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- When and how did you hear about paraconsistent logic and start your work?
First I wish to thank Jean-Yves for interviewing me for this newsletter. My research is not exactly in paraconsistent logic but it is related to it. The difference is that paraconsistent logics allow for reasoning in the presence of inconsistencies (without proving everything) while my work focuses on a quantitative approach to inconsistencies. So I will interpret the questions in that sense.
First I wish to thank Jean-Yves for interviewing me for this newsletter. My research is not exactly in paraconsistent logic but it is related to it. The difference is that paraconsistent logics allow for reasoning in the presence of inconsistencies (without proving everything) while my work focuses on a quantitative approach to inconsistencies. So I will interpret the questions in that sense.
When I was a graduate student in mathematics in the late 1960s I became interested in model theory, an area of mathematical logic that deals with the interplay of (typically infinite) relational structures with their representation in a logical language, usually first-order logic. A relational structure consists of a domain and a set of relations, each with a fixed arity. So, for example, if R is a binary relation, a and b are in the domain, then either R(a,b) or ~R(a,b) holds, but not both. In the early 1970s I became interested in extending this concept in one or both of two directions. The case where it is possible that for some tuple neither R(a,b) nor ~R(a,b) holds is an incomplete structure; the case where both R(a,b) and ~R(a,b) hold is an inconsistent structure.
When I was a graduate student in mathematics in the late 1960s I became interested in model theory, an area of mathematical logic that deals with the interplay of (typically infinite) relational structures with their representation in a logical language, usually first-order logic. A relational structure consists of a domain and a set of relations, each with a fixed arity. So, for example, if R is a binary relation, a and b are in the domain, then either R(a,b) or ~R(a,b) holds, but not both. In the early 1970s I became interested in extending this concept in one or both of two directions. The case where it is possible that for some tuple neither R(a,b) nor ~R(a,b) holds is an incomplete structure; the case where both R(a,b) and ~R(a,b) hold is an inconsistent structure.
Relational databases were introduced at that time where a database is a relational structure; each table is a relation, and a row in a table is a tuple. The difference is that in a relational database the domain is not some arbitrary set but data about various objects such id value, name, and so on. The data has to be inserted into the tables and then questions can be asked that return some portion of the database as the answer.
Relational databases were introduced at that time where a database is a relational structure; each table is a relation, and a row in a table is a tuple. The difference is that in a relational database the domain is not some arbitrary set but data about various objects such id value, name, and so on. The data has to be inserted into the tables and then questions can be asked that return some portion of the database as the answer.
The issue of incompleteness in databases is called the null value problem and I made some contributions to solving it. A database contains only positive information, so it cannot by itself be inconsistent. But there are usually some rules about the data to which the database must conform. For example, an identification value for some item must be unique. An inconsistency occurs when such a rule is broken. Although some rules can be formulated as part of the definition of the database so that, for example, in the case of unique identification numbers a second row with the same identifier cannot be inserted, I became convinced that inconsistencies would eventually become a problem. I thought that it would be useful to measure the amount of inconsistency in a database. Soon it became clear to me that database researchers were not interested in this issue. So I decided to find a way to measure inconsistency for a set of statements in first-order logic. I came up with several classifications that I was able to publish as a logic paper in 1978.
The issue of incompleteness in databases is called the null value problem and I made some contributions to solving it. A database contains only positive information, so it cannot by itself be inconsistent. But there are usually some rules about the data to which the database must conform. For example, an identification value for some item must be unique. An inconsistency occurs when such a rule is broken. Although some rules can be formulated as part of the definition of the database so that, for example, in the case of unique identification numbers a second row with the same identifier cannot be inserted, I became convinced that inconsistencies would eventually become a problem. I thought that it would be useful to measure the amount of inconsistency in a database. Soon it became clear to me that database researchers were not interested in this issue. So I decided to find a way to measure inconsistency for a set of statements in first-order logic. I came up with several classifications that I was able to publish as a logic paper in 1978.
Born May 31, 1944, Budapest, Hungary.
Born May 31, 1944, Budapest, Hungary.
2. How did you further develop your work on paraconsistent logic ?
2. How did you further develop your work on paraconsistent logic ?
After that paper, I joined a research group where we worked on applying logic to databases but did not deal with inconsistencies. Then in the early 2000s I started seeing some papers about measuring inconsistency in propositional logic. Back in the 1970s I was thinking about relational databases and first-order logic; it did not occur to me to measure inconsistency in propositional logic. It turns out to be quite complicated. I joined the research effort and wrote several papers with Anthony Hunter proposing new measures and related items in the following decade.
After that paper, I joined a research group where we worked on applying logic to databases but did not deal with inconsistencies. Then in the early 2000s I started seeing some papers about measuring inconsistency in propositional logic. Back in the 1970s I was thinking about relational databases and first-order logic; it did not occur to me to measure inconsistency in propositional logic. It turns out to be quite complicated. I joined the research effort and wrote several papers with Anthony Hunter proposing new measures and related items in the following decade.
By 2016 I thought that it would be worthwhile to put together a book on inconsistency measures and worked on it for over a year with Maria Vanina Martinez. After the book was published I started reading about various extensions of propositional logic, such as epistemic, modal, and temporal logics. I realized that there is a commonality about some of these logics that could be used to devise a method to measure inconsistency in all of them. These logics add a pair of unary operators to propositional logic, such as “necessary” and “possible” for modal and “always” and “sometime” for temporal. I used the name “generalized propositional logic” for the class of logics with any number of such unary operator pairs over propositional logic and gave a general method for defining inconsistency measures for them. This paper appeared in Logica Universalis.
By 2016 I thought that it would be worthwhile to put together a book on inconsistency measures and worked on it for over a year with Maria Vanina Martinez. After the book was published I started reading about various extensions of propositional logic, such as epistemic, modal, and temporal logics. I realized that there is a commonality about some of these logics that could be used to devise a method to measure inconsistency in all of them. These logics add a pair of unary operators to propositional logic, such as “necessary” and “possible” for modal and “always” and “sometime” for temporal. I used the name “generalized propositional logic” for the class of logics with any number of such unary operator pairs over propositional logic and gave a general method for defining inconsistency measures for them. This paper appeared in Logica Universalis.
But some logics have other types of operators. Some temporal logics have the binary “Until” operator. So I extended the previous work to the case where any number of pairs of any arity are added to propositional logic. This article has just been published in Logica Universalis.
But some logics have other types of operators. Some temporal logics have the binary “Until” operator. So I extended the previous work to the case where any number of pairs of any arity are added to propositional logic. This article has just been published in Logica Universalis.
3. How do you see the evolution and further challenges for paraconsistent logic ?
3. How do you see the evolution and further challenges for paraconsistent logic ?
Just about all the research on inconsistency measures has been done by computer scientists. I hope that some logicians and mathematicians become interested in this topic. I think they might bring a different perspective to this subject and be able to consider additional issues. In terms of logic, not much has been done to connect inconsistency measures to paraconsistent logics.
Just about all the research on inconsistency measures has been done by computer scientists. I hope that some logicians and mathematicians become interested in this topic. I think they might bring a different perspective to this subject and be able to consider additional issues. In terms of logic, not much has been done to connect inconsistency measures to paraconsistent logics.
Also, considering the close connection between logic and algebra, I wonder if some of that could be useful here. In terms of mathematics, many questions can be asked about inconsistency in mathematical, particularly algebraic structures. An algebra such as a group or lattice consists of a set with certain operations that satisfy some given properties. What would an inconsistent algebra look like? It may depend on how inconsistent it is. For a given k, can we say something about the algebras of that type that have k inconsistencies? It would be interesting to get answers to such questions.
Also, considering the close connection between logic and algebra, I wonder if some of that could be useful here. In terms of mathematics, many questions can be asked about inconsistency in mathematical, particularly algebraic structures. An algebra such as a group or lattice consists of a set with certain operations that satisfy some given properties. What would an inconsistent algebra look like? It may depend on how inconsistent it is. For a given k, can we say something about the algebras of that type that have k inconsistencies? It would be interesting to get answers to such questions.
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