In this position paper we discuss the importance of giving a proper account of unknowability in argument-based reasoning, suggest that existing formal tools are not fully adequate in this respect and lay down the basics of a research program in this direction.
Human knowledge is obviously limited and anyone has to admit that there are
things whose knowledge is beyond the present and possibly also future
capabilities of any human being, due to various kinds of limitations. For instance, the
fact that a new drug is e ective for a given disease cannot be known in absence
of an adequate testing. While such a limitation appears to be temporary and
can be overcome with adequate actions and resources, other can be more radical.
For instance, the answer to some computational problems in the worst case may
require an amount of resources which overcomes Bremermann's limit[
Theoretical limitations apart, unknowability plays a signi cant role in many practical debates about important questions.
Consider discussions about very complex global issues like climate change or world-level economic policies. Questions like \Are the causes of global warming natural or human related?" or \Are free trade policies good or bad for developing countries?" may receive both quite bold answers on either side, but one can also put forward the standpoint that any de nite position on such issues is unacceptable, since it is actually impossible to provide well-founded answers to such questions. This standpoint, while possibly unsatisfactory from a psychological point of view, might turn out to be the most reasonable one, in the light of the currently available knowledge and evidences, and of the fact that such problems are \untamable" due to their intrinsic huge complexity.
In this paper, expanding a preliminary discussion in [
The paper is organised as follows. Section 2 discusses the concept of unknowability and provides some motivation for dealing with it in argumentation, and Section 3 sketches a solution by adapting the well-known ASPIC+formalism. The solution is brie y illustrated by means of an example in Section 4. Finally, Section 5 draws some conclusions focusing on future work. 2
In most argumentation systems the language adopted to construct arguments allows one to assert two truth-values for each proposition, i.e. a proposition can either be (asserted as) true or (asserted as) false. Usually this is obtained by equipping the language with classical negation, so that asserting that p is false can be expressed as the assertion that :p is true. Accordingly, each argument takes a de nite position on its conclusion, supporting the acceptance of a proposition p or the acceptance of its negation :p.
As a consequence, undecidedness on propositions is a derived concept which
arises from the evaluation of relevant arguments. For the sake of explanation, let
is consider a Dung's style evaluation of arguments by means of a unique-status
semantics [
As mentioned in Section 1, a di erent kind of undecidedness, called epistemological undecidedness, relies on an explicit reason supporting the standpoint that any de nite answer on a proposition p is unacceptable, i.e. that p is unknowable. Examples of reasons why a proposition p is unknowable include: { Universal principles (like Heisenberg's) saying that you cannot know everything at the same time. For instance, if you know the speed of a particle with a given precision then its momentum is unknowable with \the same" precision. { Evidence from common knowledge: if there is a question which is known to be very debated with many disagreements about it, or a question that is still unsolved despite many attempts, one can infer that it is unknowable to everybody (e.g. the Millennium Prize Problems). { Something which is not unknowable in principle but it is impossible to verify in any reasonable time horizon, e.g. whether there is life on a potentially habitable planet outside the Solar System.
It is important to point out the di erence between epistemological undecidedness and the undecided justi cation states introduced above. The main di erence is that epistemological undecidedness is not the result of argument evaluation, but originates from single arguments that can independently support the conclusion that a proposition p is unknowable. Thus, it clearly di ers both from the status of \contradictory" and from that of \unknown". More speci cally, di erently from the case where p is contradictory, it does not arise from contradictory information: a single argument may support unknowability for a speci c non contradictory reason, and in case this argument has no counterarguments then unknowability of p is accepted. Moreover, di erently from the case where p is unknown, there is an explicit valid argument supporting unknowability, which is able to attack all those arguments that say that p is true or p is false.
In a nutshell, epistemological undecidedness and derived undecidedness are orthogonal concepts: whereas epistemological undecidedness concerns the level of assertions (language level), the justi cation states mentioned above concern the evaluation of arguments and their conclusions. Thus, it may well be possible that e.g. unknowability of p is contradictory since an argument supporting that p is unknowable is counterattacked by an equally preferable argument supporting that p is true.
As a nal comment, it is worth noting that epistemological undecidedness
cannot be encompassed by Pollock's undercutting attack [
While the above considerations may suggest to model unknowability as a sort of universal undercut, i.e. attacking all relevant rules, it has to be noted that also defeasible premises of an argument can be a ected by unknowability. 3
According to the considerations in the previous section, in order to capture
unknowability one may exploit the general model of structured argumentation
ASPIC+, which abstracts from the speci c language used to construct
arguments and generalizes classical negation between formulas to a generic, possibly
asymmetric, relation of contrariness . In this model, arguments are built from
a knowledge base, which includes a set of language elements partitioned into
axioms and ordinary premises, by applying two kinds of rules, i.e. strict and
defeasible rules (depicted as single arrows ! and double arrows ), respectively).
Arguments can be represented as trees with roots corresponding to their
conclusions and leaves to premises. The model is able to capture di erent kinds of
con ict, including undercutting, and to deal with a preference order between
arguments: we refer the reader to [
In order to capture unknowability, we can require that for each language symbol a there are also two other symbols, i.e. :a and a, where a means that a is unknowable. Then a strict or defeasible rule has a set of premises each of them of the kind p, :p or p, where p ranges over the considered language, and similarly for its consequent. As a consequence, chaining rules of inference yields arguments that may have conclusions of the kind c (besides c and :c), i.e. they can support unknowability.
The adoption of a third truth value makes it necessary to handle the relevant contradictions between the language elements. One option would be to include in the set of rules for any language element a the strict rules a ! :a and a ! :(:a), i.e. if a proposition is unknowable then it cannot be true and it cannot be false. However, since :a and :(:a) are in contradiction, this would mean that any argument for unknowability would give rise to a contradiction.
Another option is to exploit the generic relation of contrariness provided in ASPIC+, by establishing that for any language element a the relations a = f:a; ag, :a = fa; ag and a = fa; :ag hold, i.e. that a, :a and a are mutually contradictory. However, a technical di culty in ASPIC+prevents a direct application of the formalism with a contrariness relation like this. More specifically, in order to satisfy a set of rationality postulates, a closure requirement on strict rules is required such that whenever there is a strict rule p ! c then c ` p, where a denotes a language element a0 such that a0 2 a and a 2 a0, and ` denotes derivation under strict rules only. In our context, this entails that whenever there is a strict rule p ! c then :c strictly entails :p and p, leading to a contradiction.
Another way to model unknowability is to exploit modal operators, i.e. expressing the fact that p is known as K(p), the fact that :p is known as K(:p) and the fact that p is unknown as :K(p) ^ :K(:p) (i.e. it is not known that p is true and it is not known that :p is true). It is worth noticing that this would not avoid the need for a non binary relation of contrariness, e.g. K(p) = fK(:p); :K(p)g. As a consequence, the technical di culty mentioned above should still be dealt with.
A solution to this problem can be obtained by closing the relation of
contrariness under strict rules: a technical treatment of the resulting formalism is
however outside the scope of the present paper, and the reader is referred to [
In order to illustrate the approach, let us consider a simple football example where a set of defeasible rules model the relationship between players quality and the plausibility of winning football matches. In particular, a simple modelling may include a defeasible rule GP i ) ST i, i.e. if a team T i has good players (GP i) then normally it is strong (ST i), a defeasible rule BP i ) W T i, i.e. if a team T i has bad players (BP i) then normally it is weak (W T i), a rule ST i; W T j ) T iW IN S prescribing that if a team T i is strong and another team T j is weak then T i tends to win the match against T j (T iW IN S), and another defeasible rule ST i; W T j ) T iW IN S stating that the result of a match between a team with unknown strength and a weak team is unknowable.
Now, assume there is a match between two teams T 1 and T 2 that are known to have good and bad players, respectively. We can then build the arguments 1 : GP 1, 2 : 1 ) ST 1, 1 : BP 2, 2 : 1 ) W T 2, 1 : 2; 2 ) T 1W IN S. Since there are no attacks between arguments, all of them are justi ed and we may conclude that T 1 should win the match.
Assume now that T 1 su ers some injuries (IN J 1), and there is a defeasible rule stating that in this case the strength of a strong team becomes unknowable: GP i; IN J i ) (ST i). Then we can construct the following arguments in addition to the above ones: 3 : IN J 1, 4 : 1; 3 ) (ST 1), 2 : 4; 2 ) T 1W IN S. According to the relation of contrariness, 2 and 4 have contradictory conclusions and the same holds for 1 and 2. If we adopt a speci city criterion to compare arguments, 4 attacks 2 since it is based on a greater set of premises, thus it also attacks 1. As a consequence, 2 emerges as justi ed as well as its conclusion that the result of the match is unknowable. 5
In this position paper we have discussed the importance of dealing with unknowability in formal argumentation. Several research steps may then be performed.
First, a complete formal treatment based on ASPIC+has to be carried out. This may also require to distinguish between the concept of \unknowability to a single agent" vs \general unknowability", which may a ect the relation of attack between arguments.
A further development may account for di erent attitudes concerning unknowable facts, i.e. one may accept or refuse an argument based on them according to pragmatic considerations on values, risks, and so on. In line with this, it might be interesting to generalize the approach to continuous truth-values, i.e. fuzzy propositions.
From a more conceptual point of view, similar issues concerning
unknowability have been investigated in formal epistemology and epistemic logics, while
recent work discusses some connections between abstract argumentation and
epistemic logics [
The research reported in this paper was partially supported by the University of Brescia under the WAT CHALLENGE project.