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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-.

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Hurley's proof system is based on a set of rules (18 plus Conditional Proof and Indirect Proof) called improperly Natural Deduction.

A "standard" natural deduction proof system for propositional logic, like : Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007), has 13 rules : 2 for each connectives (included ), plus RAA.

A proof system for classical logic must be sound (i.e. able to prove only tautologies) and complete (i.e. able to prove all tautologies). The Natural Deduction proof system is sound and complete.

It seems to me that Hurley does not treat completeness of his proof system, but clearly he using a redundant system with enough rules and it would be possible to prove it.

Some rules are "critical", in the sense that if we remove them, we are cannot prove all the tautologies.

For example, if we remove RAA rule from Chiswell & Hodeges' system, what we get is a proof system for Intuitionistic Logic.

In terms of tautologies, this means that into the "reduced" proof system we cannot derive Excluded Midlle : p ∨ ¬ p, because this tautology is not intuitionistically valid.


Having said that, I think that Hurley's statement :

Moreover, some arguments have conclusions that cannot be derived by the direct method [...]. Conditional proof may thus be seen as completing the rules of inference.

means that all 18 rules of Implication and Replacement allow us only to derive new formulae from other ones, i.e. to produce derivations of form : A ⊢ B. Thus, we cannot prove a logical truth, i.e. we cannot produce a derivation of form : ⊢ A.

In order to do this, we need some way to "discharge assumptions", and thus the need for the CP and IP rules. In particular, CP allows us to transform a derivation A ⊢ B into a new derivation : ⊢ A → B.


In conclusion, the issue is not with "weaker or stronger": Hurley speaks of "shorter and simpler" derivations, but of "completing the [set of] rules of inference" in order to prove tautologies.

See §7.7 Proving Logical Truths [page 438] :

Both conditional and indirect proof can be used to establish the truth of a logical truth (tautology). Tautological statements can be treated as if they were the conclusions of arguments having no premises.

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    Complete does not mean minimal, as you seem to suggest. Redundant rules are just fine and do not threaten completeness. Considering a minimal system more 'standard' is a little biased. – user9166 Jan 2 '16 at 2:08

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