## Tuesday, 11 June 2013

### Deductive proofs: from premises to conclusion, or from conclusion to premises?

(Cross-posted at NewAPPS)

It is fair to say that the ‘received view’ about deductive inference, and about inference in general, is that it proceeds from premises to conclusion so as to produce new information (the conclusion) from previously available information (the premises). It is this conception of deductive inference that gives rise to the so-called ‘scandal of deduction’, which concerns the apparent lack of usefulness of a deductive inference, given that in a valid deductive inference the conclusion is already ‘contained’, in some sense or another, in the premises. This is also the conception of inference underpinning e.g. Frege’s logicist project, and much (if not all) of the discussions in the philosophy of logic of the last many decades. (In fact, it is also the conception of deduction of the most famous ‘deducer’ of all times, Sherlock Holmes.)

That an inference, and a deductive inference in particular, proceeds from premises to conclusion may appear to be such an obvious truism that no one in their sane mind would want to question it. But is this really how it works when an agent is formulating a deductive argument, say a mathematical demonstration?

The contrast between (deductive) demonstration and calculation may be illuminating here. When calculating, one starts with some known parameters (say, the total number of candies and the number of children among whom the candies have to be distributed) and seeks to determine the solution to a problem by determining the relevant unknown value (say, the number of candies each child will receive). By analogy, one might say that the premises of a deductive inference are (like) the known parameters and the conclusion is (like) the unknown value.

Now, I’ve been struggling for years to make sense of this conception of deductive reasoning, but to no avail. It just doesn’t seem to do justice to how deductive arguments are in fact formulated and used. So now I’ve decided to adopt a different starting point: what if, in a deductive argument, she who formulates the argument in fact proceeds from conclusion to premises? This may seem absurd at first sight, but once you start thinking about it, it makes a lot of sense (or so I claim!).

Consider for example how mathematical proofs are formulated. Is it the case that the mathematician looks at e.g. the axioms of number theory, and then starts ‘playing around’ with them trying to deduce non-trivial conclusions? I’m pretty sure everyone will agree with me that this is not how it works. Instead, mathematicians usually take conjectures as their starting point: Fermat’s last theorem, the twin-prime conjecture, the ABC conjecture etc. Starting with the ‘conclusion’, they try to establish, by reverse-engineering as it were, which premises are required to establish the conclusion, and by which argumentative paths. So in a sense, what is discovered in a mathematical proof is everything but the conclusion: instead, the mathematician discovers the necessary premises and the proof itself. (Luis Carlos Pereira once suggested to me that, in terms of the analogy with calculation, the ‘unknown value’ in a proof is the proof itself.) Of course, there is a sense in which the truth of the conclusion is ‘discovered’ (established) by means of the proof, but the content of the conclusion is what guides the mathematician in her search for the proof from the start.

Much of my thinking on these matters is yet again prompted by the close reading of the Prior Analytics that we are undertaking with our reading group in Groningen. As it turns out, the bulk of the text, and in any case about half of Book A, is dedicated to techniques on how to find the necessary premises to establish a given conclusion, in particular finding the right ‘middle term’ for it. Again, the starting point is the conclusion, and through reverse engineering, the required premises are found. (My post-doc Matt Duncombe is just finishing a terrific paper exactly on this aspect of the Prior Analytics, in connection with the so-called scandal of deduction; if anyone is interested in reading the draft, perhaps he could be persuaded to share it in the near future.)

So how come the conception of deductive inference as going from premises to conclusion became so widespread? Here again, the dialogical conceptualization of deduction that I have been developing seems to offer a plausible explanation. When a deductive argument is presented to opponent by proponent, proponent indeed starts with the premises, seeking to get opponent to grant them, and then slowly but surely moves towards the conclusion, which opponent will be forced to grant if he has granted the premises and the intermediate inferential steps. Hence, from the perspective of opponent, from-premises-to-conclusion is indeed the correct order of events in a deductive argument; however, from the perspective of proponent, the right order is from-conclusion-to-premises. In other words, a proponent, or whoever formulates a deductive argument, always knows where she is heading.

Another way of conceptualizing this dichotomy is in terms of the good old distinction between context of justification and context of discovery: for justification, the right path is from premises to conclusion; for discovery (of the proof), the right path is from conclusion to premises.

As it so happens, Descartes was already well aware of this fact when disdainfully commenting on ‘the logic of the Schools’ (he means scholastic logic, but my claim is that this applies to deductive logic in general):
[T]he logic of the Schools […] is strictly speaking nothing but a dialectic which teaches ways of expounding to others what one already knows […] I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant. (Preface to French edition of the Principles of Philosophy, in (Descartes 1988, 186); emphasis added)

Well, Descartes, in this case deductive logic is not for you.

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UPDATE: In my Google Plus feed, Timothy Gowers writes that mathematicians make use both of what he calls backwards reasoning (from conclusion to premises) and of forwards reasoning (from premises to conclusions). This seems absolutely right to me, and a wise caveat to my overly-unifying claims in this post! So the right answer to the question in my title is: both.

1. Perhaps what characterises the 'deductive journey' from premises to conclusion or from conclusion to premises is, nevertheless, unidirectional. By analogy, in physics one can evolve a system forwards in time, and then one can reverse the procedure, and evolve the system 'backwards' in time. Despite the terminology, one is always evolving the system unidirectionally irrespective of 'sense'.

2. Yes, that's how it works. Math is done inductively; but presented (to students) deductively. This leaves the students -- even at the graduate level -- ignorant of how math is done.

A couple of historical illustrations. Newton invented calculus in the mid to late 1600's. The Principia dates from 1687, so take that as the official date if you like. Now, Newton well understood that he could not make logical sense of the limit of the difference quotient (what he called the "fluxion"). It's an expression of the form 0/0, which makes no sense. But calculus worked, spectacularly so. It wasn't till the work of Weirstrass and other's in the 1800's that the logical definition of the limit was finally arrived at; and not till Zermelo, in the early 1900's, that we finally had a completely rigorous account of the real numbers and limiting processes starting from the axioms of set theory all the way up to calculus. This process took over 200 years!

But today, in real analysis class taught to math major undergrads, we start from the axioms of set theory, construct the real numbers, and then rigorously prove the basic theorems of calculus. This is a complete inversion of how the subject was discovered and developed. But undergrads -- and, sorry to say, most philosophers -- take the axiomatic, deductive *presentation* and confuse it with the actual practice of mathematics.

Another striking example is group theory. Mathematicians were trying to figure out how to solve polynomial equations. Abel showed that the 5th degree equation did not have a general solution. Galois showed that the underlying reason for this had to do with the mathematical structure of the set of permutations of the roots.

It wasn't till much later that someone came along and defined a "group" as a set with a binary operation satisfying such-and-so axioms; and then was able to re-derive Galois's and Abel's proof from the axioms of group theory.

Today, we teach undergrads that a group is such-and-so; then we spend the rest of the semester deriving consequences. The thoughtful undergrads wonder: How did they come up with these particular axioms? And they get the impression that math is about writing down axioms and mechanically deriving logical conclusions.

But this is of course a complete inversion of how math is done. In math, you first suspect the theorem; then -- after centuries, sometimes -- you eventually figure out the right axioms.

Math is done inductively and presented deductively. Math actually goes from theorems to axioms; not the other way round.

1. Agreed! :) And thanks for the helpful illustrations, anonymous.

3. There is much true here, but as far as logicism goes, it is completely wrong. Logicism is sometimes an epistemological view, and sometimes a metaphysical view, but, either way, it is a view about what is based upon what, NOT a view about what is INFERRED from what. To interpret it the latter way is to think of it as a psychological view, and that could not be further from the intentions of either Frege or Russell.

1. Well, at least I am not alone in thinking that (deductive) inference is the key concept in Frege's conception of logic:

http://www-personal.usyd.edu.au/~njjsmith/papers/smith-freges-js-logic.pdf

(I've been much influenced by my former supervisor G. Sundholm for my thinking about Frege.)

You may quibble with my use of the term 'logicism' instead of 'logic', but other than that there are very respectable Frege scholars (to mention one more: Danielle Macbeth) who hold this interpretation. You may well disagree with them, but it's not a view that is obviously so 'completely wrong' as you claim.

4. The same dichotomy between forward and backward proofs happens in type theory (in the computing tradition) where functional programmers normalize proofs written from premisses to conclusions, while logic programmers build proofs from conclusions to axioms.

5. Aristotle does seem to talk about deducing or inferring premises through conclusions in the section on reciprocal proofs. I have a theory that induction is precisely this. Hope the grant approval board thinks likewise!