Quine vs. Kripke on Necessary Truths

In our earlier summary of Quine's paper, "On What There Is," we saw that in hinting at his own ontological views he raised various difficulties with-and therefore expressed strong reservations about-the need to include possible objects among `those that there are'. In the present section we shall examine some further refinements of Quine's views on one the general topic of modality-that area of philosophy and logic that undertakes to clarify the concepts of necessity and possibility, or the use of the expressions `necessarily' and `possibly' The basic groundwork for a classical sentential and predicate logic was established by Frege. This logic involved two sets of central ideas: -hose having to do with a two-valued truth-functional system for dealing with propositions, and that of generality, as reflected in the use of quantifiers. The publication of Whitehead and Russell's Principia Mathematica marks a major milestone in the history of logic that built on these ideas. Quine's work, for all its innovative contributions, is essentially a working out of some further details within this tradition initiated by Frege and Russell. In recent decades, variousdepartures from and rivals to the classical system of Principia have characterized the lively and richly flowering field of logic. Various types of formal calculi have been developedmany-valued logics, tense-logics, deontic logics, epistemic logics, erotetic (interrogative) logics,and so on.  Among these developments is the construction of systems of modal logic. Various logicians, including Clarence I. Lewis, Jaakko Hintikka, Ruth BarcanMarcus, Saul Kripke, and others have contributed to the development of such modal calculi, including the use of devices of quantification, in order to give preciseexpression to the notions of necessity and possibility. In addition, they have sought to work out a `semantics' (schemes for determining the application and interpretation) of these formal calculi. Furthermore, an extensive philosophical literature surrounding these formal developments has grown up in which diverse viewsare presented for evaluating the significance or merit of what has thereby been achieved. It is in the context of these recent developments that Quine's comments onmodality are to be understood. He has expressed his strong misgivings about the logical and philosophical soundness of the claims made in behalf of quantified modallogic. Although the topic is a complex and technical one, with details beyond the scope of the present book, it will nevertheless be of some importance to anunderstanding both of Quine's views and of some recent lines of thought in analytic philosophy (e.g., in the work of Saul Kripke) to briefly explore some generalfeatures of the controversy thus engendered.

Let us begin by considering the following sentences, the combination of which illustrates some of the difficulties that arise, Quine would say, when we try to combine modal concepts and those of the standard classical predicate logic.*

1. 9 = the number of the planets,
2. Necessarily (9 > 7),
3. Necessarily (the number of the planets > 7).
* In modal logic, the two key ideas are 'necessarily' and 'possibly'. The former is symbolized as 'Necessarily' and the latter as 'Possibly'. Alternative symbolizations in common use are 'L' for 'necessarily', and 'M' for 'possibly'.

The first sentence can be read '9 is (the same as) the number of the planets'; it is true. The second can be read as 'Necessarily 9 is greater than T, and is also true. However, the third sentence, which can be read 'Necessarily, the number of planets is greater than T, is false, for it is logically possible (non-necessary) that the number of planets is not greater than 7. It would seem to be required, however, that sentence 3 follows from sentences I and 2, once we perform the relevant substitutions. Yet clearly something is wrong if we can obtain as a conclusion a false sentence from two true premisses.

In carrying out his critical analysis of this situation, Quine shows the role played by the principle of identity, taken as a principle of substitutivity. If we have a statement of identity, with the identity sign flanked on either side by a singular term, then the principle of substitutivity asserts that "given a true statement of identity, one of its two terms may be substituted for the other in any true statement and the result will be true. "4' This principle is sometimes also expressed by saying that terms shown to be identical are intersubstitutive salva veritate. (The expression `salva veritate' means, effectively, `without change of truth-value': that is, if a statement is true, and we replace a singular referring expression in it by one of its equivalents, the truth of the statement as a whole is unaffected-its truth is 'saved' or preserved-despite this replacement.)

Two expressions will designate the same individual if each express on is used directly or in a purely referential way to refer to its object (its referent). On the other hand, if a term is not used in a direct or purely referential way it will have what Quine calls referential opacity. Thus in order for the principle of substitutivity to hold, the singular expressions (e.g., names) must be used in a purely referential way and not be referentially opaque. "For it is clear that whatever can be affirmed about the object remains true when we refer to the object by any other name."

Thus, if `Socrates = the teacher of Plato = the husband of Xanthippe', and if the sentence `Socrates was made to drink the hemlock' is true, in accordance with the statement of identity one could replace the expression `Socrates' by either of the expressions 'the teacher of Plato' or 'the husband of Xanthippe', and the sentence as a whole will retain its truth. The predicate `was made to drink the hemlock' would continue to be true of the individual referred to by any one of these equivalent, intersubstitutive, purely referential expressions.

However, there are some sentences in which names (or other singular expressions) are not used in a purely referential way. In these cases, the principle of substitutivity will not hold, i.e., it will not generally or necessarily be the case that by replacing some singular term by its equivalent the truth of the entire sentence will be preserved. Thus, take Quine's example of the true identity

Tegucigalpa=the capital of Honduras
and consider the true sentence concerning the belief of a certain person, namely Philip.
Philip believes that Tegucigalpa is in Nicaragua.
On the assumption that Philip realizes that Honduras and Nicaragua are different countries, it would be false that
Philip believes that the capital of Honduras is in Nicaragua.
The replacement of the intersubstitutive terms `Tegucigalpa' by 'the capital of Honduras' does not preserve the truth of the sentence that describes a person's belief. In such a sentence ('Philip believes that Tegucigalpa is in Nicaragua') the use of the expressions `Tegucigalpa' and 'Nicaragua' is not purely referential. They occur in this kind of sentence in a referentially opaque way. (Frege called such occurrences `indirect', ungerade, as contrasted with 'direct', gerade, uses). In sentences containing terms marked by referential opacity, the principle of substitutivity is not applicable; it does not guarantee truth preservation.

There are many other examples of sentences in which the singular terms present in them have referential opacity in those sentences. This is the case, for example, with the whole group of sentences that convey various types of 'propositional attitude', for example, 'x fears that_____', `x believes that_____', 'x knows that_____', 'x expects that_____', 'x says that_____, 'x doubts that_____', `x is surprised that_____, and so on. In all of these, if singular terms are embedded in the clauses that follow the 'that ', they suffer from referential opacity.

Other examples where referential opacity is present occur wherever the distinction needs to be made between the use and mention of a term, and where the latter is marked by the presence of quotation marks around the term in question. Thus in the sentence

Cicero was an orator.
one is using the proper name to refer to a particular individual. On the basis of the identity sentence
Tully = Cicero
we can replace a sentence in which one of these proper names is used by another sentence in which the other name is used without affecting the truth-value. Now, however, consider the sentence
'Tully' has five letters.
While this sentence is true, we cannot replace 'Tully' by 'Cicero' since we should then get a false sentence. By employing quotation marks, as in 'Tully', one is no longer using the proper name in a purely referential way. Instead, one is mentioning Tully by means of the quoted expression. The quoted expression is referentially opaque; it does not refer to the person named.

Quine's principal argument is that one of the major weaknesses of modal sentences involving the expressions `necessarily' or `possibly', as codified and symbolized in quantified modal logic, is that they too suffer from or induce referential opacity, and thereby defeat the legitimate use of quantified variables.

As a preparation and background for understanding Quine's criticisms, let us pause to take note, first, of a distinction commonly made by many philosophers with respect to the use of the terms `necessary' and `possible'. Each of these terms, it will be said, can be used in either of two ways, de dicto and de re. (These latter expressions can be translated, respectively, as `having to do with the dictum or proposition' and `having to do with the thing'). Thus consider the following sentences:

Necessarily, a = a
And
Necessarily, if `All men are mortal, and Socrates is a man, then Socrates is mortal'.
In each case one can say that the term `necessarily' is used in such a way that it applies to the proposition as a whole that follows immediately after the term `necessarily'. If one says of the proposition `a = a' that it is necessary, this is to say that, in conveying the principle of identity, it is true universally, without exception. Similarly, the entire statement `If all men are mortal, and Socrates is a man, then Socrates is mortal', is necessarily true as a whole because it conforms to the schema `(p => q) . (p) => q', i.e., `If (if p then q) and (p), then q'. This pattern states a logically true principle that holds without exception.

On the other hand, if one says `A man is necessarily a rational animal', the term `necessarily' describes a property belonging to an individual essentially. Thus on the traditional view, God is a being whose existence is necessary (it is not possible for God not to exist), whereas the existence of his creatures (of any finite entity, or of the world itself) is nonnecessary or contingent. Or again, the property of being a rational animal is necessary (essential) to what it is to be a man; however, it is not essential to being a man that he speak Spanish; that property belongs to some individuals, not to others. A man possibly speaks Spanish, but necessarily is a rational animal.

Our first group of examples, those having to do with the use of 'necessarrily' in connection with propositions as a whole, belong to the de dicto use of 'necessarily'; the second group of examples, those having to do with the essential properties of an individual object (or person) have to do with the use of `necessarily' de re.

Analogously, one could make a distinction between two uses of the term `possibly'. Consider the two sentences

Possibly Socrates was a vegetarian.

Socrates was possibly a vegetarian.

In the first case, the use of the term `possibly' governs the entire proposition `Socrates was a vegetarian' and its truth-value. It may be true or it may be false. On the other hand, in the sentence `Socrates was possibly a vegetarian', the use of the term `possibly' tells us something about the applicability of the predicate `was a vegetarian' (the property of being a vegetarian) in connection with Socrates. As a person who has the capacity to adopt a particular kind of diet or change it, he may or may not have had that property at a particular time. The first use of `possibly' is de dicto, the second use de re.

A de dicto use of `necessarily' or `possibly' has to do with the truth or falsity of some proposition as a whole; the de re use of modal terms has to do with the thing (res) or state of affairs, with whether the property in question belongs to the object essentially or not.

There is 'one further point we need to introduce in connection with the use of quantified variables, before we put our various preliminary reminders together and return to Quine's indictment of quantified modal logic. This point concerns the use of the principle of existential generalization. According to this standard rule of classical predicate logic, if we are given a sentence in which a predicate expression is attached to a name and predicated of the object referred to by the name, we may apply the rule of existential generalization by dropping the use of the name and replacing it by a variable that is then quantified (bound) by the existential quantifier. Thus, given the sentence

Socrates is a man
we may use the rule of existential generalization to obtain the sentence
(\$x) x is a man.
Or, again, if we are given the sentence
9>7,
we may apply the rule of existential generalization to obtain
(\$x) x > 7.
In each of the above cases, the expression following the quantified variable is said to lie within the scope of the quantified variable. Thus 'x is a man' or 'x > 7' lies within the scope of the quantified variable (\$x).

We have seen previously that, as used in ordinary language, the expressions 'necessarily' and 'possibly' suffer from certain ambiguities of scope. A modal term may in some cases be used as a sentential operator and apply to the entire sentence, which may be said to lie within its scope. On the other hand, the modal expression may be used to apply to only a fragment of the sentence, and thereby have only that fragment within its scope. The introduction of quantifiers in predicate logic is a device one of whose principal uses is the elimination of such troublesome ambiguities of scope.

Let us next consider the complication presented by the attempt to combine the use of quantifiers and modal expressions. Take the case, previously mentioned, in which the following expression is obtained as a result of applying the rule of existential generalization:

(\$x)Fx
where 'F' is a predicate expression, and x is an individual variable. If, now, we introduce the use of modal expressions, such as 'necessarily' alongside the standard use of quantifiers, individual variables, predicate expressions and other logical constants, it is clearly important, in the light of what we said earlier, to be able to distinguish
(\$x) Fx
and
(\$x) ? Fx.
In the first case, the modal expression lies outside the range of the quantifier and serves as a sentential operator. In the second case, the modal expression lies within the scope of the quantified variable. The first sentence may be read
Necessarily, for some x, Fx.
And the second sentence may be read
For some x, necessarily Fx.
Quine's questions and criticisms of quantifed modal logic have to do primarily with the use of modal expressions within the scope of quantified variables, although he also has some qualms about the other uses as well. In order to see what Quine's objections are, I shall summarize these as falling within a two-pronged dilemma, each of whose horns, as we may interpret Quine, presents serious difficulties. Hence (to anticipate) instead of searching for a way of 'escaping between the horns' (as some have sought to do), he would maintain that the whole enterprise and program of quantified modal logic was "conceived in sin." It is philosophically misbegotten and should be abandoned.

In examining a typical example of quantified modal logic, let us go back to the case with thich we started our discussion: that involving 9, the number of the planets, and the property of being greater than 7. Let us now introduce the use both of quantifiers (through the use of existential generalization) and the use of the modal expression 'necessarily' within the scope of the quantified variable. Thus from

9>7
we obtain (by existential generalization)
(\$x) x > 7.
And from
9 is necessarily greater than 7
where 'necessarily' is taken as applying to a part of the entire sentence, i.e., as lying within the scope of the quantifier, we obtain
(\$x) ? x>7.
The problem, for Quine, may now be put in the form of a dilemma, each of whose horns has unpalatable consequences in his eyes.

1. Let us go back to the standard interpretation of the .use of quantifiers as we find it in the classical predicate calculus. Let us take the case of where, for a quantified variable, we insert as an argument for the variable a singular referring expression; and let us assume that as a result we obtain a true statement. (For '(\$x) x is a man' we replace 'x' by 'Socrates', and obtain the true statement 'Socrates is a man'.) According to the principle of substitutivity, we should also obtain the same truth-value (salva veritate) by substituting for 'Socrates' any singular referring expression that refers-as a purely referential term-to the same individual, in other words is intersubstitutive with 'Socrates' (e.g., 'the teacher of Plato'). According to the rules of classical predicate logic, one could always use the principle of substitutivity in this way for terms within the range of values (or arguments) of a bound variable.

However, we find that by adding the use of modal expressions within the scope of bound variables the situation changes drastically. Recall the argument in which we obtained the false conclusion 'The number of the planets is necessarily greater than 7' from the two true premises `9 is necessarily greater than 7' and `9 is the number of the planets'. We assumed, in obtaining this result, that we could rely with confidence on the principle of substitutivity, which sanctioned the substitution of `the number of the planets' for `9'. In making this substitution, we assumed that each of these expressions is here being used in a purely referential way. If now, however, as a result of this we obtain the unwelcome outcome of obtaining a false conclusion from two true premisses, we must go back and question what we have accepted and assumed along the way. The suspicion falls on the assumption that the singular terms were used in a purely referential way and therefore subject to the principle of substitutivity. If we surrender this assumption, it means that far from the singular terms being used in a purely referential way, they are in fact referentially opaque! What has brought about this change is the use of the modal expression joined to the singular term within the scope of the quantifier. Unlike the situation in classical quantificational logic that is unencumbered by modal expressions, the introduction of these expressions (like the use of verbs of propositional attitudes, or quotation marks) transforms the role and status of a singular term used in a purely referential way to one where it is now referentially opaque. Of course, under these conditions one can no longer use the principle of substitutivity to sanction replacement terms by -their equivalents. Clearly, we cannot complacently accommodate ourselves to this situation. Either we must discard ,or change our standard rules for the use of the quantifiers, or else we must surrender the use of modal terms in the way they appear here. The choice, for Quine, is obvious. He thinks the enormous strength, clarity, consistency, and successes of classical first-order predicate logic (including the use of the principle of substitutivity) are so well established that it is preferable by all odds to give up the attempt to combine the use of quantifiers and modal terms.

2. Now let us turn to the other possible route, the other horn of the dilemma. It might be suggested that it is really not required that we give up the combined use of quantifiers and modal terms, since in the case of the term `necessarily', as it appears in `an x which is necessarily F, we are appealing to the notion of essential properties, and this is a wholly respectable philosophic concept. We could call upon the philosophical doctrine of essentialism, of the sort that Aristotle or others influenced by him espouse. Such a doctrine commits one to speaking of objects as having essential or nonessential (contingent) properties; in short, to a conception of de re modalities. However, this `way out' is equally unappealing to Quine.

The only course open to the champion of quantified modal logic is to meet my strictures head on: to argue in the case of  9 and the number of the planets that this number is, of itself and independently of mode of specification, something that necessarily, not contingently, exceeds 7. This means adopting a frankly inequalitarian attitude toward the various ways of specifying the number. One of the determining traits, the succeeding of 8, is counted as a necessary trait of the number. So are any traits that follow from that one, notably the exceeding of 7. Other uniquely determining traits of the number, notably its numbering of the planets, are discounted as contingent traits of the number and held not to belie the fact that the number does still necessarily exceed 7.

This is how essentialism comes in: the invidious distinction between some traits of an object as essential to it (by whatever name) and other traits of it as accidental. I do not say that such essentialism, however uncongenial to me, should be uncongenial to the champion of quantified modal logic. On the contrary, it should be every bit as congenial as quantified modal logic itself.43

Quine confesses that he finds it bewildering to talk about the difference between necessary and contingent (accidental, non-necessary) properties:
Perhaps I can evoke the appropriate sense of bewilderment as follows. Mathematicians may conceivably be said to be necessarily rational and not necessarily two-legged; and cyclists necessarily two-legged and not necessarily rational. But what of an individual who counts among his eccentricities both mathematics and cycling? Is this concrete individual necessarily rational and contingently two-legged or vice versa? Just insofar as we are talking referentially of the object, with no special bias toward a background grouping of mathematicians as against cyclists or vice versa, there is no semblance of sense in rating some of his attributes as necessary and others as contingent. Some of his attributes count as important and others as unimportant, yes: some as enduring and others as fleeting; but none as necessary or contingent. Curiously, a philosophical tradition does exist for just such a distinction between necessary and contingent attributes. It lives on in the terms `essence' and `accident', `internal relation' and `external relation'. It is a distinction that one attributes to Aristotle (subject to contradiction by scholars, such being the penalty for attributions to Aristotle). But, however venerable the distinction, it is surely indefensible.
Quine's strictures concerning modal logic have provoked much discussion. Various lines of defense and support have been proposed in order to show the philosophic soundness of the basic ideas of modal logic. Some have insisted on retaining the traditional doctrine of essentialism and de re modalities. Others have adopted a special interpretation of the quantifiers as used in modal quantification-calling it 'substitutional quantification'-as a way of reconciling the use of quantifiers and modal concepts. A major development in recent decades has been the working out of a semantics (a way of specifying the possible applications and interpretations of the formal, syntactic formulae of modal logic) that appeals to the notion of `models' or `possible worlds'.45 Such accounts of `possible worlds semantics' have been very much in the forefront of recent discussions of quantified modal logic, and serve as the framework within which most of the responses to Quine have been expressed. This type of semantics has been elaborated largely since the time when Quine first expressed his doubts about modal logic. Many philosophers and logicians think they see in these recent developments a way of answering Quine. However, it should be noted that those who rely on these recent developments do not always agree among themselves how it would be best to interpret what we are to understand by a 'possible world', and thus differ among themselves about what such an appeal amounts to. Quine, let it be remarked, is not convinced that any one or a combination of these defenses, whether in terms of essentialism, substitutional quantification, or some form of possible-w lds semantics, is really adequate to or successful in meeting his criticisms.

Let us examine' one example of these recent moves by modal logicians to defend `themselves against Quine, the views of Saul Kripke. He is one of the chief contributors to and pioneers in formulating a possibleworlds semantics (on the formal side); also, in subsequent philosophical articles-especially "Identity and Necessity" and "Naming and Necessity"-he shows how one may support essentialism.46 One of the main ideas Kripke works out for these philosophic purposes is that of 'rigid designators'. His proposals have in turn sparked a good deal of recent discussion. It will be of interest to explore these suggestions.

In order to understand the main lines of Kripke's views and wherein they differ from those of Quine, we need first to follow Kripke as he draws a number of distinctions. It is Quine's failure (as well as that of many other philosophers) to draw the following distinctions, Kripke would maintain, that accounts for the considerable confusion and disagreements that persist in the discussion of modal concepts.

Why can one say that 9 is necessarily greater than 7, whereas we cannot say that the number of the planets is necessarily greater than 7? The intuitive response is that while the number of the planets might have been different from what in fact it is, 9 could not be different from what, it is. This difference, intuitively sensed, is, for Kripke, captured and made explicit by the distinction between rigid and nonrigid designators. What, then, is this distinction? What do these terms mean?

As an example of a nonrigid designator, I can give an expression such as 'the inventor of bifocals'. Let us suppose it was Benjamin Franklin who invented bifocals, and so the expression, 'the inventor of bifocals', designates or refers to a certain man, namely, Benjamin Franklin. However, we can easily imagine that the world could have been different, that under different circumstances someone else would have come upon this invention before Benjamin Franklin did, and in that case, he would have been the inventor of bifocals. So, in this sense, the expression 'the inventor of bifocals' is nonrigid: Under certain circumstances one man would have been the inventor of bifocals; under other circumstances, another man would have. In contrast, consider the expression 'the square root of 25'. Independently of the empirical facts, we can give an arithmetical proof that the square root of 25 is in fact the number 5, and because we have proved this mathematically, what we have proved is necessary. If we think of numbers as entities at all, and let us suppose, at least for the purpose of this lecture, that we do, then the expression 'the square root of 25' necessarily designates a certain number, namely 5. Such an expression I call 'a rigid designator' . . . . What do I mean by a 'rigid designator'? I mean a term that designates the same object in all possible worlds.'
If we accept the foregoing distinction between rigid and nonrigid designators, we should say that the sentence '9 = the number of the planets', although presented as an identity, is one that contains two different types of referring expressions: '9', as a numeral referring to the number 9, is a rigid designator; on the other hand, the phrase 'the number of the planets' i's a nonrigid designator. One of the reasons Quine is able to point out the difficulties arising from combining this sentence with modal expressions has nothing to do with the use of modal expressions as such. The difficulties result from a failure to realize that if a true identity statement is to be employed in an argument and as expressing a necessary intersubstitutivity of the expressions on either side of the identity sign, it must be one in which these expressions are both rigid designators. Thus 'the square root of 25 is 4 + 1' is a necessarily true identity statement in which both referring expressions are rigid designators. So is the sentence
"Tully is Cicero".
Kripke offers further clarifications of what he means by a rigid designator as distinguished from a nonrigid designator. The difference is not to be assigned to some difference in the conventional (arbitrary) way in which certain expressions are used in one linguistic community as compared to another.
To get rid of one confusion which certainly is not mine, I do not use "might have designated a different object" to refer to the fact that language might have been used differently. For example, the expression 'the inventor of bifocals' might have been used by inhabitants of this planet always to refer to the man who corrupted Hadleyburg. This would have been the case, if, first, the people on this planet had not spoken English, but some other language, which phonetically overlapped with English; and if, second, in that language the expression 'the inventor of bifocals' meant the 'man who corrupted Hadleyburg'. Then it would refer. of course, in their language, to whoever in fact corrupted Hadleyburg in this counterfactual situation. That is not what I mean. What I mean by saying that a description might have referred to something different, I mean that in our language as we use it in describing a counterfactual situation, there might have been a different object satisfying the descriptive conditions we give for reference. So, for example, we use the phrase 'the inventor of bifocals', when we are talking about another possible world or a counterfactual situation, to refer to whoever in that counterfactual situation would have invented bifocals, not to the person whom people in that counterfactual situation would have called 'the inventor of bifocals'. They might have spoken a different language which phonetically overlapped with English in which `the inventor of bifocals' is used in some other way. I am not concerned with that question here. For that matter, they might have been deaf and dumb, or there might have been no people at all. (There still could have been an inventor of bifocals even if there were no people-God, or Satan, will do).
One way in which Kripke characterizes a rigid designator is by saying that it is an expression that designates the same object in all possible worlds. How, then, are we to understand the notion of a possible world? Very roughly, a possible world is a counterfactual situation. On Kripke's approach, the key idea in all talk about `possible worlds' is that we start with the notion of this actual world, with the objects existent in it, and build our conception of a possible world by describing or stipulating a counterfactual situation. In our description of that possible world we retain some entities or features from the actual world but not all. A possible world is not something having its own independent mode of being that we can discover apart from all reference to the actual world.
So, we do not begin with worlds (which are supposed somehow to be real, and whose qualities, but not whose objects, are perceptible to us), and then ask about criteria of transworld identification; on the contrary, we begin with the objects which we have, and can identify, in the actual world. We can then ask whether certain things might have been true of the objects.
If one accepts this approach to the concept of a possible world-and possible objects-then one of Quine's objections to the notion of possible objects can be removed. For, it will be recalled, Quine asks how one can provide a criterion of identity for distinguishing, say, the number of possible fat and/or bald. men in that doorway! The reply would be that one cannot meaningfully discuss the abstract notion of a fat or a bald man in the doorway, or for that matter several fat and/or bald men in the doorway, as abstract entities described wholly in terms of certain qualities, and still apply a criterion of identity for individual objects to these. Quine is justified in raising doubts about that conception of a possible object. However, if one starts by referring to a given (existent) fat person (or a given person who could become fat or bald) or several such actual persons, who are not at a given time standing simultaneously in that particular doorway, and asks counterfactually; "How many of these persons could fit into that doorway?" then the matter of providing a criterion of identity for these `possible' individuals is no longer unanswerable.

In the actual world, things are as they are and have the properties and relations they do. A possible world is different from the actual world either because in part the constituents of the world would be different, or, if having the same constituents, would differ in terms of some of their properties or relations. Let us consider the case where we identify an individual as existing in the actual world. There may be some possible worlds to which that individual does not belong as a constituent. For example, we can conceive (i.e., stipulate, describe) a possible world containing the two persons whom you in this actual world call your parents, but who in that possible world never do meet, never get married, and never produce you as their child. In that possible world you would not `exist', although the individual persons whom you now call your parents would `exist'. Or again, we may think of a possible world as one to which certain individual objects or persons belong, but do not have certain properties or relations that they do in the actual world. We may conceive of a possible world in which Shakespeare did not write Hamlet, or a possible world in which, as a child, I would have been given lessons in how to play the clarinet rather than the violin.
If there is some object that exists in all possible worlds, Kripke would say it is a necessary existent, or has necessary existence. On the other hand, there are some individual entities that exist in some worlds but not necessarily in all possible worlds; in that case such individual entities will be said to have contingent existence.

If we identify an individual object as existing in this world, and also conceive of one or more possible worlds to which that same individual also belongs, then the expression that serves to designate that individual both in the actual and in those possible worlds will be said to be a rigid designator.

In talking about the notion of a rigid designator, I do not mean to imply that the object referred to has to exist in all possible worlds, that is, that it has to necessarily exist. ,Some things, perhaps mathematical entities such as the positive integers, if they exist at all, necessarily exist. Some people have held that God both exists and necessarily exists; others, that He contingently exists; others, that He contingently fails to exist; and others, that He necessarily fails to exist: all four options have been tried. But at any rate, when I use the notion of rigid designator, I do not imply that the object referred to necessarily exists. All I mean is that in any possible world where the object in question does exist, in any situation where the object would exist, we use the designator in question to designate that object. In a situation where the object does not exist, then we should say that the designator has no referent and the object in question so designated does not exist.
For Kripke, a proper name is always a rigid designator, whereas other ways of referring to the individual need not be rigid designators. He claims there is a simple intuitive test for distinguishing rigid and nonrigid designators.
We can say, for example, that the number of planets might have been a different number from the number it in fact is. For example, there might have been only seven planets. We can say that the inventor of bifocals might have been someone other than the man who in fact invented bifocals. We cannot say, though, that the square root of 81 might have been a different number from the number it in fact is, for that number just has to be 9. If we apply this intuitive test to proper names, such as for example `Richard Nixon', they would seem intuitively to come out to be rigid designators. First, when we talk even about the counterfactual situation in which we suppose Nixon to have done different things, we assume we are still talking about Nixon himself. We say, "If Nixon had bribed a certain Senator, he would have gotten Carswell through," and we assume that by 'Nixon' and 'Carswell' we are still referring to the very same people as in the actual world. And it seems that we cannot say "Nixon might have been a different man from the man he in fact was," unless, of couse, we mean it metaphorically: He might have been a different sort of person (if you believe in free will and that people are not inherently corrupt). You might think the statement true in that sense, but Nixon could not have been in the other literal sense a different person from the person he, in fact, , is, even though the thirty-seventh President of the United States might have been 1~y. So the phrase "the thirty-seventh President" is nonrigid, but `Nixon , it would seem, is rigid.
Another set of distinctions overlooked by most philosophers, according to Kripke, but important in all discussions of modal logic, has to do with the contrasting pairs of terms, 'a priori / a posteriors' and 'necessary / contingent'. He points out that for many philosophers the terms 'a priori', `necessary', `analytic', and `certain' are used interchangeably to characterize one whole group of statements, whereas, it is claimed, those which are a posteriori are contingent, synthetic, and uncertain (probable). Orthodox logical positivists (empiricists) would fall into this group. On the other hand, philosophers such as Quine would deny the meaningfulness of this sharp distinction. Kripke believes both groups of philosophers (the empiricists and their opponents) to be equally at fault in not drawing certain distinctions. If accepted, the separation of a priori statements from necessary ones will provide one crucial way of dealing with the concepts of modal logic, and will also be of value in judging the elements of soundness in essentialist doctrines.
What do we mean by calling a statement necessary? We simply mean that the statement in question, first, is true, and second, that it could not have been otherwise. When we say that something is contingently true, we mean that, though it is in fact the case, it could have been the case that things would have been otherwise. If we wish to assign this distinction to a branch of philosophy, we should assign it to metaphysics. To the contrary, there is the notion of an a priori truth. An a priori truth is supposed to be one which can be known to be true independently of all experience. Notice that this does not in and of itself say anything about all possible worlds, unless this is put into the definition. All that it says is that it can be known to be true of the actual world, independently of all experience. It may, by some philosophical argument, follow from our knowing, independently of experience, that something is true of the actual world, that it has to be known to be true also of all possible worlds. But if this is to be established, it requires some philosophical argument to establish it. Now, this notion, if we were to assign it to a branch of philosophy, belongs, not to metaphysics, but to epistemology. It has to do with the way we can know certain things to be in fact true. Now, it may be the case, of course, that anything which is necessary is something which can be known a priori . . . these two notions are by no means trivially the same. If they are coextensive, it takes some philosophical argument to establish it. As stated, they belong to different domains of philosophy. One of them has something to do with knowledge, of what can be known in certain ways about the actual world. The other one has to do with metaphysics, how the world could have been; given that it is the way it is, could it have been otherwise, in certain ways? Now I hold, as a matter of fact, that neither class of statements is contained in the other.
Having drawn the foregoing distinctions-between rigid and nonrigid designators, between the actual world and possible worlds, and between a prioricity and necessity-Kripke puts these distinctions to work in dealing with a number of controversial themes arising from developments in modal logic. As examples of these applications, and to show specifically how Kripke's approach differs from that of Quine, we turn now to examine his views on essentialism (the doctrine of de re necessities) and the analysis he gives of identity statements.

With respect to the question of essentialism, Kripke makes it clear that since the question of necessity is a metaphysical question, and therefore to be kept separate from epistemological questions concerning our knowledge about something (whether our account is a priori, a posteriors, certain or uncertain, and so on), there is no reason why one should not say that, for example, a particular object has, necessarily and as belonging to it essentially (de re), certain properties.

P => Necessarily P
P
Necessarily P
The conclusion- `Necessarily P ' -is that it is necessary that the table not be made of ice, and this conclusion is known a posteriori. So, the notion of essential properties can be maintained only by distinguishing between the notions of a priori and necessary truth, and I do maintain it.53
One way in which Kripke seeks to show the value of making the distinctions he does has to do with the understanding of identity statements, particularly those that are called `contingent identity statements' and that have been a matter of great puzzlement to many philosophers. Consider the following standard argument by which some philosophers claim we are obliged to recognize contingent identity statements, whereas other philosophers consider such a result highly paradoxical and therefore argue there cannot be such contingent identity statements.

First, the law of the substitutivity of identity says that, for any objects x and y, if x is identical to y, then if x has a certain property F, so does y:

(1) (x) (y) [(x = y) => (Fx => Fy)]
On the other hand, every object surely is necessarily self-identical:
(2) (x) Necessarily (x = x)
But
(3) (x) (y) (x =y) => [Necessarily (x =x) => Necessarily (x=y)]
is a substitution instance of (1), the substitutivity law. From (2) and (3), we can conclude that, for every x and y, if x equals y, then, it is necessary that x equals y:
(4) (x) (y)((x=y) =>Necessarily (x=y))
This is because the clause ? (x = x) of the conditional drops out because it is known to be true.

According to Kripke, if we are going to make a proper assessment of the kind of identity statement some particular combination of expressions compose, we must first determine how the expressions flanking the identity sign are being used, and if we say the entire statement is true, what warrant there is for this. Are the expressions used to fix the reference of the referent? Are the terms rigid designators or nonrigid designators? Are the expressions names or descriptive phrases? Is the warrant for accepting the truth of the statement as a whole a priori or a posteriori? Finally (bearing in mind that necessity is not the same as a prioricity, or contingency the same as empirical warrantedness), is the identity necessary or contingent? Kripke's view is that some identity statements are necessary, and cannot be contingent if they consist of the use of names as rigid designators, or indeed if they consist of any type of rigid designator. Such identity statements are necessary insofar as they have to do with an objective, metaphysical de re situation, e.g., the identity of an object with itself. If, on the other hand, an identity statement is contingent, this could only be the case if the expressions used to fix the reference are nonrigid designators and do not refer to an essential property. The epistemic character of a statement-the way in which the truth of an identity statement comes to be known and warranted-does not as such tell us whether the particular identity statement is of a necessary or contingent variety since the latter difference is a metaphysical one, not an epistemic one. Some necessary identity statements are known a priori, some a posteriori; some contingent identity statements may be known a priori, some a posteriori.

Concerning the statement 'Hesperus is Phosphorus' or the statement 'Cicero is Tully', one can find all of these out by empirical investigation, and we might turn out to be wrong in our empirical beliefs. So, it is usually argued, such statements must therefore be contingent. Some have embraced the other side of the coin and have held "Because of this argument about necessity, identity statements between names have to be knowable a priori, so, only a very special category of names, possibly, really works as names; the other things are bogus names, disguised descriptions, or something of the sort. However, a certain very narrow class of statements of identity are known a priori, and these are the ones which contain the genuine names." If one accepts the distinctions that I have made, one need not jump to either conclusion. One can hold that certain statements of identity between names, though often known a posteriori, and maybe not knowable a priori, are in fact necessary, if true. So, we have some room to hold this. But, of course, to have some room to hold it does not mean that we should hold it. So let us see what the evidence is. First, recall the remark that I made that proper names seem to be rigid designators, as when we use the name 'Nixon' to talk about a certain man, even in counterfactual situations. If we say, "If Nixon had not written the letter to Saxbe, maybe he would have gotten Carswell through," we are in this statement talking about Nixon, Saxbe, and Carswell, the very same men as in the actual world, and what would have happened to them under certain counterfactual circumstances. If names are rigid designators then there can be no question about identities being necessary, because `a' and 'b' will be rigid designators of a certain man or thing x. Then even in every possible world, a and b will both refer to this same object x, and to no other, and so there will be no situation in which a might not have been b. That would have to be a situation in which the object which we are also now calling `x' would not have been identical with itself. Then one could not possibly have a situation in which Cicero would not have been Tully or Hesperus would not have been Phosphorus.
To further clarify the difference between. a necessary identity statement and a contingent one, consider the difference between 'Cicero is Tully' and `The man who denounced Cataline is the author of certain books read in third year Latin courses'. The first is a necessary identity statement, the second a contingent one.
Suppose someone uses 'Tully' to refer to the Roman orator who denounced Cataline and uses the name 'Cicero' to refer to the man whose works he had to study in third-year Latin in high school. Of course, he may not know in advance that the very same man who denounced Cataline wrote these works, and that is a contingent statement. But the fact that this statement is contingent should not make us think that the statement that Cicero is Tully, if it is true, and it is in fact true, is contingent. Suppose, for example, that Cicero actually did denounce Cataline, but thought that this political achievement was so great that he should not bother writing any literary works. Would we say that these would be circumstances under which he would not have been Cicero? It seems to me that the answer is no, that instead we would say that, under such circumstances, Cicero would not have written any literary works. It is not a necessary property of Cicero-the way the shadow follows the man-that he should have written certain works; we can easily imagine a situation in which Shakespeare would not have written the works of Shakespeare, or one in which Cicero would not have written the works of Cicero. What may be the case is that we fix the reference of the term 'Cicero' by use of some descriptive phrase, such as 'the author of these works'. But once we have this reference fixed, we then use the name 'Cicero' rigidly to designate the man who in fact we have identified by his authorship of these works. We do not use it to designate whoever would have written these works in place of Cicero, if someone else wrote them. It might have been the case that the man who wrote these works was not the man who denounced Cataline. Cassius might have written these works. But we would not then say that Cicero would have been Cassius, unless we were speaking in a very loose and metaphorical way. We would say that Cicero, whom we may have identified and come to know by his works, would not have written them, and that someone else, say Cassius, would have written them in his place.
Such examples are not grounds for thinking that identity statements are contingent. To take them as such grounds is to misconstrue the relation between a name and a description used to fix its reference, to take them to be synonyms. Even if we fix the reference of such a name as 'Cicero' as the man who wrote such and such works, in speaking of counterfactual situations, when we speak of Cicero, we do not then speak of whoever in such counterfactual situations would have written such and such works, but rather of Cicero, whom we have identified by the contingent property that he is the man who in fact, that is, in the actual world, wrote certain works.

In the foregoing we have followed Kripke's efforts to provide the kind of philosophic distinctions and analyses that would lend support to the use of such terms as `necessarily' and `possibly' as these occur characteristically in the formulae of modal logic, and their applications to ordinary discourse. Needless to say, these efforts have not themselves met with universal acceptance, and the debate goes on. In particular, Quine himself remains unconvinced; he has written:

The notion of possible world did indeed contribute to the semantics of modal logic, and it behooves us to recognize the nature of its contribution: it led to Kripke's precocious and significant theory of modal logic. Models afford consistency proofs; also they have heuristic value; but they do not constitute explication. Models, however clear they be in themselves, may leave us still at a loss for the primary, intended interpretation. When modal logic has been paraphrased in terms of such notions as possible world or rigid designator, where the displaced fog settles is on the question when to identify objects between worlds, or when to treat a designator as rigid, or where to attribute metaphysical necessity.

Our survey of analytic philosophy from Frege to Quine has concluded with a brief account of recent controversies concerning modal logic. These controversies are not to be taken, of course, as either exhausting or dominating the entire range of discussions and philosophical preoccupations of contemporary analytical philosophers. They are, indeed, only one strand in a complex web whose fabric is still very much in the process of being woven.