Natural Deduction: Conditional Proofs

The ideaof hypothetical rules is to capture certain modes of reasoning that appeal to hypothetical statements, by which we mean statements are temporarily assumed to be true. In informal reasoning, this is usually signaled by phrases such as 'for the sake of argument, let's assume','suppose...'.

The rulesIt is not a surprise then, that our system of deduction will not be completed until we capture this aspect of reasoning. For SL, we have 3 hypothetical rules: Conditional Introduction, Reductio Ad Absurdum, and Argument by Cases.

Considerthe following argument and notice how it corresponds to the derivation to the right: if WWIII occurs, it will be the end of civilization. Surely you agree that if WWIII were to occur, nuclear weapons would be used. Also, you believe that if nuclear weapons are used, then it will be the end of civilization. So, if WWIII occurs, it will be the end of civilization.

Intuitivelythe argument is valid (if you don't believe me, check it semantically). It is however not as explicit as logicians would like, since it's skipping over exactly how we go from 2 to 6. This leaves the hypothetical basis of the argument implicit. Ultimately, the argument seeks to introduce a conditional statemet (line 6). To codify inferences like this one, we use the conditional introduction.

subproofss are essentially a derivation within a derivation. The basic idea is that we can introduce temporary assumption. Lines 3-5 constitute the subproofs in this proof.

HypothesisA subproofs begins by introducing the hypothesis - a temporary assumption to be used strictly within the subproofs. The key thing about the hypothesis is that once the subproofs is over, we need to discharge it

PurposeWithin the subproofs, we can draw inferences based on accessible formulas as usual. The idea of a conditional proof is that we prove a conditional statement such as $W \to E$ by first introducing its antecedent as an assumption. And then, we can close the subproofs by introducing an conditional statement using the last line of the subproofs as its consequent. Once it is closed, anything within the closed subproofs is no longer available.

Rule 1You can start a subproof on any line, except the last one, and introduce any assumptions with that subproof.

Rule 2All subproofs must be closed by the time the proof is over.

Rule 3Subproofs may be closed at any time. Once close, they can be used to justify hypothetical inferences.

Rule 4Nested subproofs must be closed before the outer subproof is closed.

Rule 5Once the subproof is closed, lines in the subproof cannot be used in later justiļ¬cations.