Preface: If interested, please see this ELU post on my (more archaic) placement of 'not' in this question's title.
Source: p 437, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley
Conditional proof is a method for deriving a conditional statement (either the conclusion or some intermediate line) that offers the usual advantage of being both shorter and simpler to use than the direct method. Moreover, some arguments have conclusions that cannot be derived by the direct method, so some form of conditional proof must be used on them. Conditional proof may thus be seen as completing the rules of inference. The method consists of assuming the antecedent of the required conditional statement on one line, deriving the consequent on a subsequent line, and then “discharging” this sequence of lines in a conditional statement that exactly replicates the one that was to be obtained.
This question concerns only arguments that can be proven both by
the Direct Method (abbreviated to DM) and Conditional Proof (abbreviated to CP),
which arguments I abbreviate to 'ARG'♦,
and which are exemplified by the arguments within the top 30% of this website.
For ARG, how can DM and CP be equal (eg: in strength and soundness), as asserted by the above quote and website?
The DM assumes nothing; all premises exist and are given.
A CP obliges you to assume at least one Antecedent of the Conclusion, which may or may NOT exist. The necessity of assuming something in CP: does it prove CP weaker than DM?
♦My abbreviation is inspired by the PIE root for 'argument': *arg-.