Suppose we have a database or computer program that contains, among other information, the following logically encoded data:
- All North American coyotes are gray.
- Rescued animal is a coyote
- The coyote found is from Washington.
- Washington is a North American state
If our program is able to make logical inferences, the result of the operation of the four pieces of information will be the coyote is gray. However, suppose this does not correspond to reality and the coyote has yellow fur. So we should deny the information that the coyote found is white and add it to the database, but then we create an inconsistency with the rest of the recorded information.
To maintain the reliability of the data, it would be necessary to review previously recorded data and correct some of the beliefs established before the new event. For this, it is not necessary to renounce all the previous premises, since this would mean an unnecessary and important loss of information. So you have to choose something.
The problem of revision of beliefs refers to the fact that logical information alone does not indicate which of them should be discarded, but that other means should be used to elucidate which should be deleted or modified. Also, any choice will have logical consequences when making new inferences, so when dropping a belief you also have to decide which consequences we want to keep and which are obsolete.
What is Belief Revision?
Efforts to create logical agents that act rationally ultimately involve formalizing the process of belief revision. That implies take new information into account and adjust one’s belief state, or epistemic state, accordingly.
There are several ways to alter a belief state when new information is obtained. For example, consider an agent who believes that “Pablo is a priest” and that “priests are not married”. If this agent learns that Pablo is married, the belief state becomes inconsistent and one of the statements must be rejected to resolve the inconsistency.
Belief revision, or belief change, refers to the act of changing or adding new beliefs. When adding a belief in disagreement, the conflicting beliefs should be eliminated or modified so that they remain consistent. This is possible because the beliefs are represented by a set of sentences that can be deleted or introduced, requiring consistency adjustments with other saved sentences.
Types of Changes in Belief Revision
Changing beliefs requires compliance with specific rules of rationality. These rules were tested using representation theorems derived from postulates related to specific operations involving beliefs and their representations.
1. Reviewing and Updating Information.
There are two main types of changes, which are generally divided into more specific changes.
Actualizing beliefs means adjust them to reflect new information. It involves changing old beliefs for new ones, which relate to the current situation.
New information may be inconsistent with old beliefs. Old beliefs are probably less reliable than new ones. In this case, the new information may create an inconsistency. This process of avoiding conflict is called revision, which is when new information is inserted into a set of old beliefs, without incompatibility occurring.
In order to minimize the impact of the belief review process, knowledge must remain the same before and after a revision or update. This principle is known as minimal change, which formalizes the assumption that people are unlikely to change their minds. It also requires that when updating beliefs, as much prior knowledge as possible is preserved.
The next classic example shows that the operations necessary to update and revise beliefs are not the same. It is based on two different perspectives regarding new information:
Two satellites, A and B, are in orbit around Mars with a mission to land. These emit a continuous signal about their location at the base. The earth receives a signal that one of the two satellites is in orbit; however, due to a technical problem, it does not know which satellite transmitted the data. A little later, the signal is received that satellite A has landed on Mars. According to the data update operation, the information of satellite A is adjusted and the position of B (which is not updated) is unknown. A does not interfere with B, because we don’t know if it has landed or if it is still in orbit.
In the review example, let’s say a city has two theaters, and work A will be performed in one of them. This is the first information we can record. Then, we know that on the same dates a B play will be performed in the first theatre. In this case, we can deduce that work A will be performed in the theater which remains free. There is a revision operation, when new information is introduced.
2. Other operations
In a certain situation where all the beliefs involved refer to the same scenarioadditional operations can be performed, including:
- Contraction: elimination of a belief.
- Expansion: adding a belief without checking consistency
- Extraction: elimination of a coherent set of beliefs
- Consolidation: restoring the coherence of a whole.
- Fusion: combining two or more belief systems into one, maintaining consistency. However, this differs from review in that it does not prioritize new data.
Development of research on the revision of beliefs
Since the mid-1980s, the revision of beliefs has been considered a subject in its own right. This term was created from two related areas of research: computer science and philosophy.
Since the beginning of computing, programmers have created different procedures to be able to update databases. With the development of Artificial Intelligence, more and more complex database update models have been proposed. In 1979, truth-keeping systems were born; able to revise their beliefs to maintain consistency with new information that contradicts existing ones. In 1983 he made one of the most important theoretical contributions to the revision of beliefs, the notion of priority has been introduced. Prioritization uses predefined rules to determine which data source takes precedence over other data sources at update time.
Since Antiquity, the change of beliefs has been the subject of philosophical reflection. Already in the 20th century there were more specific debates focused on changing beliefs. These discussions involve judging whether the changes in probability were rational and determining the mechanisms used for the development of scientific theories. Later, in the 1970s, debates centered on the requirements of changes in rational beliefs took place. The philosopher Issac Levi addressed many concerns in the field of linguistic research and belief revision in the 1970s.
In 1985, Carlos Alchourrón, Peter Gärdenfors, and David Makinson wrote an article that provided a formal framework for studies of belief change. Since, the AGM model was used to organize new knowledge related to belief changesthis being the theoretical core.
The AGM model
The AGM model is the revision theory most commonly accepted beliefs. He establishes that beliefs are represented by a logically closed set of sentences. It should be noted that both in the AGM model and in most belief revision models, these are represented by sentences written in a formal language. The sentences are not able to capture all the nuances and aspects of the belief, but they are currently considered the best representation of a general character.
The set of creencias is capable of expansion, contraction and revision. Expansion occurs when a new proposition is added to the set, which then logically closes. In the contraction, one of the propositions of the set is eliminated, logically closing. Finally, revision refers to the same thing as expansion to add a new proposition to the set, but it should remain consistent.
The expansion operation is simple; reduction and revision operations can be multiple. The authors only discuss multiple possible operations which must obey certain postulates of rationality. Contraction operations require:
One of the most discussed topics in belief revision theory is the retrieval postulate. This states that if a belief is deleted and then reinserted, all original beliefs will be recovered. However, this this does not apply to belief bases, which are similar models that use logically closed sets of beliefs. There has been much discussion about how the AGM model represents changes in beliefs. Several alternative models have been proposed that claim to provide a more realistic explanation of belief shifts than the AGM model.
- Alchourron, EC; Gardenfors, P.; Makinson D. (1985). On the logic of theory change: contraction and revision functions of partial reunion. Journal of Symbolic Logic, 50: 510–530.
- Antoniou, G. and MA. Williams (1997) Non-Montonic Reasoning, MIT Press.
- Gardenfors, Peter. (1992). Revision of beliefs: an introduction. 10.1017/CBO9780511526664.001.