assumption for CP)
4.     $B$ (Modus Ponens, 1, 3)
5.     $C$ (Modus Ponens, 2, 4)
6. $A \rightarrow C$ (CP, 3–5)

Lines 3–5 form the subproof. You assumed $A$, derived $C$, and then discharged the assumption to conclude $A \rightarrow C$ at line 6. Notice that after line 6, you can no longer cite lines 3, 4, or 5 individually in later steps.