Many conclusions are conditionals: $p \to r$, $p \to (q \land r)$. Conditional proof (CP) proves them directly, the way people argue every day: "Suppose $p$. Then … so $r$. So if $p$, then $r$." It extends the rules of Proofs with All Eighteen Rules.
🔑 Key idea
To prove $A \to B$ by CP: assume $A$ on a new line (justified ACP), indented in a subproof. Derive $B$ inside the subproof, using the assumption and any earlier lines. Then close the subproof and write $A \to B$, justified "CP i–j", where $i$ is the assumption line and $j$ the last line of the subproof.
💡 Why it works
Inside the subproof you have shown: whenever $A$ (and the premises) are true, $B$ is true. That is exactly what $A \to B$ claims, given the premises. The assumption is not a new premise; it is a supposition for the sake of argument. Once the subproof is closed, the assumption is dropped, and only the conditional it produced remains.
Conditional proof: the indented subproof runs from the assumption to the line before CP.
Answer: Four new lines, where the eighteen rules alone needed many more.
✏️ Worked example: What to assume
You want $\neg s \to t$. What is the assumption?
Assume the antecedent of the conditional you want.
Answer: $\neg s$, justified ACP.
✏️ Worked example: Closing
A subproof assumes $p \lor q$ on line 4 and ends with $r$ on line 7. What does CP give on line 8?
Assumption → last line.
Answer: $(p \lor q) \to r$, justified CP 4–7.
⚠️ Common mistake
Using a line from inside a closed subproof later. Once CP closes the subproof, its lines depended on an assumption that is no longer in force; citing them afterward would let you use the supposition as if it were a fact. After the CP line, only lines outside the subproof (and the new conditional) may be cited.
CP is safe because the assumption cannot leak out. Everything derived inside the box is marked by the indentation as depending on it, and the only thing that leaves the box is a conditional that states that dependence openly. You may open a CP subproof at any time, assuming anything you like; the price is that you must close it before the proof ends.
The assumption may be any formula, not only a letter. To prove $(p \lor q) \to r$, assume $p \lor q$.
The conclusion is a conditional? Assume its antecedent.
Work toward its consequent inside the subproof.
Close with CP, citing the range of lines.
Do not cite lines inside the closed subproof again.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.