One of the basic forms of reasoning, inference is closely tied to the relation of implication, or its converse, consequence. If one proposition implies another, then the latter proposition is a consequence of the former. In its most basic sense, inference is the drawing of consequences. It is useful therefore to distinguish between the consequences that a proposition has, and the consequences that it is necessary (or permissible, or reasonable) to draw. There is some disagreement among logicians as to whether the distinction is a tenable one. Some are of the view that an ideally rational individual will draw all consequences of any proposition he or she holds. Others hold that the rational thing to do is to draw only those consequences that are relevant to the agent's interests and circumstances there and then. On the first view, every rule of implication (or what is the same thing, rule of consequence) is a rule of inference. On the second, the rules of inference are disjoint from the rules of implication. Even so, it is nowhere in doubt that implication furnishes a constraint upon inference, as follows. An agent's inference of proposition B from proposition A requires that A implies B.
Also embedded in this second position is the claim that, whereas implication is strictly a pro-positional relation, inference is a linguistic and/ or mental act performed on propositions by reasoning agents., (There was a time when every school child was told ‘Propositions imply. People infer.’) Accordingly, implication can be adequately characterized by its syntactic and semantic properties alone. But getting inference right requires that we also take account of pragmatic factors, that is to say, factors affecting an agent's use of the propositions involved. These factors include the agent's goals and the cognitive resources he or she possesses for their attainment - resources such as information, time and computational capacity. So seen, inference is a matter of belief or commitment change. The tension between these two positions on consequence-having and consequence-drawing is also reproduced within present-day belief revision theory. In what is regarded as the classical or Alchourrón, Gärdenfors and Makinson (AGM) approach to belief revision, belief is closed under consequence (Alchourrón et al. 1985). In other words, a perfectly rational agent will draw all consequences of anything he believes. Against this approach is the objection that requiring belief to be closed under consequence imposes on reasoning agents a task that is both computationally too complex for them and irrelevant to their interests (Harman 1986).
Given that implication is a condition on correct inference, inference will vary in kind with variations in implication type. Accordingly, it is customary to recognize a distinction between deductive and inductive inferences, governed by implications that are deductive in the first case, and probabilistic in the second case. If A deductively implies (or entails) B, then the inference of B from A is likewise deductive. Deductive inference is truth-preserving. This means that if A deductively implies B and A is true, then in drawing B as a consequence we may be wholly assured of B's truth as well. However, if A probabilistically implies B, the inference from A to B is inductive. Though not truth-preserving, correct inductive inferences are likelihood-enhancing. What this means is that if A probabilistically implies B, then although A does not verify B, it does offer it some degree of positive confirmation. Another way of saying this is that if the probability of B given A is sufficiently high, then A is inductively evidential for B. The literature also reveals a growing tendency among theorists to acknowledge a third category of inference called abduction. Whereas deductive inference is truth-preserving and inductive inference is likelihood-enhancing, abduction is sometimes described as ignorance-preserving. Abductive inference is typified by (though not limited to) what is called ‘inference to the best explanation’. This is a form of reasoning in which from the fact, if true, that a certain proposition H would explain a certain known state of affairs, it is concluded that it is reasonable to conjecture that H and (provisionally and defeasibly) to make it a ‘working hypothesis’. The idea that abduction is ignorance-preserving derives from C.S. Peirce's insistence that the fact that H has been successfully abduced is no reason to believe it to be true. In other words, Peirce thinks that abductive success is not inductively evidential for H.
It is well to emphasize that as implication varies as to type, so too does inference. Over the past forty years, studies in logic, computer science, linguistics and psychology have discussed types of implication not adequately elucidated by the deductive-inductive-abductive trichotomy, each associated with a corresponding type of inference. Prominently included here are inferences variously characterized as defeasible, nonmomotonic, default and plausibilistic. These inferences in turn form a kind of hierarchy, with defeasible inference at the top. An inference is defeasible when, although reasonably drawn, it embeds the possibility of error (Hart 1994; McCarthy and Hayes 1969). Non-monotonic inference is a major species of defeasible inference. It is a presently reasonable inference whose reasonability may be lost upon admittance of new information (Makinson 1994). Default reasoning is a particular kind of nonmonotonic reasoning (Reiter 1980). It is typified by what computer scientists call ‘negation as failure’. For example, from the ‘failure’ of the airport's departures board to list a late-night flight to London, it may be inferred that there is no such flight.
Negation as failure is linked to a species of implication of particular interest to students of pragmatics. It was dubbed by H.P. Grice (1975) ‘conversational implicature’, which is an instance of presumptive reasoning like default inference. For example, the information that one speaker imparts to another during conversation implicates that it is relevant to the topic of the discussion. So, in default of information to the contrary, one may presume that what one's interlocutor tells one is relevant to the issue at hand. A further type of presumptive reasoning arises from the pragmatic notion of presupposition (Kempson 1975; Stalnaker 1973). If A presupposes B, we may state that in saying that A, a speaker is implicitly also saying that B. A hearer may take B as a default. That is, he may presume it to be the case unless he learns otherwise. Plausibilistic reasoning is weaker than probabilistic reasoning (Rescher 1976). If you hold that something is probable, you signal (or conversationally implicate) that it is something that warrants a positive degree of belief. But if you say only that it is plausible, you signal that it is worthy of consideration. In some respects, abduction may also be classified as plausibilistic.
The present hierarchy of inference types does not fit smoothly into the deduction-induction-abduction trichotomy, and in some respects cuts across it. To take just two examples, inductive inference is typically both defeasible and nonmonotonic, and deductive inference is nonmonotonic with regard to soundness even if monotonic with regard to validity. The monotonicity of validity provides that new information can never change a deductively valid argument into a deductively invalid argument. But since that same information might falsify a prior premise, thus rendering the original inference unsound, deductive soundness is not monotonic.
Except for those who see no difference between consequences-had and consequences-to-be-drawn, the systematic study of reasoning has two main components. One is a theory of the implication relation that governs the inference process. Theories of implication also typically include accounts of conditionality (Lycan 2001). This is motivated by the fact that whenever A implies B, there is a sense of ‘if … then’ in which the corresponding conditional sentence ‘If A then B’ is true. The second component of the study of reasoning investigates the various additional factors that are involved in consequence-driven changes of mind - factors, again, such as an agent's goals, the cognitive standards required to attain them, the cognitive resources needed to meet the standard, and so on. Broadly speaking, logicians have concentrated on the first task, and linguists and psychologists have focused on the second. But the border is far from sealed. Psychologists who study implication include Daniel Kahneman and Amos Tversky (1974), Philip Johnson-Laird and his colleagues (Johnson-Laird and Byrne 1991), John Mac-Namara (1994), Lance Rips (1994) and Gerd Gigerenzer (2004) and his colleagues. Likewise, among linguists who study implication relations one might mention George Lakoff (1970), Dan Sperber and Deirdre Wilson (1995), and Gregory Carlson and Francis Jeffry Pelletier (1995) and their colleagues. In contrast, logicians who study the non-implicational aspects of inference, such as the goals of the agent, the agent's resource limitations, the relevance of information to goals, and so on, include (in addition to Harman) Bernd van Linder and his colleagues (1995, 2002), Frank Veltman (2000), Raymond Reiter (2001), and Dov Gabbay and John Woods (2003, 2005).
See also: Abduction; computational pragmatics; entailment; Grice, H.P.; implicature; negation; Peirce, C.S.; reasoning
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