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Input/output logics

David Makinson and Leendert van der Torre

ESSLLI’01

August, 2001Helsinki, Finland

Input/output logicsDavid Makinson and Leendert van der Torre

This reader contains the following three articles:

Page

2 D. Makinson and L. van der Torre, What is input-output logic?

11 D. Makinson and L. van der Torre, Input-output logics. Journal ofPhilosophical Logic, 29: 383-408, 2000.

35 D. Makinson and L. van der Torre, Constraints for input-output logics. Journalof Philosophical Logic, 30(2):155-185, 2001.

Reproduced with permission from the copyright owner, Kluwer Academic Publishers.

Makinson and van der Torre, Input/Output Logics 2

What is Input/Output Logic?David Makinson and Leendert van der Torre

AbstractWe explain the raison d’être and basic ideas of input/output logic, sketching thecentral elements with pointers to other publications for detailed developments. Themotivation comes from the logic of norms. Unconstrained input/output operations arestraightforward to define, with relatively simple behaviour, but ignore the subtleties ofcontrary-to-duty situations. To deal with these more sensitively, we constraininput/output operations by means of consistency conditions, expressed in the conceptof an outfamily. However, this is a more complex affair, with difficult choicesbetween alternative options.

1. Motivation

Input/output logic takes its origin in the study of conditional norms. These mayexpress desired features of a situation, obligations under some legal, moral orpractical code, goals, contingency plans, advice, etc. Typically they may be expressedin terms like: In such-and-such a situation, so-and-so should be the case, or …shouldbe brought about, or …should be worked towards, or …should be followed – theselocutions corresponding roughly to the kinds of norm mentioned.

To be more accurate, input/output logic has its source in a tension between thephilosophy of norms and formal work of deontic logicians.

Philosophically, it is widely accepted that a distinction may be drawn between normson the one hand, and declarative statements on the other. Declarative statements maybear truth-values, in other words are capable of being true or false; but norms areitems of another kind. They may be respected (or not), and may also be assessed fromthe standpoint of other norms, for example when a legal norm is judged from a moralpoint of view (or vice versa). But it makes no sense to describe norms as true or asfalse.

However the formal work of deontic logicians often goes on as if such a distinctionhad never been heard of. The usual presentations of deontic logic, whether axiomaticor semantic, treat norms as if they could bear truth-values. In particular, the truth-functional connectives and, or and most spectacularly not are routinely applied toitems construed as norms, forming compound norms out of elementary ones.Semantic constructions using possible worlds go further by offering rules todetermine, in a model, the truth-values of a norm.

This anomaly was noticed more than half a century ago, by Dubislav (1937) andJørgensen (1937-8), but little was done about it. Indeed, from the 1960s onwards, thesemantic approach in terms of possible worlds deepened the gap. The first seriousattempt by a logician to face the problem appears to be due to Stenius (1963),followed by Alchourrón and Bulygin (1981) for unconditional norms, thenAlchourrón (1993) and Makinson (1999) for conditional ones. Input/output logic maybe seen as an attempt to extract the essential mathematical structure behind thisreconstruction of deontic logic.


Like every other approach to deontic logic, input/output logic must face the problemof accounting adequately for the behaviour of what are called ‘contrary-to-duty’norms. The problem may be stated thus: given a set of norms to be applied, howshould we determine which obligations are operative in a situation that alreadyviolates some among them. It appears that input/output logic provides a convenientplatform for dealing with this problem by imposing consistency constraints on thegeneration of output.

We begin by outlining the central ideas and constructions of unconstrainedinput/output logic. These are quite straightforward, and provide the basic frameworkof the theory. We then sketch a strategy for constraining those operations so as to dealmore sensitively with contrary-to-duty situations. For further details, the reader isinvited to refer to Makinson and van der Torre (2000), (2001).

2. Unconstrained Input/Output Operations

We do not treat conditional norms as bearing truth-values. They are not embedded incompound formulae using truth-functional connectives. To avoid all confusion, theyare not even treated as formulae, but simply as ordered pairs (a,x) of purely boolean(or eventually first-order) formulae.Technically, a normative code is seen as a set G of conditional norms, i.e. a set ofsuch ordered pairs (a,x). For each such pair, the body a is thought of as an input,representing some condition or situation, and the head x is thought of as an output,representing what the norm tells us to be desirable, obligatory or whatever in thatsituation. The task of logic is seen as a modest one. It is not to create or determine adistinguished set of norms, but rather to prepare information before it goes in as inputto such a set G, to unpack output as it emerges and, if needed, coordinate the two incertain ways. A set G of conditional norms is thus seen as a transformation device,and the task of logic is to act as its ‘secretarial assistant’.The simplest kind of unconstrained input/output operation is depicted in Figure 1. Aset A of propositions serves as explicit input, which is prepared by being expanded toits classical closure Cn(A). This is then passed into the ‘black box’ or ‘transformer’ G,which delivers the corresponding immediate output G(Cn(A)) = {x: for some a ∈Cn(A), (a,x) ∈ G}. Finally, this is expanded by classical closure again into the fulloutput out1(G,A) = Cn(G(Cn(A))). We call this simple-minded output.

Figure 1: Simple-Minded Outputout1(G,A) = Cn(G(Cn(A)))

Cn(G(Cn(A)))

A

G(Cn(A)) Cn(A)

G


This is already an interesting operation. As desired, it does not satisfy the principle ofidentity, which in this context we call throughput, i.e. in general we do not have a ∈out1(G,{a}) – which we write briefly, dropping the parentheses, as out1(G,a). It ischaracterized by three rules. Writing x ∈ out1(G,a) as (a,x) ∈ out1(G) and droppingthe right hand side as G is held constant, these rules are:

Strengthening Input (SI): From (a,x) to (b,x) whenever a ∈ Cn(b)Conjoining Output (AND): From (a,x), (a,y) to (a,x∧y)Weakening Output (WO): From (a,x) to (a,y) whenever y ∈ Cn(x).

But simple-minded output lacks certain features that may be desirable in somecontexts. In the first place, the preparation of inputs is not very sophisticated.Consider two inputs a and b. By classical logic, if x ∈ Cn(a) and x ∈ Cn(b) then x ∈Cn(a∨b). But there is nothing to tell us that if x ∈ out1(G,a) = Cn(G(Cn(a))) and x ∈out1(G,b) = Cn(G(Cn(b))) then x ∈ out1(G,a∨b) = Cn(G(Cn(a∨b))).

In the second place, even when we do not want inputs to be automatically carriedthrough as outputs, we may still want outputs to be reusable as inputs – which is quitea different matter.Operations satisfying each of these two features can be provided with explicitdefinitions, pictured by diagrams in the same spirit as that for simple-minded output,and characterized by straightforward rules. We thus have four very natural systems ofinput/output, which are labelled as follows: simple-minded alias out1 (as above), basic(simple-minded plus input disjunction: out2), reusable (simple-minded plusreusability: out3), and reusable basic alias out4 (all together).For example, reusable basic output may be given a diagram and definition as inFigure 2. In the definition, a complete set is one that is either maximally consistent orequal to the set of all formulae.

Figure 2: Reusable Basic Output:out4(G,A) = ∩{Cn(G(V)): A ⊆ V ⊇ G(V), V complete}

A

V1

V2

G(V1)

G(V2)

G

Cn(G(V1))

Cn(G(V2))

out4(G,A)

⊆

⊆


The three stronger systems may also be characterized by adding one or both of thefollowing rules to those for simple-minded output:

Disjoining input (OR): From (a,x), (b,x) to (a∨b,x)Cumulative transitivity (CT): From (a,x), (a∧x,y) to (a,y).

These four operations have four counterparts that also allow throughput. Intuitively,this amounts to requiring A ⊆ G(A). In terms of the definitions, it is to require that Gis expanded to contain the diagonal, i.e. all pairs (a,a). Diagrammatically it is to addarrows from G’s ear to mouth. Derivationally, it is to allow arbitrary pairs (a,a) toappear as leaves of a derivation; this is called the zero-premise identity rule ID.All eight systems are distinct, with one exception: basic throughput, which we writeas out2

+, authorizes reusability, so that out2+

= out4+. This may be shown directly in

terms of the definitions, or using the following simple derivation of CT from the otherrules.

(a,x) (a∧¬x, a∧¬x) ID (a∧x,y) SI (a∧¬x, x) .................................................. AND (a∧¬x, x∧(a∧¬x))

WO (a∧¬x,y) ........................................................................ OR

(a,y)

The application of WO here is justified by the fact that y ∈ Cn(x∧(a∧¬x)) since theright hand formula is a contradiction. Note that all rules available in basic throughput(including, in particular, identity) are needed in the derivation, reflecting the fact thatCT is not derivable in the weaker systems.This strong system indeed collapses into classical consequence, in the sense thatout4

+(G,A) = Cn(m(G)∪A) where m(G) is the materialization of G, i.e. the set of allformulae a→x where (a,x) ∈ G.The authors’ papers (2000) and (2001, section 1) investigate these systems in detail –semantically, in terms of their explicit definitions, derivationally, in terms of the rulesdetermining them, both separately and in relation to each other. We do not attempt tosummarize the results here, but hope that the reader is tempted to follow further.

3. The Need for Constraint

As mentioned in section 1, all approaches to deontic logic must face the problem ofdealing with contrary-to-duty norms. In general terms, we recall, the problem is:given a set of norms, how should we determine which obligations are operative in asituation that already violates some among them.

The following simple example is adapted from Prakken and Sergot (1996). Supposewe have the following two norms: The cottage should not have a fence or a dog; if ithas a dog it must have both a fence and a warning sign.


In the usual deontic notation, where t stands for a tautology: O(¬(f∨d)/t), O(f∧w/d); inthe notation of input/output logic: (t,¬(f∨d)), (d,f∧w). Suppose further that we are inthe situation that the cottage has a dog, thus violating the first norm. What are ourcurrent obligations? 1

Unrestricted input/output logic gives f: the cottage has a fence and w: the cottage hasa warning sign. Less convincingly, because unhelpful in the supposed situation, italso gives ¬d: the cottage does not have a dog. Even less convincingly, it gives ¬f:the cottage does not have a fence, which is the opposite of what we want.

These results hold even for simple-minded output, without reusability or disjunctionof inputs. The only rules needed are SI and WO, as shown by the following derivationof ¬f.

(t,¬(f∨d)) WO (t,¬f) SI (d,¬f)

A common reaction to examples such as these is to ask: why not just drop the rule SIof strengthening the input? In semantic terms, why not cut back the definition ofsimple-minded output from Cn(G(Cn(A))) to Cn(G(A)), and in similar (but morecomplex) fashion with the others? Indeed, this is a possible option, and the strategythat we will describe below does have the effect of disallowing certain applications ofSI. But simply to drop SI is, in the view of the authors, too heavy-handed. We need toknow why SI is not always appropriate and, especially, when it remains justified.

4. A Strategy for Constraint: Maxfamilies and their Outfamilies

Our strategy is to adapt a technique that is well known in the logic of belief change –cut back the set of norms to just below the threshold of making the current situationcontrary-to-duty. In effect, we carry out a contraction on the set G of given norms.

Specifically, we look at the maximal subsets G′ ⊆ G such that out(G′,A) is consistentwith input A. In Makinson and van der Torre (2001), the family of such G′ is calledthe maxfamily of (G,A), and the family of outputs out(G′,A) for G′ in the maxfamily,is called the outfamily of (G,A). 2

To illustrate this consider the cottage example, where G = (t,¬(f∨d)), (d,f∧w)}, withthe contrary-to-duty input d. Using simple-minded output, maxfamily(G,d) has justone element {(d,f∧w)}, and so outfamily(G,d) has one element, namely Cn(f∧w).

Although the outfamily strategy is designed to deal with contrary-to-duty norms, itsapplication turns out to be closely related to belief revision and nonmonotonicreasoning when the underlying input/output operation authorizes throughput.

When all elements of G are of the form (t,x), then for the degenerate input/outputoperation out4

+(G,a) = Cn(m(G)∪{ a}), the elements of outfamily(G,a) are just themaxichoice revisions of m(G) by a, in the sense of Alchourrón, Gärdenfors andMakinson (1985). These coincide, in turn, with the extensions of the default system(m(G),a,∅) of Poole (1988).


More surprisingly, there are close connections with the default logic of Reiter, fallinga little short of identity. Read elements (a,x) of G as normal default rules a;x/x in thesense of Reiter (1980), and write extfamily(G,A) for the set of extensions of (G,A).Then, for reusable simple-minded throughput out3

+, it can be shown thatextfamily(G,A) ⊆ outfamily(G,A) and indeed that extfamily(G,A) consists of preciselythe maximal elements (under set inclusion) of outfamily(G,A).

These results and related ones are proven in Makinson and van der Torre (2001). Butin accord with the motivation from the logic of norms, the main focus in that paper ison input/output logics without throughput. Two kinds of question are investigated indetail there.The search for truth-functional reductions of the consistency constraintFrom the point of view of computation, it is convenient to make consistency checks assimple as possible, and executable using no more than already existing programs. Forthis reason, it is of interest to ask: under what conditions is the consistency of A without(G,A) reducible to the consistency of A with the materialization m(G) of G, i.e.with the set of all formulae a→x where (a,x) ∈ G?

It is easy to check that the latter consistency implies the former for all seven of ourinput/output operations. It turns out that we have equivalence for just two of them(reusable basic with and without identity).

On the level of derivations, the question can take a rather different form, withdifferent answers. Given a derivation of (a,x) with leaves L, under what conditions isthe consistency of a with out(L,a) equivalent to its consistency with m(L)? Curiously,this holds for a wider selection of our input/output operations – in fact, for all of themexcept basic output. Even more surprisingly, for some of the operations (those withoutOR), the same reduction also holds with respect to the set h(L) of heads x, and the setf(L) of fulfilments a∧x, of elements (a,x) of L.

From this result on derivations, we can go back and sharpen the semantic one. WhenG is a minimal set with x ∈ out(G,a) then, for each of out input/output operationsother than basic output, a is consistent with out(G,a) iff it is consistent with m(G) –and for the operations without OR, with h(G), f(G).

More severe applications of the consistency check

From a practical point of view, whenever we constrain an operation to avoid excessproduction, the question arises: how cautious (timid) or brave (foolhardy) do we wantto be? For input/output operations, this issue arises in different ways on the semanticand derivational levels. On the semantic level, once we have formed an outfamily wemay ask: should we intersect, join, or choose from its elements to obtain a uniquerestrained output? On the level of derivations, it is natural to ask: do we want to applythe consistency check only at the root of a derivation, or at every step within it?

The policy of checking only at the root corresponds to the option, on the semanticlevel, of forming the join of the outfamily; while the stricter policy of checking atevery step is an essentially derivational requirement. But whichever of the two wechoose, it is of interest to know under what conditions they coincide. In other words,given a derivation of (a,x) with leaves L such that a is consistent with out(L,a), underwhat conditions does it follow that for every node (b,y) in the derivation, b isconsistent with out(L,b)? It turns out that for certain of the seven input/outputoperations (again, those without the OR rule) this result holds. For operations withOR but without the rule CT, a rather subtler result may be obtained.


One lesson of these rather intricate investigations is that the behaviour of theconsistency constraint depends very much on the choice of input/output operation; inparticular, the presence of the rule OR destroys some properties. Another lesson isthat questions can take different forms, with different answers, on the semantic andderivational levels. Thirdly, a detour through derivations can sometimes sharpensemantic results.

5. Doubts and Queries

The investigation of constrained output is a much more complex matter than that ofunconstrained output. It is also more open to doubts and queries. We put the mainones on the table.

Dependence on the formulation of G.

The outfamily construction, at least in its present form, depends heavily on theformulation of the generating set G. To illustrate this, we go back to the cottageexample of Prakken and Sergot (1996) considered in sections 3 and 4. Here G ={( t,¬(f∨d)), (d,f∧w)}, and we consider the contrary-to-duty input d. As we have seen,using simple-minded output, maxfamily(G,d) has unique element {(d,f∧w)} andoutfamily(G,d) has unique element Cn(f∧w). But if we split the first element of G into(t,¬f), (t,¬d) then we get a different result. The maxfamily has two elements {(t,¬f)},{( d,f∧w)} and the outfamily has two elements Cn(¬f ) and Cn(f∧w). Is thisdependence on formulation of G a virtue, or a vice?

Are we cutting too deeply?

This problem is related to the first one. In some cases, the outfamily construction cutsdeeply, perhaps too much. Consider again the cottage example, but this time with justone rule (t,¬(f∨d)) in G. Consider the same contrary-to-duty input d. Then themaxfamily has the empty set as its unique element, and so the outfamily has Cn(∅) asits unique element. Is this cutting too deeply? Shouldn’t Cn(¬f ) be retained?

Should we pre-process G?

If we wish to cut less deeply, then a possible procedure might be to ‘pre-process’ G.In the example, when we decompose the sole element (t,¬(f∨d)) of G into (t,¬f),(t,¬d) then Cn(¬f) becomes the unique element of outfamily in the contrary-to-dutysituation d. In general, for each element (a,x) of G, we could rewrite the head x inconjunctive normal form x1∧…∧xn, and then split (a,x) into (a,x1), …,(a,xn). Thismanoeuvre certainly meets the particular example. But is it appropriate for otherexamples of the same form with different content? And does it suffice for morecomplex examples? It looks suspiciously like hacking.

Avoid inconsistency with what?

On our definition, maxfamily(G,A) is the family of maximal subsets G′ ⊆ G such thatout(G′,A) is consistent with input A. It may be suggested that this is too radical – solong as out(G,A) is consistent we should apply it without constraint.

To illustrate this, take another variation on the cottage example. Put G = {(t,¬(f∨d)),(d,w)}. The second norm no longer requires a fence when there is a dog, only awarning sign. Consider again the contrary-to-duty input d. Now out(G,d) =Cn({( ¬f,¬d,w}) which is inconsistent with the input d, but itself perfectly consistent.Should we cut it at all?


6. ConclusionsDrawing together the threads of this paper, we emphasize three general conclusions.

• On the one hand, unconstrained input/output provides us with a simple andelegant construction with straightforward behaviour, but whose application tonorms totally ignores the subtleties of contrary-to-duty obligations.

• On the other hand, output constrained using the outfamily strategy provides away of dealing with contrary-to-duty obligations. Its behaviour is quite subtle,and depends considerably on the choice of background input/output operation,in particular on whether or not it authorizes the rule of disjunction of inputs.

• However, our definition of an outfamily has some features that may perhapsbe regarded as shortcomings. Its effect depends on the formulation of thegenerating set of norms; in some examples it gives what may be regarded as awrong result unless some pre-processing as carried out on the generating set;and in some examples the requirement of consistency of output with input maybe too strong. These are delicate issues, and it remains possible that they haveno unique solution definable in purely formal terms.

Notes

1. There are many examples in the literature. Most of them involve ingredients that,while perfectly natural in ordinary discourse, are extraneous to the essential problemand thus invite false analyses. These ingredients include defeasibility, causality, thepassage of time, and the use of questionable rules such as CT and OR in deriving

output. We have chosen a very simple example that avoids all those elements. Thereis one respect in which it could perhaps be further purified: under input d, the outputCn({ ¬(f∨d), f∧w} is not only inconsistent with the input, but also itself inconsistent.

This matter is discussed at the end of section 5.

2. So defined, the outfamily is not in general the same as the family of all maximalvalues of out(G′,A) consistent with A, for G′ ranging over subsets of G. Everymaximal value of out(G′,A) is in the outfamily, but not always conversely. For certainof our output operations, the two families do coincide, but not for others.

This can be shown by simple examples, such as the Möbius strip of Makinson (1994),(1999). Put G = {(a,x), (x,y), (y,¬a)}. Then, for out = out3 or out = out4,maxfamily(G,a) has three elements, namely the three two-element subsets of G. As aresult, outfamily(G,a) also has three elements – Cn(∅), Cn(x), and Cn({ x,y}). Ofthese, only the last is a maximal value of out(G′,A) consistent with A for G′ rangingover subsets of G.

We add that in this example, not even Cn({ x,y}) is a maximal subset of out(G,a) thatis consistent with a, for clearly Cn({ x,y}) ⊂ Cn({ x,y,¬a∨z}) ⊂ out(G,a). Care is thusneeded to avoid confusing maxfamilies with related maximal sets.

References

Alchourrón, Carlos (1993). Philosophical foundations of deontic logic and the logic ofdefeasible conditionals. In Meyer, J.J. and R.J. Wieringa (eds) Deontic Logic inComputer Science, 43-84. Wiley: New York.


Alchourrón, Carlos and Eugenio Bulygin (1981). The expressive conception ofnorms. In Hilpinen, R. (ed.) New Essays in Deontic Logic, 95-124. Reidel: Dordrecht.

Alchourrón, Carlos, Peter Gärdenfors and David Makinson. On the logic of theorychange: partial meet contraction and revision functions. The Journal of SymbolicLogic 50 (1985) 510-530.

Dubislav, Walter (1937). Zur Unbegründbarkeit der Forderungsätze. Theoria 3: 330-342.

Jørgensen, Jørgen (1937-8). Imperatives and logic. Erkenntnis 7: 288-296.

Makinson, David (1994). General Patterns in Nonmonotonic Reasoning. In Handbookof Logic in Artificial Intelligence and Logic Programming, vol. 3, ed. Gabbay, Hoggerand Robinson, Oxford University Press, pages 35-110.

Makinson, David (1999). On a fundamental problem of deontic logic, In PaulMcNamara and Henry Prakken eds Norms, Logics and Information Systems. NewStudies in Deontic Logic and Computer Science, Amsterdam: IOS Press, Series:Frontiers in Artificial Intelligence and Applications, Volume 49, pp. 29-53.Makinson, David and Leendert van der Torre (2000). Input/output logics, J.Philosophical Logic 29: 383-408.Makinson, David and Leendert van der Torre (2001). Constraints for input/outputlogics, to appear in J. Philosophical Logic.

Poole, David (1988). A logical framework for default reasoning, ArtificialIntelligence 36: 27-47.

Prakken, Henry and Marek Sergot. Contrary-to-duty obligations. Studia Logica 57:91-115.

Reiter, Ray (1980). A logic for default reasoning, Artificial Intelligence 13: 81-132.

Stenius, Erik (1963). Principles of a logic of normative systems. Acta PhilosophicaFennica 16: 247-260.

David MakinsonSenior Research Fellow, Department of ComputingKing’s College LondonEmail: [email protected]

Leendert van der TorreDepartment of Artificial IntelligenceVrije Universiteit AmsterdamDe Boelelaan 1081a, 1081 HV Amsterdam, the NetherlandsEmail: [email protected]

Last revised 17.04.01Wordcount: 3,967


DAVID MAKINSON AND LEENDERT VAN DER TORRE

INPUT/OUTPUT LOGICS

ABSTRACT. In a range of contexts, one comes across processes resemblinginference, but where input propositions are not in general included among outputs,and the operation is not in any way reversible. Examples arise in contexts ofconditional obligations, goals, ideals, preferences, actions, and beliefs. Our purpose isto develop a theory of such input/output operations. Four are singled out: simple-minded, basic (making intelligent use of disjunctive inputs), simple-minded reusable(in which outputs may be recycled as inputs), and basic reusable. They are definedsemantically and characterised by derivationrules, as well as in terms of relabelingprocedures and modal operators. Their behaviour is studied on both semantic andsyntactic levels.

KEY WORDS: input/output logic, reusability, identity, conditional goals, conditionalobligations, deontic logic

1. INTRODUCTION

Imagine a black box into which we may feed propositions as input, and that alsoproduces propositions as output. Of course, classical consequence may itself be seenin this way, but it is a very special case, with additional features - inputs are alsothemselves outputs, since any proposition classically implies itself, and the operationis in a certain sense reversible, since contraposition is valid. However, there are manyexamples without those features. Roughly speaking, they are of two main kinds.

The box may stop some inputs, while letting others through, perhaps in modifiedform. Inputs may record reports of agents, of the kind ‘according to source i, x istrue’, while the box may give as output either x itself, a qualified version of x, ornothing at all, according to the identity of i. Or it might give output x only when atleast two distinct sources vouch for it, and so on. Inputs might be facts about theperformance of the stock-market today, and outputs an analyst’s commentary; or factsabout your date and place of birth, with output your horoscope readings. In theseexamples, the outputs express some kind of belief or expectation.

Again, inputs may be conditions, with outputs expressing what is deemed desirable inthose conditions. The desiderata may be obligations of a normative system, ideals,goals, intentions or preferences. In general, a fact entertained as a condition may itselfbe far from desirable, so that inputs are not always outputs; and as is widelyrecognised, contraposition is inappropriate for conditional goals.

Our purpose is to develop a general theory of propositional input/output operations,covering both kinds of example. Particular attention is given to the case where outputsmay be recycled as inputs. In a companion paper (Makinson and van der Torre, toappear), we examine the imposition of consistency constraints on output.


From a very general perspective, logic is often seen as an ‘inference motor’, withpremises as inputs and conclusions as outputs (figure 1). But it may also be seen inanother role, as ‘secretarial assistant’ to some other, perhaps non-logical,transformation engine (figure 2). From this point of view, the task of logic is one ofpreparing inputs before they go into the machine, unpacking outputs as they emergeand, less obviously, co-ordinating the two. The process as a whole is one of ‘logicallyassisted transformation’, and is an inference only when the central transformation isso. This is the general perspective underlying the present paper. It is one of ‘logic atwork’ rather than ‘logic in isolation’; we are not studying some kind of non-classicallogic, but a way of using the classical one.

Figure 2Picture of Logic Assisting a Transformation Engine

Output Input

TRANSFORMATIONENGINE

LOGICunpacks prepares

coordinates

Figure 1

Traditional Picture of Logic as an Inference Motor

Output Input

LOGIC


On a pre-logical level, this picture is perfectly familiar from elementary set theory.Consider any universe L, not necessarily of propositions, and any relation G ⊆ L². Forexample, L may be the set of humans, and G the parent/child relation. Given an inputA ⊆ L, the output of A under G may be understood simply as G(A) = {x: (a,x) ∈ G forsome a ∈ A} - in the example, the set of all children of persons in A.

The present paper may be seen as investigating what happens to this basic picturewhen we pass to the logical level, i.e. when L is the set of propositions of somelanguage, and input and output are both under the sway of the operation Cn ofclassical consequence. These are in a certain sense frills, but give rise to subtle andinteresting behaviour.

2. LOGICAL LEVEL: THE PROBLEM

Consider a propositional language L, closed under at least the usual truth-functionalconnectives; its elements are called formulae. Let G be a set of ordered pairs (a,x) offormulae in L; the letter chosen serves as reminder of the interpretation (amongothers) of the pairs as conditional goals. We call G a generating set. We read a pair(a,x) forwards, i.e. with a as body and x as head; and we call the corresponding truth-functional formula a→x its materialisation, echoing the old name ‘materialimplication’ for the connective involved.

Suppose that we are also given a set A of formulae. Our problem is: how may wereasonably define the set of propositions x making up the output of A under G, or onemight also say, of G given A, which we write out(G,A)? Alternatively, suppose we aregiven only the generating set G: how may we define the set of input/output pairs (A,x)arising from G, which we write out(G)?

These questions are the same, for we may define (A,x) ∈ out(G) iff x ∈ out(G,A) orconversely. But the two formulations give a rather different gestalt, and one issometimes more convenient rather than the other. As we shall see, the latter tends tobe clearer in semantic contexts, whilst the former is easier to work with whenconsidering derivations in a syntactic context. We shall move freely from one to theother, just as one moves between Cn and • for classical consequence.

3. SIMPLE-MINDED OUTPUT

3.1. Semantic definition

The simplest response to our problem is to put out(G,A) = Cn(G(Cn(A))), where thefunction G(.) is defined as on the pre-logical level, and Cn alias • is classicalconsequence. In other words, given a set A of formulae as input, we first collect all ofits consequences, then apply G to them, and finally consider all of the consequencesof what is thus obtained (figure 3).


Under this definition, which we call simple-minded output and write as out1(G,A),inputs are not in general outputs; that is, we do not have A ⊆ out1(G,A).

Example 1. Put G = {(a,x)}where a,x are distinct elementary letters, and put A = {a}.Then G(Cn(a)) = {x} so a ∉ out1(G,a) = Cn(G(Cn(a))) = Cn(x). Contraposition alsofails, for although x ∈ out1(G,a) we have ¬a ∉ out1(G,¬x): since a ∉ Cn(¬x) we haveG(Cn(¬x)) = ∅ so that ¬a ∉ out1(G,¬x) = Cn(G(Cn(¬x))) = Cn(∅).

Clearly, this operation is inadequate for some purposes, for it is unable to handledisjunctive inputs intelligently.

Example 2. Put G = {(a,x), (b,x)}and A = {a∨b}. Then Cn(A)∩b(G) = ∅ where wewrite b(G) for the set of all bodies of elements of G, i.e. in this example the set {a,b}.Hence also G(Cn(A)) = ∅ so that Cn(G(Cn(A))) = Cn(∅). However, in many contextswe would want to put x in the output, as it can be obtained from each of the twodisjuncts of the input.

Nevertheless, the operation of simple-minded output has an interest of its own, and itsstudy also helps prepare the way for more sophisticated ones.

3.2. Syntactic characterisation

Our definition of simple-minded output is, in a broad sense of the term, semantic. It isnot difficult to give it a syntactic characterisation in terms of derivation rules.

In general, for any set of rules, we say that a pair (a,x) of formulae is derivable usingthose rules from a set G of such pairs iff (a,x) is in the least set that includes G,contains the pair (t,t) where t is a tautology, and is closed under the rules. In thesystems studied here, it will make no difference which tautology t is chosen. Our

Figure 3Simple-Minded Output: out1(G,A) = Cn(G(Cn(A)))

Cn(G(Cn(A)))

Input A

G(Cn(A)) Cn(A)

G


notations are (a,x) ∈ deriv(G) or equivalently x ∈ deriv(G,a), with a subscript toindicate the set of rules employed.

When A is a set of formulae, derivability of (A,x) from G is defined as derivability of(a,x) from G for some conjunction a = a1∧…∧an of elements of A. We understand theconjunction of zero formulae to be a tautology, so that (∅,x) is derivable from G iff(t,x) is for some tautology t.

In the particular case of simple-minded output, we use the following three rulesdetermining an operation deriv1. Of these, the first governs the use of inputs(strengthening the input: SI), while the other two deal with the management of outputs(conjunction in the output: AND; weakening the output: WO).

SI: From (a,x) to (b,x) whenever b• aAND: From (a,x), (a,y) to (a,x∧y)WO: From (a,x) to (a,y) whenever x• y.

OBSERVATION 1. Out1(G,A) = deriv1(G,A).

Outline of proof. The inclusion from right to left is straightforward by induction onlength of derivation. From left to right, suppose x ∈ Cn(G(Cn(A))). Then bycompactness of Cn there are x1,…,xn ∈ G(Cn(A)) with x ∈ Cn(x1∧…∧xn). In the casethat n = 0, x is a tautology t and we can also put a = t giving us a one-step derivationof (t,t). In the case that n ≠ 0 we proceed as follows. For each i ≤ n, since xi ∈G(Cn(A)) there is a bi ∈ Cn(A) with (bi,xi) ∈ G. Putting b = b1∧…∧bn we note that b∈ Cn(A), and so by compactness b ∈ Cn(a) for some conjunction a = a1∧…∧am ofelements of A. We can thus construct a derivation whose leaves are the pairs (bi,xi),followed by applications of SI to get the pairs (a,xi), followed by applications of ANDto get (a, x1∧…∧xn), followed finally by WO to get (a,x).

Evidently, the proof of Observation 1 also provides a ‘universal order’ for derivationsof simple-minded output: SI, AND, WO. More on this in section 8.

4. BASIC OUTPUT

4.1. Semantic definition and syntactic characterisation

As already remarked, simple-minded output is unable to process disjunctive inputsintelligently. How may this be done? On the syntactic level, the answer is obvious:define deriv2(G) by adding the following rule to those for simple-minded derivations:

OR: From (a,x), (b,x) to (a∨b,x).

On the semantic level, we define basic output, out2(G,A), as ∩{ Cn(G(V)): v(A) = 1},in the principal case that A is classically consistent (see figure 4). Here, v ranges overboolean valuations and V = {b: v(b) = 1}. In the limiting case that there is no such v(which by classical logic happens iff A is inconsistent) we put out2(G,A) to be


Cn(G(L)) where L is the set of all boolean formulae; equivalently Cn(h(G)) whereh(G) is the set of all heads of elements of G.

Equivalently: out2(G,A) = ∩{ Cn(G(V)): A ⊆V, V complete}. Here, by a complete setwe mean one that is either maxiconsistent or equal to L. There is always at least onecomplete V that includes A, namely L, and so there is no need for a separate limitingcase. The same trick could be done with the first formulation, by allowing v to beeither a boolean valuation or the function that puts v(b) = 1 for all formulae b.

Note that as classical consequence Cn is monotonic, and the transformation G(X) isalso monotonic in each of X and G, both simple-minded and basic output aremonotonic in each of their arguments.

To compare basic with simple-minded output, notice that simple-minded output canalso be expressed as an intersection. Trivially, out1(G,A) = ∩{ Cn(G(B)): A ⊆ B =Cn(B)}. As is well known, Cn(V) = V for any complete V, so we can say that basicoutput is like simple-minded output except that is restricts the choice of B to completesets.


Outline of Proof. We begin by disposing of the limiting case that A is inconsistent. Inthat case out2(G,A) = Cn(h(G)) = deriv2(G,A) by definition on the left and easyverification on the right. Next, we dispose of another limiting case, that x ∉ Cn(G(L)).

Figure 4

Basic Output: out2(G,A) = ∩{Cn(G(V)): v(A) = 1}

= ∩{Cn(G(V)): A ⊆ V}

Input A

V1

V2

G(V1)

G(V2)

G

Cn(G(V1))

Cn(G(V2))

out2(G,A)


Since L is complete and includes A, this gives immediately x ∉ out2(G,A); and it isalso easy to show by induction that deriv2(G,A) ⊆ Cn(G(L)) so that x ∉ deriv2(G,A).So consider finally the principal case that A is consistent and x ∈ Cn(G(L)).

The verification from right to left (soundness) is effected by first observing that itsuffices to prove the result for individual formulae a and then carrying out astraightforward induction on length of derivation. The interesting case in the inductionis that for the rule OR. Suppose x ∉ out2(G,b∨c). Then there is a boolean valuation vwith v(b∨c) =1 and x ∉ Cn(G(V)). But then either v(b) =1 or v(c) =1 so either x ∉out2(G,b) or x ∉ out2(G,c).

For the converse (completeness), we can use a maximality argument, similar to thatfamiliar for proving completeness in classical propositional logic, but with moreverifications at each step. In sketch: suppose x ∉ deriv2(G,A). Then by the monotonyand compactness of the derivability operation in its right argument (both immediatefrom its definition) there is a maximal A′ ⊇ A with x ∉ deriv2(G,A′). It is easy toverify that A′ is well-behaved with respect to conjunction and disjunction. Using thesupposition x ∈ Cn(G(L)) we can also verify that it is well-behaved with respect tonegation. Hence there is a boolean valuation v with A′ = V. To complete the proof, oneneed only show that x ∉ out1(G,V) = Cn(G(Cn(V))) = Cn(G(V)) since V is closedunder consequence. But this is immediate since by Observation 1 out1(G,V) =deriv1(G,V) ⊆ deriv2(G,V) and we have x ∉ deriv2(G,V).

Evidently, Observation 2 implies the compactness of out2, a fact rather difficult toverify directly from the semantic definition (in contrast to the situation for simple-minded output, where compactness is almost immediate).

We present two further characterisations of basic output. One uses relabeling ofelementary letters, the other translates into modal logic. They have very similarstructures. We regard these two characterisations as interesting curiosities more thanuseful tools, and they are not re-employed in subsequent sections. Hence sections 4.2and 4.3 may be skipped without loss of continuity.

4.2. Account in terms of relabeling

The basic idea of this approach is to relabel the letters in the heads. This has the effectof isolating the heads from the bodies, so that information about one cannot be carriedforwards or backwards to the other. Technically, alongside the existing language,introduce a fresh set of elementary letters, with one new letter p* for each old letter p.For arbitrary old formulae x, define x* in the natural way, by substituting the lettersp* for p in x. Write G* for { b→y*: (b,y) ∈ G}, i.e. as the set of all materialisations ofpairs (b,y*) obtained by starring heads only of elements of G.

OBSERVATION 3. x ∈ out2(G,A) iff x ∈ Cn(G(L)) and G*∪A • x*.

Proof. We dispose of the limiting cases that A is inconsistent and that x ∉ Cn(G(L)) inthe same manner as for Observation 2. So suppose for the principal case that A isconsistent and x ∈ Cn(G(L)).


Suppose first that the left side fails. Since A is consistent, there is a valuation v onunstarred letters with v(A) = 1 and x ∉ Cn(G(V)). From the latter, there is a valuationw (also on unstarred letters) with w(G(V)) = 1 and w(x) = 0. Define a valuation w* onformulae generated by starred letters by putting w*(p*) = w(p) for each starred letterp*. Write v+w* for the valuation on starred and unstarred letters determined by thetwo together. We claim that v+w*(G*∪A) = 1 and v+w*(x*) = 0. The latter isimmediate from w(x) = 0 since all letters in x* are starred. Similarly, we havev+w*(A) = 1 from v(A) = 1 since all letters in A are unstarred. It remains to check thatv+w*(G*) = 1. Let (b,y) ∈ G and suppose v+w*(b) = 1; we need to show thatv+w*(y*) = 1. Since b is unstarred the supposition tells us that v(b) = 1 so b ∈ V, so y∈ G(V) so by hypothesis w(y) = 1 so w*(y*) = 1 and finally v+w*(y*) = 1.

To show the converse, we could use the identity out2(G,A) = deriv2(G,A) establishedby Observation 2, and proceed by induction on length of derivation, but we give adirect argument, as follows.

Suppose that the right side fails. Since we are assuming that x ∈ Cn(G(L)), there is avaluation defined on both starred and unstarred letters that satisfies G*∪A and failsx*. Without loss of generality, we may write this valuation as v+w* where v,w aredefined on unstarred letters and w* is defined from w as before. Thus v+w*(G*∪A) =1 and v+w*(x*) = 0. We show that v(A) = 1 and x ∉ Cn(G(V)). We have v(A) = 1immediately from v+w*(A) = 1 since A contains only unstarred letters. For x ∉Cn(G(V)), it suffices to show that w(x) = 0 while w(G(V)) = 1. The former isimmediate from v+w*(x*) = 0. For the latter, suppose y ∈ G(V); we need to show w(y)= 1. Since y ∈ G(V) there is an unstarred formula b with (b,y) ∈ G and b ∈ V so that 1= v(b) = v+w*(b). Since v+w*(G*) = 1 we have v+w*(b→y*) = 1 so that 1 = v+w*(y*)= w*(y*) = w(y) as desired.

4.3. Modal formulation

The modal characterisation has strong formal parallels with the relabeling one. Itsessential idea is to prefix heads with boxes and apply a suitable modal logic. Indeedany modal logic from a broad interval will do the job.

Consider the modal propositional language formed by adding a unary box operator tothe classical language, and consider the modal calculus K0, serving as a lower boundon the interval, defined axiomatically as follows. Take as axioms all classicaltautologies in that language and all formulae of the form (a→x)→( a→ x); andtake as rules passage from a, a→x to x (detachment), and passage from t to t forevery classical tautology t. Evidently, we could reformulate the last rule as axioms tfor every classical tautology t.

K0 is a subsystem of the familiar modal logic K ; the latter also allows passage from ato a for every thesis a. We recall the well-known fact that for first-degree formulae(i.e. formulae without iteration of the box) all systems from K0 to K45 agree.


Write G for the set of all modal formulae b→ y with (b,y) ∈ G, and Z • S z to meanthat (∧Y → z) ∈ S for some finite Y ⊆ Z.

OBSERVATION 4. x ∈ out2(G,A) iff x ∈ Cn(G(L)) and G ∪A • S x, for any modallogic S with K0 ⊆ S ⊆ K45.

Proof. Since all systems from K0 to K45 agree on first-degree formulae, we need onlyprove the Observation for K . In the limiting case that A is classically inconsistent bothsides are equivalent to x ∈ Cn(G(L)) and we are done. So suppose that A is consistent.

Suppose x ∈ out2(G,A). Then by Observation 2, (A,x) ∈ deriv2(G) so we need onlyshow by induction that whenever (a,x) ∈ deriv2(G) then G ∪{ a} • K x, which isstraightforward.

Conversely, suppose x ∉ out2(G,A). Since A is assumed consistent, there is a valuationv of boolean formulae with v(A) = 1 and x ∉ Cn(G(V)). Fix one such v, and define arelational model (M,R,ϕ) by putting M to be the set of all purely boolean valuationsand for u,w ∈ M put (u,w) ∈ R iff for every (b,y) ∈ G, if u(b) = 1 then w(y) = 1. Putϕ(w,p) = w(p) for all elementary letters p and all w ∈ M.

To complete the proof, it suffices to check that ϕ(v, G ∪A) = 1 while ϕ(v, x) = 0.Since v(A) = 1 and A is purely boolean, ϕ(v,A) = 1. Suppose b→ y ∈ G and ϕ(v,b) =1; then (b,y) ∈ G and b is purely boolean so v(b) = 1 and also whenever (v,w) ∈ Rthen by the definition of R, w(y) = 1; thus ϕ(v, b→ y) = 1. This shows ϕ(v, G ) = 1.To show ϕ(v, x) = 0 we need to find a w with (v,w) ∈ R and ϕ(w,x) = 0. But byhypothesis, x ∉ Cn(G(V)) so there is a w with w(G(V)) = 1 and w(x) = ϕ(w,x) = 0. Itremains only to check that (v,w) ∈ R. But if (b,y) ∈ G and v(b) = 1 then immediately y∈ G(V) so w(y) = 1 and by the definition of R we are done.

5. REUSABLE OUTPUT

5.1. Idea and definitions

In certain situations, it may be appropriate for outputs to be available for recycling asinputs. For example, the elements (a,x) of G may be conditional norms of a kind thatsay that any configuration in which a is true is one in which x is desirable. In somecontexts, we may wish to entertain hypothetically the items already seen as desirable,in order to determine what is in turn so. How may such a principle of reusability beexpressed formally?

On the syntactic level, the answer again suggests itself naturally: add the followingrule of ‘cumulative transitivity’ to those already available for simple-minded output,or those for basic output:

CT: From (a,x), (a∧x,y) to (a,y).


Given SI, this immediately implies transitivity (from (a,x), (x,y) to (a,y)) but notconversely.

On the semantic level, we define simple-minded reusable output, written out3(G,A), asfollows:

out3(G,A) = ∩{ Cn(G(B)): A ⊆ B = Cn(B) ⊇ G(B)}.

There is always at least one set B with A ⊆ B = Cn(B) ⊇ G(B), namely L, and theintersection of any non-empty family of such sets satisfies the same condition.

Recalling again that simple-minded output can be expressed as an intersection, without1(G,A) = ∩{ Cn(G(B)): A ⊆ B = Cn(B)}, we can say that reusable simple-mindedoutput is like plain simple-minded output, except that it restricts the choice of B tosets that are included in their own image under G.

Since each of the operations G and Cn is monotone, their composition is alsomonotone. Hence the definition may also be expressed thus: out3(G,A) = Cn(G(A*))where A* is the least superset of A that is closed under both Cn and G.

We define basic reusable output, written out4(G,A), as follows in the principal casethat A is classically consistent:

out4(G,A) = ∩{ Cn(G(V)): v(A) = 1 and G(V) ⊆ V}.

Here as before, v ranges over boolean valuations and V = {b: v(b) = 1}. In the limitingcase that there is no such v, we proceed as for basic output, putting out4(G,A) to beCn(G(L)) where L is the set of all boolean formulae; equal to Cn(h(G)) where h(G) isthe set of all heads of elements of G. Equivalently,

out4(G,A) = ∩{ Cn(G(V)): A ⊆ V ⊇ G(V), V complete}.

Clearly out3(G,A) ⊆ out4(G,A) ⊆ Cn(G(L)). The diagrams for the two notions areessentially the same. For basic reusable output, see figure 5. For the simple-mindedversion, replace the captions Vi by Xi = Cn(Xi).


5.2. Simple-minded reusable output: properties and syntactic characterisation

As in the non-reusable case, the simple-minded reusable operation is less satisfyingthan the basic one, given its inability to deal intelligently with disjunctive inputs.Nevertheless, the simple-minded version has a certain interest, and we indicate someof its basic properties.

OBSERVATION 5 (cumulativity on the right). Out3(G,A) = out3(G,A∪D) whenever D⊆ out3(G,A).

Proof. The left is included in the right, by monotony in the right argument (immediatefrom the definition). For the converse, suppose x ∉ out3(G,A). Then by the definitionof out3 there is a B with A ⊆ B = Cn(B) ⊇ G(B) and x ∉ Cn(G(B). To show x ∉out3(G,A∪D), it suffices to show A∪D ⊆ B, and so since A ⊆ B and using thehypothesis D ⊆ out3(G,A), it is enough to show out3(G,A) ⊆ B. But by its definition,out3(G,A) ⊆ Cn(G(B)) ⊆ B and we are done.

From cumulativity and monotony it follows immediately that simple-minded reusableoutput satisfies one half of idempotence on the right: out3(G,out3(G,A)) ⊆out3(G,A∪out3(G,A)) = out3(G,A). However, the converse half of idempotence fails.

Figure 5Basic Reusable Output: out4(G,A) = ∩{Cn(G(V)): A ⊆ V ⊇ G(V)}

Input A

V1

V2

G(V1)

G(V2)

G

Cn(G(V1))

Cn(G(V2))

out2(G,A)

⊆

⊆


Example 3. Put G = {(a,x)} and A = {a} where a,x are distinct elementary letters.Then out3(G,a) = Cn(x) whereas out3(G,out(G,a)) = out3(G,Cn(x)) = Cn(∅), so thatthe former is not included in the latter.

Thus for each G, the right projection function outG,3(A), defined as out3(G,A), is insome respects like a Tarski consequence operation (that is, a closure operation on setsof propositions) and in some respects different. It is monotonic and cumulative, anditerated output is included in single output; but in general it fails inclusion and theother half of idempotence.

These remarks about the right projection function of simple-minded reusable outputshould not be confused with the fact that all of our input/output operations,understood as taking sets G of pairs (A,x) to sets outi(G) of pairs, are quite trivially,closure operations - inclusion, monotony, and idempotence all hold.

We sketch a proof of the equivalence of its semantic and syntactic definitions ofreusable simple-minded output, writing deriv3(G,A) for the latter.


Outline of proof. It suffices to prove the result for singleton A. The inclusion fromright to left is straightforward by induction on length of derivation. The interestingclause is that for CT. Suppose that x ∈ out3(G,a) and y ∉ out3(G,a); we need to showthat y ∉ out3(G,a∧x). From the second hypothesis, there is a B with a ∈ B = Cn(B) ⊇G(B) and y ∉ Cn(G(B)). By the first hypothesis, x ∈ Cn(G(B)). But since G(B) ⊆Cn(B) we have Cn(G(B)) ⊆ Cn(B) so x ∈ Cn(B). Thus a∧x ∈ Cn(B) and so y ∉out3(G,a∧x) as desired.

For the converse, suppose x ∉ deriv3(G,a); we need to find a B with a ∈ B = Cn(B) ⊇G(B) and x ∉ Cn(G(B)).

Put B = Cn({ a} ∪ deriv3(G,a)). Clearly a ∈ B = Cn(B). To show G(B) ⊆ B, suppose y∈ G(B). Then there is a b ∈ B with (b,y) ∈ G. We need to show y ∈ B, i.e. deriv3(G,a)• a→y. But since b ∈ B we have deriv3(G,a) • a→b so since deriv3(G,a) is closedunder classical consequence (by the rules AND,WO and the compactness of classicalconsequence) we have a→b ∈ deriv3(G,a), i.e. (a, a→b) ∈ deriv3(G). But since (b,y)∈ G we also have (b,y) ∈ deriv3(G) so by SI, (a∧b, y) ∈ deriv3(G), so by CT, (a,y) ∈deriv3(G), i.e. y ∈ deriv3(G,a) so by WO, a→y ∈ deriv3(G,a) so deriv3(G,a) • a→y asdesired.

It remains to check that x ∉ Cn(G(B)), i.e. x ∉ Cn(G(Cn({ a} ∪ deriv3(G,a)))) =out1(G, {a} ∪ deriv3(G,a)) = deriv1(G, {a} ∪ deriv3(G,a)) using the completenesstheorem for simple-minded output (Observation 1). Suppose the contrary. Then, usingSI, there are x1,…,xn ∈ deriv3(G,a) with x ∈ deriv1(G, a∧x1∧…∧xn) i.e. (a∧x1∧…∧xn,,x) ∈ deriv1(G) ⊆ deriv3(G). But since each xi ∈ deriv3(G,a), i.e. (a,xi) ∈ deriv3(G) wehave by AND and WO that (a, x1∧…∧xn) ∈ deriv3(G). Hence by CT, (a,x) ∈deriv3(G) i.e. x ∈ deriv3(G,a) contradicting our initial supposition.


5.3. Basic reusable output: first properties

We now focus on basic reusable output, better motivated than its simple-mindedcounterpart and also more interesting formally. To lighten terminology, from now onwe refer to it simply as reusable output. We show in section 5.4 that out4(G,A) =deriv4(G,A), where the latter is defined by the rules for basic output(SI,AND,WO,OR) plus CT. But before doing so we draw attention to some propertiesof the semantic construction.

Reusable output may equivalently be defined in the following manner, which is ratherless intuitive, but establishes a link with basic output and simplifies proofs.

OBSERVATION 7. Out4(G,A) = ∩{ Cn(G(V)): A∪m(G) ⊆ V, V complete}.

Proof. It suffices to show that for any complete set V, we have G(V) ⊆ V iff m(G) ⊆V, where m(G) is the materialisation of G, that is, the set of all formulae b→y with(b,y) ∈ G.

In one direction, suppose m(G) ⊆ V and let y ∈ G(V); we need to show that y ∈ V.Since y ∈ G(V) there is a b ∈ V with (b,y) ∈ G, so b→y ∈ V and so since V iscomplete, y ∈ V as desired.

Conversely, suppose G(V) ⊆ V and suppose b→y ∈ m(G); we need to show b→y ∈ V.Suppose b∈ V; since V is complete, it suffices to show that y ∈ V. But since b→y ∈m(G) we have (b,y) ∈ G so since b ∈ V we have y ∈ G(V) ⊆ V and we are done.

This observation immediately allows us to express reusable basic output in terms ofits non-reusable counterpart, a fact that will be useful later.

COROLLARY TO OBSERVATION 7. Out4(G,A) = out2(G,A∪m(G)).

It also permits a simplification of Figure 5: drop the backward-reaching lines withtheir inclusion signs, and alongside the input circle insert a circle for m(G), alsoincluded within the Vi ellipses.

First, we note that although out1(G,A) ⊆ {out2(G,A), out3(G,A)} ⊆ out4(G,A) ⊆Cn(A∪ m(G)), still out4(G,A) ≠ Cn(A∪ m(G)); in particular, inputs are still not ingeneral outputs, and contraposition still fails, as Example 1 continues to show.Nevertheless, contraposition plays a curious ‘ghostly’ role for reusable basic output.

Example 4 (ghost contraposition). Put G = {(¬x,¬a),(a∧x,y)}. On the one hand, x ∉out4(G,a) since x ∉ Cn(G(L)) = Cn(h(G)) = Cn(¬a,y). On the other hand, y ∈out4(G,a), since y ∈ Cn(G(L)) and also for every valuation v satisfying {a} ∪m(G),v(x) = 1, so y ∈ G(V).

Expressed more generally, this example tells us that y ∈ out4(G,a) whenever y ∈out4(G, a∧x) and ¬a ∈ out4(G,¬x). In other words, for basic reusable output we havethe rule:


GC: From (¬x,¬a), (a∧x,y) to (a,y).

Intuitively: although we cannot contrapose the premise (¬x,¬a), we can ‘use’ thecontraposition for an application of cumulative transitivity. This can be verifieddirectly, or from the following principle of input sufficiency:

OBSERVATION 8 (input sufficiency). Whenever {a} ∪m(G)• x, then if y ∈ out4(G,a∧x) then y ∈ out4(G,a). More generally, whenever A∪m(G)• X, then if y ∈ out4(G,A∪X) then y ∈ out4(G,A).

Proof. Immediate from Observation 7, for if A∪m(G)• X and A∪m(G) ⊆ V where V isa complete set, then A∪X∪m(G) ⊆ V.

This is a powerful principle, with a number of consequences. Expressed syntactically,it is the rule:

IS: From (a∧x,y) to (a,y) whenever {a} ∪m(G)• x.

This implies ghost contraposition, for ¬a ∈ out(G,¬x) implies {¬x} ∪m(G)• ¬a sothat {a} ∪m(G)• x. Again, since x ∈ out(G,a) implies {a} ∪m(G)• x, input sufficiencyalso implies CT, which we recall authorises passage from (a,x), (a∧x,y) to (a,y).

Essentially the same property may be expressed as follows: for reusable output wemay add to the input the materialisations of some or all of the generators, withoutchanging the output.

OBSERVATION 9 (shadow input). Out4(G,A) = out4(G, A∪ m(G′)) whenever G′ ⊆ G.

Proof. Immediate from Observation 7, since A∪m(G) = (A∪m(G′)∪m(G) wheneverG′ ⊆ G. It may also be seen as the case of Observation 8 in which B = A∪m(G′).

From Observation 9 we may say that for reusable output, generators are in a certainsense stronger than inputs. But only in a limited sense: we can copy from generatorsto inputs without altering output, but if we transfer from generators to inputs then wemay in general lose and gain output, as can be shown by trivial examples. Simpleexamples also show that copying from inputs to generators may change output.

Finally, we note that basic reusable output is cumulative and satisfies half ofidempotence (iterated output included in single output). The proof is the same as forsimple-minded output (Observation 5). However, these properties fail for plainsimple-minded and basic output (i.e. without reusability). This is as one would expect:cumulativity of the output operation is closely associated on the syntactic level withthe rule CT, and on the semantic level with reusability.

5.4. Basic reusable output: syntactic characterisation

We now show that out4(G,A) = deriv4(G,A), where the latter is defined by the rules forbasic output (SI,AND,WO,OR) plus CT.


OBSERVATION 10 (soundness). Deriv4(G,A) ⊆ out4(G,A).

Outline of proof. We need only add to the verification of the corresponding result forbasic output (Observation 2, first part of proof) a verification of the rule CT. Thatverification follows the same pattern as in the soundness proof for simple-mindedreusable output (Observation 6).

OBSERVATION 11 (completeness). Out4(G,A) ⊆ deriv4(G,A).

Proof. By the Corollary to Observation 7 we have out4(G,A) = out2(G,A∪m(G)) =deriv2(G,A∪ m(G)) by Observation 2, so it suffices to show deriv2(G,A∪m(G)) ⊆deriv4(G,A). That is, we need only show that the shadow input property, already notedfor the semantic operation out4 (Observation 9), also holds for the syntactic onederiv4. We do this in two steps: first, we prove the property for singleton input withsingleton generating set, and then show that it follows in the general form.

LEMMA 11a. If (b,x) ∈ G then deriv4(G,a∧(b→x)) ⊆ deriv4(G,a).

Proof. Let (b,x) ∈ G and suppose y ∈ deriv4(G,a∧(b→x)); we want to show that y ∈deriv4(G,a)). The desired derivation can be displayed as a tree diagram, as follows:

(b,x) (a∧(b→x),y) (a∧(b→x),y) SI SI (a∧b,x) (a∧b∧x,y) -------------------------------- CT

(a∧b,y) ------------------------------------- OR

(a,y)

LEMMA 11b. Deriv4(G,A∪m(G)) ⊆ deriv4(G,A).

Proof. Suppose y ∈ deriv4(G,A∪m(G)). Clearly the operation deriv4 is monotonic andcompact on left and right. By definition, there is a conjunction a of formulae in A, anda conjunction g = ∧(bi→xi) of formulae in m(G), such that y ∈ deriv4(G,a∧g).Applying Lemma 11a finitely many times according to the number of conjuncts in g,we have y ∈ deriv4(G,a) so by definition y ∈ deriv4(G,A). This completes the proof ofthe Lemma and of Observation 11.

The above proof of completeness makes use of the reduction of reusable basic outputto plain basic output, in the Corollary to Observation 7. If one prefers to argue fromfirst principles, one can re-run the same maximality construction as in the proof ofObservation 2, but ensuring that A∪m(G) ⊆ A′. For this it suffices to show thatwhenever x ∉ deriv4(G,A) then x ∉ deriv4(G,A∪m(G)), i.e. the same shadow inputproperty deriv4(G,A∪m(G)) ⊆ deriv4(G,A) proven as Lemma 11b above.

OBSERVATION 12 (semantic characterisation). Out4(G,A) = deriv4(G,A).


Proof. Immediate from Observations 10, 11. As a corollary, we may note that sincederiv4 is compact on both left and right, out4 is too.

5.5. Relabeling and modal formulations

Like basic output, its reusable extension can be characterised by means of relabeling,and also in modal terms.

OBSERVATION 13. x ∈ out4(G,A) iff x ∈ Cn(G(L)) and G*∪A∪m(G) • x*.

Proof. Immediate from the reduction of basic reusable output to out2 in the Corollaryto Observation 7, i.e. out4(G,A) = out2(G,A∪m(G)), together with Observation 3.

OBSERVATION 14. x ∈ out4(G,A) iff x ∈ Cn(G(L)) and G ∪A∪m(G) • S x, for anymodal logic S with K0 ⊆ S ⊆ K45.

Proof: Immediate from the same reduction, with Observation 4.

A more interesting modal reduction gets rid of the ‘additional premise’ m(G) infavour of the additional modal axiom a→a, known as T.

OBSERVATION 15. x ∈ out4(G,A) iff x ∈ Cn(G(L)) and G ∪A • S x, for any modallogic S with K0T ⊆ S ⊆ KT45.

Outline of proof. Since all systems from K0T to KT45 agree on first-degree formulae,we need only prove the observation for KT . The argument follows the same lines asfor Observation 4, with the following additions and modifications.

From left to right, we need to show that the modal translation satisfies the rule CT.This amounts to showing that for any formula g, if (g∧a)→ x and (g∧a∧x)→ y are inKT then so is (g∧a)→ y. But this is immediate given the availability of x→x in KT .

From right to left, we suppose x ∉ out4(G,A). As before, it follows from the definitionof the output operation that there is a valuation v of boolean formulae with v(A) = 1and x ∉ Cn(G(V)), and this time we also have G(V) ⊆ V. Fix one such v, and definethe relational model (M,R,ϕ) as before, but with a modified relation R. For the chosenvaluation v put (v,w) ∈ R iff for every (b,y) ∈ G, if v(b) = 1 then w(y) = 1; for everyvaluation u ≠ v, put (u,u) ∈ R. Note that when v(b) = 1 and (b,y) ∈ G then y ∈ G(V) ⊆V so that v(y) = 1; this shows (v,v) ∈ R. Combining this with the other part of thedefinition of R, we have its reflexivity, so that the model validates the modal systemKT . To complete the proof, it suffices to check that ϕ(v, G ∪A) = 1 while ϕ(v, x) =0. This is done exactly as in the proof of Observation 4.

We note in passing that in modal logics satisfying the modal axiom T, G impliesm(G), so that given observations 14 and 15, we can also push the upper bound of theformer up to KT45.


As far as the authors are aware, it is not possible to characterise the system of simple-minded output (with or without reusability) by relabeling or modal logic in astraightforward way. The OR rule appears to be needed, so that we can work withcomplete sets.

6. ACCEPTING INPUTS AS OUTPUTS

What happens if we strengthen the logic of some kind of output by accepting inputs asoutputs? Syntactically, add the rule: From no premises to (y,y). Evidently, such a rulecan always be applied first, so the semantic counterpart amplifies outi(G,A) toouti(G∪I,A) where I = {(y,y): y a formula}. Of these, basic reusable output plusidentity collapses into classical consequence.

OBSERVATION 16: out4(G∪I,A) = Cn(A∪m(G)).

Proof. Given Observation 11, the left in right inclusion is a trivial induction. For theconverse, write G+ for G∪I, and suppose x ∉ out4(G

+,A). Then by the definition ofout4, there is a valuation complete set V with A∪m(G+) ⊆ V and x ∉ Cn(G+(V)).Clearly V ⊆ I(V) ⊆ G+(V). Also since m(G+) ⊆ V, we have G+(V) ⊆ V. Thus V =G+(V). Since x ∉ Cn(G+(V)) we thus have x ∉ Cn(V) = V, so that the complete set V isa maxiconsistent set, corresponding to a classical valuation v, with v(x) = 0. Since alsoA∪m(G) ⊆ A∪m(G+) ⊆ V, we have v(A∪m(G)) = 1. Putting these together, x ∉Cn(A∪m(G)). •

Alternatively, one may re-run the argument for Observation 11, observing that since x∉ out4(G

+,A′) (defined as in that proof) and I ⊆ G+, we have x ∉ A′ so that v(x) = 0.Since A∪m(G+) ⊆ A′ = V we also have v(A∪m(G)) =1.

Simple-minded reusable output plus identity does not collapse into classical logic, butmay be simplified.

OBSERVATION 17: out3(G∪I,A) = ∩{ B: A ⊆ B = Cn(B) ⊇ G(B)}.

Proof. By the definition of out3 it suffices to check that whenever B = Cn(B) ⊇ G(B)we have Cn(G+(B)) = B . Left in right: if G(B) ⊆ B then since also I(B) ⊆ B we haveG+(B) ⊆ B so Cn(G+(B)) ⊆ Cn(B) = B by hypothesis. Right in left: since I ⊆ G+ wehave B ⊆ G+(B) ⊆ Cn(G+(B)). •

Note that this verification makes essential use of reusability, i.e. that G(B) ⊆ B, and ofidentity, i.e. that the generating set includes I, so that the argument does not apply toweaker kinds of output.

From our perspective, operations that accept all inputs as outputs are a limiting caseof ‘logically assisted transformations’. However, Observation 17 draws attention to aninteresting connection with a construction underlying normal default logic.Specifically: Reiter’s default logic, stripped of its consistency constraint, is the sameas simple-minded reusable output with identity. To see this, take the quasi-inductivedefinition of an extension of a normal default system as given in (Reiter 1980,


theorem 2.1) or (Makinson 1994, section 3.2), and take out the consistency constraint.This puts ext(G,a) = ∪{ Ei: 0 ≤ i < ω} where E1 = {a} and Ei+ 1 = Cn(Ei)∪G(Ei). It iseasy to check that ext(G,a) = ∩{ B: a ∈ B = Cn(B) ⊇ G(B)}, so by Observation 17,ext(G,a) = out3(G∪I,a).

7. REVERSIBILITY OF RULES IN A DERIVATION

We finally consider briefly some questions arising for the syntactic formulations ofthe four input/output operations: reversibility of rules (this section) and ‘universality’of certain orders of derivation of output (following section).

Note that all four input/output operations satisfy replacement of input, and of output,by classically equivalent propositions. That is, if (a,x) ∈ out(G) then (a′,x′) ∈ out(G)whenever Cn(a) = Cn(a′) and Cn(x) = Cn(x′). From this point on, we treatreplacement of logically equivalent propositions as a ‘silent rule’, that may be appliedat any step without explicit justification.

With this understanding, the order of application of two derivation rules is often‘reversible’. In some cases, we may simply permute the application of two successiverules, independently of the choice of the formulae to which they are applied. Forexample, any application of AND followed by SI may be replaced by one in which SIis followed by AND. In other cases, the order may be reversed, but with additional(and prior) use of a third rule - often SI and in one instance WO. Finally, there aresome configurations for which no transformation appears to be available.

Observation 18 displays in a table the transformations that the authors have noted tobe possible. The table should be read as follows:

• An entry in the cell determined by the row for rule R and the column for R′tells us to what extent the sequence R,R′ may be reversed to R′,R.

• In an application of the asymmetric rule CT, taking us from (a,x) and (a∧x,y) to(a,y), we call (a,x) the ‘minor’ premise and (a∧x,y) the ‘major’ premise. In thecolumn for CT, the left (resp. right) sub-column represents the case where theoutput of the previous rule feeds in as the minor (resp. major) premise of therule CT.

• The entry D indicates that simple permutation is possible.• When only a more complex reversal is known to be possible, it is written

explicitly. Thus for example in the cell for CT,AND we have writtenSI,AND,CT to indicate that the former order may be transformed into the latter.

• The entry none? means that no transformation is known to the authors.• The empty spaces in the diagonal mean that the question does not arise there.


OBSERVATION 18 (reversibility of rules).

SI WO CT AND OR

SI D none? none? none? D

WO D SI,CT D D none?

CT D D SI,AND,CT none?

AND D D SI,CT D WO,OR,AND

OR D D SI,CT,OR SI,CT,OR SI,AND,OR

Not to overburden the paper, we omit the verifications of the reversals in the table,giving only the least immediate among them as an example. This is the transformationOR,CT ⇒ SI,CT,OR, where the left hand configuration takes two forms according asthe conclusion of OR feeds in as ‘minor’ or ‘major’ premise of the non-symmetrictwo-premise rule CT.

OR,CT (Case 1) ⇒ SI,CT,OR

(a,x) (b,x) ((a∨b)∧x,y)-------------- OR (a∨b,x) ------------------------------ CT (a∨b,y)

(a,x) ((a∨b)∧x,y) (b,x) ((a∨b)∧x,y) SI SI (a∧x,y) (b∧x,y)CT ------------------ ------------------- CT (a,y) (b,y) ------------------------------- OR (a∨b,y)


OR,CT (Case 2) ⇒ SI,CT,OR

(z,y) (a,x) (b,x) ----------------- OR (a∨b,x) ------------------- CT: z∧y ≈ a∨b (z,x)

(z,y) (a,x) (z,y) (b,x) SI SI (z∧(¬b∨a),y) (z∧(¬a∨b),y) ------------------ CT --------------------- CT(z∧(¬b∨a),x) (z∧(¬a∨b),x) -------------------------------------------------- OR (z,x)

Here ≈ stands for classical equivalence. In the second display, the given application ofCT (on the left) is allowable iff z∧y ≈ a∨b, in which case z∧(¬b∨a)∧y ≈ a andz∧(¬a∨b)∧y ≈ b so that we can apply CT as indicated on the right.

8. UNIVERSAL ORDERS OF DERIVATION

Consider any system with n derivation rules (e.g. basic output with its four rules SI,AND, WO, OR). We say that a derivation respects an order R1,...,Rn of those rules iffa rule Rj is never applied in it before (i.e. leafwards of) a rule Ri for i < j. In otherwords, rules may be skipped or repeated (and moreover, as already mentioned earlier,it is understood that classically equivalent formulae may replace each other wheneverdesired), but the rules must never be applied contrary to the indicated order. Ofcourse, many derivations do not respect any order at all; in particular, if an applicationof R is made before one of a distinct rule R′, but also an application of R′ is madebefore one of R, then no order is respected in the derivation.

We say that an order is universal (for a given set of rules defining an input/outputoperation) iff whenever (a,x) ∈ out(G) then there is a derivation of (a,x) from Grespecting that order. The question naturally arises: are there any universal orders?Repeated application of Observation 18 tells us that there are several.

OBSERVATION 19.

(a) For simple-minded output, with the rules SI, AND, WO, there are (at least) threeuniversal orders of derivation: SI,AND,WO, and (SI,WO),AND.

(b) For basic output, with the rules SI, AND, WO, OR, there are (at least) sixuniversal orders: SI,AND,WO, OR, and (SI,WO),(AND,OR) and WO,OR,SI,AND.

(c) For simple-minded reusable output, with the rules SI, AND, WO, CT, there are (atleast) eight universal orders: SI,(WO,CT,AND) and WO,SI,(CT,AND).

(d) For reusable output, with the rules SI, AND, WO, OR, CT, there are (at least)eleven universal orders: SI,(WO,CT,AND),OR and SI,(WO,CT),OR,AND andWO,SI,(CT,AND),OR and WO,SI,CT,OR,AND.


Here the parentheses indicate that every arrangement within them is counted. The firstorder for simple-minded output also emerged from its completeness proof(Observation 1). Of course, Observation 19 depends very much on the particularchoice of rules made, not only their joint force. For the rules that we have used, weconjecture that in each case there are no universal orders of derivation other thanthose listed.

Remark. In Observation 18, there are just four non-reversible orders: SI,CT; SI,AND;WO,OR; CT,OR. Thus all orders listed as universal in Observation 19 satisfy theproperty: SI before (immediately or separated by other rules) CT, SI before AND,WO before OR, and CT before OR. More surprisingly, it can be checked byenumeration that the converse is also true: every order satisfying that property isuniversal. It is not clear whether this fact points to a deeper pattern.

9. OTHER SYSTEMS

One might consider strengthening, weakening, or otherwise modifying the systemsstudied in this paper, with either a purely formal motivation or an eye to possibleapplications.

For example, with an interest in defeasible conditionals, one might drop the SI rule,perhaps replacing it by a rule of replacement of equivalent input propositions.Semantically, the operations are Cn(G(a)) and Cn(G(E(a))) for individual formulae a,although the definition for infinite sets A (section 3.2) becomes problematic.

Again, one might consider modifying certain of the rules employed. For example, weknow (section 5.1) that given SI, cumulative transitivity CT implies transitivity T, butnot conversely. What happens if in the system of simple-minded reusable output, say,we replace CT by T? Given SI and AND, it is easy to show that T is equivalent to thefollowing principle of ‘ghost cumulative transitivity’ GCT, which the authors havenot seen in the literature: From (p,a), (a,b), (a∧b,c) to (p,c).

We conjecture that this system may be defined as out(G,A) = Cn(out3(G∪I,out1(G,A))). Diagrammatically: two G boxes, one under the other, with the sameordered pairs inside. The input A comes into the first box; whose output is input to thesecond box. Input to the second box reappears in its output; and output of the secondbox is reusable in its input. The final output is closure under Cn of the second box.

Finally, one could consider adding various rules to one or more of the systemsstudied. For example, one could look at:

Contraposition CP: from (a,x) to (¬x,¬a)Dual cumulative transitivity DCT: from (a,x∨y), (x,y) to (a,y)Conditionalisation CND: from (a,x) to (t, a→x).

We see these three rules of relatively minor interest, as they have little motivation interms of the underlying vision of input/output processes outlined in the first section ofthis paper. Nevertheless, we note some facts about them. Observe, first, that we mayadd any or all of these three rules to those for basic reusable output without collapse


into classical consequence. For if G = ∅, then whenever (a,x) ∈ out(G), where out isthe enlarged operation, then a simple induction shows that either a is a contradictionor x is a tautology, so that in particular (a,a) ∉ out(G), for contingent a, whereas a ∈Cn({ a} ∪m(G)). We also have the following equivalences.

OBSERVATION 20. Given CP, the rules CT and DCT are equivalent. Also, given therules of basic output, DCT and CND are equivalent.

Outline of proof. The only verification that is not trivial is that for Basic + DCT ⇒CND, as follows.

(t,t) (a,x) (a,x) SI WO (¬a, a∨(¬a∨x)) (a, ¬a∨x) WO-------------------------------------- DCT (¬a, ¬a∨x) (a, ¬a∨x)

------------------------------------------------- OR(t, a→x)

On the semantic level, it is difficult to see any input/output semantics for CP or DCT.However, in the case of the rule CND, we do have a semantics, indirectly.

OBSERVATION 21. For each of the systems outi (i = 1,…,4), if we add the rule CNDthen we have a semantics like that for the source system, except that the set G isreplaced by G∪{( t, a→x): (a,x) ∈ G}. If we add both CND and the identity rule, thenwe replace G in the semantics by G∪I∪{( t, a→x): (a,x) ∈ G∪I}.

Proof. It is easy to check that the rule CND may always be applied first in anyderivation using at most SI,AND,WO,OR,CT,CND.

10. SUMMARY

The investigations in this paper are inspired by a view of logic as ‘secretarialassistant’ to an arbitrary process transforming propositional inputs into propositionaloutputs. Its task is to prepare the inputs, unpack the outputs, and co-ordinate the twoin various ways. In this perspective, we introduced four principal input/outputoperations: simple-minded, basic (making intelligent use of disjunctive inputs),simple-minded reusable (in which outputs may be recycled as inputs), and basicreusable output. These are doubled by corresponding systems in which inputsreappear among the outputs. The systems are defined semantically, and arecharacterised syntactically by derivation rules. We recall the four basic systems.

• Simple-minded output, written out1(G,A), is defined as Cn(G(Cn(A))), and ischaracterised by the rules SI,AND,WO.

• Basic output, written out2(G,A), is defined as ∩{ Cn(G(V)): A ⊆ V, V complete},where a complete set is one that is either maxiconsistent or equal to the set L of allformulae of the language. It is characterised by SI,AND,WO,OR.


• Simple-minded reusable output, written out3(G,A), is defined as ∩{ Cn(G(X)): A ⊆X = Cn(X) ⊇ G(X)}. It is characterised by SI,AND,WO,CT.

• Basic reusable output, written out4(G,A), is defined as ∩{ Cn(G(V)): A ⊆ V ⊇G(V), V complete}. Equivalently, as: ∩{ Cn(G(V)): A∪m(G) ⊆ V, V complete}. Itis reducible to basic output by the equality out4(G,A) = out2(G,A∪m(G)), and ischaracterised by SI,AND,WO,OR,CT.

In none of the systems are inputs automatically outputs, that is, we do not have ingeneral a ∈ out(G,a). Nor do the systems guarantee contraposition: we may have x ∈out(G,a) without ¬a ∈ out(G,¬x). Of the four systems, basic reusable output revealsthe most subtle behaviour, for example the ‘input sufficiency’ and ‘shadow input’properties (Observations 8 and 9).

Basic output and its reusable extension may also be characterised in terms ofrelabeling procedures and modal operators. The account in terms of relabelingsubstitutes fresh elementary letters for old ones in heads, and applies classicalconsequence. The modal characterisation prefixes boxes to heads, and applies anymodal logic from within a broad interval.

On a syntactic level, it is shown that in a surprising number of cases, the applicationof rules in a derivation may be reversed (Observation 18), giving rise to certain‘universal orders’ of derivation for each of the four systems studied (Observation 19).

Given that an intended area of application of input/output logic is the study of systemsof conditional goals or obligations, it is natural to ask how one might introduceconsistency constraints into them, to deal with ‘contrary to duty’ conditions. Thisquestion is investigated systematically in a sequel (Makinson and van der Torre, toappear).

REFERENCES

Makinson, David, 1999. On a fundamental problem of deontic logic. In Norms,Logics and Information Systems. New Studies in Deontic Logic and ComputerScience, ed. Paul McNamara and Henry Prakken. Amsterdam: IOS Press, Series:Frontiers in Artificial Intelligence and Applications, Volume 49, pp 29-53.

Makinson 1994. General patterns in nonmonotonic reasoning. In Handbook of Logicin Artificial Intelligence and Logic Programming, vol. 3, ed. Gabbay, Hogger andRobinson. Oxford University Press, pages 35-110.

Makinson, David, and Leendert W.N. van der Torre (to appear). Consistencyconstraints for input/output logic.

Reiter, R. 1980. A logic for default reasoning. Artificial Intelligence, 13: 81-132.


van der Torre, Leendert W.N., 1997. Reasoning about Obligations: Defeasibility inPreference-Based Deontic Logic. Ph.D. thesis, Erasmus University of Rotterdam.Tinbergen Institute Research Series n° 140. Thesis Publishers: Amsterdam.

van der Torre, Leendert W.N., 1998. Phased labeled logics of conditional goals.Logics in Artificial Intelligence. Proceedings of the Sixth European Workshop onLogics in AI (JELIA’98). Berlin: Springer, LNCS 1489, pp 92-106.

ACKNOWLEDGEMENTS

Thanks to Veronica Becher, Salem Benferhat and anonymous referees for DEON2000 for comments on drafts. Research for this paper was begun when the secondauthor was working at IRIT, Université Paul Sabatier, Toulouse, France, and at theMax Planck Institute for Computer Science, Saarbrücken, Germany.

David MakinsonVisiting Professor, Department of ComputingKing’s College LondonPermanent address:Les Etangs B2, Domaine de la Ronce92410 Ville d’Avray, FranceEmail: [email protected]


Last revised: 06.03.00Wordcount: 9,173


DAVID MAKINSON AND LEENDERT VAN DER TORRE

CONSTRAINTS FOR INPUT/OUTPUT LOGICS

ABSTRACT. In a previous paper we developed a general theory of input/outputlogics. These are operations resembling inference, but where inputs need not beincluded among outputs, and outputs need not be reusable as inputs. In the presentpaper we study what happens when they are constrained to render output consistentwith input. This is of interest for deontic logic, where it provides a manner ofhandling contrary-to-duty obligations. Our procedure is to constrain the set ofgenerators of the input/output system, considering only the maximal subsets that donot yield output conflicting with a given input. When inputs are authorised to reappearas outputs, both maxichoice revision in the sense of Alchourrón/Makinson and thedefault logic of Poole emerge as special cases, and there is a close relation with Reiterdefault logic. However, our focus is on the general case where inputs need not beoutputs. We show in what contexts the consistency of input with output may bereduced to its consistency with a truth-functional combination of components ofgenerators, and under what conditions constrained output may be obtained by aderivation that is constrained at every step.

KEY WORDS. input/output logic, contrary-to-duty obligations, deontic logic,consistency constraints, default logics, revision.

1. BACKGROUND: INPUT/OUTPUT LOGICS

1.1. Explicit definitions

We assume familiarity with (Makinson and van der Torre, 2000), which studiesunrestricted output operations. Nevertheless, for convenience, we briefly recall itscentral points.

We consider a Boolean context, that is, a propositional language closed under theusual truth-functional connectives. The central objects of attention are ordered pairs(a,x) of formulae, which we read forwards. Intuitively, we think of each pair (a,x) as arule, with body a representing a possible input, and head x for a corresponding output.We call a set G of such pairs a generating set. The letter G also serves as a reminderof the interpretation (among others) of the pairs as conditional goals or obligations.When A is a set of formulae, we write G(A) for {x: (a,x) ∈ G for some a ∈ A}.

The operation out(G,A) takes as argument a generating set G, and an input set A offormulae, delivering as value an output set of formulae. We focus on four operations,which we define explicitly by equations. In so far as the definitions make no appeal toderivations or inductive processes, they may be thought of as semantic in a broadsense of the term.

• Simple-minded output: out1(G,A) = Cn(G(Cn(A)))


• Basic output: out2(G,A) = ∩{ Cn(G(V)): A ⊆ V, V complete}

• Reusable simple-minded output: out3(G,A) = ∩{ Cn(G(B)): A ⊆ B = Cn(B) ⊇G(B)}

• Reusable basic output: out4(G,A) = ∩{ Cn(G(V)): A ⊆ V ⊇ G(V), V complete}

Here, Cn (alias |−) is classical consequence, and whenever terms like ‘consequence’,‘equivalence’ and ‘independence’ are used in this paper, they are understood in theirclassical sense. A complete set is one that is either maxiconsistent or equal to the set Λof all formulae of the language. When A is a singleton {a}, we write simply outi(G,a),and similarly for other operations defined in the paper.

We have the inclusions out1(G,A) ⊆ {out2(G,A), out3(G,A)} ⊆ out4(G,A) ⊆Cn(A∪m(G)), but not in general conversely. Here m(G) is the set of allmaterialisations of elements of G, i.e. the set of all formulae b→y with (b,y) ∈ G. Innone of these four systems are inputs automatically outputs, that is, we do not ingeneral have A ⊆ out(G,A). Nor do the systems validate contraposition: we may havex ∈ out(G,a) without ¬a ∈ out(G,¬x).

For each of these four principal operations, we may also consider a throughputversion that also allows inputs to reappear as outputs. These are the operationsoutn

+(G,A) = outn(G+,A), where G+ = G∪I and I is the set of all pairs (a,a) for

formulae a.

It turns out that out4+ = Cn(A∪m(G)), thus collapsing into classical logic. Out3

+(G,A)does not collapse in this way, but may be expressed more simply as ∩{ B: A ⊆ B =Cn(B) ⊇ G(B)}.

These operations are distinct, with the exception that out2+ = out4

+. This identity, notmentioned in (Makinson and van der Torre, 2000), may be verified as follows. Theleft-in-right inclusion is immediate. To show the converse, suppose x ∉ out2

+(G,A).Then there is a complete set V with A ⊆ V and x ∉ Cn(G+(V)). To prove that x ∉out4

+(G,A) we need only show that G+(V) ⊆ V. But V ⊆ G+(V) so if the converse failsthen Cn(G+(V)) = Λ contradicting x ∉ Cn(G+(V)).

We write out without a subscript to cover indifferently all these seven distinctinput/output operations. We move freely between the notations x ∈ out(G,A) and (A,x)∈ out(G). The former is more useful when working directly with the above explicitdefinitions of the various kinds of output; the latter is more convenient whenconsidering derivations.

1.2. Characterizations in terms of derivability

In derivations, we work with singleton inputs, defining derivability from an input setA as derivability from the conjunction of finitely many elements of A. For any set ofderivation rules, we say that a pair (a,x) of formulae is derivable from G using thoserules, and write (a,x) ∈ deriv(G), iff (a,x) is in the least set that includes G, is closedunder the rules, and contains every pair (t,t) where t is a tautology. The specific rulesconsidered are:

SI (strengthening input): From (a,x) to (b,x) whenever b |− a


AND (conjunction of output): From (a,x), (a,y) to (a,x∧y)WO (weakening output): From (a,x) to (a,y) whenever x |− yOR (disjunction of input): From (a,x), (b,x) to (a∨b,x)CT (cumulative transitivity): From (a,x), (a∧x,y) to (a,y)

The reason why (t,t) is mentioned separately above is to ensure full correspondencewith the explicit definition, even in limiting cases. When t is a tautology, we have t ∈out(G,a) even when G is empty. To derive (a,t) from G it suffices to have (t,t) andapply SI. Evidently, given SI and WO, in the above definition of deriv(G) it wouldsuffice to require (t,t) ∈ deriv(G) for some tautology t.

As shown in the cited paper, simple-minded output coincides with derivability usingSI, AND, WO; basic output to those plus OR; simple-minded reusable output to thefirst three plus CT; and reusable basic output to all five. In other words, x ∈ out(G,a)iff (a,x) ∈ deriv(G), where the rules defining deriv are those mentioned ascorresponding to out. For the augmented throughput versions, authorising inputs toreappear as outputs, add the zero-premise rule:

ID: From no premises to (a,a)

All of our systems of derivation admit the rules SI and WO, and so satisfyreplacement of input, and of output, by classically equivalent propositions. That is, if(a,x) ∈ deriv(G) then (a′,x′) ∈ deriv(G) whenever Cn(a) = Cn(a′) and Cn(x) = Cn(x′).In derivations, it is convenient to treat replacement of logically equivalentpropositions as a ‘silent rule’ that may be applied at any step without explicitjustification.

1.3. Out+/out reductions

In two cases, the operations with throughput may be reduced to their counterpartswithout it. In one case a reverse reduction is possible. These facts are not mentionedin (Makinson and van der Torre, 2000), so we outline the proofs.

Immediately from its definition, out1+ may be reduced to out1 by the equation

out1+(G,A) = Cn(A∪out1(G,A)).

The same identity holds for out3, as can be shown by the following argument. Theinclusion Cn(A∪out3(G,A)) ⊆ out3

+(G,A) is immediate, so we need only prove theconverse. Given the compactness of out3 and out3

+ as established in (Makinson andvan der Torre, 2000), it suffices to do this for singleton values of A. Thus we needonly show that whenever x ∈ out3

+(G,a) then a→x ∈ out3(G,a), i.e. whenever (a,x) ∈deriv3

+(G) then (a,a→x) ∈ deriv3(G). But this is easily verified by an induction on thederivation, recalling for the basis that (t,t) ∈ deriv(G) in all our systems.

These reductions do not hold for out2 or out4. In the following counterexample, and allothers in the paper, a,b,x,y… are understood to be distinct elementary letters ofclassical propositional logic, and thus logically independent of each other, while t isany tautology. Put G = {(a,x)} and A = {¬x}. Then ¬a ∈ out2

+(G,A) = out4+(G,A) =

Cn(A∪m(G)) = Cn({ ¬x} ∪{ a→x}). But ¬a ∉ Cn(A∪out4(G,A)) ⊇ Cn(A∪out2(G,A)).


To see this, consider any complete set V that contains neither a nor x, so that G(V) =∅ ⊆ V ⊇ {¬x} = A and Cn(G(V)) = Cn(∅), so out4(G,A) = Cn(∅) and clearly ¬a ∉Cn({ ¬x} ∪ Cn(∅)).

In the special case of out3 we also have a converse reduction: out3(G,A) =Cn(G(out3

+(G,A)). The left in right inclusion is immediate from the definition of out3,since out3

+(G,A) satisfies the three conditions imposed on B in that definition. For theright-in-left inclusion, it suffices to show that G(out3

+(G,A)) is included in out3(G,A),and by compactness it suffices to do this for singletons. Suppose x ∈ G(out3

+(G,a)).Then there is a b with (b,x) ∈ G and b ∈ out3

+(G,a), i.e. (a,b) ∈ deriv3+(G). Hence as

shown above, (a,a→b) ∈ deriv3(G). Since (b,x) ∈ G ⊆ deriv3(G) we have by SI(a∧b,x) ∈ deriv3(G), so by CT (a,x) ∈ deriv3(G), i.e. x ∈ out3(G,a) as desired.

The reduction outi(G,A) = Cn(G(outi+(G,A)) does not hold for any of the other output

operations outi , i = 1,2,4, as can be shown by simple examples. For out1, out2, put G= {(a,x),(x,y)} and A = {a}; then the left side is Cn(x) while the right side isCn({ x,y}). For out4, put G = {(a,x),(¬a,x)} and A = {t}; then the left side is Cn(x)while the right side is Cn(∅).

2. EXCESS OUTPUT AND ITS ELIMINATION

2.1. Two kinds of excess output

Two kinds of excess are of particular interest for output: inconsistency of output perse, and its inconsistency with input. Since inputs are not in general authorised toreappear as outputs, these are not the same.

• Given a generating set G and input A, the output out(G,A) is inconsistent iff ⊥∈ Cn(out(G,A)). Equivalently, since out(G,A) = Cn(out(G,A)) for all of ourinput/output operations, iff ⊥ ∈ out(G,A).

• On the other hand, output out(G,A) is inconsistent with input A iff ⊥ ∈Cn(out(G,A)∪A). Equivalently, iff ¬∧A0 ∈ out(G,A) for some finite A0 ⊆ A.When A is a singleton {a}, this comes to ¬a ∈ out(G,a).

Here ⊥ is the falsum (negation of a tautology). Clearly, inconsistency of outputimplies its inconsistency with input, but not conversely, as illustrated by the followingsimple example, well known from discussions of conditional norms in deontic logic.

EXAMPLE 1 (broken promise). Let G = {(t,¬a), (a,x)} where t is a tautology. Togive it flesh, read a as ‘you break your promise’ and x as ‘you apologize’. Thenout(G,a) = Cn({ ¬a,x}) for each of the four principal input/output operations outn (n =1,2,3,4). Thus ⊥ ∉ out(G,a), i.e. output is consistent. However ¬a ∈ out(G,a), i.e.output is inconsistent with input.

It is clear why we may wish to ensure consistency of output. But why might we alsowant to guarantee its consistency with input? The motivation comes from deonticlogic. Suppose we are given a code G of conditional norms. Imagine that we are


presented with a condition (input) that is unalterably true, and ask what obligations(output) it gives rise to. It may happen that the condition is something that should nothave been true in the first place. But that is now water under the bridge; we have to“make the best out of the sad circumstances” as (Hansson 1969) put it. We thereforeabstract from the deontic status of the condition, and focus on the obligations that areconsistent with its presence. In the above example, if the person has not kept apromise, then we want to know what should be done consistent with that situation.How to determine this in general terms, and if possible in formal ones, is the well-known problem of contrary-to-duty conditions.

2.2. Avoiding excess output: maxfamilies and outfamilies

Our strategy for eliminating excess output is to cut back the set of generators to justbelow the threshold of yielding excess. To do that, we adapt a technique that is wellknown in the more specific areas of belief change and nonmonotonic inference – lookat the maximal non-excessive subsets.

The formal definition is general, covering as special cases both inconsistency ofoutput and its inconsistency with input. Let G be a generating set, and let C be anarbitrary set of formulae, which we will call the ‘constraint set’. For any input set A,we define maxfamily(G,A,C) to be the family of all maximal H ⊆ G such thatout(H,A) is consistent with C.

The cases C = ∅ and C = A express consistency of output, and its consistency withinput. In other words with C = ∅ (respectively C = A), maxfamily(G,A,C) gathers themaximal H ⊆ G such that out(H,A) is consistent (respectively consistent with inputA). For throughput operations out = outn

+ we have A ⊆ out(G,A), so thatmaxfamily(G,A,∅) = maxfamily(G,A,A). But for the operations out = outn withoutthroughput, they are quite different.

Care should be taken when applying the definition to the throughput operations.Maxfamily(G,A,C) is understood to be the family of all maximal H ⊆ G such thatoutn

+(H,A) = outn(H∪I,A) is consistent with C. It is not the family of all maximal H ⊆G∪I such that outn(H,A) is consistent with C. In other words, the set I is protectedfrom attrition.

We define outfamily(G,A,C) to be the family of outputs under input A, generated byelements of maxfamily(G,A,C). In other words, outfamily(G,A,C) is the family of allsets out(H,A) such that H is maximal among the subsets H′ ⊆ G with out(H′,A)consistent with C.

2.3. Meets and joins of outfamilies

As one would expect from the analogous constructions in the logics of belief changeand nonmonotonic inference, the definition of an outfamily gives rise to notions offull meet and full join constrained output, i.e. ∩(outfamily(G,A,C)) and∪(outfamily(G,A,C)). Some special cases of these operations have been studied inwork on conditional norms in deontic logic. For example, (Hansson and Makinson1997) in effect give a way of constructing ∩(outfamily(G,a,∅)) for out = out2. As we


shall see in Sections 5 and 6, certain truth-functional approximations to∪(outfamily(G,A,A)) have also been studied in connection with contrary-to-dutyconditional norms.

As in the logics of belief change and nonmonotonic reasoning, in addition to the fullmeet and join operations, there are partial ones, ∩(γ(outfamily(G,A,C))) and∪(γ(outfamily(G,A,C))). Here γ is any selection function, with γ(Y) being a subset ofY, non-empty if Y is. Thus the full meet and full join operations are uniquely defined,while partial meet and join are schemas whose value depends on γ. The meetoperations are closed under classical consequence since the intersected sets out(H,A)are all closed, but in general the join operations are not closed under classicalconsequence.

In this paper we focus on the outfamilies themselves, without investigatingsystematically their meets and joins. Nevertheless, it will be useful when we considerconstrained derivations in Sections 6 and 7, to note that full join constrained outputmay be characterized more directly, without mention of maximality.

OBSERVATION 1. x ∈ ∪(outfamily(G,A,C)) iff there is a H ⊆ G such that x ∈out(H,A) and out(H,A) is consistent with C.

Proof. Left to right is immediate from the definition of ∪(outfamily(G,A,C)). For theconverse, suppose there is a H ⊆ G such that out(H,A) contains x and is alsoconsistent with C. Now each of our seven input/output operations is compact in its leftargument G; this is immediate from their characterizations by derivation rules inSection 1.2. Hence by Zorn’s Lemma, there is a maximal H′ with H ⊆ H′ ⊆ G suchthat out(H′,A) is consistent with C. Thus H′ ∈ maxfamily(G,A,C). By the monotony ofthe unrestricted output operation out(G,A) in its left argument, since x ∈ out(H,A) wehave x ∈ out(H′,A). Thus x ∈ ∪(outfamily(G,A,C)).

In Appendix #1 we note the monotony/antitony properties of the full join and meetoperations, with respect to arguments G,A,C.

3. SOME EXAMPLES FROM DEONTIC LOGIC

We give some examples that are familiar from discussions of contrary-to-duty normsin deontic logic. We calculate separately for C = ∅ and C = A. In Examples 3.1 and3.2 the calculation holds indifferently for the four principal input/output operationsnot authorising throughput. In Example 3.3 we calculate for out3 and out4.

EXAMPLE 3.1. We return to the broken promise (Example 1) and reconsider it in thelight of our formal definitions. Recall that G = {(t,¬a), (a,x)} where t is a tautology, ais ‘you break your promise’ and x is ‘you apologize’. Then as already noted, out(G,a)= Cn(¬a,x) which is consistent. Thus on the one hand, for C = ∅:

maxfamily(G,a,∅) = {G}outfamily(G,a,∅) = {Cn(¬a,x)}∩(outfamily(G,a,∅)) = ∪(outfamily(G,a,∅)) = Cn(¬a,x)


On the other hand, out(G,a) is not consistent with the input a. There is a just onemaximal subset of G whose output is consistent with input a, namely the singleton{( a,x)}. Thus for C = {a} we have:

maxfamily(G,a,a) = {{( a,x)}}outfamily(G,a,a) = {Cn(x)}∩(outfamily(G,a,a)) = ∪(outfamily(G,a,a)) = Cn(x)

This agrees with the intuitive assessment of the example, where the elements of G areunderstood as conditional obligations. Given that one has broken the promise, theobligation to apologize becomes operative, while the obligation not to violate thepromise is no longer in play.

EXAMPLE 3.2. (broken promise without apology). Multiple levels of violation maybe analysed in the same way. For example, put G = {(t,¬a), (a,x), (a∧¬x,y)} wheret,a,x are as in Example 3.1 and y is ‘you are ashamed’. Consider the input a∧¬x.

Then out(G,a∧¬x) = Cn(¬a,x,y), which is consistent, so that maxfamily(G, a∧¬x, ∅)= {G} and outfamily(G, a∧¬x, ∅) = {Cn(¬a,x,y)}. On the other hand, out(G,a∧¬x) isinconsistent with input a∧¬x, so that maxfamily(G, a∧¬x, a∧¬x) = {(a∧¬x, y)} andoutfamily(G, a∧¬x, a∧¬x) = {Cn(y)}.

EXAMPLE 3.3. (Möbius strip, Makinson 1994, 1999). Put G = {(a,b), (b,c), (c,¬a)}.For instance, a,b,c could represent ‘Alice (resp. Bob,Carol) is invited to dinner’. Thepair (a,b) then tells us that if Alice is invited then Bob should be, and so on. Considera as input. Calculating for out ∈ {out3, out4}, we have out(G,a) = Cn(b,c,¬a) which isconsistent. Hence:

maxfamily(G,a,∅) = {G}outfamily(G,a,∅) = {Cn(b,c,¬a)}∩(outfamily(G,a,∅)) = ∪(outfamily(G,a,∅)) = Cn(b,c,¬a).

But out(G,a) is not consistent with the input a. This time there are three maximalsubsets of G whose output (under input a) is consistent with a, namely the three two-element subsets. Thus:

maxfamily(G,a,a) = {{( a,b),(b,c)}, {( a,b),(c,¬a)}, {( b,c),(c,¬a)}outfamily(G,a,a) = {Cn(b,c), Cn(b), Cn(∅)}∩(outfamily(G,a,a)) = Cn(∅)∪(outfamily(G,a,a)) = Cn(b,c).

The Möbius strip is a fascinating example. Since Cn(∅) ⊂ Cn(b) ⊂ Cn(b,c), it showsthat for out3 elements of the outfamily are not always maximal. In Appendix #2 weinvestigate this matter further.

4. SPECIAL CASES FOR DEFAULT LOGIC AND REVISION


We show how Poole systems and maxichoice revision may both be seen as a specialcase of constrained input/output logic, and that normal Reiter default systems areclosely related to another one. Specifically, Poole systems and maxichoice revisioncorrespond to constrained reusable basic throughput out4

+, while normal Reiterdefaults are closely related to constrained out3

+. This section may be omitted withoutloss of continuity for the general theory, but those readers already familiar with theabove systems may find the connections revealing.

OBSERVATION 4. Let (D,A,C) be a Poole default system, where D is the set of itsdefault formulae, A its set of premises, and C its set of constraining formulae. Letextfamily(D,A,C) be the family of its extensions in the sense of Poole. Thenextfamily(D,A,C) = outfamily(G,A,C), where G = {(t,x): x ∈ D} and outfamily isdefined using reusable basic throughput out4

+.

Proof. Recall again from Section 1.1 that out4+(G,A) = Cn(A∪m(G)). Note also that

since t→x is classically equivalent to x, m(G) is equivalent to D.

By the definition of a Poole default system – see (Poole 1988) or the exposition in(Makinson 1994, Section 3.3) – extfamily(D,A,C) is the family of all sets Cn(A∪D′)with D′ a maximal subset of D such that Cn(A∪D′) is consistent with C. By theconstruction of G, this is identical to the family of all sets Cn(A∪m(H)) such that H isa maximal subset of G with Cn(A∪m(H)) consistent with C. Since out4

+(H,A) =Cn(A∪m(H)), we conclude that extfamily(D,A,C) is the family of all sets out4

+(H,A)such that H is a maximal subset of G with out4

+(H,A) consistent with C, i.e. it isoutfamily(G,A,C).

In the logic of belief change, the well-known partial meet revisions of (Alchourrón,Gärdenfors and Makinson 1985) have maxichoice revisions as a special case, alreadystudied by (Alchourrón and Makinson 1982). It is well-known that maxichoicerevisions may also be regarded as Poole default extensions: the maxichoice revisionsK∗a are precisely the extensions of the Poole default system (K,{a},∅). Thus fromObservation 4 (or by direct verification) we also have:

COROLLARY TO OBSERVATION 4. Let K be a belief set and a any formula. Letrevfamily(K,a) be the set of all maxichoice revisions of K by a. Then revfamily(K,a) =outfamily(G,a,∅) = outfamily(G,a,a), where G = {(t,x): x ∈ K} and outfamily isdefined using out4

+.

Normal default systems in the sense of (Reiter 1980) are closely related toinput/output logics with out3

+ (reusable simple-minded throughput) as the underlyingunconstrained operation. However, the correspondence is less complete than for Poolesystems. The family of Reiter extensions forms a distinguished subset of thecorresponding outfamily. Roughly speaking, Reiter extensions maximize output,while constrained outputs maximize subsets of G, whose outputs need not be maximal(cf. Appendix #2).

This may be illustrated by looking again at Example 3.3, the Möbius strip, where G ={( a,b), (b,c), (c,¬a)}, and which we have already calculated for out ∈ {out3, out4}.


Recalculating for out = out3+, we get a very similar pattern. On the one hand, there are

three elements in the outfamily, forming a chain, as follows:

out(G,a) = Cn(a,b,c,¬a)maxfamily(G,a,∅) = maxfamily (G,a,a) = {{( a,b),(b,c)}, {( a,b),(c,¬a)},{( b,c),(c,¬a)}outfamily(G,a,∅) = outfamily(G,a,a) = {Cn(a,b,c), Cn(a,b), Cn(a)}

On the other hand, if we read the elements of G as normal Reiter default rules a;b/betc., then (G,a) has a unique Reiter extension Cn(a,b,c), which is the largest elementof the outfamily.

The general relationship between outfamily and extfamily is given by the followingobservation. To simplify notation, we write outfamily(G,A) indifferently foroutfamily(G,A,A) and outfamily(G,A,∅), knowing that these are identical forinput/output operations outn

+ admitting throughput. For simplicity of notation weidentify a normal rule a;x/x with the corresponding pair (a,x).

OBSERVATION 5. Let (G,A) be any normal Reiter default system, with G the set ofits default rules and A the set of its premises. Suppose that A is consistent. Writeextfamily(G,A) for the family of all its extensions in the sense of Reiter. Let outfamilybe defined using reusable simple-minded throughput out3

+. Then:

(a) extfamily(G,A) ⊆ outfamily(G,A)(b) for every X ∈ outfamily(G,A), there is an E ∈ extfamily(G,A) with X ⊆ E.

Proof. See Appendix #3.

COROLLARY TO OBSERVATION 5. Under the same conditions as Observation 5,extfamily(G,A) consists of exactly the maximal elements of outfamily(G,A). In brief:extfamily(G,A) = max(outfamily(G,A)).

Proof. Immediate from Observation 5, using the fact that no extension is properlyincluded in any other (Reiter 1980, Theorem 2.4).

SECOND COROLLARY TO OBSERVATION 5. Under the same conditions asObservation 5, ∪(extfamily(G,A)) = ∪(outfamily(G,A)).

Proof. Immediate from Observation 5.

Observation 5 also implies that for out = out3+ and consistent A, we have the

following ‘embedding property’: every element of outfamily(G,A) is included in amaximal such element. However, this is a rather roundabout way to prove theproperty, via Reiter default systems. It also leaves several questions open. Can wedrop the hypothesis of the consistency of A? Can we prove the property moregenerally for an arbitrary constraint set C? Can we prove it for other values of outi?

We can prove it for out1, out1+, and out4

+ (alias out2+). Indeed for those operations we

have a much stronger property: no element of outfamily(G,A,C) is properly includedin any other (Observation 3 in Appendix #2). But for the remaining operations out2,


out3, out4, this stronger property fails and the status of the embedding propertyremains open.

We end this section with a terminological warning. The term ‘generating set’ is usedin this paper in a sense quite different from that of (Reiter 1980, Definition 2). For us,a generating set is any set of pairs (a,x) of formulae. Reiter uses the term in a quitedifferent sense. Given a default system (G,A) and a set X of formulae, the generatingset for X, in Reiter’s sense, is the set of all default rules in G whose prerequisites arein X and whose justifications are consistent with X.

5. TRUTH-FUNCTIONAL REDUCTIONS OF THE INPUT/OUTPUTCONSTRAINT

5.1. Materialisations, heads, and fulfilments

We recall from Section 2.1 that a principal motivation for studying constrainedinput/output logics is their application to deontic contexts, and in particular tocontrary-to-duty conditional norms. For this reason, from this point on we restrictattention to the requirement that output out(G,A) is consistent with input A, i.e. thecase that C = A. We call this the input/output constraint. It gives the setsmaxfamily(G,A,A) and outfamily(G,A,A).

One may imagine truth-functional counterparts to this constraint. For example, onecould require that input A is consistent with the set m(G) of materialisations ofelements of G, i.e. the set of all formulae a→x with (a,x) ∈ G. Again, we couldrequire A to be consistent with the set h(G) of heads x of elements (a,x) of G, or withthe set f(G) of its fulfilments a∧x. The authors have studied some of these truth-functional approximations in earlier work on conditional norms. (Van der Torre 1997,1998) examined the fulfilment constraint as applied to derivations, first for out3 andthen for out4. (Makinson 1999) investigated the head constraint for out4 in the samecontext.

What is the relationship between the input/output constraint and its truth-functionalcounterparts? In one direction, we have the following simple fact.

OBSERVATION 6. Let out be any of our seven unrestricted input/output operations.For all C ⊇ A, if C is consistent with a set in the list out(G,A), m(G), h(G), f(G) then itis consistent with any set earlier in the list.

Proof. Recall from Section 1.1 that out(G,A) ⊆ Cn(A∪m(G)). It follows that for all C⊇ A, if C is consistent with m(G) and thus also with Cn(A∪m(G)), then it is consistentwith out(G,A). For the remainder, simply note that by classical logic f(G) |− h(G) |−m(G).

In the converse direction, the situation depends on the choice of the backgroundinput/output operation, as we now show.

5.2. Truth-functional reductions


We begin by noting that the converse of Observation 6 fails for outi when i ∈{1,1+,2,3,3+}, even in the case that C = A and even for the ‘nearest’ truth-functionalcounterpart m(G). We give two examples that together cover the operations.

EXAMPLE 4.1. For i ∈ {1,1+,2}, put G = {(t,x), (x,y)} and A = {¬y}. Then A isinconsistent with m(G) and so with h(G) and f(G). On the other hand, A is consistentwith outi(G,A), since out1(G,A) ⊆ out1

+(G,A) = Cn(G+(Cn(A))) = Cn({ x,¬y}) whileout2(G,A) = Cn({ x}), both of which are consistent with ¬y.

EXAMPLE 4.2. For i ∈ {1,1+,3,3+}, Put G = {(a,x), (¬a,x)} and A = {¬x}. Then A isinconsistent with m(G) and so with h(G) and f(G). On the other hand, A is consistentwith out3

+(G,A) ⊇ outi(G,A). To see this, put B = Cn(¬x). Then A ⊆ B = Cn(B) =G+(B) = Cn(¬x) so out3

+(G,A) ⊆ Cn(¬x) which is consistent with ¬x.

On the other hand, we have a converse for out4+ (which, we recall, coincides with

out2+), and another one for out4 in the case C = A.

OBSERVATION 7. For all C ⊇ A, C is consistent with out4+(G,A) iff it is consistent

with m(G).

Proof. Right to left is already given by Observation 6. For left to right, recall fromSection 1.1 that out4

+ = Cn(A∪m(G)) and conclude.

OBSERVATION 8. A is consistent with out4(G,A) iff it is consistent with m(G).

Proof. Right to left is already given by Observation 6. For left to right, suppose A isinconsistent with m(G); we need to show that it is inconsistent with out4(G,A). Let Vbe any complete set with A ⊆ V ⊇ G(V). Then m(G) ⊆ V, for otherwise there is a pair(a,x) ∈ G with a→x ∉ V so a ∈ V and ¬x ∈ V contradicting G(V) ⊆ V. Since both Aand m(G) are included in V, V = Λ. It follows that out4(G,A) = Cn(G(Λ)) = Cn(h(G))⊇ Cn(m(G)) ⊇ m(G). Since A is inconsistent with the last, it is inconsistent with thefirst, and we are done.

COROLLARY TO OBSERVATIONS 7,8. For out ∈ {out4, out4+}, maxfamily(G,A,A)

is the family of all maximal H ⊆ G such that A is consistent with m(H); thus alsooutfamily(G,A,A) is the family of sets out(H,A) for maximal H ⊆ G such that A isconsistent with m(H).

5.3. The effect of throughput on maxfamilies

From Observations 7 and 8, together with results in Section 1.3, we obtain a furthercorollary, on the effect of throughput on consistency of output with input, and thus inturn on the identity of maxfamilies.

SECOND COROLLARY TO OBSERVATIONS 7,8. For out ∈ {out1, out3, out4}, thefollowing three conditions are equivalent: A is consistent with out(G,A), A isconsistent with out+(G,A), out+(G,A) is consistent.


Proof. For out1 and out3 this immediate from the reduction out+(G,A) =Cn(A∪out(G,A)) established for those operations in Section 1.3, recalling that A ⊆out+(G,A). For out4, apply Observations 7 and 8.

THIRD COROLLARY TO OBSERVATIONS 7,8. For out ∈ {out1, out3, out4},maxfamily(G,A,A) = maxfamily+(G,A,A).

Proof. Immediate from the Second Corollary. Notation: when maxfamily isdetermined by out, then maxfamily+ is understood to be determined by out+.

Note however that for the same operations out1, out3, out4, we may haveoutfamily(G,A,A) ≠ outfamily+(G,A,A) since in general out(H,A) ≠ out+(H,A).

Moreover, for out2 the identity maxfamily(G,A,A) = maxfamily+(G,A,A) fails. For acounterexample, put G = {(a,x), (x,¬a)}. Then using characterizations in Section 1.1,¬a ∈ out2

+(G,a) = out4+(G,a) = Cn({ a} ∪m(G)). But ¬a ∉ out2(G,a), as witnessed by

any complete set containing a but not x, for then G(V) = {x} and ¬a ∉ Cn(x).

6. CONSTRAINED DERIVATIONS

6.1. Definitions

Up to this point, we have considered constrained output in terms of its explicitdefinition, i.e. from a perspective that may be called, in a broad sense of the word,semantic. We now examine it in terms of derivations, where a number of newquestions arise.

We know (Section 1.2) that each of our unconstrained input/output operations may begiven a characterization in terms of derivations: x ∈ out(G,a) iff (a,x) is derivablefrom G using appropriate rules. In this section we consider constraint as a requirementon the root of a derivation, relative to its leaves. In Section 7, we examine a moredemanding way of applying it, as a requirement on every step of a derivation.

In Section 1.2 we sketched the notion of derivability, but leaving implicit thedefinitions of a rule and of a derivation. We now need to be more explicit. A rule r (ofarity n ≥ 0) is a subset of Pn+1 where P is the set of all ordered pairs of formulae.When ((a1,x1),…, (an,xn), (an+1,xn+1)) ∈ r, then (a1,x1),…, (an,xn) are called its premisesand (an+1,xn+1) its conclusion.

A derivation of a pair (a,x) from a set G of pairs of formulae, given a set R of rules, isunderstood to be a tree with (a,x) at the root, each non-leaf node related to itsimmediate parents by the inverse of a rule in R, and each leaf node either theconclusion of a zero-premise rule in R, or an element of G, or of the form (t,t). It isunderstood that not all elements of G need to appear as leaves. Nor do all rules in Rneed to be applied in the derivation, but no others may be employed.

A pair (a,x) of formulae is said to be derivable from G given rule-set R, and we write(a,x) ∈ deriv(G) or x ∈ deriv(G,a), iff there is some derivation of (a,x) from G given


R. Equivalently, recalling the formulation in Section 1.2, deriv(G) is the least set thatincludes G, contains the pair (t,t) where t is any tautology, and is closed under therules.

We say that a derivation ∆ is constrained iff the body of the root is consistent with itsown derivability set. To be precise, let ∆ be a derivation of (a,x) from G, given a rule-set R. Let L ⊆ G be the set of the leaves of ∆. We say that ∆ is constrained withrespect to rule-set R iff (a,¬a) ∉ deriv(L) where deriv is derivability using only rulesin R. Equivalently, when R contains the rules AND, WO, iff deriv(L,a) is consistentwith a.

EXAMPLE 5. Put G = {(a,x), (a∧x,y)} and let R = R3 = {SI, AND, WO, CT}.Consider the following derivation ∆ of (a∧¬x,y) from G.

(a,x) (a∧x,y) ------------------------ CT (a,y)

SI *(a∧¬x,y)

∆ is not constrained with respect to R because (a∧¬x,¬(a∧¬x)) ∈ deriv(L) where L isthe set of leaves of ∆, as witnessed by the derivation:

(a,x) SI (a∧¬x,x) WO(a∧¬x,¬(a∧¬x))

It is immediate from the definition that no derivation with root of the form (a,¬a) isconstrained (with respect to the set of rules used in the derivation). If R also containsWO, then no derivation whose root (a,x) has an inconsistent fulfillment, isconstrained. For if a is inconsistent with x then x |− ¬a so that by an application ofWO we have (a,¬a) ∈ deriv(L).

In the definition of a constrained derivation ∆ of (a,x) from G given a rule-set R, wehave considered only the set L ⊆ G of leaves of ∆, but the whole set R of rules ratherthan those actually employed in ∆. It may be asked why we proceed in thisasymmetric manner. We envisage the agent as working with a fixed stock R ofderivation rules that it regards as acceptable. In general, a derivation will appeal toonly some of the allowed stock of rules, but we regard the agent as remainingcommitted to all of them. On the other hand, the set G of pairs that the agent assumesas premises is regarded as variable. In general, a derivation will use only some ofthem as leaves, and they are the only ones to which the agent is committed. So whenwe test to see whether the body of the root is consistent with its own output under therules, we do so with respect to the entire set R, but only the set L ⊆ G of leaves of thederivation.


Evidently, this is a delicate point on which other perspectives are possible, withdifferent consequences, for as L and R grow so does deriv(L). We compare them inAppendix #4.

We say that (a,x) is derivable with constraint from G given rule-set R iff there is somederivation of (a,x) from G given R that is constrained with respect to R. As one wouldexpect, for each of our input/output operations, this is equivalent to full joinconstrained output, where the constraint is the same as the input.

OBSERVATION 9. Let out be any one of the operations outi or outi+ (i = 1,..,4), and

let R be the corresponding set of derivation rules. Then x ∈ ∪(outfamily(G,a,a)) iffthere is a derivation of (a,x) from G (given rules from R) that is constrained (withrespect to R).

Proof. Recall from Observation 1 that x ∈ ∪(outfamily(G,a,a)) iff x ∈ out(H,a) forsome H ⊆ G such that a is consistent with out(H,a). Since out = deriv and the rulesAND, WO are available, we need only show that the following are equivalent, whereL∆ is the set of all leaves of ∆.

(1) ∆ is a derivation of (a,x) from some H with H ⊆ G, such that ¬a ∉deriv(H,a),(2) ∆ is a derivation of (a,x) from G such that ¬a ∉ deriv(L∆,a).

But (1) immediately implies L∆ ⊆ H ⊆ G and so implies (2). Also (2) implies (1)taking H to be the set of all leaves of ∆ other than those of the form (t,t) and, for outi

+,of the form (a,a).

6.2. Truth-functional reductions of the constraint on derivations

We return to the question of truth-functional reductions of the input/output constraint,but now in terms of derivations and their leaves. Observation 8 gave a positive resultfor out4 on the semantic level, which we can translate to the language of derivations.

OBSERVATION 10. Let ∆ be any derivation of (a,x) from G, with leaves L. Then ∆is constrained with respect to R4 iff a is consistent with m(L).

Proof. By definition, ∆ is constrained with respect to R4 iff ¬a ∉ deriv4(L,a) =out4(L,a) = Cn(out4(L,a)), i.e. iff a is consistent with out4(L,a), i.e. (using Observation8) iff a is consistent with m(L).

Evidently, since out2+(L,a) = out4

+(L,a) = Cn(a∪m(L)) (section 1.1), Observation 10also holds for R2

+ and R4+.

For out1 and out3, the truth-functional reduction fails on the semantic level, as we sawin Examples 4.1 and 4.2. Nevertheless, on the level of derivations we have a positiveresult.

OBSERVATION 11. Let R ∈ {R1,R3}. Let ∆ be any derivation of (a,x) from G withleaves L, given rules R. Then ∆ is constrained with respect to R iff a is consistent(indifferently) with m(L), h(L), f(L).


Proof. Given Observation 6, we need only show that if a is inconsistent with f(L) thena is inconsistent with deriv(L,a). For this, it suffices to show that {a} ∪ deriv(L,a) |−f(L).

For each node n:(b,y) in the derivation, write Ln for the set of all leaves in the subtreedetermined by n. We show by induction that {b} ∪ deriv(L,b) |− f(Ln).

Basis: Suppose n:(b,y) is a leaf of the tree. Then y ∈ deriv(L,b), f(Ln) = {b∧y}, soclearly {b} ∪ deriv(L,b) |− f(Ln).

SI: Suppose n:(b,y) is derived by SI from p:(c,z). Then b |− c. By the inductionhypothesis, {c} ∪ deriv(L,c) |− f(Lp), so {b} ∪ deriv(L,b) |− f(Lp) = f(Ln) since Ln = Lp.

AND: Suppose n:(b,y) is derived by AND from p:(b,z) and q:(b,w). By the inductionhypothesis, {b} ∪ deriv(L,b) |− f(Lp), f(Lq) so {b} ∪ deriv(L,b) |− f(Lp) ∪ f(Lq) = f(Ln)since Ln = Lp∪Lq.

WO: Suppose n:(b,y) is derived by WO from p:(b,z). By the induction hypothesis,{ b} ∪ deriv(L,b) |− f(Lp) = f(Ln) since Ln = Lp.

CT: Suppose n:(b,y) is derived by CT from p:(b,z) and q:(b∧z,y). By the former, z ∈deriv(L,b) and, using CT, deriv(L,b∧z) ⊆ deriv(L,b). By the induction hypothesis,{ b} ∪ deriv(L,b) |− f(Lp) and {b∧z} ∪ deriv(L,b∧z) |− f(Lq). Putting these together wehave {b} ∪ deriv(L,b) |− f(Lp) ∪ f(Lq) = f(Ln) since Ln = Lp∪Lq.

How does this Observation resist Examples 4.1 and 4.2? When consideringconstraints on a derivation, we look only at the set L of its leaves, and when L ⊂ Gthen m(L) may be weaker than m(G). In the case of Example 4.1, where G = {(t,x),(x,y)} and A = {¬y}, we can construct a derivation of (¬y,x) with leaf-set L = {(t,x)}⊂ G, as follows:

(t,x) SI

(¬y,x)

As noted in the presentation of Example 4.1, ¬y is consistent with out1(G,¬y) =deriv1(G,¬y) ⊇ deriv1(L, ¬y). But while ¬y is inconsistent with m(G) = {t→x, x→y},it is consistent with m(L) = {t→x}. In fact, there is no derivation of (¬y,x), using onlyrules from R1, whose leaves cover all elements of G. Similar considerations apply toExample 4.2.

Observation 11 fails for R2. Put G = {(a,¬a),(b,¬b)} and consider the derivation:

(a,¬a) (b,¬b) WO WO(a,¬a∨¬b) (b,¬a∨¬b)---------------------------- OR


(a∨b,¬a∨¬b)

Then a∨b is inconsistent with m(G). On the other hand, a∨b is consistent without2(G,a∨b), as witnessed by any complete set V containing a but not b. For then a∨b∈ V, and G(V) = {¬a} so Cn(G(V)) = Cn(¬a) which is consistent with a∨b.

We end this section by noting that from Observation 11 we can get its counterpart forR1

+ and R3+.

COROLLARY TO OBSERVATION 11. Let R+ ∈ {R1+,R3

+}. Let ∆ be any derivationof (a,x) from G with leaves L, given rules R+. Then ∆ is constrained with respect to R+

iff a is consistent (indifferently) with m(L), h(L), f(L).

Proof. Let ∆ be any derivation of (a,x) from G given rules R+. Let L be the set ofleaves of ∆. On the one hand, by Observation 6, if a is consistent with any one ofm(L), h(L), f(L) then it is consistent with outi

+(L,a), i.e. ∆ is constrained wrt R+. Forthe converse, note that since identity is a zero-premise rule, ∆ is also a derivation,using only rules in R, from G∪J where J ⊆ I. If ∆ is constrained wrt R+ then it isconstrained wrt to R and we may apply Observation 11 to conclude that a is consistentwith each of m(L), h(L), f(L).

7. APPLYING CONSTRAINTS MORE SEVERELY

7.1. General picture

If we are interested in derivations beyond their role as syntactic counterparts ofexplicitly defined operations, then further questions are suggested by the conceptsintroduced in Section 6. The definition of a constrained derivation (Section 6.1)requires that (a,¬a) ∉ deriv(L), where a is the body of the root and L is the set ofleaves of the derivation. But the root and the leaves are not the only nodes of aderivation. What happens if we apply the concept more severely? In particular:

• What if, instead of checking the body of the root node with respect to leaves,we check it with respect to all nodes in the derivation?

• What if, instead of checking the body of the root node only, we check the bodyof every node in the derivation?

The first question has an easy answer. It makes no difference whether we check theroot with respect to leaves only, or all nodes in the derivation. This is irrespective ofthe subset of rules considered to be available.

OBSERVATION 12. Consider any set R of rules from SI, AND, WO, OR, CT. Aderivation ∆ with root (a,x) is constrained (with respect to R) iff (a,¬a) ∉ deriv(H),where H is the set of all nodes of ∆ (and derivability is defined in terms of R).

Proof. Since H is the set of all nodes of ∆, we have L ⊆ H ⊆ deriv(L) where L is theset of leaves of ∆. Hence, since deriv (understood as taking sets of pairs of


propositions to sets of pairs) is clearly a closure operation, we have deriv(H) =deriv(L). In particular, (a,¬a) ∉ deriv(H) iff (a,¬a) ∉ deriv(L), i.e. iff ∆ isconstrained.

The second question is much more complex. The formulation above is rather loose.Let us say that a derivation ∆ is constrained at node n:(b,y) iff ¬b ∉ deriv(L,b), whereL is the set of leaves of the entire derivation ∆ and deriv is defined by the same set ofrules as used when checking the root node. Our question is: given a derivation of (a,x)from G that is constrained at the root, is there always some derivation of (a,x) from Gthat is constrained at every node?

In general, the answer depends on the rules allowed, for one rule may allow us tobypass steps that fall foul of a constraint which another rule cannot avoid.Specifically, for the rule sets R2 and R4, it can make a difference to derivationswhether we check the root only, or all nodes in the derivation. But for the rule sets R1

and R3 (which lack OR), it makes no difference. Moreover, for the rule set R2 (whichlacks CT), if the given derivation satisfies the constraint at both the root and allleaves, then some derivation of the same root from the same leaves satisfies theconstraint at every step. We now prove these results.

7.2. Results on severe application of the constraint on derivations

We begin with the negative result, which reveals the limits of the positive ones thatfollow.

OBSERVATION 13. For rule set R ∈ {R2, R4}, there is a derivation that isconstrained with respect to R, but such that no derivation given R of the same rootfrom the same generators is constrained at every node with respect to R.

Proof. We re-use Example 2.1 of Appendix #1. Put G = {(a,⊥), (¬a,x)} and considerthe following derivation.

*(a,⊥) (¬a,x) WO *(a,x)

-------------------------- OR (a∨¬a,x)

This derivation is constrained with respect to R2, since ¬(a∨b) ∉ deriv2(G,a∨¬a) =deriv4(G,a∨¬a) = Cn(x). But it fails the constraint at the starred nodes. Moreover,there is no other derivation of the same root from the same (or fewer) leaves thatsatisfies the constraint at all nodes. For if the constraint is to be satisfied at all nodes,the leaf (a,⊥) cannot be used; but the root is not in even the unrestricted output of theother leaf (¬a,x) taken alone, as is easily checked from its definition (Section 1.1).

We now pass to the positive results. We begin by noting some preservation propertiesfor the various rules.


LEMMA 14. Let {SI, AND, WO} ⊆ R ⊆ {SI, AND, WO, OR, CT}, i.e. let R be oneof Ri (i = 1,2,3,4). Then:

(a) Satisfaction of the constraint is preserved backwards by each of the rules SI,AND, WO, CT. That is, in any derivation, if the conclusion of the rulesatisfies the constraint (with respect to R), then so do its premises.

(b) Satisfaction of the constraint is preserved forwards by each of the rules AND,WO, CT, OR. That is, in any derivation, if the premises of the rule satisfy theconstraint (with respect to R), then so does its conclusion.

Proof.SI: Suppose (b,x) is obtained from (a,x) where b |− a. Suppose the premise fails theconstraint. Then ¬a ∈ deriv(L,a) ⊆ deriv(L,b). But since b |− a we have ¬a |− ¬b sosince WO is in R, ¬b ∈ out(L,b) and the conclusion fails the check.

AND: Suppose (a,x∧y) is obtained from (a,x) and (a,y). The conclusion has the samebody as each of the two premises, and so we have preservation in both directions.

WO: Suppose (a,y) is obtained from (a,x) where x |− y. Again, the conclusion has thesame body as the premise, and so we have preservation in both directions.

CT: Suppose (a,y) is obtained from (a,x) and (a∧x,y). Preservation forwards isimmediate, since the conclusion has the same body as one of the two premises. Forpreservation backwards, we have two cases to consider. Suppose first that (a,x) failsthe constraint. Then ¬a ∈ deriv(L,a) and the conclusion fails the constraint. Supposenext that (a∧x,y) fails the constraint. Then ¬(a∧x) ∈ deriv(L,a∧x). But also x ∈deriv(L,a), so by CT, ¬(a∧x) ∈ deriv(L,a). So by AND, WO which are in R, ¬a ∈deriv(L,a) and the conclusion fails the constraint.

OR: Suppose (a∨b,x) is obtained from (a,x) and (b,x). Suppose that the conclusionfails the constraint, i.e. ¬(a∨b) ∈ deriv(L,a∨b). Then since SI is in R, ¬(a∨b) ∈deriv(L,a) say, so since WO is in R, ¬a ∈ deriv(L,a) and the premise fails theconstraint.

We note in passing that that Lemma 14(a) continues to apply if the value of L is notfixed, but is taken to be the set of leaves in the derivation up to the node beingconsidered. However, parts of Lemma 14(b) would then fail. As shown by Examples7.1-7.3 in Appendix #5, satisfaction would no longer be preserved forwards for AND,CT, OR.

From Lemma 14 we have immediately the following for derivations without OR:

OBSERVATION 15. For R ∈ {R1,R3}, if the constraint is satisfied at the root then itis satisfied at every node. More explicitly: let R ∈ {R1,R3}, and let ∆ be any derivationusing only rules from R that satisfies the constraint (with respect to R) applied to theroot. Then ∆ satisfies the same constraint (with respect to R) at every step.


The example used in the proof of Observation 13 already shows that Observation 15can fail when OR is present. But we can also obtain a weaker result for derivationswith OR, so long as they do not contain CT.

OBSERVATION 16. For derivations without CT, satisfaction of the constraint at bothroot and leaves suffices to ensure its satisfaction everywhere in some derivation of thesame root from the same leaves. More explicitly: let R ∈ {R1,R2}, and let ∆ be anyderivation given R, that satisfies the constraint (with respect to R) applied to the rootand also applied to each leaf. Then there is a derivation ∆′ given R with the same rootand leaves that satisfies the constraint at every node.

Proof. By Observation 19(b) of (Makinson and van der Torre, 2000) any unrestrictedderivation using only the given rules may be rewritten as one in which they areapplied in the order WO, OR, SI, AND. Here it is understood that some of the rulesmay be skipped, and some may be applied several times, but never applied contrary tothe indicated order. Put ∆′ to be a derivation ordered in this way. As the leaves and theroot have not changed, the constraint continues to be satisfied for ∆′. It remains toshow that the constraint is satisfied at every node of ∆′. For this we need only applyLemma 14(b) forwards from the leaves through WO, OR, and Lemma 14(a)backwards from the root through AND, SI, thus covering all nodes in ∆′.

Observation 16 cannot be extended to derivations admitting both OR and CT. Moreexplicitly:

OBSERVATION 17. For the rule-set R4: there is a derivation that satisfies theconstraint applied to the root and also applied to each leaf, but such that neither it norany other derivation given R4 of the same root from the same generators (or a subsetof them) satisfies the constraint at every node.

Proof. See Appendix #6.

8. SUMMARY AND PROSPECTS

8.1. Summary

In this paper, we have studied what happens when input/output operations areconstrained to avoid excess output. Our strategy for eliminating excess output is to cutback the set of generators to just below the threshold of yielding excess. To do that,we adapt a technique that is well known in the more specific areas of belief changeand nonmonotonic inference – look at the maximal subsets of the generator set whoseoutput is not excessive.

Outfamily(G,A,C) is defined as the family of all sets out(H,A) where H is a maximalsubset of G such that out(H,A) is consistent with C. The case C = ∅ corresponds tothe requirement that output is consistent; the case C = A to consistency of output withinput. When the underlying unrestricted input/output operation is reusable basicthroughput (out4

+), this gives exactly the default extensions of Poole and themaxichoice revisions of Alchourrón and Makinson (Observation 4). When it isreusable simple-minded throughput (out3

+), the result is very closely related to the


normal default extensions of Reiter (Observation 5).

Our main focus in the paper is on the case where output is required to be consistentwith input. We call this the input/output constraint. In the context of reusable basicoutput out4, the definition of maxfamily(G,A,A) can be given a ‘truth-functionalreduction’ in terms of the materialisation of the generating set (Observation 8).

Constraints may also be approached in terms of derivations. Here, the naturalconstraint is to require the body a of the root (a,x) of the derivation to be consistentwith its own unrestricted output under the leaves of the derivation. This amounts torequiring that x ∈ ∪(outfamily(L,a,a)) (Observation 9). In the context of derivations,we can give more truth-functional reductions of the constraint than we could on thesemantic level (Observations 10 and 11).

Finally, we investigated the consequences of applying the consistency constraint moreseverely than at the root only and with respect to the leaves only. On the one hand, ifthe constraint is applied at the root only, but with respect to all nodes, this makes nodifference (Observation 12). On the other hand, if we apply the constraint at all nodeswith respect to the leaves, it can make a difference (Observation 13).

Nevertheless, when the choice of derivation rules is limited in certain ways,application of the constraint at all nodes of a derivation no longer increases itsseverity. This is so for derivations without OR (Observation 15), because all rulesfrom our palette other than OR preserve backwards satisfaction of the constraint(Lemma 14a). For derivations without CT, the situation is subtler. The rules otherthan CT may be partitioned into those that preserve forward satisfaction of theconstraint and those that preserve backward satisfaction (Lemma 14), and derivationswithout CT can always be rewritten with the former rules applied first. Thus checkingthe constraint at both root and leaves suffices to guarantee its satisfaction at every stepof the rewritten derivation (Observation 16).

8.2. Prospects

A technical problem that has been left open is the order structure of outfamilies forouti where i = 2,3,4, and to a lesser extent for out3

+, specifically the role of theirmaximal elements (see end of Section 4 and Appendix #2).

Some general lines of investigation deserve further exploration. These include theproperties of full and partial meets and unions of outsets (cf. the observations onmonotony and antitony in Appendix #1), and constraints with respect to values of Cother than A and ∅. One might also consider possible refinements in the definition ofan outfamily. For example, adapting an idea already studied in the logic of beliefrevision (Makinson 1997), one could designate a certain part of the unrestrictedoutput as protected from attrition in the construction of outfamilies.

On the level of derivations, it could be of interest to investigate ways of applying theconstraint at every step, but with respect to a variable set L of leaves (cf. the remarksafter Lemma 14, and the examples in Appendix #5). One might also try to ascertainwhether our choice of rules SI, WO, AND, OR, CT provides the best possible palettefor the analysis of derivations, or whether there are other sets of rules, collectively


equivalent but separately not so, that give more insight into the powers of consistencyconstraints.

One could also go back to unconstrained systems, and ask whether there are others,different from those studied here, that merit examination. For application to defeasibleobligations, it may be of interest to study systems that do not validate SI. Forapplication to permissive norms (even on the indefeasible level) one could considersystems failing both SI and AND.

Finally, we recall that the paper focuses on the creation and study of an abstractstructure. It remains to consider its application in practical contexts. Some well-known examples of contrary-to-duty conditional norms are analysed in Section 3(Examples 3.1-3.3), but there are many more – see for instance the tables in(Makinson 1999) and (van der Torre and Tan 1999). Application elsewhere, such asthe logics of action and belief, remains open.

APPENDICES

#1. Monotony/antitony Properties for the Full Join and Meet Operations

It is immediate from Observation 1 that the full join operation is monotonic inargument G and antitonic in C. For argument A, the situation is less stable, dependingon the choice of the underlying unrestricted input/output operation.

On the one hand, the full join operation based on out1 is monotonic in A. For supposex ∈ ∪(outfamily(G,A,C)). By Observation 1, there is a H ⊆ G such that out1(H,A) =Cn(H(Cn(A))) contains x and is consistent with C. We may assume without loss ofgenerality that all bodies of elements of H are in Cn(A), since cutting them out makesno difference to the value of H(Cn(A)). Thus when A ⊆ B we have H(Cn(A)) =H(Cn(B)), so that out1(H,B) = Cn(H(Cn(B))) = Cn(H(Cn(A))) is consistent with C andcontains x, so we may apply Observation 1 again and conclude. A more complexversion of this argument shows the same for underlying out3.

On the other hand, when the underlying unconstrained operation is out2 or out4, thenthe full join operation fails monotony in A, as shown in the following example.

EXAMPLE 2.1. Put G = {(a,⊥), (¬a,x)} and let out ∈ {out2, out4}. We show x ∈∪(outfamily(G,Cn(t),∅)) while x ∉ ∪(outfamily(G,Cn(a),∅)) although Cn(t) ⊆Cn(a). On the one hand, out(G,t) = Cn(x) which is consistent, so maxfamily(G,t,∅) ={ G} and outfamily(G,t,∅) = {Cn(x)} and thus ∪(outfamily(G,t,∅)) = Cn(x). On theother hand out(G,a) = Cn(⊥), which is inconsistent so maxfamily(G,a,∅) = {(¬a,x)},so outfamily(G,a,∅) = {Cn(∅)} and thus x ∉ ∪(outfamily(G,a,∅)) = Cn(∅).

Again, monotony in A fails when the underlying unconstrained operation is any ofoutn

+ authorising input to reappear as output.

EXAMPLE 2.2. Put G = {(t,x)}. For out = outn+ we have out(G,t) = Cn(x) which is

consistent so ∪(outfamily(G,a,∅)) = Cn(x), while out(G,¬x) = Cn(⊥), so


maxfamily(G,¬x,∅) = {∅} and outfamily(G,¬x,∅)) = {Cn(∅)} and thus∪(outfamily(G,¬x,∅)) = Cn(∅).

The full meet operation is even less well behaved. It fails all three properties:monotony in G, antitony in C, and monotony in A. The following examples arecalculated indifferently for our four input/output operations without throughput.

EXAMPLE 2.3. To illustrate non-monotony in argument G for the full meetoperation, put G = {(a,x)}. Then x ∈ Cn(x) = ∩(outfamily(G,a,∅)). But for H =G∪{( a,¬x)}, the set out(H,a) is inconsistent and maxfamily(H,a,∅) ={{( a,x)},{( a,¬x)}}, so that ∩(outfamily(H,a,∅)) = Cn(x)∩Cn(¬x) = Cn(∅), whichdoes not contain x.

EXAMPLE 2.4. To illustrate non-monotony in argument A, put G = {(a,x), (b,¬x)}.Then x ∈ Cn(x) = ∩(outfamily(G,a,∅)). But ∩(outfamily(G,{a,b},∅)) =Cn(x)∩Cn(¬x) = Cn(∅), which does not contain x.

EXAMPLE 2.5. To illustrate failure of antitony in argument C, put G = {(a,x), (a,y)},C = {¬x∨¬y }, D = C∪{ ¬x}. Then C ⊆ D, and y ∈ Cn(y) = ∩(outfamily(G,a,D)), buty ∉ Cn(x)∩Cn(y) = ∩(outfamily(G,a,C)).

#2. Elements of outfamily versus maximal outputs

It is important to distinguish between an outfamily, i.e. the set of outputs determinedby maximal subsets of the generators, and the set of maximal outputs. The latter is asubset of the former, but for certain of the input/output operations, the two sets are notidentical.

OBSERVATION 2. Let X be a maximal value of out(H,A), for H ranging over subsetsof G such that out(H,A) is consistent with C. Then X ∈ outfamily(G,A).

Sketch of Proof. Straightforward, using the compactness of each of our input/outputoperations, established in (Makinson and van der Torre, 2000).

On the other hand, the converse fails for certain of the input/output operations. For out∈ {out3, out3

+, out4} the Möbius strip provides a counterexample, as shown inExample 3.3 and in the discussion of normal Reiter default systems (Section 4).

For out2, the same phenomenon can be illustrated by a different example. Put G ={( a,⊥), (¬a,⊥), (¬a,x)}. Then, calculating for out = out2 with t as input and ∅ asconstraint, out(G,t) = Cn(⊥), which is inconsistent, and maxfamily(G,t,∅) ={{( a,⊥),(¬a,x)}, {( ¬a,⊥),(¬a,x)}} so that outfamily(G,t,∅) = {Cn(x), Cn(∅)}, andevidently Cn(x) ⊃ Cn(∅).

But for out1, out1+, and out4

+ (alias out2+) we get the opposite. In those cases, every

element of outfamily is maximal, as we now show.

OBSERVATION 3. Let out = outi for i ∈ {1,1+,2+,4+}. Then no element ofoutfamily(G,A,C) properly includes any other.


Proof. Let X, Y be elements of outfamily(G,A,C) and suppose X ⊂ Y; we derive acontradiction. Since X, Y ∈ outfamily(G,A,C) we have:

X = out(G1,A), where G1 ⊆ G, X is consistent with C; and for all G1′ with G1 ⊂G1′ ⊆ G, out(G1′,A) is inconsistent with C

Y = out(G2,A), where G2 ⊆ G, Y is consistent with C; and for all G2′ with G2 ⊂G2′ ⊆ G, out(G2′,A) is inconsistent with C.

Since X ⊂ Y, we have X ⊆ Y, i.e. out(G1,A) ⊆ out(G2,A). Also since X ⊂ Y, we havenot: Y ⊆ X, so not: G2 ⊆ G1, so G1 ⊂ G1∪G2. Hence by maximality of G1,out(G1∪G2,A) is inconsistent with C. Thus to obtain a contradiction, it suffices toshow that out(G1∪G2,A) ⊆ out(G2,A) which by hypothesis is consistent with C.

For out1, we branch the argument as follows. Since out(G1,A) ⊆ out(G2,A) we have bythe definition of out1 in Section 1.1 that Cn(G1(Cn(A))) ⊆ Cn(G2(Cn(A))) soG1(Cn(A)) ⊆ Cn(G2(Cn(A))). Clearly also G2(Cn(A)) ⊆ Cn(G2(Cn(A))). ThusG1∪G2(Cn(A)) ⊆ Cn(G2(Cn(A))) and so finally Cn(G1∪G2(Cn(A))) ⊆ Cn(G2(Cn(A)))i.e. out(G1∪G2,A) ⊆ out(G2,A) as desired. For out1

+, simply replace G1,G2 byG1∪I,G2∪I in this argument.

For out4+ alias out2

+, we branch as follows. Since out(G1,A) ⊆ out(G2,A) we have bythe characterization in Section 1.1 of out4

+(G,A) as Cn(A∪m(G)), that Cn(A∪m(G1))⊆ Cn(A∪m(G2)) so out(G1∪G2,A) = Cn(A∪m(G1)∪m(G2)) ⊆ Cn(A∪m(G2)) =out(G2,A) as desired.

#3. Normal Reiter Default Systems: Proof of Observation 5.

We prove Observation 5 from first principles, not having seen any result in theliterature from which it would follow directly. The argument is not difficult, but rathercomplex when set out rigorously.

Proof. We need to prove assertions (a) and (b) of the Observation. For (a), let E beany extension of the normal Reiter default system (G,A). To show that E ∈outfamily(G,A) it suffices to show that E = out3

+(H,A) for some maximal H ⊆ G suchthat out3

+(H,A) is consistent.

Put H = {(a,x) ∈ G: either a ∉ E or x is consistent with E}. Then H ⊆ G, and it is easyto show using the quasi-inductive characterization of extensions in Theorem 2.1 of(Reiter 1980) that E is also an extension of the normal Reiter default system (H,A).Since A is consistent, we know from Corollary 2.2 of (Reiter 1980) that E isconsistent. It remains to show that (1) E = out3

+(H,A), (2) out3+(H′,A) is inconsistent

for all H′ with H ⊂ H′ ⊆ G.

For (1), by Reiter’s fixed-point definition of an extension, E = ∩{ B: A ⊆ B = Cn(B),such that x ∈ B whenever (a,x) ∈ H and a ∈ B and x is consistent with E}. On the onehand, since out3

+ is reusable throughput, out3+(H,A) is such a B (without even


appealing to the consistency condition), so E ⊆ out3+(H,A). On the other hand, E is

also such a B, i.e. A ⊆ E = Cn(E), and x ∈ E whenever (a,x) ∈ H and a ∈ E and x isconsistent with E. But by the definition of H, (a,x) ∈ H and a ∈ E together imply thatx is consistent with E. Thus A ⊆ E = Cn(E), and x ∈ E whenever (a,x) ∈ H and a ∈E, i.e. A ⊆ E = Cn(E) ⊇ H(E), so E ⊇ ∩{ B: A ⊆ B = Cn(B) ⊇ H(B)} = out3

+(H,A) by acharacterization of the latter (Section 1.1).

To show (2), suppose H ⊂ H′ ⊆ G with (a,x) ∈ H′−H, so that by the definition of H, a∈ E and x is inconsistent with E. By (1), E = out3

+(H,A) ⊆ out3+(H′,A), so x is

inconsistent with out3+(H′,A). But since (a,x) ∈ H′ and a ∈ E ⊆ out3

+(H′,A) we have x∈ out3

+(H′,A), since out3+ satisfies reusability. Putting these together, out3

+(H′,A) isitself inconsistent as desired.

To show assertion (b), let X ∈ outfamily(G,A). Then X = out3+(H,A) for some maximal

H ⊆ G such that out3+(H,A) is consistent. It will suffice to show that X is an extension

of the normal Reiter default system (H,A), for as shown in the ‘semi-monotonicity’Theorem 3.2 of (Reiter 1980), when H,G are sets of normal default rules with H ⊆ G,then every extension of system (H,A) is included in some extension of (G,A).

By the hypothesis, X is consistent. Consider the sequence E0, E1, … where E0 = A andEi+1 = Cn(Ei) ∪ {x: (a,x) ∈ H and a ∈ Ei and x is consistent with X}. By Theorem 2.1of (Reiter 1980), to show that X is an extension of (H,A), it suffices to show that ∪{ Ei

: 0 ≤ i < ω}, written briefly ∪Ei , equals X.

By induction, we have each Ei ⊆ out3+(H,A) = X so that ∪Ei ⊆ X. For the converse, we

have by a characterization of out3+ (Section 1.1) that out3

+(H,A) = ∩{ B: A ⊆ B =Cn(B) ⊇ H(B)} so it suffices to show that ∪Ei is such a B, i.e. that A ⊆ ∪Ei = Cn(∪Ei)⊇ H(∪Ei). The first inclusion and the equality are trivial. To show the secondinclusion, suppose x ∈ H(∪Ei) so that (a,x) ∈ H for some a ∈ ∪Ei, so a ∈ Ei for somei. It will suffice to show x ∈ Ei+1. By the definition of Ei+1 it suffices to show that x isconsistent with X. But since a ∈ Ei ⊆ out3

+(H,A) = X and (a,x) ∈ H we have x ∈ X byreusability, so since X is consistent, x is consistent with X as desired and we are done.

#4. Remarks on the definition of a constrained derivation

Let ∆ be a derivation of (a,x) from G given a rule-set R, and let L ⊆ G be the set ofleaves of ∆. In Section 6.1 we defined ∆ to be constrained with respect to rule-set Riff (a,¬a) ∉ deriv(L) where deriv is derivability using only rules in R.

To illustrate the effect of using deriv(L) rather than deriv(G) in this definition, put G ={( a,x), (a,y)} and consider the one-step derivation:

(a,x) SI(a∧¬y,x).


This derivation is constrained under our definition, since (a∧¬y, ¬(a∧¬y) ∉ deriv(L).But (a∧¬y, ¬(a∧¬y) ∈ deriv(G), by applying SI and WO to the other element of G.

To illustrate the effect of using the whole of R rather than the subset of its rulesactually applied in ∆, consider again Example 5 in Section 6.1. On the one hand, it isnot difficult to show that (a∧¬x,¬(a∧¬x)) ∉ deriv(L) when deriv is determined by theset {SI, CT} of rules actually applied in ∆. On the other hand, as noted in Example 5,∆ is not constrained with respect R3 = {SI, AND, WO, CT}, since (a∧¬x,¬(a∧¬x)) ∈deriv(L) where deriv is determined by that rule-set. A more interesting illustration ofthe same point is the following.

EXAMPLE 6. Put G = {(a, x∧(a→y)), (¬a, x∧(¬a→y))} and consider the derivation:

(a, x∧(a→y)) (¬a, x∧(¬a→y)) WO WO (a,x) (¬a,x) -------------------------------------- OR (t,x) SI

(¬y,x)

This derivation is constrained with respect to the set {SI, WO, OR} of rules appliedwithin it. On the other hand, it is not constrained with respect to the larger set R4 ={SI, WO, AND, OR, CT}. Indeed, it is not difficult to show that there is noderivation, given R4, of the same root from the same leaves (or a subset of them), thatis constrained with respect to R4. We omit the verifications.

#5. Varying the set L and its effect on Lemma 14

Suppose that when constraining a node n:(a,x) of a derivation we do so with respect tothe set Ln of leaves in its subtree, rather than the set L of all leaves of the tree (seeremark after Lemma 14). Then parts of Lemma 14(b) fail: satisfaction of theconstraint is no longer preserved forwards by the rules AND, CT, OR. We give anexample for each. In each example, the derivation fails the constraint at theconclusion of the rule in question, marked by an asterisk. It also fails the constraint atthe premises of that rule, if L is held constant. But it would satisfy the constraint ateach premise of the rule if L were allowed to diminish. For instance in Example 7.1,node n:(a,x), a premise of the AND rule, is a leaf and so Ln = {(a,x)}, and (a,¬a) ∉deriv(Ln).

EXAMPLE 7.1 (for AND). Let R ∈ {R1,..,R4} and consider the following derivation.

(a, x) (a, ¬x) --------------------------- AND

*(a,⊥)


EXAMPLE 7.2 (for CT). Let R ∈ {R3,R4} and consider the following derivation.

(a, x∧¬y) WO (a,x) (a∧x,y) ---------------------------- CT

*(a,y)

EXAMPLE 7.3 (for OR). Let R ∈ {R2,R4} and consider the following derivation.

(a, x) (a,¬x) WO WO (a, t) (a, t) ----------------------------- OR

*(a, t)

#6. Derivations using both OR and CT: proof of Observation 17

Proof. Consider the following derivation of ((a∧¬x)∨b, x→¬(a∧b)) from thegenerator set G = {(a,x), (b, x→¬(a∧b))}.

(a,x) (b, x→¬(a∧b)) (b, x→¬(a∧b)) SI SI

* (a∧(x→b)), x) *(a∧b∧x), x→¬(a∧b)) ----------------------------------------------------------- CT

*( a∧(x→b)), x→¬(a∧b)) ----------------------------------------------------------- OR

((a∧¬x)∨b, x→¬(a∧b))

On the one hand, the derivation satisfies the constraint at the root and at the leaves. ByObservation 10, to show this it suffices to check that the body of the root, and thebody of each leaf, is consistent with m(L) = m(G). On the other hand, the derivationfails the constraint at each of the starred nodes. By Observation 10 again, it suffices tocheck that the body of each starred node is inconsistent with m(L). We now show thatthere is no derivation with the same root and the same (or fewer) leaves that satisfiesthe constraint at all nodes. This is the challenging part of the proof.

Call a node of a derivation small if it is in the unrestricted output of a single leaf;otherwise big. Quite generally, if a node is small, then it is of the form (c+,z−) for someleaf (c,z) where c+ |− c and z |− z−, as can be checked by simple induction on thederivation.

Consider any derivation, given rule-set R4, of the root node from G. Its leaves will allbe from G, or else of the form (t,t) where t is a tautology. The root node is big, since itis not of the above form. Hence there is a first big node in the derivation. This neednot be unique, but we choose one and call it m. By construction, m is not one of theleaves, so it must be obtained using one of the rules SI, WO, AND, CT, OR. Since mis a first big node, it is not obtained by the single-premise rules SI or WO, so it must


be obtained by one of AND, CT, OR from two small premises. If either of these twopremises is of the form (d,t) then m will be equivalent to the other premise, or elseitself of the form (e,t) as is easily checked by cases; and so will itself be small,contrary to supposition. So m is obtained by one of AND, CT, OR from two smallpremises, one of which is of the form (a+,x−) and the other of the form (b+,(x→¬(a∧b))−).

Application of OR to two such premises gives a conclusion of the form (d,t) which issmall, leaving only the rules AND, CT to consider.

Application of AND to two such premises is possible only if a+ is equivalent to b+, inwhich case a+ |− a∧b. But it is easily checked that ¬(a∧b) ∈ out4(G, a∧b), so by SIand WO, ¬ a+ ∈ out4(G, a+), so node m fails the constraint (as also its two premises).

For CT there are two cases to consider, as it is an asymmetric rule. In the first case, b+

is equivalent to a+∧x−, so that b+ |− a∧b so by the same argument as for AND, thepremise with body b+ fails the constraint. In the second case, a+ is equivalent tob+∧(x→¬(a∧b))− so a+ |− a∧b and by the same argument the premise with body a+

fails the constraint.

REFERENCES

Alchourrón, Carlos, Peter Gärdenfors and David Makinson, 1985. On the logic oftheory change: partial meet contraction and revision functions, The Journal ofSymbolic Logic 50: 510-530.

Alchourrón, Carlos and David Makinson, 1982. On the logic of theory change:contraction functions and their associated revision functions, Theoria 48: 14-37.

Hansson, Bengt, 1969. An analysis of some deontic logics, Nous 3: 373-398.Reprinted in R. Hilpinen, R. ed Deontic Logic: Introductory and Systematic Readings,Dordrecht: Reidel, 1971 and 1981, pp. 121-147.

Hansson, Sven Ove and David Makinson, 1997. Applying normative rules withrestraint, in M.L. Dalla Chiara et al eds Logic and Scientific Methods, Dordrecht:Kluwer, pp. 313-332.

Makinson, David, 1994. General patterns in nonmonotonic reasoning. In Dov Gabbayet al eds Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3,Oxford University Press, pp. 35-110.

Makinson, David, 1997. Screened revision, Theoria 63: 14-23.

Makinson, David, 1999. On a fundamental problem of deontic logic, In PaulMcNamara and Henry Prakken eds Norms, Logics and Information Systems. NewStudies in Deontic Logic and Computer Science, Amsterdam: IOS Press, Series:Frontiers in Artificial Intelligence and Applications, Volume 49, pp. 29-53.


Makinson, David and Leendert van der Torre, 2000. Input/output logics, J.Philosophical Logic 29: 383-408.

Poole, David, 1988. A logical framework for default reasoning, Artificial Intelligence36: 27-47.

Reiter, Ray. 1980. A logic for default reasoning, Artificial Intelligence 13: 81-132.

van der Torre, Leendert W.N., 1997. Reasoning about Obligations: Defeasibility inPreference-Based Deontic Logic. Ph.D. thesis, Erasmus University of Rotterdam.Tinbergen Institute Research Series n° 140. Thesis Publishers: Amsterdam.

van der Torre, Leendert W.N., 1998. Phased labeled logics of conditional goals, inLogics in Artificial Intelligence, Proceedings of the Sixth European Workshop onLogics in AI (JELIA’98). Berlin: Springer, LNCS 1489, pp 92-106.

van der Torre, Leendert W.N. and Yao-Hua Tan, 1999. Contrary-to-duty reasoningwith preference-based dyadic obligations. Annals of Mathematics of ArtificialIntelligence 27: 49-78.

ACKNOWLEDGEMENTS

Thanks for helpful comments to Salem Benferhat, the referees for DEON 2000 wherean early version of this paper was presented, and the referee for this journal. Researchfor the paper was begun when the second author was working at IRIT, Université PaulSabatier, Toulouse, France, and at the Max Planck Institute for Computer Science,Saarbrücken, Germany.

David MakinsonVisiting Professor, Department of ComputingKing’s College LondonPermanent address:Les Etangs B2, Domaine de la Ronce92410 Ville d’Avray, FranceEmail: [email protected]


Last revised 11.01.01Typos corrected 20.01.01

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