In: Culture in Retrospect pp 194-214, Roux APJ and Coetzee PH (editors), Pretoria: UNISA Press 2001 (ISBN 1-86888-166-0)

Knowledge and belief : the agent-oriented view

Johannes Heidema and Willem Labuschagne

Introduction

From our vantage at the end of the 20th century it is possible to trace an evolution in the use of the word ‘knowledge’, a journey from the absolute to the relative. This paper sketches the main steps in this evolution and describes a notion of information template, in terms of which the beliefs of an agent may be ordered in a spectrum ranging from those to which the agent attaches the greatest weight (and which fulfil the operational role of knowledge) to those the agent is readily willing to retract.

Our perspective is a logical one, and more broadly a scientific one. No apology for this is necessary–the chief advances in philosophy during the past century have occurred in logic and the philosophy of science. Indeed, much of postmodernism is merely a distorted reflection of ideas that have contributed usefully to the development of logic and science. We resist the temptation to deliver yet another critique [Ki], [Gal], [Ho], [GL], [SB] of the intellectual frivolity perpetrated by French philosophers during this century (and the humourless mangling of their jocund absurdities by sanctimonious epigones), choosing a more constructive goal. By giving due prominence to semantic issues, we draw together apparently disparate strands from modal logic, information theory, and non-monotonic logic. The unifying framework that results embodies the current agent-oriented paradigm in logic, from which information templates emerge naturally.

Milestones in the journey from knowledge to belief

At the start of the 20th century, Bertrand Russell inherited from Gottlob Frege the notion of a formal system and a mature quantification theory. Implicit in Russell's approach was the assumption that logic is about ‘the’ universe of mathematical and physical objects and that well-formed sentences have significance (meaning) to the extent that they report on facts or beliefs about this universe ([Ru] pp168-172). To fearlessly oversimplify, there was assumed to be just one way to assign denotations of constants and hence meaning to the sentences of a formal system. In terms of this logical atomist paradigm, it made sense to speak of a sentence as being true without qualifying the description by making explicit the universe of discourse and the denotations in it relative to which truth obtains. And it therefore made sense to claim, as Russell famously did, that ‘Quadruplicity drinks procrastination’ has no meaning.

This approach did not long survive the rise of model theory following Tarski's publications on truth and logical consequence [Ta]. For an array of reasons that deserve, but cannot here be granted, extended treatment, the model-theoretic paradigm supplanted the logical atomist paradigm. Truth came to be seen in terms of a relation between a sentence and an interpretation of the language. An interpretation is a set-theoretic structure, and the relation determining the assignment of truth values is established by specifying the denotations of terms and predicates in the domain of the interpretation [En], [BS]. Under the model-theoretic paradigm, there is no difficulty at all in giving a meaning to ‘Quadruplicity drinks procrastination’, for it is necessary only to select a set-theoretic structure and assign denotations to the terms ‘Quadruplicity’ and ‘procrastination’ and to the predicate ‘drinks’. Whether the resulting interpretation turns out to be useful is another matter entirely.

The new paradigm helped undermine the connotation of absoluteness carried by the word ‘knowledge’. A tradition traceable back to Plato [Sch] holds that knowledge is justified true belief. For an unqualified ‘true’, it now became prudent to substitute reference to the interpretations relative to which truth obtained. The journey towards relativity was carried further by difficulties surrounding the notion of justification, several of which are pointed out by Musgrave [Mu].

If we take a sentence to be (conclusively) justified whenever it is entailed by some sentence that is true (in all the interpretations deemed to be of relevance), then all sentences that are true (in the relevant class of interpretations) are justified, and so too are all the logically weaker statements entailed by them. This is problematic on two accounts. Not only does it require the agent whose belief is thus justified to be logically omniscient, but now ‘true’ has subsumed ‘justified’, a subsumption that is undesirable because two agents may both believe a true statement, but one may be ‘justified’, in some intuitive sense, in believing it, while the other agent may not. Thus the notion of justification should not only involve some added value beyond truth but should be a relative concept taking an agent as a parameter and subject to resource bounds in the agent.

In the justified-true-belief view of knowledge, belief is of interest only as a germ from which knowledge may spring. What if only partially justified beliefs could spring from the germ? Consider scientific reasoning. During the 1930s Popper's revival of Hume's demolition of induction as a source of knowledge [Poa] interacted with contemporaneous developments in probability and statistics to bring about Carnap's logical reconstruction of probability theory [Caa]. A long controversy ensued between Popper and Carnap (and their adherents) over such questions as whether Carnap's explication of degree of confirmation was supposed to (or could possibly) be an adequate measure of the acceptability of scientific theories, and whether probabilistic support qualified to be regarded as inductive support [Cab], [Pob], [BHa], [Poc], [BHc], [Cac], [Cad], [PMa], [Le], [Je], [Gob], [PMb], [Red], [Gi], [Ee]. Two insights emerged almost in passing from the sustained scrutiny to which scientific reasoning was subjected.

Firstly, there is general acceptance that scientific reasoning is defeasible. Suppose that one were to adopt a conjecture because, in the light of evidence, it is more probable than some competitor (say, its negation). Then this would be only a tentative adoption, for it might be necessary to retract the conjecture in the light of new evidence relative to which its probability is less (the new evidence thus undermining or defeating the conjecture). Indeed, it might be necessary to revise the whole probability measure underpinning the initial inference (the key idea of Bayesian inference [HU]). However one may view the role of testing in distinguishing between theories, scientific ‘knowledge’ occupies a middle ground. A scientific theory is a set of beliefs judged reliable because they have survived a process of testing, but this reliability does not possess the mystical absoluteness that is one connotation of the word ‘knowledge’ and that used to perfuse reference to ‘the laws of nature’.

Secondly, it has come to be seen that the naïve view of induction, as a process by which instances (such as observations of black ravens) partially justify a universally quantified sentence (such as ‘All ravens are black’), is mistaken. This insight is due not only to Hempel's paradox (which asserts that observation of any non-black non-raven would have to count as a confirming instance of the generalisation) and Goodman's paradox (which asserts that a given instance can be regarded as confirming a range of different generalisations [Goa]), but also to a growing perception that probability constitutes merely one of many mechanisms capable of supporting defeasible conjectures. In terms of the class of alternatives known as non-monotonic logics, the sentence ‘All ravens are black’ of the object language would not be the vehicle used to reflect a preponderance of black ravens in the world. Instead, a default rule would be used to represent this information, and such default rules would appear either as special-purpose rules of inference (in the default logic of Reiter [Rei]), or as semantic orderings in logics based on minimal model semantics (which we shall examine in detail later). Whatever the mechanism of formalisation, the formation of defeasible beliefs could no longer be straitjacketed into a pattern of instances confirming a universal generalisation.

If the analysis of scientific reasoning drew attention to the idea of defeasibility and the inadequacy of naïve induction, then the final step in construing belief as the concept of primary utility had its origin in the advent of the computer during the 1940s and Turing's 1950 paper [Tu] on the possibility of machine intelligence. Discussions of scientific reasoning were concerned only with human agents, and there have never been lacking those willing to endow humans with special epistemic powers, from Plato to Penrose [Pe]. Thus the defeasibility of scientific reasoning could be interpreted as a necessity imposed by a restrictive methodology. However, the new fields of artificial intelligence and cognitive science found it natural to adopt an agent-oriented stance which broadened the range of application of such concepts as rationality beyond merely human agents.

What is an agent? Pollock [Pol] memorably contrasts agents and rocks. Each is a stable structure in a dynamically changing world. But whereas a rock achieves its stability passively, by virtue of being hard to destroy, an agent achieves its stability by interacting with its surroundings in order to render these more congenial to continued survival. Whether animal, vegetable, or mineral, a rational agent is one which has beliefs reflecting the state of its environment and constraining its actions. As for the source of these beliefs, experience in building robots shows that knowledge, in any absolute sense, is unobtainable even by observation. Sensory input requires disambiguation. For instance, the two-dimensional image upon the retina of an agent may in principle be the projection of an infinite number of different three-dimensional configurations in the field of vision [Ma], [Pi], and the agent will defeasibly have to choose between these explanations (although the mechanism of choice is typically unconscious and performed by a specialised module). Even at the conscious level, common-sense reasoning goes far beyond the constraints of valid inference while maintaining a greater or lesser degree of plausibility. It is not merely scientific reasoning that is defeasible; defeasibility is ubiquitous.

An illustrative application

In the sections that follow, we shall repeatedly refer to the following example. Consider an agent observing a simple fan-heater system (FHS). The FHS has just two components: a fan and a heater. (Lest the system appear wholly trivial, reflect that it is a coarse-grained view of a nuclear powerplant in which the heater symbolises the pile and the fan the cooling system.) Of interest to the agent is the state of the system, and the state of the system is determined by two elementary facts: whether the fan is on, and whether the heater is on. The agent represents his beliefs about the system in a propositional language with two atoms: p represents the fact that in English would be expressed by ‘The fan is on’ and q the fact expressed in English by ‘The heater is on’. The propositional language is equipped with the standard connectives such as  for ‘it is not the case that’,  for ‘and’, and  for ‘either … or … or both’.

The FHS may at any time be in one of four states. Each state may conveniently be treated as a vector of two co-ordinates. According as p is true or not, the first co-ordinate is either 1 or 0, and similarly the second co-ordinate is 1 if q is true, 0 otherwise. Thus 11 is the state in which both the fan and the heater are on, 10 the state in which the fan is on but the heater is off, 01 the state in which only the heater is on, and 00 the state in which both are off.

The agent wishes to answer, with the aid of observations of the system, the question ‘What is the current state of the system?’ The fanciful reader may imagine that the system is really a nuclear powerplant and therefore potentially dangerous. The agent sits in the control room and has the responsibility, should the system enter a state which may lead to meltdown or some other disaster, of averting calamity by pulling a large red lever, which shuts down the system. Ideally, the agent would have complete information about the state of the system, perhaps by being equipped with sensors enabling it to observe clearly both the fan and the heater. However, the interesting cases are those in which the agent has incomplete information

The problem of incomplete information

As Davis [Da] demonstrates, everyday life requires us to cope with partial information. One believes that the water in the tea-kettle will come to a boil soon after one presses the switch, but does not know how soon, or in fact whether all the preconditions for boiling are satisfied (is the kettle in good working order?).

This incompleteness cannot be overcome by trying harder. Every agent has limited resources. In particular every agent has a limited sensory apparatus, and as in the case of vision must disambiguate input by selecting from amongst a large collection of external circumstances that would produce the same sensory input. Furthermore the preconditions that have to be satisfied in order for the water to boil (or one's car to start or indeed for any action to bring about the intended effect) may be multiplied endlessly. Should one wish to check in the newspaper how one's investments in the stock market are doing, it is not even feasible to enumerate, let alone to check the satisfaction of, preconditions such as that the vendor is willing to sell one a paper, that the newspaper is in a language one understands, that there is sufficient light to read it by, that no lunatic passerby will tear the paper from one's hands, that the newspaper is not a fake distributed so as to spread disinformation, that the editor has not decided to stop printing stock prices, that one will not suffer an hallucination instead of actually reading the paper, … The infeasibility of enumerating and checking preconditions is known in artificial intelligence as the qualification problem, and it is merely one amongst a number of similar problems like the frame problem [Br], [Py].

One copes with incomplete information by invoking some mechanism for the formation of defeasible beliefs. Such a mechanism may apply a default rule (a rule having general but not universal validity). Default rules are sometimes founded on judgements of relative frequency (quantitative in the case of probability theory but quite likely to be qualitative in other contexts). Of this kind would be a rule cautioning that to enter a lottery is to lose your investment because of the very small chance of winning. But default rules may be founded on a notion of typicality (or normality): the psychologists Kahneman and Tversky have shown that ordinary reasoning often violates the principles of probability theory and seems instead to have recourse to stereotypes [KST].

These considerations raise several questions for the logician. How can one measure the information available to an agent? How can one represent constraints on an agent's sensory apparatus (or more general resource bounds) that may compel it to operate with incomplete information? How can one implement mechanisms for the formation of defeasible beliefs? One strand in the progress of logic during the 20th century is tied to precisely these questions, and its course is traced in the next three sections, which deal respectively with semantic information theory [CBH], [Hib], [Hic], [Hid]; modal logic, and non-monotonic logic [Mca], [Mcb], [Sho], [KLM], [LM].

Theories of information

How can one measure the information available to an agent? The term information is usually assumed satisfactorily to have been explicated by Hartley [Ha] and Shannon [SW] in the context of the transmission of messages, for instance over a telephone line. Shannon himself was aware that the informal use of the term was not fully captured by Hartley's explication and cautioned: “Frequently the messages have meaning; …These semantic aspects of communication are irrelevant to the engineering problem”. Information is of course associated with the elimination of uncertainty, but two kinds of uncertainty may be involved. There is the unexpectedness of the choice of a particular formulation of a message, and there is the unexpectedness of the events reported by the message. Under very special circumstances there may be an isomorphism between the two, as when each relevant event is represented by an unique string of symbols, but in general, languages rich enough to describe non-trivial systems will contain multiple strings that are semantically equivalent, so that the same event may be represented by two or more strings of which one is perhaps a more conventional or ordinary usage than another.

That the Hartley-Shannon definition addresses the uncertainty of formulation rather than the uncertainty of event, may be seen as follows. Suppose that a communication channel is able to transmit n messages (each message being some string of symbols from a given alphabet). Assume, for simplicity, that each of the messages is equally likely to be selected by a sender, so that each message s has a probability p(s) = 1/n of being chosen. Motivated by a desire “to set up a quantitative measure whereby the capacities of various systems to transmit information may be compared,” Hartley defined the amount of information produced when a message s is selected from n equally likely messages  to be log 1/p(s). Since 1/p(s) = n, the connection with the capacity of the system is clear. Use of the logarithm is inessential; it merely confers the benefit that, as Shannon puts it: “Parameters of engineering importance such as time, bandwidth, number of relays, etc., tend to vary linearly with the logarithm of the number of possibilities.”

From the formula it is apparent that the Hartley-Shannon definition relies on the relative frequencies of occurrence of symbols rather than on the meanings of messages, and therefore would ascribe a greater information content to an unexpected formulation. For example, the string of symbols ‘A rotating sphere of compressed silicaceous material accumulates no accretion of bryophitic vegetation’ would be adjudged to have greater information content than ‘A rolling stone gathers no moss’, merely because the former is in practice uttered less frequently. While the emphasis on relative frequency of symbols is useful for many important purposes such as error-detection and correction, it is a doubtful principle on which to base an explication of information.

An account of the mismatch between symbol-frequency and event-frequency was first set out by Bar-Hillel [BHb], who went on (with Carnap) to formulate a theory of semantic information [CBH]. Semantic information theory focuses on the elimination not of the uncertainty pertaining to the choice of message but that related to the question ‘What is the state of the world?’ As suggested by Popper [Poa], the more information we have about a system, the more states we are able to rule out. Thus the intuition underlying semantic information theory is model-theoretic: the set of possible states of the world (or of some system) is partitioned into two blocks consisting firstly of those states that, as far as is known, remain possible candidates for being the actual state of the world, and secondly those states that may be excluded from consideration. (The special case of no information corresponds not to a genuine partition but to a pair consisting of the set of all states, these being viewed as possible, and the empty set of states, these being the excluded states.

This may be illustrated by the FHS. Suppose the agent observing the system is equipped with a sound sensor only, and so can detect whether the fan is on but not whether the heater is on. If the actual state is 11, the information available to the agent may be represented by the partition that follows, in which the lower block comprises the possible candidates and the upper block the states that the agent is able to rule out:
01, 00	
	
11, 10	
Strictly speaking, we have here not merely a partition but an ordered partition: to encode the idea that one block contains excluded states and the other not, we have adopted a convention which positions the excluded states above the remainder.

This dichotomous ordered partition (which orders the states by assigning them to one of two levels) is a local representation of the agent's information, in the sense that the state of the FHS may change and a new ordered partition be appropriate. For example, suppose that the actual state changes from 11 to 01, by virtue of the fan switching off. Given the agent able to hear the fan but not feel the heater, a suitable representation of the available information in the new state would be given by the ordered partition
11, 10	
	
00, 01	
We shall see in the next section how epistemic logic links together the local representations of an agent's information into a global representation.

The semantic information represented by a dichotomous ordered partition of states may be expressed sententially by exploiting the finiteness of the system. Every state has a corresponding state description in the appropriate propositional language; a state description is a conjunction in which every atom appears exactly once and is either negated or unnegated. The significance is that a state description is true in exactly one state, namely that which it describes. For example, the state 10 of the FHS has the state description pq. Now exclusion of a state may be indicated by negating its state description. The result is a sentence (called a content element) falsified only by the relevant state and equivalent to a disjunction in which every atom appears once, negated or unnegated. Every sentence is equivalent to a set (conjunction) of content elements and every dichotomous ordered partition of states may therefore be represented by the sentence whose content elements correspond to the excluded states. (Equivalently, by the logically strongest sentence true in the bottom states.)

Summarising, semantic information theory model-theoretically represents the information available to an agent as a dichotomous ordered partition of states, and can also represent this information sententially via a correspondence between content elements and excluded states.

Epistemic logic

How can one represent constraints on an agent's sensory apparatus (or more general resource bounds) that may compel it to operate with incomplete information? The possible world semantics of epistemic logic gives one method.

Epistemic logic uses a modal language to sententially express the information available to an agent, and does so for two reasons. Firstly, by extending the simple propositional language of the previous section so as to include a modal operator , and by defining the semantics in such a manner that  expresses ‘According to the information available to the agent,  is the case’, it becomes possible to distinguish the agent's perhaps limited information from the information that an omniscient oracle would have. (In the case of multiple agents, include multiple operators [1], [2], …, [n] indexed by identifiers for the agents.) Secondly, it becomes possible to represent information acquired by introspection or the beliefs one agent may have about other agents' beliefs.

For our purposes, the use of modal operators is less pertinent than the possible world semantics itself. A set of possible worlds (for our purposes, these may be taken to be the states of a system like FHS) is structured by a binary accessibility relation [Ch]. The accessibility relation encodes the constraints which limit the agent's access to facts about the system. For example, in the case of an agent observing the FHS and able to hear the fan but not detect whether the heater is on, the set of possible worlds would be W = {11, 10, 01, 00} and the appropriate accessibility relation
R = {(11, 11), (11, 10), (10, 11), (10, 10), (00, 00), (00, 01), (01, 00), (01, 01)}
which is clearly an equivalence relation having as its equivalence classes the blocks {11, 10} and {01, 00} respectively. The former consists of the states regarded as possible candidates when the system is actually in state 11 (or in 10), and thus conveys the same information as the ordered partition

01, 00	
	
11, 10	

previously did. The second equivalence class, namely {01, 00}, consists of the states regarded as possible candidates when the system is actually in state 01 (or 00), and is equivalent to the ordered partition

11, 10	
	
01, 00	

Recalling that these ordered partitions could be viewed as local model-theoretic representations of the information available to the agent, it follows that the accessibility relation may be viewed as a global model-theoretic representation. When the system is in a state w, the accessibility relation gives an equivalence class of states each of which is, because of the limitations to which the agent is subject, indistinguishable from w and hence also regarded as a possible candidate to be the actual state. This equivalence class suffices to determine a dichotomous ordered partition. (The extreme case of complete information would be encoded by an accessibility relation which is the identity relation on the set of possible worlds.)

The sentential representation of the information available to the agent is governed by the accessibility relation according to the following truth definition: a sentence  is true at a state w if  is true at every state in the equivalence class of w. The set of sentences of the form  that are true at w constitute the sentential expression of the information available to the agent when the system is actually in state w, and it is customary to speak of the sentences  as the knowledge or more generally the beliefs of the agent. (There is a tradition according to which the term knowledge is used when the accessibility relation is an equivalence relation and belief is used if the criteria are relaxed, but this is not relevant for our purposes.)

Summarising, epistemic logic and semantic information theory are compatible inasmuch as the accessibility relation may be taken to be a device that assigns, at every state w, a set of states that, by virtue of being candidates for the actual state, define a dichotomous ordered partition of the familiar sort.

What about resource bounds? The possible world semantics described above does not lend itself to the encoding of resource bounds, but there are alternative approaches that do, such as that of Konolige [Ko], who essentially replaces accessibility relations by pairs (, ) in which  is a set of sentences (initial beliefs) and  is a deduction algorithm which may be rendered incomplete by constraints on time and memory.

Non-monotonic logic

How can one implement mechanisms that support the formation of defeasible beliefs? An attractive option is to use default rules. One way to think of a default rule, characteristic of the approach known as minimal model semantics, and specifically the more modern formulations of that approach [Sho], [KLM], [LM], is to see it as a semantic ordering, which is to say an arrangement of the states into levels (an arbitrary, as opposed to dichotomous, ordered partition). The states at the lowest level are the most normal or most preferred states, those a level higher are less normal, and so on. This is a natural way to reflect statistical or heuristic information. If, for example, it is evident from past history that the FHS is typically in state 11, less usually in state 00, and very seldom in states 10 or 01, then this suggests a default rule represented as follows:

10, 01	
00	
11	

The default rule supports the formation of defeasible beliefs by refining the information available to the agent via observations. Given the agent able to hear the fan but not detect the heater, and assuming the FHS to be, for the sake of argument, in state 11, the information available via observation is represented by

01, 00	
	
11, 10	

which may be refined by the default rule (in a manner we shall gloss over) to produce 

00, 01	
10	
11	

If we take the agent's beliefs to be sententially represented by the logically strongest sentence true in the states at the bottom level, then the agent now believes, defeasibly but accurately, that 11 is the actual state of the system (i.e. has the belief pq). Lest this seem too good to be true, reflect that had the FHS been in the actual state 10, the agent would have had exactly the same information available via observation, and would have applied exactly the same default rule, and would have ended up with exactly the same defeasible belief–which in this case would have been false. But the circumstances would have been exceptional.

Summarising, minimal model semantics uses arbitrary ordered partitions to refine the information available to the agent via its sensors, and sententially represents the defeasible beliefs of the agent by sentences true in all the states occupying the bottom level of the resultant ordered partition.

Information templates

Default rules have a natural representation as not-necessarily-dichotomous ordered partitions of states, and once a default rule is used to refine the information available to an agent via observation, the result may very well also be an ordered partition consisting of more than two blocks. There is an upper limit, however. A system consisting of k components (and whose states are therefore determined by the status of k elementary facts, corresponding to k atoms in a propositional language) has 2k possible states. Thus a partition of the set of states can have no more than 2k blocks.

By an information template we understand (loosely speaking) the arrangement of 2k levels, visualised perhaps as empty boxes stacked on one another. Having in mind that the question of overriding interest to the agent is ‘What is the actual state of the system?’ we adopt the convention that the lowest box is for states that are considered most normal or most likely candidates to be the actual state, and that the uppermost box is reserved for states that the agent definitely rules out as candidates. In between, the intuition is that moving upwards corresponds to being regarded as less normal or less likely to be the actual state. For the moment, of course, the boxes are empty.

By a templated ordering of the set of states we understand an allocation of states to the boxes of an information template. This allocation must exhaust the set of states and allocate each state to no more than one box. Formally, a templated ordering is (induced by) a function from the set of states into the set {1, 2, …, 2k}. State u is below state w if the value assigned to u is less than the value assigned to w. Templated orderings are a special case of the modular partial orderings of [LM].

A templated ordering may allocate one state to each box of the relevant information template, but more usually would allocate several states to the same box, thus leaving some other boxes empty. Even if empty, the boxes exist and can continue to play a role in various ways. Thus a templated ordering has more structure than an ordered partition of the sort encountered in the previous section. The key step forward is the availability of an uppermost box for definitely excluded states. The ordered partitions of the previous section could not represent definite exclusion, and so were able to support only the sentential expression of an agent's defeasible beliefs (the sentences true in all states at the bottom level). With templated orderings, it becomes possible to distinguish between those beliefs to which the agent attaches the greatest weight (the cautious beliefs, or knowledge of the agent) and those beliefs which go to the limits of plausibility (the brave or defeasible beliefs of the agent). The agent's knowledge consists of all sentences true in the states below the uppermost level, and the agent's defeasible beliefs comprise all sentences true in the states occupying the bottom level. (For ‘all sentences’ we may substitute ‘the logically strongest sentence’.)

By way of illustration, suppose the FHS is in state 11, that the agent can hear that the fan is on but cannot detect whether the heater is on, and that the agent is equipped with a default rule according to which state 10 is more normal than 11. The agent is definitely able to rule out states 01 and 00, because able to observe that the fan is on and thus that the first co-ordinate must be 1. If the agent views 11 as only moderately less normal than 10, the following templated ordering would be a suitable model-theoretic representation of the information available to the agent:
01, 00	
	
11	
10	
The agent's knowledge, in this case, may be represented by the sentence p (essentially the sentence whose content elements correspond with the states in the topmost box, or equivalently the sentence true in all states below the topmost box). The agent's defeasible belief is that pq, this being the sentence true at exactly the states in the bottommost box. (The defeasible belief happens to be false inasmuch as the actual state is 11, but such is at times the fate of defeasible beliefs.)

If the agent's default rule led to the view that 11 is grossly, rather than moderately, less normal than 10, but not entirely ruled out, then the mechanism of refining observational information by default rules gives the following templated ordering:

01, 00	
11	
	
10	
We note that the agent's knowledge and defeasible beliefs (in their sentential form) would be precisely the same as before. However, in the larger picture, there is a difference. Agents must act, and in the case of rational agents, their beliefs guide their actions. Often actions have risks. Pulling the red lever to shut down the nuclear powerplant may, if judged later to have been unwarranted, cost the company money and the agent a job. Not pulling the lever may have even worse consequences. It follows that in a model that links the epistemic state of an agent with actions, it is desirable to take into account the intensity of conviction (or degree of justification) of a belief. The templated ordering allows nuances of intensity to be recorded and used in determining whether thresholds for various actions have been exceeded. Thus the intermediate levels of a template do have an important contribution to make.

The road ahead

If information templates are used to represent the epistemic states of an agent, then the journey towards relativism in the use of the term ‘knowledge’ jars to a halt long before the extreme is reached in which all knowledge is opinion and one opinion is as good as any other. By specifying criteria according to which states are shifted upwards in a templated ordering, grounds may be established for differentiating between knowledge and less well-justified beliefs. Such criteria would range from the operation by which a default rule refines observational information to the incorporation of judgements as to reliability of information from other agents.

We have not touched on the interesting question of revision or update of information. This is an active field, and the notion of information templates holds promise of contributing to an understanding of the distinction between various forms of belief change. For example, should a new observation provide information that is inconsistent with a current defeasible belief but not with the agent's current knowledge, then the agent should judge that the system continues to be in the same state as before, and merely revise the defeasible belief. But should the new observation be inconsistent with the agent's knowledge, then a judgement that the system has changed state would be in order, and the agent should be prepared to make a more radical change (update) to its spectrum of beliefs, possibly changing its knowledge too.

Also promising is the reformulation of epistemic logic so that accessibility relations may be replaced by functions that assign, to each state, a templated ordering of states. This would be a proper generalisation of the present assignment, by the accessibility relation, of what is effectively a dichotomous ordered partition to each state.

Finally, not much has been said about default rules. An entire industry is engaged in the classification of different kinds of default rules, some numerical (probabilistic, or fuzzy, or other) and some non-numerical (often based, as in the minimal model semantics, on stereotypes or a notion of typicality, but sometimes based on other qualitative methods). How to revise default rules in the light of evidence brings us to one of the most active fields of artificial intelligence, learning. There are many different kinds of learning, some modelled by neural nets, some by symbolic methods. The question of learning applied to default rules in the context of information templates is still largely untouched.

References
 [BHa]	Bar-Hillel Y (1955) Comments on ‘Degree of Confirmation’ by Professor KR Popper, British Journal for the Philosophy of Science VI 155-157.
[BHb]	Bar-Hillel Y (1955) An examination of information theory, Philosophy of Science 22 86-105.
[BHc]	Bar-Hillel Y (1956) Further comments on ‘Degree of Confirmation’ by Professor KR Popper, British Journal for the Philosophy of Science VII 245-248.
[Br]	Brown FM (ed.) (1987) Proceedings of the 1987 Workshop, The Frame Problem in Artificial intelligence, Morgan Kaufmann.
[BS]	Bell JL and AB Slomson (1969) Models and Ultraproducts, North-Holland.
[Caa]	Carnap R (1950) Logical Foundations of Probability, University of Chicago Press.
[Cab]	Carnap R (1953] The two concepts of probability, in Readings in the Philosophy of Science, H Feigl and M Brodbeck (ed.), Appleton-Century-Crofts.
[Cac]	Carnap R (1960) Statistical and inductive probability, in The Structure of Scientific Thought, EH Madden (ed.), Houghton Mifflin.
[Cad]	Carnap R (1962) The aim of inductive logic, in Logic, Methodology and Philosophy of Science, E Nagel, P Suppes and A Tarski (ed.), Stanford University Press.
[CBH]	Carnap R and Y Bar-Hillel (1952) An outline of a theory of semantic information, Technical Report no. 247, Research Laboratory of Electronics, MIT, reprinted as Chapter 15 of Y Bar-Hillel (1964) Language and Information, Addison-Wesley.
[Ch]	Chellas BF (1980) Modal Logic, Cambridge University Press.
[Da]	Davis E (1990) Representations of Commonsense Knowledge, Morgan Kaufmann.
[Ee]	Eells E (1988) Discussion: On the alleged impossibility of inductive probability, British Journal for the Philosophy of Science 39 111-116.
[En]	Enderton HB (1972) A Mathematical Introduction to Logic, Academic Press.
[Gal]	Gallo E (1991) Nature faking in the Humanities, Skeptical Inquirer 15 371-375.
[Gi]	Gillies D (1986) Discussion: In defense of the Popper-Miller argument, Philosophy of Science 53 110-113.
[GL]	Gross P and N Levitt (1994) Higher Superstition, Johns Hopkins University Press.
[Goa]	Goodman N (1965) Fact, Fiction, and Forecast, Bobbs-Merrill.
[Gob]	Good IJ (1984) Letter in Matters Arising: The impossibility of inductive probability, Nature 310 434.
[Ha]	Hartley RVL (1928) Transmission of information, Bell System Technical Journal 7 525-563.
[Hia]	Hintikka J (1962) Knowledge and Belief, Cornell University Press.
[Hib]	Hintikka J (1968) The varieties of information and scientific explanation, in Logic, Methodology and Philosophy of Science III, Proceedings of the 1967 International Congress, B van Rootselaar and JF Staal (ed.), North-Holland.
[Hic]	Hintikka J (1970) On semantic information, in Physics, Logic, and History, W Yourgrau (ed.), Plenum Press.
[Hid]	Hintikka J (1970) Surface information and depth information, in Information and Inference, J Hintikka and P Suppes (ed.), D Reidel.
[Ho]	Holton GJ (1993) Science and Anti-science, Harvard University Press.
[HU]	Howson C and P Urbach (1989) Scientific Reasoning: The Bayesian Approach, Open Court.
[Je]	Jeffrey RC (1984) Letter in Matters Arising: The impossibility of inductive probability, Nature 310 433.
[Ko]	Konolige K (1986) A Deduction Model of Belief, Pitman.
[Ki]	Kimball R (1990) Tenured Radicals–How Politics has Corrupted our Higher Education, HarperCollins.
[KLM]	Kraus S, D Lehmann and M Magidor (1990) Nonmonotonic reasoning, preferential models and cumulative logics, Artificial Intelligence 44 167-207.
[KST]	Kahneman D, P Slovic and A Tversky (ed.) (1982) Judgment under Uncertainty: Heuristics and Biases, Cambridge University Press.
[Le]	Levi I (1984) Letter in Matters Arising: The impossibility of inductive probability, Nature 310 433.
[LM]	Lehmann D and M Magidor (1992) What does a conditional knowledge base entail? Artificial Intelligence 55 1-60.
[Ma]	Marr D (1982) Vision, WH Freeman.
[Mca]	McCarthy J (1980) Circumscription–a form of non-monotonic reasoning, Artificial Intelligence 13 27-39.
[Mcb]	McCarthy J (1986) Applications of circumscription to formalizing common-sense knowledge, Artificial Intelligence 28 89-116.
[Mu]	Musgrave A (1999) Rationality and Reliability, unpublished seminar, University of Otago, 14 April 1999.
[Pe]	Penrose R (1989) The Emperor's New Mind, Oxford University Press.
[Pi]	Pinker S (1997) How the Mind Works, Penguin.
[PMa]	Popper KR and D Miller (1983) A proof of the impossibility of inductive probability, Nature 302 687-688.
[PMb]	Popper KR and Miller D (1984) Letter in Matters Arising: The impossibility of inductive probability, Nature 310 434.
[Poa]	Popper KR (1959) The Logic of Scientific Discovery, Hutchinson. (Translation of Logik der Forschung (1934) Vienna.)
[Pob]	Popper KR (1954) Degree of confirmation, British Journal for the Philosophy of Science V 143-149.
[Poc]	Popper KR (1955) ‘Content’ and ‘Degree of Confirmation’: A reply to Dr Bar-Hillel, British Journal for the Philosophy of Science VI 157-163.
[Pol]	Pollock JL (1995) Cognitive Carpentry, MIT Press.
[Py]	Pylyshyn ZW (ed.) (1987) The Robot's Dilemma, Ablex.
[Red]	Redhead M (1985) On the impossibility of inductive probability, British Journal for the Philosophy of Science 36 185-191.
[Rei]	Reiter R (1980) A logic for default reasoning, Artificial Intelligence 13 81-132.
[Ru]	Russell B (1940) An Inquiry into Meaning and Truth, Penguin.
[SB]	Sokal A and J Bricmont (1998) Intellectual Impostures, Profile.
[Sch]	Schmitt FF (1992) Knowledge and Belief, Routledge.
[Sho]	Shoham Y (1987) Reasoning about Change, MIT Press.
[SW]	Shannon CE and W Weaver (1975) The Mathematical Theory of Communication, University of Illinois Press.
[Ta]	Tarski A (1956) Logic, Semantics, Metamathematics: Papers from 1923 to 1938, translated by JH Woodger, Clarendon Press.
[Tu]	Turing A (1950) Computing machinery and intelligence, Mind 59 433-460. Reprinted in Computers and Thought (1963), EA Feigenbaum and J Feldman (ed.), McGraw-Hill.