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What is Inference? 1 Introduction In some previous work, I tried to give a concept-based account of the nature of our entitlement to certain very basic inferences (see the papers in Part III of Boghossian 2008b ). In this previous work, I took it for granted, along with many other philosophers, that we understood well enough what it is for a person to infer. In this paper, I turn to thinking about the nature of inference itself. This topic is of great interest in its own right and surprisingly understudied by philosophers. A correct understanding of inference promises to shed light on a number of important topics. In particular, it threatens to undermine the sort of concept-based story about entitlement to which I had previously been attracted. 2 Preliminaries We will need to spend some time making sure that we zero in on the topic I mean to be discussing. By “inference” I mean reasoning with beliefs. Specifically, I mean the sort of “reasoned change in view” that Harman (1986 ) discusses, in which you start off with some beliefs and then, after a process of reasoning, end up either adding some new beliefs, or giving up some old beliefs, or both. I, therefore, explicitly leave aside practical reasoning. Within the sphere of theoretical reasoning, it is becoming customary to distinguish between two kinds, dubbed “System 1” and “System 2” by Daniel Kahneman. As Kahneman (2011 , pp. 20–21) characterizes them, • System 1 operates automatically and quickly, with little or no effort and no sense of voluntary control. • System 2 allocates attention to the effortful mental activities that demand it, including complex computations. The operations of System 2 are often associated with the subjective experience of agency, choice, and concentration. Examples of System 1 thinking are detecting that one object is more distant than another, orienting to the source of a sudden sound, responding to a thought experiment with an intuitive verdict. Examples of System 2 thinking are searching memory to identify a surprising sound, monitoring your behavior in a social setting, checking the validity of a complex logical argument. There are many things to be said about this distinction, but I don’t have the space to say them here. I will make two brief comments. First, to the extent to which I understand the distinction, it seems to me to correspond to the distinction between reasoning that is sub-personal, sub-conscious, involuntary and automatic, on the one hand, and reasoning that is person-level, conscious, attention hogging and effortful, on the other. Second, given this understanding of the distinction, it seems to me that a lot of reasoning falls somewhere in between these two extremes. Consider, for example, the following episode of thought, which I will call (Rain): On waking up one morning I recall that: 1. (1) It rained last night.I combine this with my knowledge that 1. (2) If it rained last night, then the streets are wet. to conclude: So, 1. (3) The streets are wet. This belief then affects my choice of footwear. I judged (1) and (2) and inferred from them that (3). This is neither the sort of sub-personal, subconscious, involuntary process characteristic of System 1. Nor is it the effortful, concentrated process attributed to System 2. It resembles System 2 thinking in that it is a person-level, conscious, voluntary mental action; it resembles System 1 in that it is quick, relatively automatic and not particularly demanding on the resources of attention. We could call it System 1.5 reasoning. When I say that I am interested in inference, I mean that I am interested in reasoning that is System 1.5 and up. That is to say, I am interested in reasoning that is person-level, conscious and voluntary, not sub-personal, sub-conscious and automatic, although I shall not also assume that it is effortful and demanding. Since it will help us focus attention on the issues that interest me, I shall work with the simple and somewhat simplistic (Rain) example. I shall be asking: What is it for me to infer (3) from (1) and (2)?1 This question about inference may be thought to fall under some other more general rubrics. My inferring (3) from (1) and (2), it may be thought, is for me to judge (3) on the basis of (1) and (2). Our question about inference, then, may be seen to be a special case of the topic that is discussed in the epistemological literature under the label the ‘basing relation.’ My inferring (3) from (1) and (2), it may also be thought, is for (1) and (2) to serve as my reasons for concluding (3). It is for me to arrive at the judgment that (3) with (1) and (2) serving as my reasons for so judging. Our question about inference, then, is also a special case of the topic that is discussed in the theory of action literature under the label ‘explanatory reasons.’ Explanatory reasons are the reasons that, in some appropriate sense, you take yourself to have for what you are doing, even if they may not be good reasons for doing what you are doing. (See Lenman 2009 ) Although our topic might plausibly be thought to fall under these other more general rubrics, and although these more general notions will play a role in what follows, I want to begin by pursuing our question about inference in its own right. 3 The nature of inference So let us turn, finally, to asking: What is it for me to infer (3) from (1) and (2)? It goes without saying that it can’t simply consist in my judging (3) after I have judged (1) and (2). Lots of thoughts can succeed one another without being related by inference.It may be less obvious, but is no less true, that my inferring (3) can’t consist just in my judging (1) and (2), and in this fact causing me to judge (3). Many philosophers would agree with this verdict because of what is known as the problem of ‘deviant causal chains.’ Alvin Plantinga has given a nice example. Suppose I see Aline. This causes me to believe that I see Aline, which causes me to drop the coffee I had been holding, which causes a stain on my shirt, which leads me to believe that my shirt is stained. My belief that I see Aline is part of the causal explanation for why I believe that my shirt is stained. But we wouldn’t want to say that I inferred that my shirt is stained from the fact that I see her. It’s not sufficient for my judging (1) and (2) to cause me to judge (3) for this to be inference. The premise judgments need to have caused the conclusion judgment ‘in the right way.’ Now, this formulation is harmless enough, I suppose, because the notion of ‘the right way’ could be taken to be a placeholder for whatever the correct relation is. But it would be a mistake, I think, to embark upon a search for the correct relation merely against the backdrop of the problem of deviant causal chains. Because the examples that illustrate this sort of deviance typically involve causal chains that are indirect they suggest that the problem is merely one of finding an intimate enough causal relation between the premise judgments and the conclusion. But such a view would be mistaken, I think. A habitual depressive’s judging ‘I am having so much fun’ may routinely cause and explain his judging ‘Yet there is so much suffering in the world,’ as directly as you please, without this being a case in which he is inferring the latter thought from the earlier one. What’s missing? I think that Frege (1979 ) put his finger on it when he said: To make a judgment because we are cognisant of other truths as providing a justification for it is known as inferring. (p. 3) I agree with the gist of Frege’s thought here, which I take to be this. A transition from some beliefs to a conclusion counts as inference only if the thinker takes his conclusion to be supported by the presumed truth of those other beliefs. Here I am tweaking Frege’s characterization in certain small ways. First, we needn’t always infer from truths, if we are to count as inferring. It’s enough that we take our premises to be true, that is, judge them to be true. Second, ‘cognizant’ seems to have a success grammar built into it, and so Frege’s characterization would seem to imply that one cannot reason badly: if one is reasoning at all, one is reasoning to a conclusion that one has justification to draw. There is, of course, a substantial tradition in philosophy that maintains that there are limits to the extent to which one can be said to be reasoning badly. However, I don’t want to assume any such limitation up front. If there are limits on the extent to which one can reason badly, they should be explained by a correct account of the nature of inference. At any rate, even if there were such limits they would surely leave a great deal of room for reasoning that is bad. And even this modest room for error would not seem to be accommodated by Frege’s characterization of inference in terms of one’s being ‘cognizant’ of the justification one has for one’s conclusion. I would therefore prefer to offer the following modified version of Frege’s characterization: (Inferring) S’s inferring from p to q is for S to judge q because S takes the (presumed) truth of p to provide support for q.On this account, my inferring from (1) and (2) to (3) must involve my arriving at the judgment that (3) in part because I take the presumed truth of (1) and (2) to provide support for (3). Let us call this insistence that an account of inference must in this way incorporate a notion of “taking” the Taking Condition on inference. Any adequate account of inference, I believe, must, somehow or other, accommodate this condition. (Taking Condition): Inferring necessarily involves the thinker taking his premises to support his conclusion and drawing his conclusion because of that fact.2 The intuition behind the Taking Condition is that no causal process counts as inference, unless it consists in an attempt to arrive at a belief by figuring out what, in some suitably broad sense, is supported by other things one believes. In the relevant sense, reasoning is something we do, not just something that happens to us. And it is something we do, not just something that is done by subpersonal bits of us. And it is something that we do with an aim—that of figuring out what follows or is supported by other things one believes. It’s hard to see how to respect these features of reasoning without something like the Taking Condition. Although I think that these observations ought to be enough to underwrite the Taking Condition, it is worth noting that that condition can help us make sense of two further phenomena involving inference that might otherwise seem puzzling. 4 Deductive versus inductive inference It is tempting to think that there are two kinds of inference—deductive and inductive. But in what could the difference between these two kinds of inference consist? Of course, in some inferences the premises logically entail the conclusion and in others they merely make the conclusion more probable than it might otherwise be. That means that there are two sets of standards that we can apply to any given inference. But that only gives us two standards that we can apply to an inference, not two different kinds of inference. Intuitively, though, we are able to distinguish between a person who intends to be making a deductively valid inference versus someone who intends merely to be making an inductively valid one. A scientist need not be perturbed if we were to point out to him that some inference of his was not deductively valid, but merely inductively strong; but a mathematician would, and should, be perturbed. How, though, are we to capture the difference between the scientist and the mathematician, if not in terms of how they take their premises to be related to their respective conclusions? So, the Taking Condition enables us to distinguish, as intuitively we ought to be able to do, between deductive and inductive inferences. 5 Impossible inferences Some inferences seem not only obviously unjustified, and so not ones that rational people would perform; more strongly, they seem impossible. Even if you were willing to run the risk of irrationality, they don’t seem like inferences that one could perform. Consider someone who claims to infer Fermat’s Last Theorem (FLT) directly from the Peano axioms, without the benefit of any intervening deductions, or knowledge of Andrew Wiles’s proof of that theorem. No doubt such a person would be unjustified in performing such an inference, if he could somehow get himself to perform it. This fact about justification is itself a puzzle, for the inference would be a truth-preserving one. It is certainly a puzzle for many reliabilist views of justification. However, my main concern just now is not to explain why such a transition would be unjustified, but to explain why we have the considerable feeling (as Kripke is fond of saying) that no such transition could be an inference to begin with. The Taking Condition provides an answer. For the Peano Axioms to FLT transition to be a real inference, the thinker would have to be taking it that the Peano axioms support FLT’s being true. And no ordinary person could so take it, at least not in a way that’s unmediated by the proof of FLT from the Peano Axioms. (The qualification is there to make room for extraordinary people, like Ramanujan, for whom many more number-theoretic propositions were obvious than they are for the rest of us.) 6 The doxastic construal of taking3 How, though, should we understand the Taking Condition? What is it to believe something because one takes it to be supported by other things one judges to be true? What kind of taking are we talking about? The first thought that is likely to occur to one is that the Taking Condition requires that a thinker have a meta-belief about the relation between his premise judgments and his conclusion, a belief to the effect that his premise judgments supply him with a justification for believing his conclusion. We may call this the “full-fledged normative doxastic construal” of the Taking Condition. (FFNC) My judging (1) and (2) supports my judging (3). Such a construal would be problematic in at least two respects. First, we don’t want to say that a thinker regards the fact that he judges p to be his reason for judging q. We want his reason for judging q to be the (presumed) truth of p, not the fact that he judges p. Second, there is a worry that the content of any such normative belief will be too sophisticated for ordinary thinkers. A child, we are inclined to think, can reason. Luke and Drew are playing hide-andseek. Seeing Drew’s bicycle leaning against the tree, Luke thinks: “If he were hiding behind that tree, he would not have left his bicycle there. So, he must be behind the hedge.” That looks like reasoning. But do children have meta-beliefs about the relations between their premise judgments and their conclusions? Do children have the concepts of premises and conclusions? Do they have the normative concept of one belief justifying another? Worried, perhaps, by these sorts of consideration, some philosophers have proposed doxastic views that are “meta-propositional” rather than meta-attitudinal. Philip Pettit, for example, has written: A meta- propositional attitude is an attitude towards a proposition – if you like, a meta- proposition – in which propositions may themselves figure as objects of which properties and relations are predicated. Some meta-propositions will ascribe properties like truth and evidential support and relations like consistency and entailment to propositions, as in the claim that ‘p’ is true or that it is inconsistent with ‘q’. … The absence of meta-propositional attitudes in the robot and its more flexible counterparts means that they are subject to a salient restriction…because the robot and its counterparts don’t have meta-propositional attitudes, they cannot ask themselves similar questions about connections between propositions, say about whether they are consistent or inconsistent, and then do something – pay attention to the inter-propositional relations – out of a desire to have a belief form one way or the other. This restriction means that the robotic creatures cannot reason. (Pettit 2007 , pp. 498–499) So, Pettit’s idea is that a genuine reasoner must be able to ask itself questions about the propositions that figure in its judgments and let the answers to those questions guide his thinking, if he is to be a genuine reasoner. Applied to our Taking Condition, it would imply that a genuine reasoner would have to believe (take it) that his conclusion proposition follows, in a suitably broad sense, from his premise propositions and let that guide what conclusion he arrives at. Someone performing the seemingly simple (Rain) inference would have to have the meta-propositional belief (Meta-Rain): (3) follows from (1) and (2). Now, this proposal successfully evades the first objection to the meta-attitudinal proposal; but it seems to leave the second one in place: we still seem committed to attributing to ordinary thinkers implausibly sophisticated concepts and propositional attitudes. Do Luke and Drew, playing hide-and-seek, really need to have the concept of a proposition, and the concept of one proposition following from another? To deal with this residual problem, we could try going even more first-order, requiring only that a reasoner have a belief in the appropriate conditional: (First Order Rain Belief, FORB): If It rained last night and If it rained last night, then the streets are wet, then, the streets are wet. But this proposal only invites new objections. First, we know from Carroll’s famous note (1895 ) that we must at all costs avoid insisting that (FORB) be part of the premises from which the conclusion of the (Rain) inference is to be derived. In that direction lies a vicious regress from which no reasoning can emerge. So, if there is to be such a firstorder belief invariably involved in inference, it must act as a background condition rather than as a premise judgment.4 But what exactly is the difference between a premise belief and a belief that plays a role in the background of an inference? Presumably, it is the difference between a belief on which the conclusion is based versus a belief that, while somehow involved in the inference, is not one on which the conclusion is based. But the notion of a conclusion’s being based on a premise judgment is precisely the notion that we are trying to understand, in investigating the nature of inference. So in employing the notion of a belief that plays a role as mere background, and so not one on which the conclusion is based, we seem to have helped ourselves to the very notion at issue. Secondly, there is a pair of problems having to do with the question how any such doxastic view would mesh with a plausible account of inferential justification. An account of the nature of inference will eventually have to confront the question what conditions an inference must satisfy if it is to transmit the justification that a thinker has for its premises to its conclusion. In other words: What explains why I would be justified in concluding (3) on the basis of (1) and (2)? If we have such a doxastic account of the nature of inference, it is natural to think that the answer to the justification question will involve the relevant belief. What conditions would (FORB) have to fulfill if (Rain) is to be a justified inference? Clearly, (FORB) should be true. But is that enough? It doesn’t seem so. What if (FORB) were arrived at unjustifiably? Given that it is a belief, it can be assessed not merely for its truth but also for its epistemic properties. And given that it can be so assessed, a positive appraisal of its epistemic standing seems required for the inference in which it is involved to be a justified one.But it’s hard to see how one could acquire a justification for (FORB) without this involving some justified, logical inferences, whose justification will eventually depend on that of a basic rule (Modus Ponens, we may suppose). And, of course, if this were true, then we would have no hope of explaining what it is for (Rain) to be justified by saying that it involves (FORB)’s being justified, for we will have turned around in a tight little circle. However, let us waive this problem. Let’s assume that we have some non-inferential way of arriving at a justified belief in (FORB). Still, it’s very hard to see how such a belief could explain why I am in a position to justifiably perform the (Rain) inference. How does my belief in (FORB) If, It rained last night and If it rained last night, then the streets are wet, then the streets are wet. explain why I am in a position justifiably to infer that 1. (3) The streets are wet from 1. (1) It rained last night and 1. (2) If it rained last night, then the streets are wet. Wouldn’t anything that looked like an explanation here need to invoke the subject’s ability to infer from (FORB) that the 1-2-3 pattern of inference is valid? And wouldn’t any such explanation, therefore, be hopelessly circular?