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> Consistency is the hobgoblin of small minds.
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> Inconsistency can be inconsistent for it's own reasons.
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Paraconsistent Logic
The development of paraconsistent logic was initiated in order to challenge the logical principle that anything follows from contradictory premises, ex contradictione quodlibet (ECQ). Let be a relation of logical consequence, defined either semantically or proof-theoretically. Let us say that is explosive iff for every formula A and B, {A , ~A} B. Classical logic, intuitionistic logic, and most other standard logics are explosive. A logic is said to be paraconsistent iff its relation of logical consequence is not explosive. The modern history of paraconsistent logic is relatively short. Yet the subject has already been shown to be an important development in logic for many reasons. These involve the motivations for the subject, its philosophical implications and its applications. In the first half of this article, we will review some of these. In the second, we will give some idea of the basic technical constructions involved in paraconsistent logics. Further discussion can be found in the references given at the end of the article.
Inconsistent but Non-Trivial Theories A most telling reason for paraconsistent logic is the fact that there are theories which are inconsistent but non-trivial. Clearly, once we admit the existence of such theories, their underlying logics must be paraconsistent. Examples of inconsistent but non-trivial theories are easy to produce. An example can be derived from the history of science. (In fact, many examples can be given from this area.) Consider Bohr's theory of the atom. According to this, an electron orbits the nucleus of the atom without radiating energy. However, according to Maxwell's equations, which formed an integral part of the theory, an electron which is accelerating in orbit must radiate energy. Hence Bohr's account of the behaviour of the atom was inconsistent. Yet, patently, not everything concerning the behavior of electrons was inferred from it. Hence, whatever inference mechanism it was that underlay it, this must have been paraconsistent.
Dialetheias (True Contradictions) The importance of paraconsistent logic also follows if, more contentiously, but as some people have argued, there are true contradictions (dialetheias), i.e., there are sentences, A, such that both A and ~A are true. If there are dialetheias then some inferences of the form {A , ~A} B must fail. For only true conclusions follow validly from the true premises. Hence logic has to be paraconsistent. A plausible example of dialetheia is the liar paradox. Consider the sentence: This sentence is not true. There are two options: either the sentence is true or it is not. Suppose it is true. Then what it says is the case. Hence the sentence is not true. Suppose, on the other hand, it is not true. This is what it says. Hence the sentence is true. In either case it is both true and not true.
Automated Reasoning Paraconsistent logic is motivated not only by philosophical considerations, but also by its applications and implications. One of the applications is automated reasoning (information processing). Consider a computer which stores a large amount of information. While the computer stores the information, it is also used to operate on it, and, crucially, to infer from it. Now it is quite common for the computer to contain inconsistent information, because of mistakes by the data entry operators or because of multiple sourcing. This is certainly a problem for database operations with theorem-provers, and so has drawn much attention from computer scientists. Techniques for removing inconsistent information have been investigated. Yet all have limited applicability, and, in any case, are not guaranteed to produce consistency. (There is no algorithm for logical falsehood.) Hence, even if steps are taken to get rid of contradictions when they are found, an underlying paraconsistent logic is desirable if hidden contradictions are not to generate spurious answers to queries.
Belief Revision As a part of artificial intelligence research, belief revision is one of the areas that have been studied widely. Belief revision is the study of rationally revising bodies of belief in the light of new evidence. Notoriously, people have inconsistent beliefs. They may even be rational in doing so. For example, there may be apparently overwhelming evidence for both something and its negation. There may even be cases where it is in principle impossible to eliminate such inconsistency. For example, consider the "paradox of the preface". A rational person, after thorough research, writes a book in which they claim A1, ... , An. But they are also aware that no book of any complexity contains only truths. So they rationally believe ~(A1 & ... & An) too. Hence, principles of rational belief revision must work on inconsistent sets of beliefs. Standard accounts of belief revision, e.g., that of Gärdenfors et al., all fail to do this since they are based on classical logic. A more adequate account is based on a paraconsistent logic.
More at: < http://plato.stanford.edu/entries/logic-paraconsistent/ >
Ian