Logic is the process of drawing a conclusion from one or more premises

As an example of how logic can be abused, consider the following argument which has been widely propagated on the Internet

One way of testing the logic of an argument like this is to translate the basic terms and see if the conclusion still makes sense

This is a rather extreme example of how logic can be abused

It should be noted that a message can be illogical without being propagandistic -- we all make logical mistakes

Stupid generalization: all humans are scientists!

The tendency to make huge predictions about the future on the basis of a few small facts is a common logical fallacy

Extrapolation is what scientists call such predictions, with the warning that they must be used with caution

This logical sleight of hand often provides the basis for an effective fear-appeal

When a communicator attempts to convince you that a particular action will lead to disaster or to utopia, it may be helpful to ask the following questions:

Like any rich language, many words can be present in a language that might mean the same thing except in matters of context, size, and history etc

Good words: to be used to support a cause

Bad words: to be used to OPPOSE a cause

                The whole of 20th century was devoted to classical logics built on the notion of sets

This particularly interesting system of MVL was the result of research on

The standard approaches to these problems of inconsistency are, by and large, ones of expedience

Paraconsistent logic also has important philosophical ramifications

The foregoing discussion indicates some of the motivations for paraconsistent logic, its applications and implications

Relevance logics are non-classical logics

Among the paradoxes of strict implication are the following:

Relevance logicians claim that what is unsettling about these so-called paradoxes is that in each of them the antecedent seems irrelevant to the consequent

In addition, relevance logicians have had qualms about certain inferences that classical logic makes valid

Again here there seems to be a failure of relevance

At this point some confusion is natural about what relevant logicians have attempted to do

In this article we will give a brief and relatively non-technical overview of the field of relevant logic

Our exposition of relevant logic is backwards to most found in the literature We will begin, rather than end, with the semantics, since most philosophers at present are semantically inclined

The semantics that I present here is the ternary relation semantics due to Richard Routley and Robert K

The idea behind the ternary relation semantics is rather simple

Like the semantics of modal logic, the semantics of relevance logic relativises truth of formulae to worlds

For new customers, it takes some time to get used to this truth condition

But how should this accessibility relation be interpreted philosophically? Oddly, there has not been very much work on this, but there has been some

Another similar interpretation is given in Barwise (1993) and developed in Restall (1996)

By itself, the ternary relation is not sufficient to avoid all the paradoxes of implication

This brings us to the semantics for negation

Once again, we have the difficulty of interpreting a part of the formal semantics

There are other semantics for negation

There is now a lage variety of approaches to proof theory for relevant logics

Anderson and Belnap�s natural deduction system is based on Fitch�s natural deduction systems for classical and intuitionistic logic

1

hyp

2

hyp

3

1,2,

5

1,4,

2

hyp

3

1,2, &I

4

3, &E

5

2,4,

To a relevance logician, the first premise is completely out of place here

This rule says that two formulae must have the same index before the rule of conjunction introduction can be used

There is, of course, a lot more to the natural deduction system, but this will suffice as for our current purposes

Historically, the central systems of relevance logic have been the logic E of entailment and the system R or relevant implication

On the other hand, there are those relevance logicians who reject both R and E

On an extreme end of the spectrum is the logic S of R

Apart from the motivating applications of providing better formalisms of our pre-formal notions of implication and entailment, relevance logic has been put to various uses in philosophy and computer science

Dunn has developed a theory of intrinsic and essential properties based on relevant logic

Meyer has produced a variant of Peano arithmetic based on the relevance logic, R

In a similar vein, Ross Brady and others have used weak relevant logics as bases for set theories

Anderson (1967) formulates a system of deontic logic based on R

Mares and Fuhrmann (1995) present a theory of counterfactual conditionals based on relevant logic

Relevant logics have been used in computer science as well as in philosophy

The term Temporal Logic has been broadly used to cover all approaches to the representation of temporal information within a logical framework, and also more narrowly to refer specifically to the modal-logic type of approach introduced around 1960 by Arthur Prior under the name of Tense Logic and subsequently developed further by logicians and computer scientists

Applications of Temporal Logic include its use as a formalism for clarifying philosophical issues about time, as a framework within which to define the semantics of temporal expressions in natural language, as a language for encoding temporal knowledge in artificial intelligence, and as a tool for handling the temporal aspects of the execution of computer programs

Tense Logic was introduced by Arthur Prior (1957, 1967, 1969) as a result of an interest in the relationship between tense and modality attributed to the Megarian philosopher Diodorus Cronus (ca

The logical language of Tense Logic contains, in addition to the usual truth-functional operators, four modal operators with intended meanings as follows:

"It has at some time been the case that

F

"It will at some time be the case that

H

"It has always been the case that

G

"It will always be the case that

P and F are known as the weak tense operators , while H and G are known as the strong tense operators

Pp

Fp

On the basis of these intended meanings, Prior used the operators to build formulae expressing various philosophical theses about time, which might be taken as axioms of a formal system if so desired

Gp

Prior (1967) reports on the extensive early work on various systems of Tense Logic obtained by postulating different combination of axioms, and in particular he considered in some detail what light a logical treatment of time can throw on classic problems concerning time, necessity and existence; for example, "deterministic" arguments that have been advanced over the ages to the effect that "what will be, will necessarily be", corresponding to the modal tense-logical formula Fp

"What is, has always been going to be"

and, of course, all the rules of ordinary Propositional Logic

Tense Logic is obtained by adding the tense operators to an existing logic; above this was tacitly assumed to be the classical Propositional Calculus

The interpretation of such formulae is not unproblematic, however

These problems are related to the so-called Barcan formulae of modal logic, a temporal analogue of which is

Fp

The importance of the S and U operators is that they are expressively complete with respect to first-order temporal properties on continuous, strictly linear temporal orders (which is not true for the one-place operators on their own)

Metric tense logic

Pp

Fp

Hp

Gp

The "next time" operator O

which says that p will be true at some future time, between which and the present time nothing is true

In discrete time, the future-tense operator F is related to the next-time operator by the equivalence

p is true at all times t

Gp

(unbounded in the future)

Fp

(dense ordering)

FFp

(transitive ordering)

FPp

(linear in the past)

PFp

(linear in the future)

The homogeneity of states and inhomogeneity of events is secured by axioms such as

where "In" expresses the proper subinterval relation

The method of event-token reification was proposed by Donald Davidson (1967) as a solution to the so-called "variable polyadicity" problem

John saw Mary in London on Tuesday

Therefore, John saw Mary on Tuesday

The key idea is that each event-forming predicate is endowed with an extra argument-place to be filled with a variable ranging over event-tokens, that is, particular dated occurrences

In this form, the inference does not require any additional logical apparatus over and above standard first-order predicate logic; on that basis, the validity of the inference is considered to be explained

Prior�s motivation for inventing Tense Logic was largely philosophical, his idea being that the precision and clarity afforded by a formal logical notation was indispensible for the careful formulation and resolution of philosophical issues concerning time

The rivalry between the modal and first-order approaches to formalising the logic of time reflects an important set of underlying philosophical issues related to the work of McTaggart

There is a clear affinity between the A-series and the modal approach and between the B-series and the first-order approach

Prior (1967) lists amongst the precursors of Tense Logic Hans Reichenbach�s (1947) analysis of the tenses of English, according to which the function of each tense is to specify the temporal relationships amongst a set of three times related to the utterance, namely S, the speech time, R, the reference time, and E, the event time

Prior notes that Reichenbach�s analysis is inadequate to account for the full range of tense usage in natural language

We have already mentioned the work of Allen (1984), which is concerned with finding a general framework adequate for all the temporal representations required by AI programs

Much of the work on temporal reasoning in AI has been closely tied up with the notorious frame problem , which arises from the necessity for any automated reasoner to know, or be able to deduce, not only those properties of the world which do change as the result of any event or action, but also those properties which do not change  

  Long but essential for those I care about. Don't even try to understand it in one or 10 sittings. Do not ignore either.

What is it about? Use and abuse of language for propaganda, and techniques for destroying propaganda.

What are applications? Lives in general, as all humans communicate and think using a language. In particular buying, law, politics, advertisement, teaching, dealing with patients, users, friends, family, lawyers etc.

Is there a unifying theme to building propaganda? Yes. Always pretend you don't know, and subtly ensure the target T believes T came to the conclusion itself.

Is there a unifying theme of prevention? Yes. Always give a fair chance to competitors to say their side with opportunity for rejoinders, restricting participants and judges to their stake, not allowing debate domination by any one including the judges, and always respecting the verdict of a jury.

Is there a stupid theme of prevention? Yes. Belief in any one's, or any organized theory of truth.

Why worry about this? Next step to my technology, and the VAMPIRE unfairness visited on me.

Why should Nina worry about this? Build your life on top of mine. This effort will be useful to you in many ways.

This sounds like English teacher to me? Probably is. It is amazing to me how little good stuff there is in HSS though. Get it and then do useful things!

Acknowledgements?

Stanford, and whom they choose to. In The Fine Art of Propaganda , the IPA stated that "It is essential in a democratic society that young people and adults learn how to think, learn how to make up their minds. They must learn how to think independently, and they must learn how to think together. They must come to conclusions, but at the same time they must recognize the right of other men to come to opposite conclusions. So far as individuals are concerned, the art of democracy is the art of thinking and discussing independently together. "

Abusive Word Games

"Bad names have played a tremendously powerful role in the history of the world and in our own individual development. They have ruined reputations, stirred men and women to outstanding accomplishments, sent others to prison cells, and made men mad enough to enter battle and slaughter their fellowmen. They have been and are applied to other people, groups, gangs, tribes, colleges, political parties, neighborhoods, states, sections of the country, nations, and races." (Institute for Propaganda Analysis, 1938)

The name-calling technique links a person, or idea, to a negative symbol. The propagandist who uses this technique hopes that the audience will reject the person or the idea on the basis of the negative symbol, instead of looking at the available evidence.

The most obvious type of name-calling involves "bad names." For example, consider the following:

 &evel1 lfnbsp;       Commie

         Fascist

         Pig

         Yuppie Scum

         Bum

         Queer

         Feminazi

         social engineering

         radical

         stingy

         counter-culture

  Abuse of glittering generalities

"We believe in, fight for, live by virtue words about which we have deep-set ideas. Such words include civilization, Christianity, good, proper, right, democracy, patriotism, motherhood, fatherhood, science, medicine, health, and love.

For our purposes in propaganda analysis, we call these virtue words "Glittering Generalities" in order to focus attention upon this dangerous characteristic that they have: They mean different things to different people; they can be used in different ways.

This is not a criticism of these words as we understand them. Quite the contrary. It is a criticism of the uses to which propagandists put the cherished words and beliefs of unsuspecting people.

When someone talks to us about democracy, we immediately think of our own definite ideas about democracy, the ideas we learned at home, at school, and in church. Our first and natural reaction is to assume that the speaker is using the word in our sense, that he believes as we do on this important subject. This lowers our 'sales resistance' and makes us far less suspicious than we ought to be when the speaker begins telling us the things 'the United States must do to preserve democracy.'

The Glittering Generality is, in short, Name Calling in reverse. While Name Calling seeks to make us form a judgment to reject and condemn without examining the evidence, the Glittering Generality device seeks to make us approve and accept without examining the evidence. In acquainting ourselves with the Glittering Generality Device, therefore, all that has been said regarding Name Calling must be kept in mind..." (Institute for Propaganda Analysis, 1938)

The Institute for Propaganda Analysis suggested a number of questions that people should ask themselves when confronted with this technique:

Abuse of euphemisms

When propagandists use glittering generalities and name-calling symbols, they are attempting to arouse their audience with vivid, emotionally suggestive words. In certain situations, however, the propagandist attempts to pacify the audience in order to make an unpleasant reality more palatable. This is accomplished by using words that are bland and euphemistic.

Since war is particularly unpleasant, military discourse is full of euphemisms. In the 1940's, America changed the name of the War Department to the Department of Defense. Under the Reagan Administration, the MX-Missile was renamed "The Peacekeeper." During war-time, civilian casualties are referred to as "collateral damage," and the word "liquidation" is used as a synonym for "murder."

The comedian George Carlin notes that, in the wake of the first world war, traumatized veterans were said to be suffering from "shell shock." The short, vivid phrase conveys the horrors of battle -- one can practically hear the shells exploding overhead. After the second world war, people began to use the term "combat fatigue" to characterize the same condition. The phrase is a bit more pleasant, but it still acknowledges combat as the source of discomfort.. In the wake of the Vietnam War, people referred to "post-traumatic stress disorder": a phrase that is completely disconnected from the reality of war altogether.

Abuse of transfer

You shall not press down upon the brow of labor this crown of thorn. You shall not crucify mankind upon a cross of gold! -- William Jennings Bryan, 1896

"Transfer is a device by which the propagandist carries over the authority, sanction, and prestige of something we respect and revere to something he would have us accept. For example, most of us respect and revere our church and our nation. If the propagandist succeeds in getting church or nation to approve a campaign in behalf of some program, he thereby transfers its authority, sanction, and prestige to that program. Thus, we may accept something which otherwise we might reject.

In the Transfer device, symbols are constantly used. The cross represents the Christian Church. The flag represents the nation. Cartoons like Uncle Sam represent a consensus of public opinion. Those symbols stir emotions . At their very sight, with the speed of light, is aroused the whole complex of feelings we have with respect to church or nation. A cartoonist, by having Uncle Sam disapprove a budget for unemployment relief, would have us feel that the whole United States disapproves relief costs. By drawing an Uncle Sam who approves the same budget, the cartoonist would have us feel that the American people approve it. Thus, the Transfer device is used both for and against causes and ideas." (Institute for Propaganda Analysis, 1938)

When a political activist closes her speech with a public prayer, she is attempting to transfer religious prestige to the ideas that she is advocating. As with all propaganda devices, the use of this technique is not limited to one side of the political spectrum. It can be found in the speeches of liberation theologists on the left, and in the sermons of religious activists on the right.

In a similar fashion, propagandists may attempt to transfer the reputation of "Science" or "Medicine" to a particular project or set of beliefs. A slogan for a popular cough drop encourages audiences to "Visit the halls of medicine." On TV commercials, actors in white lab coats tell us that the "Brand X is the most important pain reliever that can be bought without a prescription." In both of these examples, the transfer technique is at work.

These techniques can also take a more ominous turn. As Alfred Lee has argued, "even the most flagrantly anti-sicentific racists are wont to dress up their arguments at times with terms and carefully selected illustrations drawn from scientific works and presented out of all accurate context." The propaganda of Nazi Germany, for example, rationalized racist policies by appealing to both science and religion.

This does not mean that religion and science have no place in discussions about social issues! The point is that an idea or program should not be accepted or rejected simply because it has been linked to a symbol such as Medicine, Science, Democracy, or Christianity. The Institute for Propaganda Analysis has argued that, when confronted with the transfer device, we should ask ourselves the following questions:

Abuse of testimonials

Bruce Jenner is on the cereal box, promoting Wheaties as part of a balanced breakfast. Cher is endorsing a new line of cosmetics, and La Toya Jackson says that the Psychic Friends Network changed her life. The lead singer of R.E.M appears on a public service announcement and encourages fans to support the "Motor Voter Bill."

"This is the classic misuse of the Testimonial Device that comes to the minds of most of us when we hear the term. We recall it indulgently and tell ourselves how much more sophisticated we are than our grandparents or even our parents.

With our next breath, we begin a sentence, 'The Times said,' 'John L. Lewis said...,' 'Herbert Hoover said...', 'The President said...', 'My doctor said...,' 'Our minister said...' Some of these Testimonials may merely give greater emphasis to a legitimate and accurate idea, a fair use of the device; others, however, may represent the sugar-coating of a distortion, a falsehood, a misunderstood notion, an anti-social suggestion..." (Institute for Propaganda Analysis, 1938)

There is nothing wrong with citing a qualified source, and the testimonial technique can be used to construct a fair, well-balanced argument. However, it is often used in ways that are unfair and misleading.

The most common misuse of the testimonial involves citing individuals who are not qualified to make judgements about a particular issue. In 1992, Barbara Streisand supported Bill Clinton, and Arnold Schwarzenegger threw his weight behind George Bush. Both are popular performers, but there is no reason to think that they know what is best for this country.

Unfair testimonials are usually obvious, and most of us have probably seen through this rhetorical trick at some time or another. However, this probably happened when the testimonial was provided by a celebrity that we did not respect. When the testimony is provided by an admired celebrity, we are much less likely to be critical.

According to the Institute for Propaganda Analysis , we should ask ourselves the following questions when we encounter this device.

You may have noticed the presence of the testimonial technique in the previous paragraph, which began by citing the Insitute for Propaganda Analysis. In this case, the technique is justified. Or is it?

Abuse of common sense

By using the plain-folks technique, speakers attempt to convince their audience that they, and their ideas, are "of the people." The device is used by advertisers and politicans alike.

America's recent presidents have all been millionaires, but they have gone to great lengths to present themselves as ordinary citizens. Bill Clinton eats at McDonald's and reads trashy spy novels. George Bush hated broccoli, and he loved to fish. Ronald Reagan was often photographed chopping wood, and Jimmy Carter presented himself as a humble peanut farmer from Georgia.

We are all familiar with candidates who campaign as political outsiders, promising to "clean out the barn" and set things straight in Washington. The political landscape is dotted with politicians who challenge a mythical "cultural elite," presumably aligning themselves with "ordinary Americans." As baby boomers enter their fifth decade, we are starting to see politicans in blue jeans who listen to rock and roll.

During the 1980s, Bartels and James appeared on television in comfortable, farm-style clothing, and, with a folksy drawl, thanked consumers for their continued support. The irony was that these two "regular guys" who pushed wine coolers were actually multi-millionaires -- hardly like you or me. In all of these examples, the plain-folks device is at work.

The Institute for Propaganda Analysis has argued that, when confronted with this device, we should suspend judgement and ask ourselves the following questions:

Abuse of bandwagon effect

"The propagandist hires a hall, rents radio stations, fills a great stadium, marches a million or at least a lot of men in a parade. He employs symbols, colors, music, movement, all the dramatic arts. He gets us to write letters, to send telegrams, to contribute to his cause. He appeals to the desire, common to most of us, to follow the crowd. Because he wants us to follow the crowd in masses, he directs his appeal to groups held together already by common ties, ties of nationality, religion, race, sex, vocation. Thus propagandists campaigning for or against a program will appeal to us as Catholics, Protestants, or Jews...as farmers or as school teachers; as housewives or as miners.

With the aid of all the other propaganda devices, all of the artifices of flattery are used to harness the fears and hatreds, prejudices and biases, convictions and ideals common to a group. Thus is emotion made to push and pull us as members of a group onto a Band Wagon." (Institute for Propaganda Analysis, 1938)

The basic theme of the Band Wagon appeal is that "everyone else is doing it, and so should you." Since few of us want to be left behind, this technique can be quite successful. However, as the IPA points out, "there is never quite as much of a rush to climb onto the Band Wagon as the propagandist tries to make us think there is." When confronted with this technique, it may be helpful to ask ourselves the following questions:

Abuse of fear

"The streets of our country are in turmoil. The universities are filled with students rebelling and rioting. Communists are seeking to destroy our country. Russia is threatening us with her might, and the Republic is in danger. Yes - danger from within and without. We need law and order! Without it our nation cannot survive." - Adolf Hitler, 1932

When a propagandist warns members of her audience that disaster will ensue if they do not follow a particular course of action, she is using the fear appeal. By playing on the audience's deep-seated fears, practitioners of this technique hope to redirect attention away from the merits of a particular proposal and toward steps that can be taken to reduce the fear.

This technique can be highly effective when wielded by a fascist demagogue, but it is usually used in less dramatic ways. Consider the following:

Ever since the end of the second world war, social psychologists and communication scholars have been conducting empirical studies in order to learn more about the effectiveness of fear appeals. Some have criticized the conceptualization of the studies, and others have found fault with the experimental methods, but the general conclusions are worth considering, if not accepting.

In summary, there are four elements to a successful fear appeal: 1) a threat, 2) a specific recommendation about how the audience should behave, 3) audience perception that the recommendation will be effective in addressing the threat, and 4) audience perception that they are capable of performing the recommended behavior.

When fear appeals do not include all four elements, they are likely to fail. Pratkanis and Aronson provide the example of the anti-nuclear movement, which successfully aroused public fear of nuclear war, but offered few specific recommendations that people perceived as effective or doable. By contrast, fall-out shelters were enormously popular during the 1950s because people believed that shelters would protect them from nuclear war, and installing a shelter was something that they could do.

In a similar fashion, during the 1964 campaign, Lyndon Johnson was said to have swayed many voters with a well-known television commercial that portrayed a young girl being annihilated in a nuclear blast. This commercial linked nuclear war to Barry Goldwater (Johnson's opponent), and proposed a vote for Johnson as an effective, doable way of avoiding the threat.

In contemporary politics, the fear-appeal continues to be widespread. When a politician agitates the public's fear of immigration, or crime, and proposes that voting for her will reduce the threat, she is using this technique. When confronted with persuasive messages that capitalize on our fear, we should ask ourselves the following questions:

Stupid logic: all humans are logicians!

Logic is the process of drawing a conclusion from one or more premises. A statement of fact, by itself, is neither logical or illogical (although it can be true or false).

As an example of how logic can be abused, consider the following argument which has been widely propagated on the Internet.

One way of testing the logic of an argument like this is to translate the basic terms and see if the conclusion still makes sense. As you can see, the premises may be correct, but the conclusion does not necessarily follow.

This is a rather extreme example of how logic can be abused. The following pages describe others.

It should be noted that a message can be illogical without being propagandistic -- we all make logical mistakes. The difference is that propagandists deliberately manipulate logic in order to promote their cause.

Stupid generalization: all humans are scientists!

The tendency to make huge predictions about the future on the basis of a few small facts is a common logical fallacy. As Stuart Chase points out, "it is easy to see the persuasiveness in this type of argument. By pushing one's case to the limit... one forces the opposition into a weaker position. The whole future is lined up against him. Driven to the defensive, he finds it hard to disprove something which has not yet happened.

Extrapolation is what scientists call such predictions, with the warning that they must be used with caution. A homely illustration is the driver who found three gas stations per mile along a stretch of the Montreal highway in Vermont, and concluded that there must be plenty of gas all the way to the North Pole. You chart two or three points, draw a curve through them, and extend it indefinitely." (Chase, 1952)

This logical sleight of hand often provides the basis for an effective fear-appeal. Consider the following contemporary examples:

Like any rich language, many words can be present in a language that might mean the same thing except in matters of context, size, and history etc. Situation is much worse in that one can usually use words from a similar context. Situation gets explosive when the other context is unfamiliar but rigorous like sciences in general, and mathematics and computer science in particular. Nothing more stupid than agreeing to a word, phrase, or event set, without realizing the import of that agreement. That is what lawyers are experientially good at. What they are not is in restricting their questions to be specific enough to allow answering, or demands to be met.

Good words: to be used to support a cause.

Bad words: to be used to OPPOSE a cause.

Logics

                The whole of 20th century was devoted to classical logics built on the notion of sets. Computers and AI have caused the introduction of more logics which are subsets of and richer than classical logic. Only some are introduced to you.

This particularly interesting system of MVL was the result of research on relevance logic , but it also has significance for computer science applications. Its truth degree set may be taken as

and the truth degrees interpreted as indicating (e.g. with respect to a database query for some particular state of affairs) that there is

This set of truth degrees has two natural (lattice) orderings:

Given the inf and the sup under the truth ordering, there are truth degree functions for a conjunction and a disjunction connective. A negation is, in a natural way, determined by a truth degree function which exchanges the degrees { } and { }, and which leaves the degrees { , } and � fixed.

Actually, there is no standard candidate for a implication connective, and the choice of the designated truth degrees depends on the intended applications:

The choice of suitable entailment relations is still an open research topic.

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.

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.

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.

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.

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 A ₁ , ... , A _n . But they are also aware that no book of any complexity contains only truths. So they rationally believe ~( A ₁ & ... & A _n ) too. Hence, principles of rational belief revision must work on inconsistent sets of beliefs. Standard accounts of belief revision, e.g., that of G�rdenfors et al. , all fail to do this since they are based on classical logic. A more adequate account is based on a paraconsistent logic.

Other applications of paraconsistent logic concern theories of mathematical significance. Examples of such theories are formal semantics and set theory .

Semantics is the study that aims to spell out a theoretical understanding of meaning. Most accounts of semantics insist that to spell out the meaning of a sentence is, in some sense, to spell out its truth-conditions. Now, prima facie at least, truth is a predicate characterised by the Tarski T-scheme:

where A is a sentence and A is its name. But given any standard means of self-reference, e.g., arithmetisation, one can construct a sentence, B , which means that ~T( B ) . The T-scheme gives that T( B ) ~T( B ) . It then follows that T( B ) & ~T( B ) . (This is, of course, just the liar paradox.)

The situation is similar in set theory. The naive, and intuitively correct, axioms of set theory are the Comprehension Schema and Extensionality Principle :

where x does not occur free in A . As was discovered by Russell, any theory that contains the Comprehension Schema is inconsistent. For putting y y for A in the Comprehension Schema and instantiating the existential quantifier to an arbitrary such object r gives:

So, instantiating the universal quantifier to r gives:

It then follows that r r & r r .

The standard approaches to these problems of inconsistency are, by and large, ones of expedience. However, a paraconsistent approach makes it possible to have theories of truth and sethood in which the fundamental intuitions about these notions are respected. The contradictions may be allowed to arise, but these need not infect the rest of the theory.

Paraconsistent logic also has important philosophical ramifications. One example of this concerns G�del�s theorem. One version of G�del�s first incompleteness theorem states that for any consistent axiomatic theory of arithmetic, which can be recognised to be sound, there will be an arithmetic truth - viz., its G�del sentence - not provable in it, but which can be established as true by intuitively correct reasoning. The heart of G�del�s theorem is, in fact, a paradox that concerns the sentence, G , �This sentence is not provable�. If G is provable, then it is true and so not provable. Thus G is proved. Hence G is true and so unprovable. If an underlying paraconsistent logic is used to formalise the arithmetic, and the theory therefore allowed to be inconsistent, the G�del sentence may well be provable in the theory (essentially by the above reasoning). So a paraconsistent approach to arithmetic overcomes the limitations of arithmetic that are supposed (by many) to follow from G�del�s theorem.

The foregoing discussion indicates some of the motivations for paraconsistent logic, its applications and implications. We will now indicate some of the main approaches to paraconsistency. There are many different paraconsistent logics. Most of them can be defined in terms of a semantics which allows both A and ~A to hold in an interpretation. Validity is then defined in terms of the preservation of holding in an interpretation, and so ECQ fails. We will illustrate this with four kinds of propositional paraconsistent logics: non-adjunctive , non-truth-functional , many-valued , and relevant . (Paraconsistent quantified logics are straightforward extensions of these.) In all the following systems, not only ECQ fails, but so does the Disjunctive Syllogism (DS), defined as the following inference rule: { A , ~A B } B . In particular, then, if one defines the material conditional, A B , as ~A B (as usual) then modus ponens for this fails.

Relevance logics are non-classical logics. Called �relevance logics� in North America and �relevant logics� in Britain and Australasia, these systems developed as attempts to avoid the paradoxes of material and strict implication. Among the paradoxes of material implication are

Among the paradoxes of strict implication are the following:

Relevance logicians claim that what is unsettling about these so-called paradoxes is that in each of them the antecedent seems irrelevant to the consequent.

In addition, relevance logicians have had qualms about certain inferences that classical logic makes valid. For example, the inference

Again here there seems to be a failure of relevance. The conclusion seems to have nothing to do with the premise. Relevance logicians have attempted to construct logics that reject theses and arguments that commit "fallacies of relevance".

At this point some confusion is natural about what relevant logicians have attempted to do. They have not given formal criteria of relevance that any true implication must meet, altough some relevant logicans have interpreted the semantics for relevance logic using informal notions of relevance (see the section "Semantics" below). Instead, relevant logic is relevant in two ways: (1) Relevance logics do not force us to accept any irrelevances. That is, they do not make valid any of the paradoxes. (2) Some relevance logics, through their proof theory, yield a relevant notion of proof (see the section "Proof Theory" below).

In this article we will give a brief and relatively non-technical overview of the field of relevant logic.

Our exposition of relevant logic is backwards to most found in the literature We will begin, rather than end, with the semantics, since most philosophers at present are semantically inclined.

The semantics that I present here is the ternary relation semantics due to Richard Routley and Robert K. Meyer. There are algebraic semantics due to J. Michael Dunn and Alasdair Urquhart and operational semantics produced by Kit Fine. These systems are interesting in their own right, but we do not have room to discuss them here.

The idea behind the ternary relation semantics is rather simple. Consider C.I. Lewis� attempt to avoid the paradoxes of material implication. He added a new connective to classical logic, that of strict implication. In post-Kripkean semantic terms, A B is true at a world w if and only if for all w such that w is accessible to w , either A fails in w or B obtains there. Now, in Kripke�s semantics for modal logic, the accessibility relation is a binary relation. It holds between pairs of worlds. Unfortunately, from a relevant point of view, the theory of strict implication is still irrelevant. That is, we still make valid formulae like p ( q q ). We can see quite easily that the Kripke truth condition forces this formula on us.

Like the semantics of modal logic, the semantics of relevance logic relativises truth of formulae to worlds. But Routley and Meyer go modal logic one better and use a three-place relation on worlds. This allows there to be worlds at which q q fails and that in turn allows worlds at which p ( q q ) fails. Their truth condition for on this semantics is the following:

For new customers, it takes some time to get used to this truth condition. But with a little work it can be seen to be just a generalisation of the Kanger-Kripke truth condition for strict implication (just set b = c ).

But how should this accessibility relation be interpreted philosophically? Oddly, there has not been very much work on this, but there has been some. Mares (1997) uses a theory of information due to David Israel and John Perry (see Israel and Perry (1990)). On this interpreation, in addition to other information a world contains informational links, such as laws of nature, conventions, and so on. Thus, for example, a Newtonian world will contain the information that all matter attracts all other matter. In information-theoretic terms, this world contains the information that two things� being material carries the information that they attract each other. On this view, Rabc if and only if, according to the links in a , all the information carried by what obtains in b is contained in c . Thus, for example, if a is a Newtonian world and the information that x and y are material is contained in b , then the information that x and y attract each other is contained in c .

Another similar interpretation is given in Barwise (1993) and developed in Restall (1996). On this view, worlds are taken to be information-theoretic "sites". Rabc means that a is an information-theoretic channel between b and c . Both this channel-theoretic interpretation of the accessibiliity relation and the other information-theoretic interpretation attempt to provide the formal semantics with an informal notion of relevance. On these interpretations, what is needed for an implication to be true is that the antecedent carry the information that the consequent obtains. The antecedent must be informationally relevant to the consequent.

By itself, the ternary relation is not sufficient to avoid all the paradoxes of implication. Given what we have said so far, it is not clear how the semantics can avoid paradoxes such as ( p & ~ p ) q and p ( q ~ q ). These paradoxes are avoided by the inclusion of inconsistent and non-bivalent worlds in the semantics. For, if there were no worlds at which p & ~ p holds, then, according to our truth condition for the arrow, ( p & ~ p ) q would also hold everywhere. Likewise, if q ~ q held at every world, then p ( q ~ q ) would be universally true.

This brings us to the semantics for negation. The use of non-bivalent and inconsistent worlds requires a non-classical truth condition for negation. In the early 1970s, Richard and Val Routley invented their "star operator" to treat negation. The operator is an operator on worlds. For each world a , there is a world a* . And

Once again, we have the difficulty of interpreting a part of the formal semantics. What may be the nicest interpretation of the Routley star is that of Dunn (1993). Dunn uses a binary relation, C , on worlds. Cab means that b is compatable with a . a* , then, is the maximal world (the world containing the most information) that is compatable with a .

There are other semantics for negation. One, due to Dunn and developed by Routley, is a four-valued semantics. This semantics is treated in the entry on paraconsistent logics .

There is now a lage variety of approaches to proof theory for relevant logics. There is a Gentzen system for the negation-free fragment of the logic R by J.M. Dunn and an elegant and very general Gentzen-style approach called "Display Logic" recently developed by Nuel Belnap. But here I will only deal with an treatment that most philosophers will find somewhat familiar, that is, the natural deduction system due to Anderson and Belnap.

Anderson and Belnap�s natural deduction system is based on Fitch�s natural deduction systems for classical and intuitionistic logic. The easiest way to understand this technique is by looking at an example.

1. A _{1}

hyp

2. ( A B ) _{2}

hyp

3. B _{1,2}

1,2, E

4. (( A B ) B ) _{1}

2,3, I

5. A (( A B ) B )

1,4, I

The numbers in set brackets indicate the assuptions used to prove the formula. We will call them �indices�. The idea here is that for an assumption to be counted as helping to generate the conclusion, an index denoting the assumption must be appear in the deduction and at some later point be discharged. This ensures that each premise is really used in the deduction. This natural deduction system gives an intuitive understanding of relevance in proofs. The indices keep track of which assumptions are used. For an argument to be valid in this system, all assumptions stated must really be used.

Now, it might seem that the system of indices allows irrelevant premises to creep in. One way in which it might appear that irrelevances can intrude is through the use of a rule of conjunction introduction. That is, it might seem that we can always add in an irrelevant premise by doing, say, the following:

1. A _{1}

hyp

2. B _{2}

hyp

3. ( A & B ) _{1,2}

1,2, &I

4. B _{1,2}

3, &E

5. ( B B ) _{1}

2,4, I

6. A ( B B )

1,5, I

To a relevance logician, the first premise is completely out of place here. To block moves like this, Anderson and Belnap give the following conjunction introduction rule:

This rule says that two formulae must have the same index before the rule of conjunction introduction can be used.

There is, of course, a lot more to the natural deduction system, but this will suffice as for our current purposes. The theory of relevance that is captured by at least some relevant logics can be understood by how the corresponding natural deduction system understands a real use of a premise and how the rules are allowed to access premises.

Historically, the central systems of relevance logic have been the logic E of entailment and the system R or relevant implication. E was supposed to capture strict relevant implication. But, when a necessity operator and the appropriate modal axioms were added to R (to produce the logic NR ), it was discovered that the resulting modal system was different from E . This has left some relevant logicians with a quandary. They have to decide whether to take NR to be the system of strict relevant implication, or to claim that NR was somehow defficient and that E stands as the system of strict relevant implication.

On the other hand, there are those relevance logicians who reject both R and E . On one hand there is Arnon Avron who has used semantic arguments to motivate logics stronger than R . On the other hand there are logicians like Ross Brady, John Slaney, Steve Giabrione, Richard Sylvan, Graham Priest and Greg Restall who have argued for the acceptance of systems weaker than R or E . Among the points in favour of weaker these systems is that, unlike R or E , many of them are decidable.

On an extreme end of the spectrum is the logic S of R.K. Meyer, Errol Martin and Robin Dwyer. This logic contains no theorems of the form A A . In other words, according to S , no proposition implies itself and no argument of the form A , therefore A is valid. Thus, this logic does not make valid any circular arguments.

Apart from the motivating applications of providing better formalisms of our pre-formal notions of implication and entailment, relevance logic has been put to various uses in philosophy and computer science. Here I will list just a few.

Dunn has developed a theory of intrinsic and essential properties based on relevant logic. This is his theory of relevant predication . Briefly put, a thing i has a property F relevantly if x ( x=i F ( x )). Informally, an object has a property relevantly if being that thing relevantly implies having that property. Since the truth of the consequent of a relevant implication is by itself insufficient for the truth of that implication, things can have properties irrelevantly as well as relevantly. Dunn�s formulation would seem to capture at least one sense in which we use the notion of an intrisic property. Adding modality to the language allows for a formilisation of the notion of an essential property.

Meyer has produced a variant of Peano arithmetic based on the relevance logic, R . Meyer gives a finitary proof that his relevant arithmetic does not have 0 = 1 as a theorem. Thus Meyer solves one of Hilbert�s central problems in the context of relevant arithmetic: He shows using finitary means that relevant arithmetic is absolutely consistent. Unfortunately, as Meyer and Friedman have shown, relevant arithmetic does not contain all of the theorems of classical Peano arithmetic. Hence we cannot infer from this that classical Peano arithmetic is absolutely consistent (see Meyer and Friedman (1992)).

In a similar vein, Ross Brady and others have used weak relevant logics as bases for set theories. Brady shows that a weak relevant logic together with some set-theoretic axioms that include a naive comprehension principle is not trivial. That is, not every proposition can be proved in this system.

Anderson (1967) formulates a system of deontic logic based on R . This system avoids some of the standard problems with more traditional deontic logics. For example, the rule of necessitation from A s being a theorem to OA s being a theorem is rejected. Thus, it does not say that all theorems ought to be the case.

Mares and Fuhrmann (1995) present a theory of counterfactual conditionals based on relevant logic. This theory avoids the analogs of the paradoxes of implication that appear in standard logics of counterfactuals.

Relevant logics have been used in computer science as well as in philosophy. Linear logics, a branch of logic discoverd by the French logician Girard, is a logic of computational resources. Linear logic is, in fact, a weak relevant logic with the addition of two operators

The term Temporal Logic has been broadly used to cover all approaches to the representation of temporal information within a logical framework, and also more narrowly to refer specifically to the modal-logic type of approach introduced around 1960 by Arthur Prior under the name of Tense Logic and subsequently developed further by logicians and computer scientists.

Applications of Temporal Logic include its use as a formalism for clarifying philosophical issues about time, as a framework within which to define the semantics of temporal expressions in natural language, as a language for encoding temporal knowledge in artificial intelligence, and as a tool for handling the temporal aspects of the execution of computer programs.

Tense Logic was introduced by Arthur Prior (1957, 1967, 1969) as a result of an interest in the relationship between tense and modality attributed to the Megarian philosopher Diodorus Cronus (ca. 340-280 B.C.). For the the historical context leading up to the introduction of Tense Logic, as well as its subsequent developments, see �hrstr�m and Hasle, 1995.

The logical language of Tense Logic contains, in addition to the usual truth-functional operators, four modal operators with intended meanings as follows:

P

"It has at some time been the case that ..."

F

"It will at some time be the case that ..."

H

"It has always been the case that ..."

G

"It will always be the case that ..."

P and F are known as the weak tense operators , while H and G are known as the strong tense operators . The two pairs are generally regarded as interdefinable by way of the equivalences

Pp

H p

Fp

G p

On the basis of these intended meanings, Prior used the operators to build formulae expressing various philosophical theses about time, which might be taken as axioms of a formal system if so desired. Some examples of such formulae, with Prior�s own glosses (from Prior 1967), are:

Gp Fp

"What will always be, will be"

G(p q) (Gp Gq)

"If p will always imply q, then if p will always be the case, so will q"

Fp FFp

"If it will be the case that p, it will be --- in between --- that it will be"

Fp F Fp

"If it will never be that p then it will be that it will never be that p"

Prior (1967) reports on the extensive early work on various systems of Tense Logic obtained by postulating different combination of axioms, and in particular he considered in some detail what light a logical treatment of time can throw on classic problems concerning time, necessity and existence; for example, "deterministic" arguments that have been advanced over the ages to the effect that "what will be, will necessarily be", corresponding to the modal tense-logical formula Fp Fp.

Of particular significance is the system of Minimal Tense Logic K _t , which is generated by the four axioms

p HFp

"What is, has always been going to be"

p GPp

"What is, will always have been"

H(p q) (Hp Hq)

"Whatever always follows from what always has been, always has been"

G(p q) (Gp Gq)

"Whatever always follows from what always will be, always will be"

together with the two rules of temporal inference:

RH:

From a proof of p, derive a proof of Hp

RG:

From a proof of p, derive a proof of Gp

and, of course, all the rules of ordinary Propositional Logic. The theorems of K _t express, essentially, those properties of the tense operators which do not depend on any specific assumptions about the temporal order. This characterisation is made more precise below.

Tense Logic is obtained by adding the tense operators to an existing logic; above this was tacitly assumed to be the classical Propositional Calculus. Other tense-logical systems are obtained by taking different logical bases. Of obvious interest is tensed predicate logic, where the tense operators are added to classical First-order Predicate Calculus. This enables us to express important distinctions concerning the logic of time and existence. For example, the statement A philosopher will be a king can be interpreted in several different ways, such as

x(Philosopher(x) & F King(x))

Someone who is now a philosopher will be a king at some future time

xF(Philosopher(x) & King(x))

There now exists someone who will at some future time be both a philosopher and a king

F x(Philosopher(x) & F King(x))

There will exist someone who is a philosopher and later will be a king

F x(Philosopher(x) & King(x))

There will exist someone who is at the same time both a philosopher and a king

The interpretation of such formulae is not unproblematic, however. The problem concerns the domain of quantification. For the second two formulae above to bear the interpretations given to them, it is necessary that the domain of quantification is always relative to a time: thus in the semantics it will be necessary to introduce a domain of quantification D(t) for each time t. But this can lead to problems if we want to establish relations between objects existing at different times, as for example in the statement "One of my friends is descended from a follower of William the Conqueror".

These problems are related to the so-called Barcan formulae of modal logic, a temporal analogue of which is

For this formula to be true, we require the "domain cumulation" principle, which says that the whole domain of quantification D(t) at time t is included in all the domains D(t ) for times t later than t. For more on this and related matters, see van Benthem, 1995, Section 7.

Soon after its introduction, the basic "PFGH" syntax of Tense Logic was extended in various ways, and such extensions have continued to this day. Some important examples are the following:

The binary temporal operators S and U ("since" and "until") . These were introduced by Kamp (1968). The intended meanings are

Spq

"q has been true since a time when p was true"

Upq

"q will be true until a time when p is true"

It is possible to define the one-place tense operators in terms of S and U as follows:

Pp

Sp(p p)

Fp

Up(p p)

The importance of the S and U operators is that they are expressively complete with respect to first-order temporal properties on continuous, strictly linear temporal orders (which is not true for the one-place operators on their own).

Metric tense logic . Prior introduced the notation Fnp to mean "It will be the case the interval n hence that p". We do not need a separate notation Pnp since we can write F(-n)p for "It was the case the interval n ago that P". The case n=0 gives us the present tense. We can define the general, non-metric operators by

Pp

n(n<0 & Fnp)

Fp

n(n>0 & Fnp)

Hp

n(n<0 Fnp)

Gp

n(n>0 Fnp)

The "next time" operator O . This operator assumes that the time series consists of a discrete sequence of atomic times. The formula Op is then intended to mean that p is true at the immediately succeeding time step. Given that time is discrete, it can be defined in terms of the "until" operator U by

which says that p will be true at some future time, between which and the present time nothing is true. This can only mean the time immediately following the present in a discrete temporal order.

In discrete time, the future-tense operator F is related to the next-time operator by the equivalence

Indeed, F can here be defined as the least fixed point of the transformation which maps an arbitrary propositional operator X onto the operator p.Op OXp.

One could similarly define a past-time version of O; but since the main usefulness of this particular operator has been in relation to the logic of computer programming, where one is mainly interested in execution sequences of programs extending into the future, this has not so often been done.

The standard model-theoretic semantics of Tense Logic is closely modelled on that of Modal Logic. A temporal frame consists of a set T of entities called times together with an ordering relation < on T. This defines the "flow of time" over which the meanings of the tense operators are to be defined. An interpretation of the tense-logical language assigns a truth value to each atomic formula at each time in the temporal frame. Given such an interpretation, the meanings of the weak tense operators can be defined using the rules

Pp is true at t

if and only if

p is true at some time t such that t <t

Fp is true at t

if and only if

p is true at some time t such that t<t

from which it follows that the meanings of the strong operators are given by

Hp is true at t

if and only if

p is true at all times t such that t <t

Gp is true at t

if and only if

p is true at all times t such that t<t

We can now provide a precise characterisation of system K _t of Minimal Tense Logic. The theorems of K _t are precisely those formulae which are true at all times under all interpretations over all temporal frames.

Many tense-logical axioms have been suggested as expressing this or that property of the flow of time, and the semantics gives us a precise way of defining this correspondence between tense-logical formulae and properties of temporal frames. A formula p is said to characterise a set of frames F if

Thus any theorem of K _t characterises the class of all frames.

A first-order formula in < determines a class of frames, namely those in which the formula is true. A tense-logical formula p corresponds to a first-order formula q just so long as p characterises the class of frames for which q is true. Some well-known examples of such pairs of formulae are:

Hp Pp

t t (t <t)

(unbounded in the past)

Gp Fp

t t (t<t )

(unbounded in the future)

Fp FFp

t,t (t<t t (t<t <t ))

(dense ordering)

FFp Fp

t,t ( t (t<t <t ) t<t )

(transitive ordering)

FPp Pp p Fp

t,t ,t ((t<t & t <t ) (t<t t=t t <t))

(linear in the past)

PFp Pp p Fp

t,t ,t ((t <t & t <t ) (t<t t=t t <t))

(linear in the future)

However, there are tense-logical formulae (such as GFp FGp) which do not correspond to any first-order temporal frame properties, and there are first-order temporal frame properties (such as irreflexivity , expressed by t (t<t)) which do not correspond to any tense-logical formula. For details, see van Benthem (1983).

In this method, the temporal dimension is captured by augmenting each time-variable proposition or predicate with an extra argument-place, to be filled by an expression designating a time, as for example

If we introduce into the first-order language a binary infix predicate < denoting the temporal ordering relation "earlier than", and a constant "now" denoting the present moment, then the tense operators can be readily simulated by means of the following correspondences, which not surprisingly bear more than a passing resemblance to the formal semantics for Tense Logic given above. Where p(t) represents the result of introducing an extra temporal argument place to the time-variable predicates occurring in p, we have:

Pp

t(t<now & p(t))

Fp

t(now<t & p(t))

Gp

t(t<now p(t))

Hp

t(now<t p(t))

Before the advent of Tense Logic, the method of temporal arguments was the natural choice of formalism for the logical expression of temporal information.

The method of temporal arguments encounters difficulties if it is desired to model aspectual distinctions between, for example, states, events and processes. Propositions reporting states (such as "Mary is asleep") have homogeneous temporal incidence, in that they must hold over any subintervals of an interval over which they hold (e.g., if Mary is asleep from 1 o�clock to 6 o�clock then she is asleep from 1 o�clock to 2 o�clock, from 2 o�clock to 3 o�clock, and so on). By contrast, propositions reporting events (such as "John walks to the station") have inhomogeneous temporal incidence; more precisely, such a proposition is not true of any proper subinterval of an interval of which it is true (e.g., if John walks to the station over the interval from 1 o�clock to a quarter past one, then it is not the case that he walks to the station over the interval from 1 o�clock to five past one --- rather, over that interval he walks part of the way to the station).

The method of state and event-type reification was introduced to cater for distinctions of this kind. It is an approach that has been especially popular in Artificial Intelligence, where it is particularly associated with the name of James Allen, whose influential paper (Allen 1984) is often cited in this connection. In this approach, state and event types are denoted by terms in a first-order theory; their temporal incidence is expressed using relational predicates "Holds" and "Occurs", as for example,

where terms of the form (t,t ) denote time intervals in the obvious way.

The homogeneity of states and inhomogeneity of events is secured by axioms such as

where "In" expresses the proper subinterval relation.

The method of event-token reification was proposed by Donald Davidson (1967) as a solution to the so-called "variable polyadicity" problem. The problem is to give a formal account of the validity of such inferences as

John saw Mary in London on Tuesday.

Therefore, John saw Mary on Tuesday.

The key idea is that each event-forming predicate is endowed with an extra argument-place to be filled with a variable ranging over event-tokens, that is, particular dated occurrences. The inference above is then cast in logical form as

e(See(John,Mary,e) & Place(e,London) & Time(e,Tuesday)),

Therefore, e(See(John,Mary,e) & Time(e,Tuesday)).

In this form, the inference does not require any additional logical apparatus over and above standard first-order predicate logic; on that basis, the validity of the inference is considered to be explained. This approach has also been used in a computational context in the Event Calculus of Kowalski and Sergot (1986).

Prior�s motivation for inventing Tense Logic was largely philosophical, his idea being that the precision and clarity afforded by a formal logical notation was indispensible for the careful formulation and resolution of philosophical issues concerning time. See the article on Arthur Prior for a discussion of some of these.

The rivalry between the modal and first-order approaches to formalising the logic of time reflects an important set of underlying philosophical issues related to the work of McTaggart. This work is especially well-known, in the context of temporal logic, for introducing the distinction between the "A-series" and the "B-series". By the "A-series" is meant, essentially, the characterisation of events as Past, Present, or Future. By contrast, the "B-series" involves their characterisation as relatively "Earlier" or "Later". A-series representations of time inescapably single out some particular moment as present; of course, at different times, different moments are present --- a circumstance which, followed to what appeared to be its logical conclusion, led McTaggart to assert that time itself was unreal (see Mellor, 1981). B-series representations have no place for a concept of the present, instead taking the form of a synoptic view of all time and the (timeless) interrelations between its parts.

There is a clear affinity between the A-series and the modal approach and between the B-series and the first-order approach. In the terminology of Massey (1969), adherents of the former approach are called "tensers" while adherents of the latter are called "detensers". This issue is related in turn to the question of how seriously to take the representation of space-time as a single four-dimensional entity in which the four dimensions are at least in some respects on a similar footing. In view of the Theory of Relativity, it can be argued that this issue is not so much a matter for Philosophy as for Physics.

Prior (1967) lists amongst the precursors of Tense Logic Hans Reichenbach�s (1947) analysis of the tenses of English, according to which the function of each tense is to specify the temporal relationships amongst a set of three times related to the utterance, namely S, the speech time, R, the reference time, and E, the event time. In this way Reichenbach was neatly able to distinguish between the simple past "I saw John", for which R=E<S, and the present perfect "I have seen John", for which E<R=S, the former statement referring to a past time coincident with the event of my seeing John, the latter referring to the present time, relative to which my seeing John is past.

Prior notes that Reichenbach�s analysis is inadequate to account for the full range of tense usage in natural language. Subsequently much work has been done to refine the analysis, not only of tenses but also other temporal expressions in language such as the temporal prepositions and connectives ("before", "after", "since", "during", "until"), using the many varieties of temporal logic. For some examples, see Dowty (1979), Galton (1984), Taylor (1985), Richards et al. (1989).

We have already mentioned the work of Allen (1984), which is concerned with finding a general framework adequate for all the temporal representations required by AI programs. The Event Calculus of Kowalski and Sergot (1986) is pursued more specifically within the framework of logic programming, but is otherwise similarly general in character. A useful survey of issues involving time and temporal reasoning in AI is Galton (1995).

Much of the work on temporal reasoning in AI has been closely tied up with the notorious frame problem , which arises from the necessity for any automated reasoner to know, or be able to deduce, not only those properties of the world which do change as the result of any event or action, but also those properties which do not change. In everyday life, we normally handle such facts fluently without consciously adverting to them: we take for granted without thinking about it, for example, that the colour of a car does not normally change when one changes gear. The frame problem is concerned with how to formalise the logic of actions and events in such a way that indefinitely many inferences of this kind are made available without our having to encode them all explicitly. A seminal work in this area is McCarthy and Hayes (1969). A useful recent reference for the frame problem is Shanahan, 1997.