1 Introduction
2 Background
2.1 Alternate action based transition systems
2.1.1 Formal definitions
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Q is a finite, non-empty set of states;
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\(q_{0}\) \(\in\) Q is the initial state;
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Ag = {1,...,n} is a finite, non-empty set of agents;
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\(Ac_{i}\) is a finite, non-empty set of actions, for each \(ag_{i}\) \(\in\) Ag where \(Ac_{i}\) \(\cap\) \(Ac_{j}\) = \(\emptyset\) for all \(ag_{i}\) \(\ne\) \(ag_{j}\) \(\in\) Ag;
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\(\rho\) : \(Ac_{ag}\) \(\rightarrow\) 2\(^Q\) is an action pre-condition function, which for each action \(\alpha \in\) \(Ac_{ag}\) defines the set of states \(\rho (\alpha )\) from which \(\alpha\) may be executed;
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\(\tau\) : Q \(\times\) \(J_{Ag}\rightarrow\) Q is a partial system transition function, which defines the state \(\tau\)(q, j) that would result by the performance of j from state q. This function is partial as not all joint actions are possible in all states;
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\(\varPhi\) is a finite, non-empty set of atomic propositions; and
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\(\pi\) : Q \(\rightarrow\) 2\(^\varPhi\) is an interpretation function, which gives the set of primitive propositions satisfied in each state: if p \(\in\) \(\pi\)(q), then this means that the propositional variable p is satisfied (equivalently, true) in state q.
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V is a finite, non-empty set of values.
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\(\delta\) : Q \(\times\) Q \(\times\) V \(\rightarrow\) {+, −, =} is a valuation function which defines the status (promoted (+), demoted (−) or neutral (=)) of a value \(v_{u}\) \(\in\) V ascribed to the transition between two states: \(\delta\)(\(q_{x}\), \(q_{y}\), \(v_{u}\)) labels the transition between \(q_{x}\) and \(q_{y}\) with one of {+, −, =} with respect to the value \(v_{u}\) \(\in\) V.
2.2 Reasons for action
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R1 We should participate in j in q in which \(\phi\) holds to maintain \(\phi\) and so promote v.
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R2N We should not participate in j in q in which \(\phi\) holds since it would remove \(\phi\) and so demote v.
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R3 We should participate in j in q in which \(\lnot {\phi }\) holds to achieve \(\phi\) and so promote v.
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R4N We should not participate in j in q in which \(\lnot {\phi }\) holds since it would avoid \(\phi\) and so fail to promote v.
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R5 We should participate in j in q to ensure \(\phi\) and so promote v. Note that \(\phi\) may be contingently realised or unrealised in q and that, in some variants, the promotion of v might not be immediate, or permanent. This also applies to R5N and R6.
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R5N We should not participate in j in q which would ensure \(\lnot {\phi }\) and so demote v.
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R6 We should participate in j in q to prevent \(\lnot {\phi }\) and so promote v. Note that \(\lnot \phi\) may be contingently realised or unrealised in q.
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R6N We should not participate in j in q which would prevent \(\phi\) and so fail to promote v. We suggest that to make the reason worth consideration we should only use variants which prevent \(\phi\) immediately and permanently.
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R7 We should participate in j in q in which \(\lnot {\phi }\) to enable \(\phi\) to be achieved and v to be promoted on the next move.
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R8N We should not participate in j in q in which \(\phi\) which will risk \(\phi\) being removed on the next move which would demote v.
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R9 We should participate in j in q because performing \(j^{ag}\) promotes v.
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R9N We should not participate in j in q because performing \(j^{ag}\) demotes v.
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Problem formulation essentially the construction of an AATS+V for the particular problem situation. The AATS+V will reflect the views of the agent engaged in the reasoning, and so will embody that agent’s causal model (to determine the transitions) and its values (to enable the labelling of transitions), as is demonstrated in Atkinson and Bench-Capon (2016). As Atkinson and Bench-Capon (2016) indicates there can be arguments justifying the formulation of the problem, before the AATS+V is used in the subsequent stages.
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Epistemic stage this involves determination of what the agent engaged in the reasoning believes (or chooses to assume) about the current state and the joint action that will result from the choice of a particular individual action by the agent concerned;
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Option selection the arguments generated from the AATS+V based on the reasons given above, and objections and counterexamples to these arguments, are formed into a VAF and their acceptability status determined according to the preferences of the agent engaged in the reasoning. The acceptability status determines which actions, according to its beliefs, values and preferences, the agent can justifiably choose to perform.
2.3 Example
3 Justifying norms with value based argumentation
3.1 Arguments in q1
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A You should not play since it will risk \(L_{s}\) being demoted (R4N)
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B You should work since it will enable \(P_{s}\) to be promoted (R7)
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C You should play to promote \(P_{s}\) (R9)
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D You should not work since it will demote \(P_{s}\)(R9N)
3.2 A first set of norms
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SN1 7 It is forbidden to play
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SN2 It is forbidden to give food
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SN2a It is allowed to give food only once,
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SN2b It is obligatory to meet the first request for food.
3.3 Critique
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There is no net pleasure in the society: the displeasure of working is offset by the pleasures of feasting at carnival, but there is no net gain. Such a society lacks progress and reward and any point beyond it own continuance.
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There is no choice or diversity in the society: the path taken is determined at all times by the norms, and deviation is effectively punishable by death.
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The pleasure enjoyed in this society is of a rather basic kind, whereas the pleasure it denies itself might be seen as a higher pleasure. The hard line utilitarian might adopt the view of Bentham (1825) that “Prejudice apart, the game of push-pin is of equal value with the arts and sciences of music and poetry”, but others, like Mill would disagree: “it is better to be a human being dissatisfied than a pig satisfied; better to be Socrates dissatisfied than a fool satisfied” (Mill 1871). Such higher pleasures can only be provided by (certain forms of) play, not by feasting.
3.4 Allowing for play and the evolution of more complex societies
3.4.1 Power
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SN3 It is forbidden to demand non-surplus food.
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C1: You should \(play_{a}\) to promote culture (C).
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C2: You should \(play_{d}\) to promote frivolity (F).
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MN4 It is forbidden to \(play_{d}\).
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MN4a It is obligatory to \(play_{a}\)
This equates with a preference ordering \(L \succ C \succ P \succ F\) that can be advocated within a Virtue Ethics approach or a Consequentialist approach (indeed given Mill’s view that not all pleasures are of equal worth, the consequences are an improvement: since only the powerful can act to promote culture, it is good that they do so, even if it is at the expense of the powerless, since culture is preferred to pleasure).The rich man in his castle, The poor man at his gate, God made them high and lowly And ordered their estate.
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MN5 It is obligatory to be generous in your treatment of the less fortunate
3.4.2 Wealth
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The wealthy may own the land (or the means of production) and be in a position to take some proportion of the labour of others in the form of rent or profit. The situation is little different from the feudal, except that payment is now in money, not in kind. The flexibility afforded by money is more suitable to an Industrial society where production requires more than land and labour, and where produce is not bread alone, but a whole range of manufactured goods.
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The wealthy may choose to lend money at interest. Since many will regard a “bird in the hand as worth two in the bush”, there is likely to be takers for such loans, allowing for people with initial wealth to pay for their needs from the interest and maintain their wealth, and perhaps even, given sufficient borrowers or high enough interest rates, to increase it. Note, however, this requires some way of ensuring that the lenders can be confident that the interest will be paid, and the debt repaid. This in turn requires some kind of norm, e.g.This would be associated with a new value of Trustworthiness or Honesty (H), promoted by observance of debts (and contracts and agreements generally) and demoted by reneging on such agreements. Such agreements are typically governed not by moral norms alone, but are supported by laws and enforced by sanctions. In order to make this more general we might prefer to use the formulation:
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SN6a It is obligatory to repay debts.
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SN6 It is obligatory to honour agreements.
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Some people may have access to wealth from outside. For example, in the sixteenth century the Spanish rulers had a seemingly inexhaustible supply of gold and silver from the Americas.
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Deference or Generosity may mean that some agents are not required to work or pay but are simply given some kind of tribute. For example monks or priests may be supported by tithes or donations, or the infirm by alms. The latter, where the motivating value is generosity, are perhaps covered by MN5, but modified to be applicable to all, and to cover all those considered worthy of alms, not just the unfortunate. Moreover, the notions of tithes (and today contributions to welfare benefits) are often the subject of law, rather than left to morality.This rephrasing as SN5a means that we broaden the notion of unable to support themselves from incapacity to include those engaged in some other, worthwhile but unremunerative, activity. This allows us to subsume Mercy under Generosity, while the qualification still acknowledges justice as a value.
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SN5a It is obligatory to give alms to those unable, for good reason, to support themselves.
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\(L \succ H \succ C \succ G \succ D \succ J \succ P \succ F\)
3.4.3 Turn taking
3.4.4 Service provision as work
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SN1a It is obligatory to \(play_{a}\) or to work.
3.4.5 Emergence of a state
Once punishment is seen as a social duty it is a small step to organise and pay for a third party to punish violators. Assuming relatively few law breakers, a small levy will enable a dedicated agent to be paid to enforce the norms. Of course, non-payment of the levy will also be subject to punishment. From this it is a small step to taxation, and the provision of services such as law enforcement by the state, paid for from these taxes. And if law enforcement, why not other duties? Thus SN5a may be better observed by contribution to a central fund responsible for identifying those who should be supported and providing that support, than by individual charity.This move from enforcement by vigilantes (those taking the law into their own hands) to seeing law enforcement as the social duty of responsible citizens is an important milestone in the development of a society that respects its laws.
4 Rules are made to be broken
4.1 Why agents might need to violate norms
What the wiser among us do is to think deeply about the crucial moral questions, especially those that face us in our own lives, but when we have arrived at an answer to a particular problem, to crystalize it into a not too specific or detailed form, so that its salient features may stand out and serve us again in a like situation without so much thought.
if the need be so manifest and urgent that it is evident that the present need must be remedied by whatever means be at hand (for instance when a person is in some imminent danger, and there is no other possible remedy), then it is lawful for a man to succor his own need by means of another’s property, by taking it either openly or secretly: nor is this properly speaking theft or robbery.20 (Aquinas 2012, Question 66, Article 6).
4.2 Rule breaking in AI and law
4.3 Road traffic example
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Whether there is an obstruction and whether it is moving or stationary (0 = clear, 1 = slow moving, 2 = stationary).
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Whether there is an on-coming vehicle and whether it can stop safely or not (0 = no on-coming, 1 = can stop safely, 2 = cannot stop).
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Whether our own vehicle can stop safely (0 = can stop safely, 1 = cannot stop safely).
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Whether there has been22 a collision (0 = no collision, 1(x, y) = x has collided with y).
Self: on-coming | Continue | Slow | Stop | Change lane | Mount pavement |
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Continue | J1 | J2 | J3 | J4 | J5 |
Slow | J6 | J7 | J8 | J9 | J10 |
Stop | J11 | J12 | J13 | J14 | J15 |
Change lane | J16 | J17 | J18 | J19 | J20 |
Mount pavement | J21 | J22 | J23 | J24 | J25 |