View
5
Download
0
Category
Preview:
Citation preview
ESSLLI’18 Lecture 4 Gabriella Pigozzi & Leon van der Torre
CHANGING NORMS
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.
• 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.
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.
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.
Autem vel eum iriure dolor in hendrerit in vulputate velit esse
AGM 1985
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.
• 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.
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.
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
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)))
RULES
EXPANSION, CONTRACTION, REVISION
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.
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.
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)
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’.
APPLYING AGM
• : a set of rules closed under an input/output logic
• Expansion:
• Contraction:
• Revision:
• Expansion unproblematic:
R
AGM RULE CONTRACTION
AGM RULE CONTRACTION
Conjunctions: not defined for rules (optional postulates)
CONTRACTION OF RULES
(a,x) in out(R), iff x in out(R,a)
Success has been criticized in BR but reasonable to impose such a
requirement when we wish to enforce a new norm or obligation. example
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.
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).
All alternatives give health insurance to all poor people (no matter if they’re old or not).
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.
However, all R2, R3 and R4 continue NOT to give house to the poor.
Heavily criticized axiom in belief revision, but intuitively desirable when laws are suspended. Yet, Recovery does not always hold for AGM rule change.
OUT3=REUSABLE OUTPUT
OUT2= BASIC OUTPUT
RULE REVISION
RULE REVISION
Same as for contraction
homonymous conditions as for contraction
define the relation between revision and expansion
undefined in input/output!
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:
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
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)
We can prove the corresponding theorem for rule change:
•
…AND BACKNot only belief revision can be defined in terms of belief contraction, but also the inverse can be defined, that’s Harper identity:
…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
…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??
…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??
…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?
…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?
RELATIONS BETWEEN THE TWO THEOREMS
RELATIONS BETWEEN THE TWO THEOREMS
But…
RELATIONS BETWEEN THE TWO THEOREMS
But…
Again, whether that holds for out2 is work in progress…
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.
• 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.
Recommended