Suppose that, of five propositions, A,B,C,D, and E, only D is seriously in doubt. One wishes to prove not D. If one can show that A,B,C,D,E => Z where Z is absurd, then one of A,B,C,D,E must be wrong. Given that only D is seriously in doubt, it must be D that is wrong. Hence not D is established.
Notice how my presentation departs from usual mathematical practice. Usually A,B,C, and E are axioms. Hence D is the only antecedent that could be wrong. I am more interested in day to day reasoning. Day to day, one does not have powerful enough axioms to permit the rigorous proof of useful results. One has to make assumptions. The knack of living logically is to be aware of the relative plausibility of one's assumptions, and to reject the weakest, in this case D, when one discovers that one has assumed too much, in this case by making such powerful assumptions that one can prove the silly conclusion Z.
What is to be done? The aim of planning is to come up with a set of actions sufficient to achieve one's end. Constructing a plan can involve much backtracking. For example, if the budget is tight, a plan that nearly fits the budget is only seen to fail when most of the details have been filled in, bringing the cost up close to the total. Then one must go back, saying can we economize there or do that another way. It is very helpful to know that certain elements are necessary, that is to say, they will be present in any plan. That reduces the number of options that must be considered.
If the various constraints, A,B,C,D,E imply Z, then Z is necessary, even if the conclusion is unwelcome. Indeed logic is seen as especially useful here. If one did not know that Z was necessary, one might draw up many plans lacking Z, little suspecting that they were all doomed from the moment they omitted Z.
Imagine that Alice has doubts about D, and regards Z as unreasonable, while Bob is committed to D and regards Z as unwelcome, but not absurd. Alice and Bob will have very different attitudes to the proof that A,B,C,D, and E imply Z.
For Alice, this is a reductio ad absurdum, confirming her doubts about D. For Bob it is a proof of necessity, showing by pure logic that it is time to stop complaining about how unwelcome conclusion Z is, and to start accepting it.
In business, D might be the proposition that a software project can be completed in 6 months, while Z might be the conclusion that it is necessary to drop testing from the schedule. Alice, the programmer, interprets the logical of the situation as proving that the schedule is impossible, while Bob, the manager, considers the logic of the situation as proving that testing must be dropped.
In politics, D might be the proposition that capitalism stinks, and Z the conclusion that we should give socialism a go. Those who consider that the fall of the Soviet Union discredits socialism will see this as a reductio ad absurdum which shows that capitalism doesn't stink that badly after all. Others, committed to their adverse judgment of capitalism, consider the conclusion proven by logic.
For another political example, D might be the claim that legal prohibition is the right way to tackle the problems posed by cannabis, while Z might be the conclusion that we must tolerate some loss of civil liberties to make this prohibition workable. Some will consider the loss of civil liberties as the final straw that convinces them that drug prohibit is wrong, while others will consider it a regrettable necessity.
Logic has very different roles in mathematics and in real life. In mathematics, the premises are certain, and thus proof provides conclusions which are certain also. In real life, the premises are assumptions, which one makes with varying and revisable degrees of confidence. Proof creates dilemmas, forcing one to use one's judgment to chose between accepting a troubling conclusion, and reconsidering one's weaker assumptions.
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