1989
	Text of unpublished philosophical note

	REVISING SEMANTICALLY EQUIVALENT SETS OF SENTENCES




The Gaerdenfors analysis of belief revision [1] presupposes that no
distinction need be made between two sets of sentences that share the same
closure.  Gaerdenfors takes revision to occur on belief sets:  sets of
sentences that are closed under familiar rules of deduction.  A set of
sentences that is not closed must stand for the belief set that is its
closure; two distinct sets of sentences with the same closures must be
different notation for the same belief set.

This view unfortunately can make no sense of the intuition that {"a", "b"}
revised by "~b" clearly should be {"a", "~b"}; while {"a", "a --> b"}
revised by "~b" is ambiguous.

A theory of revision that arises from a system of arguments or a system of
defeasible reasoning [2][3][4] can account for this intuition.

Take a set of sentences S to describe an epistemic state in which the agent
produced a warranted (defeasible) argument supporting each sentence in S.  

To say that this state is to be revised by w is to suggest either (1) that w
became part of the evidential basis upon which (defeasible) arguments must now
be constructed, or (2) that a superior or preferred argument has been produced
that now warrants w.  A superior or preferred argument is, roughly, an
argument that defeats the arguments with which it disagrees.  This is usually
because it has superior syntactic form:  better structure, or more specific
use of evidence; but the reason for its superiority does not matter here:
revision presupposes superiority.

In the case of {"a", "b"} revised by "~b", sometimes the argument for "a"
and the argument for "b" will be independent.  In such cases, a superior
argument for "~b" defeats only the argument for "b", and the argument for
"a" continues to warrant its conclusion.  

In the case of {"a", "a --> b"} revised by "~b", apparently there is an
argument for "a" and an argument for "a --> b".  Together, they form a
third argument, for "b".  This third argument is defeated by the new
superior argument, but which subargument to impugn is not clear.

For concreteness, suppose {"e1", "e2"} is the evidential basis and the
defeasible warrants are "e1" is reason for "a", i.e., "e1" >-- "a"; "e2" >--
"b"; "e2 ^ e3" >-- "~b".  There are thus arguments for "a" and for "b".
Should the agent add "e3" to the evidential basis, which provides a superior
argument for "~b", the argument for "b" is defeated.  This also would occur if
the revision involved adding "~b" to the evidential basis.

On the other hand, suppose {"e1" "e2"} is the evidential basis and the
defeasible warrants are "e1" >-- "a"; "e2" >-- "~a v b".  In [3], an argument
structure consists of the warrants used in the argument, and the proposition
the argument supports.  The structure, <{<"e1", "a">}, "a"> is an argument for
"a"; call it Arg1.  The structure <{<"e2", "~a v b">}, <"~a v b"> is an
argument for "~a v b"; call it Arg2.  And the structure, Arg3, <{<"e1", "a">,
<"e2", "~a v b">}, "b"> is an argument for "b".  

Adding "~b" to the evidential basis (this revision is the most perspicuous),
the argument for "~b" cease to be an argument.  Now there are two new
arguments that involve "~b":  <{<"e2", "~a v b">}, "~a">, or Arg4, an argument
for "~a"; and Arg5, <{<"e1", "a">}, "a ^ ~b)">, which is just an argument for
"~(~a v b)".  Arg4 disagrees with Arg1, and there is no basis for preference.
Arg5 disagrees with Arg2, and there is no basis for preference.  In such
cases, defeasible reasoning produces no conclusion when there are disagreeing
arguments with no preference.  Although this is not a minimal contraction,
it is the meet of the minimal contractions.

	First Case  {"a", "b"} revised by "~b"

	"a"          "b"          "~b"

	 ^            ^             ^
	 |            |             |
	 |            |             |

	"e1"         "e2"        "e2^e3"

	Arg1         Arg2       Arg3

	Arg3 defeats Arg2 but does not affect Arg1.



	Second Case  {"a", "a --> b"} revised by "~b"

	                                "b"
	"a"         "a-->b"          "a"   "a-->b"
	 ^            ^               ^       ^
	 |            |               |       |
	 |            |               |       |
	"e1"         "e2"            "e1"    "e2"

	Arg1         Arg2                 Arg3

	
	     "~a"
	"a-->b"  "~b"
	   ^
	   |
	   |
	  "e2"

	  Arg4


	 "~(a-->b)"
	"a"     "~b"
	 ^
	 |
	 |
	"e1"

	  Arg5

	Arg5 and Arg2 disagree, with no basis for choice.  Arg4 and Arg1,
	likewise.


Elsewhere, Gaerdenfors' axiomatic treatment of belief revision has been
faulted for disallowing the intuitions of people designing systems for
defeasible and non-monotonic reasoning [5].  

Perfectly reasonable properties such as monotonicity and closure for belief
sets must yield when more detailed constructive theories can explain why those
axioms can be violated.  


REFERENCES

[1] Peter Gaerdenfors, KNOWLEGE IN FLUX, MIT Press, 1988.
[2] John Pollock, 'Defeasible reasoning,' COGNITIVE SCIENCE 11:3, pp.
	481-518, 1987.
[3] R. P. Loui, 'Defeat among arguments,' COMPUTATIONAL INTELLIGENCE 3, pp.
	100-106, 1987.
[4] Guillermo Simari and R. P. Loui, 'A mathematical treatment of defeasible
	reasoning,' ARTIFICIAL INTELLIGENCE (to appear), 1991.
[5] Charles Cross, 'Belief revision, non-monotonic reasoning, and the Ramsey
	test,' in H. Kyburg, et al., KNOWLEDGE REPRESENTATION AND
	DEFEASIBLE REASONING, Kluwer, pp. 223-244, 1990.