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Definition of necessity with entailment.

The concept of necessity has been examined for many years but still we do not have a good theory of it. There is some variety of accounts but after all they do not seem to be convincing. Perhaps the reason is that we need a new approach. In this article I make a step towards a new direction.

After five introductory paragraphs I present my theory in the next twelve ones (6-16). The concept of necessity is defined with the concepts of truth, law and entailment, so my attempt is reductive. Paragraph 16 is a short recap, and the following paragraphs (17-23) present some further discussion. I raise such questions like kinds of necessity (logical, physical, practical etc.), unavoidability, real and fictional necessity, possible worlds, epistemic necessity, necessity de re, conceivability and implication.

1. Various Kinds of Necessity

What is striking when you analyze the concept of necessity (and possibility as well) is that there are many kinds of necessity. You have logical, conceptual, physical, psychological and many other types of necessity. Many authors concentrate only on some chosen cases or examine them separately. Some kinds of necessity are still omitted as it is with practical necessity, usually expressed with the word "must".

My idea is that a good theory of necessity is not to be a theory of a kind of necessity or a set of theories, each for a detailed case, but a theory that explains what is to be necessity in general, and only then elucidate what the reason is for the variety of necessities and how to deal with them. The idea is based on the belief that the various kinds are the kinds of one common concept, and that it is not a case of essential polysemy.

Thus from my point of view numerous authors which have worked on detailed kinds of necessity missed the general point, though their works may be valuable as case studies. For example, in The Critique of Pure Reason Kant worked on a priori and analytic (kind of logical) necessity. Carnap (1956) also worked on analytic necessity as did Quine in his famous essay on dogmas (1951), also Bealer (2006) worked on logical necessity. Many authors like Horgan (1993) or Chalmers (1996) dealt with both logical and physical necessity.

On the other hand every necessity is of a kind. So the authors, who worked on the concept of necessity without any specification, also called "metaphysical necessity", as C. I. Lewis (1932) or Kripke (1959), (1963), 1970), (1980) and their numerous followers (even of the counterfactuals approach like Stalnaker (1968), Lewis (1973a), Hill (2006), Kment (2006) and Williamson (2007)), and as it is usually in theology or similar domains, also have missed the general point. Generally Kripke puts his claims in the sense of necessity tout court (cf. p. 164), but according to the meaning of his argument he supposed that it is close to the physical necessity (pp. 99, 160, 164). Nevertheless he did not explain what "necessity" means, only he used the concept.

Compare the theory of necessity with the theory of geometric figures. Theories of circles or rectangles or of any other detailed cases will never exhaust the general theory of geometric figures. But on the other hand every figure has its shape and talking about a figure without the shape has no sense.

2. Kinds and Relativity of Necessity

The belief that there is one general concept of necessity starts to seem plausible when you realize that it is difficult to draw some evident boundaries between detailed kinds. For example Quine's challenge against analytic and synthetic dualism (1951) made it difficult to find a strict boundary between logical (or conceptual or analytic) and physical necessity.

On the other hand you can see that any kind of necessity may be divided into more detailed kinds according to what constraints are taken into account. Within physical necessity you can differentiate what is necessary form the chemical point of view, or from the point of view of Newton's theory, Quantum Mechanics or Relativity Theory. You can even speak about biological, psychological or economical necessity. For example one might say that if you want to sell a house then the faster you want to sell it the lower price you obtain. So it is economically necessary that if you sell the house tomorrow the price will be lower than if you looked for a buyer for a month. Finally you can see that any detailed kind of necessity is based upon a law with which you deduce some consequences.

Similarly you can divide logical necessity. If you ask whether a geometrical theorem is necessarily true it is important what axioms and rules of inference you take into account. So there is not any general logical necessity but only various concepts of necessity relative to logical systems. This relative aspect of logical necessity was noticed by authors that meant it to be equal to provability, like Carnap (1956) or Bealer (2006). The idea that necessity equals to provability is quite close to the theory that I present below.

3. Mental Ability Approaches

Conceivability approach, like of Descartes and Hume and their followers as Yablo (1993), Menzies (1998), or Chalmers (1996) and (2002), and imagination (Hume), understanding or intuition approaches as well (Bealer 1987, 1999 and 2002, Peacock 1997, 1998 and 2002), are of a psychological kind and tell nothing about the concept of necessity. They are to a degree circular because they explain possibility with what is possible to think, to conceive, to imagine, to understand or intuitively grasp by a human being.

4. Emptiness of Possible Worlds

The idea of possible worlds, started with works of Kripke (1959), (1963), (1970), gives a useful background for necessity calculus but on the other hand it is misleading when you search for the explanation of the concept. It is so because when you think about necessity through the possible worlds you may think that something is explained, whereas in fact, possible worlds explain nothing about the concept of necessity. It is still to be explained: what does it mean that a world is possible?

5. Constraint for Necessity

An important hint to understand necessity is that every statement of it is associated with some constraints. So it looks like an inseparable feature of the notion. Just look at the list of modalities given by Vaidya (2011). Notoriously he uses such phrases like "given everything James knows," "given propositional logic," "given classical propositional logic," "given the actual laws of physics," etc. Generally when somebody claims something to be necessary the natural question is what the basis for it is.

For illustration consider the following common situation. You say:

"John must have met Mary."

The question could be:

"Why do you say that he must?"

Your answer could be:

"If John went to Peter he would meet Mary and I know that he went."

In many contexts of necessity and possibility considerations we automatically assume some constraints and take them as the basis for any necessity or possibility claims. Take for example the following quotations from Kripke (1980):

   Although we could say cats might turn out to be demons, of a
   certain species, given that cats are in fact animals, any cat-like
   being which is not an animal, in the actual world or in a
   counterfactual one, is not a cat. (p. 126) ... one thing we cannot
   imagine happening to this thing is that it, given that it is
   composed of molecules, should still have existed and not have been
   composed of molecules. (p. 126) The objector is correct when he
   argues that if I hold that this table could not have been made of
   ice, then I must also hold that it could not have turned out to be
   made of ice. (p. 141)


6. Necessity

Finally my theory of necessity goes as follows. I say that the word "necessity" means the same as the phrase "being entailed upon a set of truths and a set of laws". For example the sentence:

"It is necessary that P"

means in fact the same as:

"It is entailed upon a set of truths and a set of laws that P."

When you say that it is necessary that P in fact you say that

1. There is a set of truths and

2. There is a set of laws such that

3. P is implied by the set of truths and the set of laws.

It means that a necessity statement is in fact always relative to a set of truths and a set of laws. One thing may be necessary upon some sets but not necessary upon some others.

The truths may be facts but they may be also some laws more detailed then the laws that are the engine for the deduction. If you say that

"It is physically necessary that E = [mc.sup.2]"

You may mean that

1. There is a set of physical laws and

2. There is a set of more general physical laws such that

3. E = [mc.sup.2] is implied by the set of laws and the set of more general laws.

7. Negation of Necessity

The sentence "It is not necessary that P" is ambiguous. In literal sense it means the entire conditionlessness, indeterminacy or rationalelessness. Literally the sentence means:

1. No set of truths and

2. No set of laws

3. Imply that P.

But such a statement is very utmost and it is very rarely used. Usually the way we use the concept of necessity is not entirely tidy from formal point of view. And this probably is a case of a petrified implicatural meaning, similarly as with "no fool" and "clever." It is possible because of the rarity of the literal meaning, and it works upon the Gricean Maxim of Manner. So in the implicatural (usual) meaning when you say that P is not necessary you do not mean that the negation refers to the pair of existential conditions (1) and (2) but only to the existence of the laws, it is, to the second condition (2). In the implicatural meaning the sentence:

"It is not necessary that P"

means:

1. There is a set of truths but

2. There is not any set of laws such that

3. P is implied by the set of truths and the set of laws.

Below I will mean the negation of necessity in this manner. Colloquially speaking when you say that P is necessary you say that there are such facts and such laws that P is entailed from the facts upon these laws. While when you say that P is not necessary you say that there are such facts that there are no such laws that P is entailed from these facts upon these laws.

To illustrate that we usually use the implicatural necessity let me consider a case. When you say that it was necessary that Nixon won the election in fact you mean:

1. There is a description of Nixon's situation in a given moment before the election and

2. There are some sociological and psychological laws such that

3. They imply that Nixon won the election.

But when you want to deny it, it is, when you want to say that it was not necessary that he won, usually you are not to say that

1. No description of Nixon's situation in a given moment before the election and

2. No sociological and psychological laws

3. Imply that Nixon won the election,

because it would be a very strong negative thesis, claiming shortly speaking that Nixon's victory did not come out from anything. It would be very counterintuitive and it is little probable that one would like to say something like that (though it might happen). It is rather obvious that shortly before election the situation in USA and the states of electors' minds made it upon the laws of psychology and sociology (even on the intuitive level) necessary that Nixon would have won. So you would not want to say that there was no such description of this situation that it was not necessary that Nixon would have won. Thus in fact when you say that it is not necessary that he won you want to say that there is such description of Nixon's situation that upon no laws it results that he won. Usually you speak about a situation which was much earlier than the election, for example when Nixon was a child. So when you say that it was not necessary that Nixon won the election you may mean:

1. There is a description of Nixon's situation in a given moment before the election but

2. There are no sociological or psychological laws such that

3. They imply from the description that Nixon won the election.

In other cases you may mean that a restricted set of Nixon's features was not enough to win, for example the feature of being a man is not enough, because not every man wins:

1. There is such a limited description of Nixon that

2. There are no sociological or psychological laws such that

3. They imply from the description that Nixon won the election.

8. Possibility

The sentence:

"It is possible that P"

means:

1. There is such a set of truths that

2. There is no set of laws such that

3. It is implied by the set of truths and the set of laws that not P.

So for example when you say that it is possible that Nixon become a musician you mean that there was such a moment in his life, maybe when he was a child, that upon no laws it was implied that he would not become a musician. More strictly speaking:

1. There is a description of Nixon's situation in a given moment in his life but

2. There are no sociological or psychological laws such that

3. They imply that Nixon did not become a musician.

Generally speaking my theory explains how it is possible that P is necessary and it is possible that P is not true in the same time, e. g. "It is necessary that Nixon won election in 1970" and "It is possible that he lost." The apparent contradiction is possible because the uses of necessity and possibility in these two statements are based on different descriptions of Nixon, that is, on different sets of facts and different laws concerning him.

9. Avoidance of Tautologies

To avoid tautological necessity the law may not be the law of identity, nor a law of propositional calculus in general. If to omit the condition everything would be necessary in a trivial way. Whereas to avoid tautological possibilities P may not be entirely general, like "Something happens," because in such cases there are obviously no laws that could forbid it to be true.

10. Knowledge of the Basis and the Context

The speaker does not have to know the truths and the laws that are to be the basis for the necessity. He claims only that they exist. He may have feeling of what kind they might be, or share the conviction that they exist after he heard it from somebody else. In the cases of Nixon you may claim that something is necessary because you yourself know some psycho-sociological rules that would be the basis for such statements but also you may refer to some laws that you only have heard about and you would not be able to quote.

Nevertheless the speaker usually has in mind a more or less clear imagination of the facts and the laws that are the basis for the necessity and once he speaks out the necessity claim they create a conversational context to speak about the given necessity. Anybody who discusses with him this necessity makes it in the sense of the sets and the laws which comes out from the context which he just has made with his utterance.

It is a very important point, and it makes discussions on the concepts of necessity and possibility difficult. When somebody speaks about possibility you automatically seek for the kind of possibility he means. When you start to feel and somehow understand it you follow him thinking what may be possible or not, within the limits of the given kind of possibility. When you lose the consciousness that the results which you obtain refer only to this particular kind of possibility (to a particular basis for possibility consideration) a mistake is possible.

That is the problem of various Kripke's examples, e.g. the table made of a hunk of wood (1980, p. 113f). Saying what is necessary he relies on his intuition. Often he uses the phrase that "it seems to me" that something is necessary (e.g., 1980, p. 10, 41, 43). It is possible that his intuitive understanding of the contexts of his utterances keeps a constant meaning and is connected with a uniform consistent kind of necessity. But it is also possible that in different contexts he speaks of different kinds, and his theory is inconsistent.

The problem may be seen also when Kripke (1980, p. 110) discusses Sprigge's (1962) case, if the Queen could have been a swan. They put it in such a way (such a context) that it seems plausible to say that she could not. If one sketches the context (Kripke does not but it is suggested) saying that if she had looked like a human, talked like a human and thought like a human, so it seems impossible that she could have been a swan. But any possibility consideration is based on the choice what facts and laws are taken into account. If you take as a fact her human appearance then the context becomes very narrow and being a swan becomes impossible. But it does not mean that it is not possible at all. It is impossible only in the very particular context, on the very particular basis. The only medicine for such situations is to step back from the given context and ask on the basis of what the given possibility is being discussed. Then you can see that there are some assumptions that make the context narrow and you may brush them off. Only then you can see that the Queen could have been a swan, because she had been a living being. Similarly you can see that atomic number of gold could be 80, because it is an element (cf. Kripke 1980, p. 123f); and that cats could be demons, because beings looking like cats could be demons (cf. Kripke 1980, p. 122f), and also that water could be something else than [H.sub.2]O, because it is a compound and compounds have various molecular structures (cf. Kripke, 1980. p. 117f).

In some cases the vagueness of the basis goes very far. It is so in the case of logical or conceptual necessity. If you say that it is logically necessary that a bachelor is an unmarried man the basis for the statement are some facts concerning concepts of bachelor, marriage and manhood, and some logical rules concerning the proper ways of putting the concepts together. But our imagination what these facts and laws would be is very vague. It is often said that analytic statements are true upon the meaning of the words. It seems that the meanings should be the logical facts. Whatever they are the necessity speaker expresses his conviction that some facts and laws which are the basis for the given necessity exist.

The problem of the vagueness is the reason that sometimes you mix the question of possibility with some other concepts. For example it seems that Kripke discussing essential properties sometimes mixes the question of possibility with the problem of diachronic identity (1980, p. 114, notes 56, 57). Perhaps this is the reason why he mixes the question of the essential properties and the structural (molecular) properties which could be the basis for identity. For example he writes:

   The molecular theory has discovered, let's say, that this object
   here is composed of molecules. ... It was something we didn't know
   in advance; maybe this might have been composed, for all we knew,
   of some ethereal entelechy. Now imagine an object occupying this
   very position in the room which was an ethereal entelechy. Would it
   be this very object here? It might have all the appearance of this
   object, but it seems to me that it could not ever be this thing.
   The vicissitudes of this thing might have been very different from
   its actual history. It might have been transported to the Kremlin.
   It might have already been hewn into bits and no longer exist at
   the present time. Various things might have happened to it. But
   whatever we imagine counterfactually having happened to it other
   than what actually did, the one thing we cannot imagine happening
   to this thing is that it, given that it is composed of molecules,
   should still have existed and not have been composed of molecules.
   (p. 126f)


It looks that the atomic structure of an object he takes as essential property whereas the spatial location (Kremlin or whatever) he does not. But he forgets that atomic structure also relies on some spatial relations and that there is a continuity between spatial relations of macroscopic objects and spatial relations of atoms. So in fact relations between atoms of an object and between the object and Kremlin are of the same kind, and there is no reason to say that the proper are essential and the latter not. It seems that his concept of essence is close to structural identity.

Let me also add that Kripke is wrong when he writes: "Ordinarily when we ask intuitively whether something might have happened to a given object, we ask whether the universe could have gone on as it actually did up to a certain time, but diverge in its history from that point forward so that the vicissitudes of that object would have been different from that time forth." (1980, p. 115, note 57)

Of course such consideration (such a meaning of possibility) might happen, but in fact it is very rare. It would rely on the question if the world is entirely deterministic, even on the quantum level. Ordinarily if we consider a counterfactual situation we are conscious that some other facts should have been different too. Nixon could have lost the election in 1970 if some "ifs" had had place. But the origins of these "ifs" could have had their places very long ago (for example USA had not ever existed), so no particular point in time is considered, and we are aware of it.

11. Classification and Kinds of Necessity

The relativization of necessity to the sets of truths and the sets of laws makes the ground for classification. You speak about kinds of the necessity according to the kinds of the laws. Besides, because it is necessary that the truths and the laws are of the same kind, for the entailment could go on, the kinds are in the same time the kinds of the truths. For example in the Nixon case you speak about psycho-sociological necessity because the facts and the laws that you have in mind belong to this domain. There is also the general kind of necessity, when no domain is determined, and all the facts and all the laws are taken into account. It is rather never used.

Usually when you speak of necessity you do not add any label. In such cases the necessity is called "necessity tout court," or "metaphysical necessity" (Vaidya 2011). However you do not mean the general necessity then.

The kind of necessity that whatever kind of laws is allowed seems to be a very weak kind of necessity, whereas usually when we mean "metaphysical necessity" (or necessity tout court) we mean something strong. Kripke calls it even "necessity in the highest degree" (1980, p. 99). If the detailed kind of necessity is not explicitly mentioned it is still quite well determined but the listener is to guess it. He has to guess the facts and the laws as well. The feeling of the strength of the metaphysical necessity is based on the strength of the kind of necessity which is to be guessed. It is possible that philosophical intuition connected with the concept is consistent and close to the epistemic, the physical or the logical necessity (for Kripke it is close to the physical one, 1980, p. 99, 160, 164). However it seems more probable that in different metaphysical contexts philosophers deal with various kinds of necessity, in particular they probably take into account different kinds of the sets of the facts.

When for example one says that Aristotle could have not been the Alexander's teacher, the listener automatically guess that all the Aristotle's features that made it necessary that he became the Alexander's teacher are to be omitted. The more easy it is because usually the listener has a very little knowledge concerning Aristotle's life and perhaps knows nothing that had made it necessary that he became the Alexander's teacher. Nevertheless he silently assumes that these facts unknown to him are to be omitted. They have to, because on the other hand if in fact Aristotle became the Alexander's teacher then for sure there was a moment in the history when upon some psycho-sociological laws it was necessary that he would have become the teacher. For example it could have been when Aristotle had agreed to be Alexander's teacher.

So Kripke is not right when he writes: "It just is not, in any intuitive sense of necessity, a necessary truth that Aristotle had the properties commonly attributed to him" (1980, p. 74).

It is fact that in the context of discussion over the philosophy of language, like in his lectures, perhaps only some conceptual principles come to our minds, so only conceptual (or logical) necessity is at stake. Thus Kripke is right in his own context, but after the context change any other kind of the laws and the facts may come into the range of consideration and many other kinds of necessity may be found.

It is worth to mention that because there are many logical laws that may be the basis for necessity considerations there are many kinds of logical necessity and possibility. Especially you may suspend most of the logical laws and leave in force only a few of them. Then even a square circle could be possible as a cluster of properties, no matter if consistent.

12. Commonness of Necessity and Possibility

In fact every true proposition is necessary upon some other truths and some laws. Even measurements in quantum physics are not any exception, because even if you could not have foreseen the result of a quantum experiment upon any physical laws, you know it upon the principles of the measurement. Also the sensual data are necessary upon physiological laws of perception. The reason is that we do not claim anything to be true without any reason, and the reason always consists of some facts and some general rules. The rules do not have to be causal laws, they may be some principles of measurement, moral or logical laws etc. It means that everything that is real is generally necessary.

On the other hand everything what is real is possible. But it does not come out automatically that reality entails possibility. Any fact which is real is possible upon the existence of some other facts and a domain of laws which enables it. When you say that P is physically possible you mean that there is a set of facts S but there are no physical laws which entail upon these facts that P is false. The principle of commonness of possibility, it is, the principle that everything what is real is possible means that for any fact you can find such facts and such a domain of laws that make it possible.

13. Real and Fictional Necessity

When you say that something is necessary or possible it is important that you assume a set of truths. If you say that something is necessary about Nixon you may take some very strange, devoid of many obvious facts from his life, description of his person but it is to be true. This is the condition for the necessity or possibility to be real. In an extreme case you may strip almost all the facts off him and leave only this that he was a man. It could be a basis to say that he could have been a Roman legionary, no matter that there would be a 2000 years gap. It would mean that there are no laws upon which it results that somebody who had been a man would not become a legionary, Nixon for example. Which would be the last property (like being a man) that you leave is not determined, so it might be any of the properties possessed by the object. In the case of Nixon it could be being a person, being a material object or having a nose. So it is possible that Nixon was made of lithium, because he is a material object, and it is possible that Nixon saw Brutus stabbing Caesar, because he was a seeing person. Also it is possible that the Queen is a swan, because she is a living being, and even she is a vertebrate, thus Sprigge is wrong (1962). Within the range of the general possibility there is everything what is not excluded by a fact upon any law.

D. Lewis is not right when he insists that possible worlds are given only qualitatively and that when one talks about possible Nixon he talks only about his counterpart (1968, p. 114). Kripke is right criticizing him (1980, p. 76). But on the other hand Kripke is wrong to insists that a given object in a counterfactual situation may not be deprived of its essential properties because otherwise it could not be the same object (1980, p. 77). There are no essential properties in this sense and there are no reasons why an object could not be deprived of almost all its properties and still be itself in a sentence like "Nixon could have been a Roman legionary."

Yet we often consider some fictional situations. They may refer to some real objects or states of affairs, and then we call them "counterfactual." They may also be entirely thought out, and then we call them simply "fiction." In general the rules referring to fictional necessity or possibility are the same as for any other kind of fiction. Fiction relies on assuming some facts (or even a very long list of facts if you write a novel). You may also assume some fictional laws, though usually in a fiction real laws are more or less silently accepted. If you write a realistic novel then lots of real facts are silently assumed too. Furthermore if you in addition assume that your fictional picture is to be logically consistent then some consequences are implied. Then one might say that some things are necessary or possible in the fiction. The general rule is that from fictional assumptions you may obtain only fictional consequences, fictional necessity or possibility. Only one fictional assumption among many real makes a consideration fictional. (The conditional character of fictional considerations makes it possible to call fictional necessity also "conditional" or "suspended." I prefer the word "fictional" because it makes clear that it is not real in any sense.)

Clearly fictional or counterfactual considerations may be very useful in education as a practice of mind, in science for better understanding the facts and the laws, and of course in art for artistic and emotional experience. They are also unavoidable in communication, because until you decide that the speaker tells the truth the content of his utterance which you recreate in your mind has the status of fiction. But in none of these situations they tell us anything about reality. If you asked what would have happened if Nixon had had a cancer five years before the election it could start perhaps an interesting consideration, but it would not tell us anything about Nixon. You may obtain some new knowledge about Nixon but only if you draw consequences from some true facts concerning him. Counterfactual assumptions may lead only to fiction. The same refers to counterfactuals in science. If you consider such counterfactual situation as physical properties of a 100 meters big iron cube (as far as it is known there is no such object in the universe), it tells us nothing about reality, and nothing about the real physical laws. It may only make us better understand them.

The condition, put by some authors (e.g., Lewis 1973b, Putnam 1975, Kripke 1980, Chalmers 1996, p. 34), that the laws of nature are to be in force even in counterfactual situations is tautological and adds nothing to the concept of a law of nature. It is obvious that if one assumes that in a counterfactual situation some given laws are to be in force then they would be. Hence the laws of nature are simply the regularities of facts, and causality means only regularity in all facts, but they may not be anything more. Any appeal to fiction cannot change it.

Let me add that the real and fictional mode of necessity speech sometimes are mixed and in some situations it is a question of gust which mode one uses to express what he wants to say. For example you may say that "It is possible that Nixon was a Roman legionary." It is true because it is based on the true premise that he was a man, and on the true rule that a man could have become a Roman legionary. The possibility is real, because it is based on a true premise, but it is counterfactual. But one may ask how it could have been possible that Nixon had become a Roman Legionary. The answer could be that it could have been possible if he had lived 2000 years ago. It means that a real counterfactual possibility requires some other counterfactual conditions. The additional conditions are to be possible too, as it is in this case, because it is possible that Nixon lived 2000 years ago, as a human, and there were conditions for humans to live 2000 years ago at the Earth.

But the same set of imaginations you may put in a different way. You may start to imagine what would have been possible if Nixon had lived 2000 years ago. The assumption that Nixon lived 2000 years ago is counterfactual, so whole the consideration would be fictional. You might say that 2000 years ago Nixon could have become a Roman legionary. But it would be a fictional possibility, because it would be based on a false premise.

14. Fiction of Possible Worlds

It is important to notice that possible worlds are fictional and differ radically from reality. The difference relies on this that a possible world is notoriously indefinite, or it is definite only as far as you define it, but any series of acts of defining me be only finite, so any possible world is indefinite in an infinite number of aspects. In this point Hartshorne (1965) is right, and Adams (1974, p. 189f) is wrong. The infinite indefinity is a general feature of any fiction.

Take the example of Nixon. Say you consider a counterfactual possible world in which Nixon lost election in 1970. But this may not be a one possible world. The state of affairs that he lost may be realized in an infinite number of possibilities. But you cannot just say that you take one of them, because it would not be clear what the world is like. So to make the world more definite you have to add some facts to explain the Nixon's failure. Of course they would be counterfactual too. For example the reason could have been that Nixon had blundered heavily speaking about equal rights for men and women. But how could this have happened? Perhaps he had a bad day, was angry and thinking about a stupid joke during a TV interview etc. The definition of a counterfactual world requires an infinite series of postulates, and every postulate requires a further explanation. All such explanations serve to make the counterfactual assumption consistent with reality. The basic difference between real and possible world is that a possible world does not wait ready-made for research. In fact it may not be researched as something unknown. It may be only postulated step by step and the only thing you may examine is the logical consistency of the set of the postulates.

Let me also mention that there is a substantial difference between a possible world thought out as a counterfactual situation placed within the real world (as in the Nixon cases) and an entirely thought out world with fictional (assumed) objects, fictional facts, and fictional laws. The difference relies on the structure of the definition of a world. An entirely fictional world is given only upon the assumptions, whereas a counterfactual situation in a "counterfactual within reality" world is defined with a set of counterfactual assumptions but also upon a set of postulates which aim is to make the counterfactual assumption consistent with reality.

15. Epistemology of Necessity

Justification of necessity relies on justification of the points of the definition, it is, justification of the existence of the set of facts and the set of laws and justification of the entailment of the necessity claim upon these facts and laws.

16. Recap

The presented idea of the concept of necessity gives one general definition for any kind of necessity (that a statement comes out form a set of truths and a set of laws) but on the other hand it points out what is the ground for the variety of necessities (according to the domain of laws taken into account). Also it explains why any necessity speech is to be based on some premises or assumptions. So it fulfills the general intuitive constraints for any explanation of the concept. Now I shall discuss some further detailed questions.

17. Epistemic Necessity

If the set of truths and the set of laws are all the truths and laws that a person knows we talk about epistemic necessity. The sentence:

"It is epistemically necessary for Mark that P"

means the same as:

1. All the propositions Mark believes in and

2. All the laws Mark believes in

3. Imply that P.

Of course it does not happen that all Mark's beliefs are used to imply that P but only a part of them. So the conditions may be formulated as follows:

1. There is a set of factual beliefs among Mark's beliefs and

2. There is a set of laws among Mark's beliefs such that

3. P is implied by the set of factual beliefs and the set of laws.

Because of the commonness of necessity in fact all that we know is epistemically necessary for us, whereas the realm of the unknown gives the basis for epistemic possibility. If we do know that Nixon was a president of USA, this is epistemically necessary for us. But when we do not know if he had a mole on his left shoulder, then it is only epistemically possible for us. It is so, because if, for example, we knew some facts and rules which exclude existence of that mole (e. g. photographs), then it would belong to our knowledge that he had not any, and it would be epistemically impossible.

A special kind of epistemic necessity uses Kripke in his lectures (1980), though he would mind perhaps that his notion is epistemic. In a few places he distinguishes the clearly epistemic possibility as if it were different from his (e.g., 1980, p. 103, 125, 141). The reason is that his meaning is epistemic but limited. Properties of an object or of a natural kind he divides into essential and contingent. Only the proper are necessary for him. The principle of the division is not clear but it is obvious that the essential properties are necessary in epistemic sense. The limitation means that only some of known facts and laws Kripke takes into account when deciding what is necessary. The division into essential and contingent properties enables him to introduce the concept of rigid designator. He defines rigid designators as expressions pointing out the same objects or the same kinds of objects in every possible world. The identity of objects and kinds in counterfactual worlds is enabled by the essential properties. Till an object or a kind possesses its essential properties it is the same object or a kind.

But as I have claimed above there are no essential properties necessary to talk about a given object in a counterfactual world, it is enough that you mean the name as pointing to the same object, and necessity or possibility is real until it is based on some real properties of the object. Even you can say that Nixon could have been a Roman legionary if he had lived 2000 years ago (since he was a man). It means that every name is rigid or in other words the concept of "rigidness" is pointless.

As I understand a simple reason to introduce rigid designators for Kripke were such expressions like "the inventor of bifocals" (1980, p. 98). The most popular example of such expressions is "the highest mountain on the Earth." He claims that they may not point to the same object in every possible world because it is possible that in a possible world another man is the inventor of bifocals. I claim that saying this he is not consistent in his critique of the classic theory of names. In the sentence of the structure "A is not A", like:

a. The inventor of bifocals is not the inventor of bifocals.

The first A plays a different role than the second one. The only function of the subject (the first A) is to inform which object the sentence is about, and not to describe it. Kripke is right that in this respect only the effectiveness counts and anything goes. In a given case of conversation it may happen that the man who is meant is not the inventor of bifocals, but still he may be pointed with the expression "the inventor of bifocals." It relies on the context. The result is that the sentence (a) is not contradictory (though literally it is), and there is not possible that there is a contradiction between literal meaning of the name and the predicate. Even if Franklin were not the inventor of bifocals the expression could be used to point to the man on the basis of literally mistaken but effective reference, like in Donellan cases (1966).

The delusion that Kripke is sometimes right about rigid designators is possible because he mixes the use of a phrase as a name and as a predicate, but he seems not to see it. For example he writes: "The reason is that 'the inventor of bifocals' is not a rigid designator; a world in which no one invented bifocals is not ipso facto a world in which Franklin did not exist." (p. 145)

But such a statement contains an obvious semantic equivocation. At the beginning of the statement Kripke speaks of a name "the inventor of bifocals" and in the second part of it he uses the phrase as a predicate. It is obvious that if a predicate calls an object the objects is not determined within the meaning of the predicate. The case of "the inventor of bifocals" shows that in fact Kripke mixes names and predicates that have a singular designate. Names point to some objects whereas predicates may be only used accessorily during an act of pointing.

Moreover Kripke seems to forget that literally taken even proper names may be used as predicates. So his "rigid designators" could be sometimes not "rigid". If you saw a woman you might ask "Is she a Kathy?". It would sound funnily but it would be alright form logical point of view. The same phrase may be used as a name and as a predicate, and there are no exceptions. The only problem is the effectiveness of communication. But if in a group of people there would be shared a stereotype of Kathyhood an utterance like "She is a Kathy" could be quite informative. The difference between names and predicates relies on the role they play in the proposition and not on the literal sound. The whole Kripke's theory of rigid designators seems to be based on this mistake.

The use of epistemic necessity requires much attention in reference to the moment in the subject's knowledge history to which the speaker refers. Unfortunately Kripke seems to mix various cognitive perspectives and this makes his meaning of necessity more difficult to interpret. Take the case of Hesperus and Phosphorus (1980, p. 20f). Kripke insists that it is necessary that Hesperus is Phosphorus, because both terms refer to the same planet Venus. But saying this he bases on our contemporary knowledge that in facts it is so. Today because it occurred to be true it is also necessary, upon the principle of commonness. But for our ancestors it was not a fact so they could have thought that it was possible that Hesperus is not Phosphorus, in the epistemic sense. But Kripke evidently wants that even then it would have been necessary, because no matter if our ancestors had known this these two names had pointed to the same planet. It is the point where he mixes the perspectives.

Of course for an ancient speaker it was possible that in future it would occur that Hesperus is Phosphorus, so it was possible for him that there would be some reasons that would make this identity necessary. Kripke even feels the problem when he writes that it is epistemically possible that "the four color theorem might turn out to be true and might turn out to be false. It might turn out either way. It still doesn't mean that the way it turns out is not necessary" (1980, p. 103).

But still it would be only a supposed necessity and not real. In his discussion of the Hesperus-Phosphorus case Kripke seems to forget that it is entirely based on his own contemporary knowledge. And this forgetfulness, that his claims are based on his own perspective, forces him to seek for some properties that are independent from the speaker's knowledge, which he calls essential. Thus finally when you use the epistemic necessity you have to be very careful not only in reference to the described speaker's knowledge but also in reference to your own one.

Of course the discussion here I limit to the epistemic sense of necessity because otherwise even today one might say that it is not necessary that the Morning Star and the Evening Star are the same object. And it could be true if the basis of the statement were devoid of these our contemporary beliefs which make the identity justified.

18. Practical Necessity

A special kind of detailed necessity is practical necessity. The realm is difficult to define but in general if a state of affairs is practically necessary it does not imply that it is real. For example from moral point of view it may be necessary that John will do P, though in fact he will not. Practical reasoning that may lead to necessities of this kind may be based on such sources like drives, needs, emotions, wants, dreams, purposes, good, evil, values, duties, morals or state law. According to the detailed kind of the practical deduction there are several kinds of practical necessity, like political, moral, legal, based on agreement, based on what is objectively good, or what is good for a given human or for a group of humans, based on somebody's needs, wants or dreams etc. Let me call the states of affairs that are practically necessary "practical aims." They may be autotelic or instrumental. The latter are deduced upon some autotelic purposes and some causal laws. The causal laws may be of physical kind but also they may be more general and belong to cybernetics. If in an academic system it is necessary first to obtain M.A. title before one may obtain Ph.D., then M.A. title may be an instrumental purpose for one's action, though it does not rely on any detailed physical causal laws. Similarly if in a computer system it is necessary first to start program P in the aim to start program Q then starting program P may be an instrumental aim in the system, even though the relation does not rely on any detailed physical laws, upon which the computer system works.

If the practical aim is a person's action then usually we use the word "must." A sentence which calls for a practically necessary action I call a "categorical imperative." (My use of the term is quite far from Kant's one. He used it only in reference to morals and he did not use the concept of necessity in reference to practical issues.)

The difference between "You must go" and "Go!" lies in this that with the former you state that there is a reasoning (some truths and laws) upon which "the going" is entailed. In other words saying "You must go" you mean that "There is a reason to go." You use the phrase toward a person that has doubts in his thinking. People who have doubts do not like to act without any reason. The simple order "Go!" you use toward a person which does not need any extra reason but only an impulse to start the action.

Besides of real practical necessity there is also the fictional kind. It appears when you consider what would be the consequences if, for example, somebody wanted P or if P would be a value or if P would be the purpose of action. A sentence which calls for an action which is necessary to obtain a fictional practical aim is called "hypothetical imperative."

19. Negative Practical Necessity (Unavoidability)

It happens during the decision making process that there are some facts that will happen in future and it is not possible to avoid them. All you can do is to accept them. For example the sentence:

"It is practically necessary (unavoidable) that the price of oil will be higher than 100 dollars a barrel tomorrow"

means the same as:

1. There is such a set of truths concerning oil market that

2. There is a set of laws (esp. economical) such that

3. It is implied by the set of truths and the set of laws that the price of oil will be higher than 100 dollars a barrel tomorrow.

But what is important the sets of truths and laws include facts and laws concerning some possible actions of a given group of people, because when you speak of unavoidability you always mean it in reference to a group of people. In fact you consider if they are able to intervene the course of events. Such statement is also relative to a given kind of actions which the members of the group could perform, for example you may say that something is unavoidable unless they use force. Thus more strictly the elucidation is:

1. There is such a set of truths concerning oil market and a given group of people that

2. There is a set of laws (esp. economical, psychological and sociological, including the range of considered actions of the given group of people) such that

3. It is implied by the set of truths and the set of laws that the price of oil will be higher than 100 dollars a barrel tomorrow.

20. Conceivability. Law of Non-contradiction

The concept of conceivability literally speaking refers to what is possible for a subject to conceive, so it is a detailed kind of possibility and it belongs to psychology. It is not the place here to discuss mind-body problem, so in general I say that a theory of human mind is to describe what a human mind is able to conceive. (Within physicalism, which I prefer, psychology is simply a causal theory of mind; in fact it belongs to cybernetics.) Nevertheless contemporary psychology is far from a good theory of conceivability. A subject may claim that something is conceivable for him upon his intuition. This intuition is in fact his intuitive autopsychology, based on what happens to him to conceive or not. But it does not tell us what "possible" means.

On the other hand our ability to conceive things as an aspect of mind relies on the conceptual and propositional content of mind. So the question if something is possible to conceive tells us something about our notions and beliefs. So thought experiments based on imagination may be used to analyze the content of our minds. Hence imagination is a way to examine concepts, though as a method it needs a deeper analysis. Simply it seems that there are lots of kinds of conceivability, for example it seems that in the case if it is conceivable that 5 + 7 = 13 the meaning of conceivability is different than in the case if it is conceivable that horses have wings. Also it is seems that one conceives himself flying in another meaning than he conceives a fifth degree equation having a solution or a 100-figured number which figures are placed without any rule. Moreover the question is to be answered if it is enough to conceive a thing when you only describe it with words, like the number above, and the description is not contradictory. Another basic question is what conceivability tells us about the construction of human mind and what it tells about the concepts and propositions which are being used in an imagination.

From the third point of view considering a thing is the same as conceiving a situation and the same as making a thought experiment or performing an imagination. So when you consider if something is possible then you cannot avoid conceiving things. Thus considering and conceiving as well is based on taking into account some facts and some laws. When they are real and true then the consideration refers to reality, when only one is fictional then whole the consideration is fictional. From this point of view you may call the result of any consideration "the conceivable." But this meaning of conceivability is not psychological, because it is not based on the idea that a conceivable state of affairs is to be graspable by human mind. This kind of conceivability is sometimes called "conceivability in principle". It means that there are possible beings that can grasp it.

In this meaning "conceivable" means "possible to obtain in the course of correct consideration," so this is the same as "thinkable" (by a human or a better thinker). But the concept is useless to explain "possible" because any possibility may be only an object of consideration. If you can think that P is possible it tells you no more about the concept of possibility than if you can think that P is red. So the concept of conceivability in principle also does not tell us what "possible" means. Generally speaking it seems that the idea that the concept of conceivability is a key to understand possibility relies on mixing a mental act with its object, and it is perhaps based on a colloquial metonymy.

The customary connection of conceivability and possibility (esp. logical possibility) is the source of various usage of the words "necessary" and "possible" in the context and the meaning of "conceivable." An example is the law of non-contradiction. The sentence:

"It is logically impossible that a proposition is true and false in the same time" (Aristotle, Metaphysics 1011b).

Literally means the same as:

1. There is such a set of truths (concerning in particular the concepts of proposition, truth and false) that

2. There is a set of logical laws such that

3. It is implied by the set of truths and the set of laws that no proposition is true and false in the same time.

The question arises what the truths and what the laws they could be, because the principle of non-contradiction seems to be an axiom believed without any reasoning (Aristotle, Met. 1006a). So you can say that you feel that it is true or that it is clearly evident. If it is so, that there is no reasoning for the principle, then I do not count it as a counterexample to my theory. It only means that you should not use the word "necessary," "possible" or "impossible" when you express it. Instead you should say something like:

"The situation never arises in which a proposition is true and false in the same time"

or:

"No proposition is true and false in the same time".

or (as I say above, that it expresses an intuitive feeling referring to human ability to grasp things with mind):

"It is inconceivable that a proposition is true and false in the same time"

Another example is the law of identity, in nominal (A = A) or propositional (P [left right arrow] P) formula. Simply because it is not thinkable that A is not A or that P is not equivalent to P we usually say that (A = A) and (P [left right arrow] P) are necessary. In fact they are not. Moreover sentences "A is A" and "P if and only if P" as pleonastic are not a correct sentences. Because of pleonasticy tautologies are not correct sentences in general, though they serve well as bases for schemata of deduction. The sentence "A is A" is not thinkable equally as "A is not A", though perhaps the mental reasons of these are different. The apparent thinkability of "A is A" comes probably out of this that "A is not A" seems to be more unthinkable, so the opposite would have to be thinkable.

21. Independence and Necessity

Some authors (more recently Bealer 1996) think that if a being exists independently from physical and psychological facts its existence is necessary. It is wrong because the fact of independence means only that the existence of such a being is physically and psychologically possible. It is so because you can only say that there is no true physical or psychological facts and laws showing that this object does not exist. To say that something exists necessarily it is not enough to say that it is independent from a domain of facts. You may say that something exists necessarily only if there are some facts and laws entailing that it exists. If existence of a being is independent from the physical and forms psychological then the reasoning is to belong to another domain.

22. Necessity de Re

Sentences in which necessity is stated about objects I treat as paraphrases of sentences in which they are stated referring to sentences. It is, the sentence

"A is necessarily B"

or:

"Necessarily A is B"

means the same as:

"It is necessary that A is B".

There are no examples in which such a rule of transformation would be wrong.

Quine's example with number 9 (1961) which is meant to be a counterexample against necessity de re is based on a mistake because the reasoning goes wrong, and also, when you take it with necessity de dicto:

1. It is necessary that 9 is greater than 7

2. 9 = the number of planets

3. It is necessary that the number of planets is greater than 7.

So it could be an argument against the concept of necessity in general. But it is not. What is wrong is the premise (2). It is not true that number 9 is exactly the same as the number of planets. The phrase "number 9" or simply "9" we use to talk about numbers; like in (1). But the phrase "the number of planets" we use to talk about planets. And in fact (3) is a sentence about planets. So (2) is wrong because a switch of reference takes place in it.

Of course there are 9 planets in the solar system (they were before the definition change). So one may suggest that he means number 9 with reference to the number of planets. So to avoid a switch you should say something like:

2'. 9 = the number which is the number of planets.

Such description still refers to a number, but the phrase "number of planets" is only an attribute (a phrase which function is to point out which object of a kind the speaker means). Finally reformulated (3) contains no problem:

3'. It is necessary that the number which is the number of planets is greater than 7.

23. Implication

Concepts of necessity and implication are closely tied together. Especially when you have a true implication (P [right arrow] Q) and you know that P is true then it is obvious that Q is necessarily true. Necessity upon modus ponens is entirely according to my theory; P and (P [right arrow] Q) are a fact and such a law that their result is Q.

Nevertheless the concept of necessity is not useful to explain implication. Implication itself contains no necessity in itself. Implication means only that if you have a sentence you can go to another one. And there is nothing more in it.

Introduction of natural implication to propositional calculus needs no modal operators, as C. I. Lewis tried (1932). It is enough to use the following rule of substitution:

In any formula of propositional calculus in any position you may substitute a sentence Q with the formula:

P and (P [right arrow] Q)

If you apply the rule to a tautology of propositional calculus, then you obtain a tautology of propositional calculus with the natural implication. For example (modus ponens):

[P and (P [right arrow] Q)] > Q

is a substitution instance of

Q > Q,

where ">" means material implication.

This rule of substitution is enough to obtain all the tautologies that are necessary as the basis for logical entailment in any reasoning in which natural implication is used. The idea of the rule fulfils all the intuitive requirements that we have toward the natural implication, it is, that the function of implication is to make shift from one sentence to another.

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ROMAN GODLEWSKI

rogodlewski@wp.pl

Institute of Philosophy

Kazimierz Wielki University, Bydgoszcz
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