(Cross-posted at M-Phi)

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 todirect our reasonwith a view to discovering the truths of which we are ignorant. (Preface to French edition of thePrinciples 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*.

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