esslli lecture 4 normchange - pigozzi.org › esslli lecture 4 normchange.pdf · motivation •...
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ESSLLI’18 Lecture 4 Gabriella Pigozzi & Leon van der Torre
CHANGING NORMS
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MOTIVATION• Norm change is an interesting open problem.
• There are various models of norm change but no consensus yet on a common framework.
• Most models focus on dynamics of obligations and permissions. Models that represent changes in underlying norms are needed.
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• 2001 J. Maranhao “Refinement - A tool to deal with inconsistencies”: Inspired by selective revision of Fermè and Hansson (only part of the input is accepted).
• 2009 Aucher, Grossi, Herzig & Lorini “Dynamic context logic”: Approach inspired by the dynamic logic paradigm. It looks at norm change as a form of model update.
• 2010, Governatori & Rotolo “Changing legal systems: legal abrogations and annulments in defeasible logic”: norm change performed on the rules contained in the code. Inspired by legal practice: annulments and abrogations. Temporal extension of DL.
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NORMS AND OBLIGATIONS• When new norms are created or old norms are
retracted, the changes have repercussions on obligations and permissions that such norms established.
• Norms must be distinguished from obligations.
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AGM THEORY OF BELIEF CHANGE• They started working on how to change a legal system but then
generalized to theory change and used propositions, losing the proper representation of norms.
• “The same concepts and techniques may be taken up in other areas, wherever problems akin to inconsistency and derogation arise”.
• In their approach a code is a non-empty and finite set of propositions and norm x is just a formula in propositional logic.
• AGM85 is more about obligation change than norm change.
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Autem vel eum iriure dolor in hendrerit in vulputate velit esse
AGM 1985
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OUR APPROACH
• AGM formal study of theory change
• Expansion, revision and contraction are the three theory change operations that Alchourron, Gärdenfors and Makinson identified.
• We take AGM theory change as a framework to evaluate the dynamics of rule based systems.
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• We do not consider one particular logic for rules, but several of them in the input/output logic framework (Makinson and van der Torre 2000).
• Distinction between norms and obligations.
• We assume that rules are represented as pairs of formulas of an arbitrary logic.
• We do not necessarily have reflexivity (“if p then p”) or transitivity.
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I/O LOGIC• A rule is a pair (a,x)
• A normative system R is a set of such pairs
• R is a set of conditional obligations (a,x). An unconditional obligation of x is (T,x), e.g. you must not be cruel to animals.
• The key idea is to make obligations relative to a given set of conditional norms.
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I/O LOGIC
• Calculates whether according to normative system R and in context a, a formula x is obligatory
a is the input
x is the output
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SIMPLE-MINDED OUTPUT• (a,x): if input implies a, then output implies x.
• Each out1(R,In) is closed under Cn
• Let R(a) ={x|(a,x) R} 2
out1(R, a) = Cn(R(Cn(a)))
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RULES
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EXPANSION, CONTRACTION, REVISION
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HOW TO CONTRACT?
• (a,x) = people owning valuable terrains must pay the tax on fortune
• Instead of having {(a,x)} − (a,x) = {}, we may wish to have {(a,x)} − (a,x) = {(a/\b, x}, where b are people with high income.
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Virtual community:
hse = house with low rent
pr = low income person
hi = free health insurance
old = elderly person
fam = big family
R = {(pr, hse /\hi)} and we want to retract (pr/\old, hse) from R => several possibilities.
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HOW TO CONTRACT?
1. (too drastic?)
2. (weaker rule)
3. (change the context)
(pr ^ ¬old, hse ^ hi)
{(pr, hse ^ hi)}� (pr ^ old, hse)
(pr, hi)
(pr ^ old ^ fam, hse ^ hi)
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HOW TO CONTRACT?
3 options:
1.
2.
3. (change the context)
(a ^ ¬b, x ^ y)
(a ^ b ^ c, x ^ y)
(a, x ^ y)� (a ^ b, x)
(a, y)
What does it mean to remove by adding rules?
How can we make sure not to remove too much? We need some
kind of ‘closure’.
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APPLYING AGM
• : a set of rules closed under an input/output logic
• Expansion:
• Contraction:
• Revision:
• Expansion unproblematic:
R
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AGM RULE CONTRACTION
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AGM RULE CONTRACTION
Conjunctions: not defined for rules (optional postulates)
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CONTRACTION OF RULES
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(a,x) in out(R), iff x in out(R,a)
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Success has been criticized in BR but reasonable to impose such a
requirement when we wish to enforce a new norm or obligation. example
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Virtual community:
hse = house with low rent
pr = low income person
hi = free health insurance
old = elderly person
fam = big family
R = {(pr, hse /\hi)} and we want to retract (pr/\old, hse) from R => several possibilities.
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We need to change R: if we don’t (R=R1) for pr/\old, R outputs hse/\hi. But for the success
postulate hse should not be inout(R’, pr/\old).
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All alternatives give health insurance to all poor people (no matter if they’re old or not).
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The difference is for pr/\not-old: R2 removes the house also to the poor. That’s too drastic. Other options are R3 and R4 which give hse/\hi to poor /\ not-old. But for this we need additional rules in R3 and R4.
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However, all R2, R3 and R4 continue NOT to give house to the poor.
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Heavily criticized axiom in belief revision, but intuitively desirable when laws are suspended. Yet, Recovery does not always hold for AGM rule change.
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OUT3=REUSABLE OUTPUT
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OUT2= BASIC OUTPUT
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RULE REVISION
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RULE REVISION
Same as for contraction
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homonymous conditions as for contraction
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define the relation between revision and expansion
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undefined in input/output!
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FROM REVISION TO CONTRACTION• Postulates for (belief and rule) revision and
postulates for (belief and rule) contraction are independent.
• However the Levi identity defines revision as a sequence of contraction and expansion:
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FROM REVISION TO CONTRACTION• Postulates for (belief and rule) revision and
postulates for (belief and rule) contraction are independent.
• However the Levi identity defines revision as a sequence of contraction and expansion:
the revision function obtained from a contraction function via
the Levy identity
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FROM REVISION TO CONTRACTION• Postulates for (belief and rule) revision and
postulates for (belief and rule) contraction are independent.
• However the Levi identity defines revision as a sequence of contraction and expansion:
no recovery postulate! Good news for us (out1 and out3 had problems
with that)
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We can prove the corresponding theorem for rule change:
•
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…AND BACKNot only belief revision can be defined in terms of belief contraction, but also the inverse can be defined, that’s Harper identity:
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…AND BACKNot only belief revision can be defined in terms of belief contraction, but also the inverse can be defined, that’s Harper identity:
the contraction function obtained from a revision
function via the Harper identity
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…AND BACKNot only belief revision can be defined in terms of belief contraction, but also the inverse can be defined, that’s Harper identity: Can we prove the same for rule change??
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…AND BACKNot only belief revision can be defined in terms of belief contraction, but also the inverse can be defined, that’s Harper identity: Can we prove the same for rule change??
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…AND BACKNot only belief revision can be defined in terms of belief contraction, but also the inverse can be defined, that’s Harper identity:
The reason is that we have seen that (R-5) does not hold for out1 and out3… can we prove it for
out2?
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…AND BACKNot only belief revision can be defined in terms of belief contraction, but also the inverse can be defined, that’s Harper identity:
The reason is that we have seen that (R-5) does not hold for out1 and out3… can we prove it for
out2?
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RELATIONS BETWEEN THE TWO THEOREMS
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RELATIONS BETWEEN THE TWO THEOREMS
But…
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RELATIONS BETWEEN THE TWO THEOREMS
But…
Again, whether that holds for out2 is work in progress…
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WRAP-UP• Norm change is an interesting problem
• No consensus yet
• We have used AGM as a framework for normative change
• We used input/output logics to represent rules
• In order not to eliminate too much => some notion of closure is needed
• AGM contraction by adding rules.
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• AGM contraction: surprising result that out1 and out3 do not satisfy the contraction postulates
• AGM revision: the translation to rule revision is more difficult: when is a set of norms inconsistent?
• Defined rule revision in terms of rule contraction using the Levi identity and showed that the operators satisfy the AGM postulates
• For Harper identity, the question about out2 is still open.
• AGM principles prove to be too general to deal with the revision of a normative system. For example, one difference between revising a set of propositions and revising a set of regulations is the following: when a new norm is added, coherence may be restored modifying some of the existing norms, not necessarily retracting some of them.