Conclusion before proof

Proofs of arguments in logic and mathematics differ from problem solving (except where the latter is used loosely and covers both situations) in that we know the answer in advance. The goal is to show that, given the information we have, and the rules we know, we can actually reach the stated conclusion.

Starting with the conclusion is not cheating. We often work backwards, from the conclusion to the premises, as far as we can go, in order to spot any clues that might help us pick up the threads when we reason in the conventional but often ineffective way. In some contexts, our success depends to a large extent on whether we can actually articulate the conclusion, and then translate it into a language we can work with.

If, for example, the task is to prove that the composition of an odd and even function is always even, and we are actually able to put down on paper what has just been said, we have as good as proved the theorem. In this case, we’d be looking at proving that f(g(x)) = f(-g(x)). The proof consists of only a few steps and relies entirely on the definition of an odd and even function.

Deductive reasoning is in fact an infinitesimally small part of all the reasoning that we do, but it sets standards that are worth emulating despite one failed attempt after another. The professions pride themselves on reasoning for a living. The professionals I teach take umbrage at me saying they don’t reason deductively (I review and proofread legal matter as part of my job), but I don’t mean it as criticism. What I criticize, and which strangely they find uncontroversial at all, is that they reason the wrong way round. If they reason at all.

Typically, legal advice will be frontloaded with facts, evidence, legislation and legal interpretations, plus some assumptions, all of which is to serve as the premises. Then, writing and thinking start and proceed at the same time. This part is usually called ‘analysis’. That’s where the problem lies. The analysis should have been done before. The writing stage should be where the outcome of the analysis is summarized, not the analysis per se.

These reports, analyses, research papers and legal opinions often lack any sort of conclusion – they concentrate on the trip, not on the destination. The numerous ‘howevers’ and ‘neverthelesses’ are ways of telling the reader how much the writer is enjoying the trip. The writer says: let’s toss the facts ‘in the instant case’ and my rhetoric and let’s see what pattern they form when they fall on paper. When I finish the writing, I will know the conclusion. You wait patiently.

‘To the extent that the strategy you pursue may be contended (and constructed) to fit within the parameters of the law, success in prevailing against a challenge based upon alleged noncompliance will be enhanced, but this office is not in a position to predict the odds of such a successful outcome,’ is a legitimate conclusion, but if I have to wade through 5 pages of legal meanderings to read this, I want my money back.