## Final Report Summary - 3VAL (Three valued logics and uncertainty management)

Traditionally, in classical logic there are just two truth-values: true and false and the main operations to connect propositions (say, p and q) are conjunction (p AND q), disjunction (p OR q), negation (NOT p) and implication (p IMPLIES q). In the last century, starting from the Polish philosopher Lukasiewicz, there has been a plethora of new, non-standard logics, which generalise classical Boolean logic. There are several lines of generalisation, mainly, by adding further truth- values, intermediate between true and false or adding new symbols or connectives, such as modalities of necessity and possibility.

These two lines give rise to many-valued logics and modal logics respectively. The present project dealt with three-valued logics, namely, logics with one more truth-value different from true and false. There are several logics of this kind, where the third truth-value can be interpreted in different ways. Indeed, we can find, at least: half-true, unknown, undefined, irrelevant, possible, inconsistent. As can be seen, the third truth-value is not always an intermediate value between true and false (that is, half-true) but it can assume other meanings, often referring to the knowledge level. Using half-true to represent an epistemic notion (such as lack of knowledge or contradictory knowledge) is open to criticism both from an intuitive point of view and from a formal one, since the truth-functionality assumption, that is the basic building block of logic, can generate paradoxes.

To overcome this difficulty and clarify this situation, we had the idea to translate all these logics in a (epistemic) modal one, that is a logic with two truth-values but with a further unary connective that may account for the meaning of the third truth-value at the syntactic level. At first, we considered the case of 'unknown' and translated the logics that deal with 'unknown' into a minimal epistemic logic. This translation is significant from different points of view:

- It clarifies the semantics of three-valued logics, especially the meaning of the different implications.

- It enables a comparison between three-valued logics and epistemic modal logic showing that even a very small fragment of the latter is more expressive than any three-valued logic.

- It bridges the gap with many-valued and modal logics, showing how to systematically translate the former into the latter.

We are now trying to act similarly with respect to other interpretations of three-valued logics and especially, paraconsistent ones, where the third truth-value is interpreted as 'inconsistent'.

As we said there are several three-valued logics, which have been defined for different purposes and with different meanings. Based on a survey of these logics, we studied all the connectives, which can be reasonably defined on three values, which extend to three values the standard Boolean connectives. We obtained 14 different monotonic conjunctions, 14 disjunctions, 14 implications and 3 negations. Surprisingly, we were able to connect all of them using some well-known transformation and, moreover, to define them all in a basic logic. So, we can say that the richness of three-valued logics is just apparent: they can all be viewed as a fragment of a unique one.

Then, we applied the above outlined results to rough set theory. Rough sets are a set of tools to deal with uncertainty management where the available vocabulary cannot accurately describe reality. A distinction between three situations naturally arises in rough sets: we are sure that something holds or that it does not hold or we are uncertain about it. So, from a formal standpoint, it is possible to use three-valued logics to manage rough sets. We showed that all the connectives we defined on three values are definable also in terms of rough sets (operations on pairs of upper and lower approximations of sets). In the rough set case, however, there is a problem with the semantics of the obtained operations since their interpretation with respect to reasoning about data tables is questionable. Indeed, all the operations on rough sets depend on the available knowledge and cannot be defined a priori based only on the operands. Further, we were able to relate the translation into rough sets with the one into epistemic logic: the former can be seen as an application of the latter.

Rough sets are the only application taken into account at the moment. Among the other possible settings, we mention the NULL-value problem in databases. This is an issue that relates to everyday life, since it concerns all the information systems (bank, registry office, airlines, etc.). It consists in how to deal with unknown (or missing for some reason) information in a collection of data.

On the side of training, which is an important part in this kind of projects, the involved researcher reached all the prescribed objectives: a better historical and philosophical knowledge of many valued logics; more technical skills in formal logic; an increase of knowledge about the formal treatment of uncertainty. Of course, these acquired skills will be helpful in the proceedings of his career both in research and in teaching.

Finally, we have to say that the collabouration among the involved researchers and institutions will continue also after the end of the project, in terms of common publications, projects, organisation of workshops, visiting, etc. contributing to a European dimension of research.

These two lines give rise to many-valued logics and modal logics respectively. The present project dealt with three-valued logics, namely, logics with one more truth-value different from true and false. There are several logics of this kind, where the third truth-value can be interpreted in different ways. Indeed, we can find, at least: half-true, unknown, undefined, irrelevant, possible, inconsistent. As can be seen, the third truth-value is not always an intermediate value between true and false (that is, half-true) but it can assume other meanings, often referring to the knowledge level. Using half-true to represent an epistemic notion (such as lack of knowledge or contradictory knowledge) is open to criticism both from an intuitive point of view and from a formal one, since the truth-functionality assumption, that is the basic building block of logic, can generate paradoxes.

To overcome this difficulty and clarify this situation, we had the idea to translate all these logics in a (epistemic) modal one, that is a logic with two truth-values but with a further unary connective that may account for the meaning of the third truth-value at the syntactic level. At first, we considered the case of 'unknown' and translated the logics that deal with 'unknown' into a minimal epistemic logic. This translation is significant from different points of view:

- It clarifies the semantics of three-valued logics, especially the meaning of the different implications.

- It enables a comparison between three-valued logics and epistemic modal logic showing that even a very small fragment of the latter is more expressive than any three-valued logic.

- It bridges the gap with many-valued and modal logics, showing how to systematically translate the former into the latter.

We are now trying to act similarly with respect to other interpretations of three-valued logics and especially, paraconsistent ones, where the third truth-value is interpreted as 'inconsistent'.

As we said there are several three-valued logics, which have been defined for different purposes and with different meanings. Based on a survey of these logics, we studied all the connectives, which can be reasonably defined on three values, which extend to three values the standard Boolean connectives. We obtained 14 different monotonic conjunctions, 14 disjunctions, 14 implications and 3 negations. Surprisingly, we were able to connect all of them using some well-known transformation and, moreover, to define them all in a basic logic. So, we can say that the richness of three-valued logics is just apparent: they can all be viewed as a fragment of a unique one.

Then, we applied the above outlined results to rough set theory. Rough sets are a set of tools to deal with uncertainty management where the available vocabulary cannot accurately describe reality. A distinction between three situations naturally arises in rough sets: we are sure that something holds or that it does not hold or we are uncertain about it. So, from a formal standpoint, it is possible to use three-valued logics to manage rough sets. We showed that all the connectives we defined on three values are definable also in terms of rough sets (operations on pairs of upper and lower approximations of sets). In the rough set case, however, there is a problem with the semantics of the obtained operations since their interpretation with respect to reasoning about data tables is questionable. Indeed, all the operations on rough sets depend on the available knowledge and cannot be defined a priori based only on the operands. Further, we were able to relate the translation into rough sets with the one into epistemic logic: the former can be seen as an application of the latter.

Rough sets are the only application taken into account at the moment. Among the other possible settings, we mention the NULL-value problem in databases. This is an issue that relates to everyday life, since it concerns all the information systems (bank, registry office, airlines, etc.). It consists in how to deal with unknown (or missing for some reason) information in a collection of data.

On the side of training, which is an important part in this kind of projects, the involved researcher reached all the prescribed objectives: a better historical and philosophical knowledge of many valued logics; more technical skills in formal logic; an increase of knowledge about the formal treatment of uncertainty. Of course, these acquired skills will be helpful in the proceedings of his career both in research and in teaching.

Finally, we have to say that the collabouration among the involved researchers and institutions will continue also after the end of the project, in terms of common publications, projects, organisation of workshops, visiting, etc. contributing to a European dimension of research.