Modality and Reference

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Modality and Reference Richmond H. Thomason; Robert C. Stalnaker Noûs, Vol. 2, No. 4. (Nov., 1968), pp. 359-372. Stable URL: http://links.jstor.org/sici?sici=0029-4624%28196811%292%3A4%3C359%3AMAR%3E2.0.CO%3B2-L Noûs is currently published by Blackwell Publishing.

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Modality and Reference1 RICHMONDH. THOMASON YALE UNIVERSITY

ROBERTC. STALNAKER UNIVERSITY OF ILLINOIS

1. In the philosophical logic of the past twenty years or so, a great deal of attention has been devoted to puzzles having to do with quantification into intensional, or modal, contexts. Recent semantical analyses of modal logic have resulted in the solution, or at least the clarification, of a number of these puzzles. Although one may still fly from intension for philosophical reasons, these analyses make it impossible to excuse the flight by pointing to logical incoherencies. Although these semantical theories of quantification and modality have been remarkably successful in clarifying the nature of reference, we believe that a puzzle or two remain in this area; this paper is devoted to their isolation and solution. Although our discussion presupposes technical material, it will remain nontechnical; we will avoid metamathematical notation as much as possible, and will provide no proofs of metamathematical claim^.^ Our primary concern will be the application of our work to a philosophical understanding of modality and reference. 2. Formalization contributes in a number of ways to the resolution of ambiguities present in natural language. For instance, formal systems are arranged so that different senses of expressions 1 This research was supported by National Science Foundation grant GS-1567. 2 See Stalnaker and Thomason [81, where the semantical theory discussed informally here is presented rigorously, and is shown to characterize an axiomatic extension of the system Q3 discussed in Thomason [9] and 1101.

which are equivocal in natural language are represented formally by syntactically different expressions. A simple example of this is the English 'or', which has two truth-functional senses. In sentential calculi, then, there are two different connectives which may serve to translate 'or'. In cases of this sort where the ambiguous expression is regarded as complex, one is inclined to speak of a difference in use of the parts of the expression; this is typical of cases where the ambiguity may be rendered by a difference of scope. A good example of this is the f~llowing.~ ( 1 ) Socrates can run while not running.

A linguist would be likely to regard (1) as having an ambiguous syntactical structure; it may result from the addition of the modal 'can' to a sentence such as 'Socrates runs while not running', or from the addition of the adverbial phrase 'while not running' to a sentence such as 'Socrates can run'. Logicians, however, habitually arrange their formal languages so that each formula can display only one syntactical structure; thus, there are two ways of rendering ( 1 ) formally, as follows.

According to this way of construing the ambiguity in ( I ) ,there is no question of different senses of 'can7; rather, two different uses of 'can' are disclosed: in particular, two ways of construing the syntactical structure of ( I ) , in which 'can' appears in different contexts. Logicians tend to subscribe to the mathematician's policy of keeping primitive notions at a minimum; a theory should have few simple notions, and achieve richness of content through combinations of these. This has the corollary that, whenever possible, ambiguity in natural language should be explained as a difference in the use of simple expressions, rather than as a difference in their sense. All things being equal, it is better to regard ( 1 ) as translated by two different formulas, rather than as determining two different possibility-signs. Such an account has advantages over and above economy; often, for instance, it sheds light on other matters related to the 3

Aristotle, De Sophisticus Elenchus, 165825.

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original problem of ambiguity. Once we see, for example, that the difference between

( 4 ) Everyone can come along with us and

(5) Anyone can come along with us corresponds to the difference in scope between

and

(7) ( x ) 0 P ( x ) , a beginning has been made in sorting out the complexities of 'every' and 'any' in English usage. 3. A logical theory, however, must do more than resolve ambiguities; this resolution must be placed within a general and systematic metatheory. This metatheory may include a syntactical treatment of inference, using the notion of a formal proof. But to be of much philosophical interest, this should be accompanied by a semantical study of the conditions under which formulas may be regarded as true and false, By showing that circumstances may arise in which ( 3 ) is true and ( 2 ) false, such a semantical theory helps us to explain w h y a syntactical distinction is needed between them; thus, a successful semantical account will also provide a general and systematic framework within which philosophical issues may be focused and clarified. In the case of modal logic, for instance, semantical considerations serve to direct issues concerning iterated modalities to questions regarding relationships among "possible worlds", and issues concerning quantification into modal contexts to the problem of "identifying individuals across worlds". In the present paper, we require a theory adequate to deal with modality, quantification, and definite descriptions. A number of such theories have been developed in recent years, which happily do not differ in respects relevant to our purposes. We will require that formulas be interpreted in a structure consisting of "possible worlds", where associated with each such world is a domain of individuals-those individuals which "exist" in the world. Truth under an interpretation is defined relative to the possible worlds in the structure. A necessity formula, n A , is true in a given possible world a just in case A is true in all worlds which are possible with

respect to a. A universally quantified formula (x)A is true in a given possible world a just in case the formula A is satisfied in a by all of the individuals that exist in a. The traditional puzzles arise in the interpretation of formulas involving both the necessity sign and the quantifier. For example, in order to evaluate the truth of ( x )O P ( x ) in a, we must ask whether an open sentence, P ( x ) , is satisfied in possible worlds other than a by the individuals which exist in a. To answer this question, we must be able to recognize the individuals of one domain (those that exist in one possible world) as the same individuals, when they appear in another possible world. We accomplish this by presupposing in the definition of a structure that its individuals are already "identified across worlds", and in fact are the same from one world to another. That is, we suppose that an individual will be invarient under change of perspective from one world to another; it will "be" in many different worlds at once. (How this identification is carried out in special cases is a pragmatic question, which may lead to philosophical problems concerning criteria of identification. But such questions can be treated independently of the general semantical theory which is our present concern.) Following the terminology of [9], we speak of individuals, in the sense in which individuals are identified across worlds, as substances. A term that refers to the same substance in each possible world we call a substance term.4 An individual constant may or may not be a substance term, depending on whether it is more like 'Socrates' or 'Miss America'; 'Socrates' refers to the same person at all times, while 'Miss America' names a different person each year. An individual variable, however, must always be a substance term. Individual variables range over substances, and hence play a more specific logical role than do singular terms in general. Thus, e.g., 'Miss America is mortal' would not be regarded as an instantiation of a formula of the sort P ( x ) , whereas 'Socrates is mortal' would.5 4 A substance term is much like what Russell called a logically pToper name, since it is a referring expression for which the relation between term and referent is not a function of any contingent fact. We would differ with Russell about what sorts of names are examples of substance terms, or logically proper names, but the disagreement depends on differences of epistemology and pragmatics rather than logic. ' "or an example where the restriction on the role of variables makes a difference formally, note that the formula x = y 1 Ox = y is valid, while the more general schema, t = s 1 Ot = s (where t and s are any singular terms) is invalid.

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In contrast with the Russellian analysis, definite descriptions are treated as genuine singular terms; but in general, they will not be substance terms. An expression like I ,P(x) is assigned a referent which may vary from world to world. If in a given world there is a .unique existing individual which has the property corresponding to P, this individual is the referent of I .P(x); otherwise, I ,P(x) refers to an arbitrarily chosen individual which does not exist in that world.6 4. With these resources mobilized, let's consider the following example. ( 8 ) The largest state in the Union might be smaller than some state in the Union. Like ( I ) , this sentence is likely to be regarded as contradictory, unitil reflection reveals a possible ambiguity. A user of ( 8 ) , speaking several years ago, might have been using 'the largest state in the Union' to refer to Texas, and have asserted ( 8 ) while contemplating the possible admission to statehood of Alaska. Similarly, in view of constitutional provisions, one would be inclined to regard ( 9 ) The President of the U.S. is necessarily a citizen of the U.S. as true. However, circumstances can be imagined in which 'The President of the U.S. might not be a U.S. citizen' would be true; e.g., it might be used to mean that the President, Johnson, was considering resigning his office, relinquishing his citizenship and fleeing to Argentina. In sorting out the ambiguity here, it is plausible to say that, on the one hand, ( 9 ) may be used to assert, roughly, that being the President of the U.S. entails being a citizen of the U.S. Nothing is said here of any one person; what is being discussed is the ofice. On the other hand, ( 9 ) may also be used to say that a certain person, LBJ, (who happens to be President of the U.S.) is essentially, or necessarily, a citizen of the U.S. We can resolve the ambiguity by paraphrasing the sentence in the following two ways. (10) It is necessarily the case that the President of the U.S. is a citizen of the U.S. 6 The theory of descriptions sketched here is spelled out more fully in 191. For other discussions of description theory, and brief criticisms of Russell's analysis, see [2], [5], and [Ill.

(11) The President of the U.S. has the property of being necessarily a citizen of the U.S. When the difference is put this way, it is seen to correspond to a traditional distinction; (10) is an assertion of modality de dicto, and (11) an assertion of modality de re. Our problem, now, is to give a formal account of this di~tinction.~ 5. The constructions involved in (10) and (11) suggest a formal difference in t$e scope of a modal operator. We want to distinguish between it being necessary that the President of the U.S. has a certain property, and the President of the U.S. having a property necessarily. This, however, cannot be expressed, using just the usual notation of the predicate calculus; [I]Q(l .P(x) ) may be regarded equally well as the result of applying C]Q to I .P(x), or as the result of applying C] to Q(1 .P(x) ). To overcome this difficulty, we need a notation in which predication is rendered without making this presupposition. A notation of this kind is ready to hand, in the so-called abstraction operator of higher-order logic. Adopting just enough of this notation to meet our present needs, we will allow formulas of the sort ?A(t), where A is a formula and t a term. This permits us to render the differencebetween (10) and (11) as follows.

The formula (12) may be rewritten using abstracts to help make the contrast with (13) clearer.

The difference between (12) and (13) (or between (12') and 13) ) is plainly a matter of scope in (12) and (12'), the singular 7 The distinction between de dicto and de re is not entirely clear, perhaps because it has never been explicated within a fully adequate logical theory. But our formal theory seems to us to capture the intuitive content of the distinction. Where H is a modal operator, HA is a de dicto formula, since in it the modal operator is applied to a formula A, representing a dictum. On the other hand, f H A ( t ) is a de re formula since in it the modal operator is used to construct a predicate ZHA, which is then applied to a singuIar term; such a formula, then, represents the ascription of a modal property to a thing. In connection with a theory of essential predicates (i.e., of predicates P used in identifying individuals across worlds, so that ( x ) ( P ( x ) 1 O P ( x ) ) holds true), one can go far in explaining many of the things Aristotle says about modal logic in terms of this reconstruction of the distinction.

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term I .P(x) falls within the scope of the necessity-sign, whereas in (13) it does not. That is, in (13) the necessity-sign applies only to the formula Q( y ) ; in (12') it applies to ijQ( y ) ( 1 .P(x) ). We have now found a syntactical way of rendering the ambiguity represented by (10) and (11). 6. In order, however, to establish that the distinction between (12) and (13) has any logical or philosophical significance, we must provide a semantical theory which shows these formulas capable of saying different things. Since, as we remarked in 3, above, we are presupposing semantical rules which suffice to account for modality, quantifiers, and descriptions, we need only to decide the truthconditions for formulas of the kind jZA(t) to have a theory capable of settling this question. Classically, ?A(t) is true (with respect to an interpretation I ) if A is true of the object I ( t ) denoted by t. This can be stated more precisely by letting I1ct)/x be the result of adjusting I so that x is assigned the value I ( t ) . Then ?A( t ) is made true by I if and only if A is made true by I1ct)/x. The modal analogue of this is that i A ( t ) should be true in a ~ossibleworld a if A is true in a of the object referred to by t. But there is an ambiguity here. In general, to determine whether A is true in a of the referent of t we will need to know what this referent is in other possible worlds. But the referent of t may vary from world to world. If we return at this point to the intuitive meaning of i A ( t ) , it will be clear that we are interested here only in the substance denoted by t in the world a; the referent of t in other possible worlds is irrelevant. The sentence (14) The man sitting in the corner can ride a horse would ordinarily be translated by a formula of the kind ?A(t). Here, in a given situation (say, a ) , certain features of this situation are used to pick out a referent, the man sitting in the corner. Then it is asserted this person, in other circumstances, and with the appropriate desires, would ride a horse.8 Now, in these circumstances, he would, of course, not be sitting in the corner; nevertheless, he would be the same man, in a sense of "same" which presupposes that individuals can be identified from situation to situation. This 8 Though this is adequate for our present purposes, it is not a satisfactory rendering of the dispositional sense of 'can' which is at stake in (14). An adequate formalization can be obtained using the conditional connective of [6] and [7].

last presupposition, as we explained in 3, above, is already built into our semantical theory. We want to say, then, that I makes i!A(t) true in a if A is true in a of the referent I a ( t ) of t in a. As in the classical case, this can be made mathematically precise. Let ITa(t)/xbe the interpretation which is like I except in assigning the object I,(t) (i.e., the substance referred to by t in a ) to the variable x. (Note that since variables must be assigned the same referent in all possible worlds, this means that I1a(t)/~ gives x the value I,(t) uniformly to x, in every world.) Then I makes ?A(t) true in a if and only if I r a ( t ) / ~ makes A true in a. This semantical rule accords with the difference in scope between, e.g., n k P ( x ) ( a ) and k n P ( x ) ( a ) . Since in the latter formula a does not occur within the scope of a modal operator, its truth-value in a should depend only on the referent of a in a. This criterion of extensionality is built into our definition, and in consequence all formulas of the sort s = t > (i!A(s) = 2 A ( t ) ) will be valid. Thus, we have to make no changes in the principle of substitutivity of identicals salva veritate in contexts not falling within the scope of a modal operator. Also, applying our theory to examples (12') and (13) we find that these formulas are not semantically equivalent; in fact, both (15) D$Q(x) (lxP(x) ) 2 ^xUQ(x)( l d ' ( x ) ) and (16 iClQ(x) (1 ,P(x) )

3

O;Q(X) (1 ,P(x))

are invalid. Both invalidations can be carried out in a model structure in which there are just two possible universes, a and (3 (both of these universes being possible relative to one another), where the domains of each of these universes consist of just two individuals, say, the numbers 1 and 2. Furthermore, we can let P be true of 2 and false of 1in a, and true of 1and false of 2 in p, so that xP(x) refers to 2 in a and to 1in (3. To make (15) false in a, let Q be true of 2 and false of 1in a, and true of 1 and false of 2 in (3. (For example, a may be the actual universe, p a possible universe in which Mars has just one moon, and P may stand for 'is a number of the moons of Mars'. And in this case, Q may also stand for 'is a number of the moons of Mars'.) To make (16) false in a, change the above interpretation so that Q is true of 2 and false of 1in both a and (3. (E.g.,let Q stand for 'is greater than l'.)

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Thus, our semantical theory justifies the distinction we have made between (12') and (13). If we are right in regarding these as alternative constructions resolving an ambiguity in, e.g., (17) The number of moons of Mars is necessarily greater than 1, the above examples also show how our theory disarms Quine's modal "paradox7' about the number of planets. It becomes a simple fallacy of a m b i g ~ i t y . ~ Our theory also renders the equivalence valid, thus justifying our equating of (12) and (12'). The validity of (18) follows from the principle that if x does not occur within the scope of a modal operator in A, then At/x E ?A(t) is valid. 7. Most of the semantical concepts developed for the interpretation of necessity and possibility have also been applicable to tense logic, deontic logic, and the logic of propositional attitudes. The ambiguities of scope which are resolved by the abstraction operator are also present in these other kinds of modal logic, and have been responsible for some traditional sophistical puzzles. We shall illustrate the application of our distinction in these areas with a number of examples. The ambiguity in question is partly responsible for the puzzles that both ancient and modern philosophers have found in the notion of a false belief. No one believes that the unjust is just, or the ugly beautiful. Yet if I believe that the Vietnam war is just, or that Medusa is beautiful, it seems plausible that these paradoxical statements correctly describe my opinions. Quine makes this kind of example precise in an argument against quantification into belief contexts. Suppose John believes that Tegucigalpa is in Nicaragua. Then there is something that John believes to be in Nicaragua, and that something is the capital of Honduras. Although John would never assent to the proposition that the capital of Honduras is in Nicaragua, there is a sense in which John believes that the capital of Honduras is in Nicaragua. This sense is formalized as follows: 9 This account supplements the one given of Thomason 191, where the force of the paradox is attributed to the fact that this invalid argument is an instantiation of a valid argument. The two accounts are closely connected, inasmuch as the failure of the principle of instantiation and of the principle of abstraction are two sides of the same coin.

The claim formalized by the following formula, however, is false: (20) B?Q(x) ( 1 ,P(x)). This last statement is false since John knows, of course, that the capital of Honduras is in Honduras, and not in Nicaragua. The ambiguity may be resolved in English by reading (19) as "John believes of the capital of Honduras that it is in Nicaragua." In a sense, tomorrow never comes, since it is always today. Yet in a sense, tomorrow will come. In fact, it will come tomorrow. The referent of a description may change with time, and thus the same kind of ambiguity arises in a language which has tenses. In one sense, I am quite positive that the President of the United States will never be a woman. I n another sense, I am willing to admit that the proposition might be debated. These distinctions can be made in the same way, by means of abstracts. We know of no stock puzzles of this kind from deontic logic, but again these distinctions prove to be useful. If, for instance, I have found a watch, there are two senses of (21) I am obliged to return this watch to the man who owns it. The more natural one would be rendered by a formula such as (22) OQ(a,q .P(x, a ) ); in this sense, (21) is true. However, in the sense which corresponds to (23) 30Q(x, a ) (lxP(x, a ) ), (21) is false. (This would be better expressed in English by 'It is true of the man who owns the watch that I am obliged to return it to him.') Indeed, unless the owner of the watch identifies himself to me as its owner, I should not return it to him. This example illustrates nicely how epistemic considerations influence obligations; what we do not know can multiply deontic possibilities, and thus render formulas such as (23) false. (I.e., (23) is false because since I do not know who the owner is, there is a deontically possible world in which the owner is Jones, a deontically possible world in which the owner is Smith, etc. Thus, I am not obliged to return it to anyone until further knowledge is obtained.)

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Contrary-to-fact conditionals may be regarded as statements about non-actual possible situations. Consequently, similar puzzles arise in connection with such conditionals. Goodman, in his early paper on counterfactuals, discusses a special class of counterfactuals which he calls counter-comparatives. For example, consider

( 2 4 ) If I had more money, I'd buy a car. "The trouble with these", Goodman wrote, is that when we try to translate the counterfactual into a statement about a relation between two tenseless non-modal sentences, we get an antecedent something like If "I have more money than I have" were true . . . which wrongly represents the original antecedent as self-contradictory.1° The problem is that only one of the amounts of money being compared is the amount of money I have in the situation referred to by the antecedent. Using abstracts, we can formalize the conditional as follows: Here, Q stands for the relation 'is greater than', P for 'is a total amount of money I have', and R for 'I buy a car'. The corner, >, is the conditional connective. 8. In most examples that some come readily to mind, the distinction of scope that we have been discussing could be represented by using identity and quantification rather than abstracts. For example, in case there is in fact a unique city which is a capital of Honduras, then the following two propositions are equivalent:

( 2 6 ) John believes of the capital of Honduras that it is in Nicaragua. ( 2 7 ) There is a city which is identical to the capital of Honduras, and John believes that it is in Nicaragua. This follows formally from the general principle that if ( 3 x ) x = t is true, then ( 3 x ) ( x = t A A ) = % A ( t )is true.ll 10 Goodman

[I], p. 16. [3], pp. 155ff. and [4], pp. 46ff.; uses identity and quanta(a). cation to make the distinction we formalize using, e.g., O P ( a ) and fuP(x) These works contain, however, no explicit theory of descriptions; we consider the theory presented in [21 unsatisfactory. 11 Hintikka

This suggests that our introduction of abstracts may be superfluous, since 2A(t) might be replaceable by a contextual definition. But there is a flaw in this; the definition is plausible only in those cases in which ( 3 x)x = t is true, i.e., only in those cases in which the referent of t exists. In other cases, ( 3 x) ( x =t tA) is false, and thus, though the "contextual definition" is satisfactory in most cases which arise in practice, it is in general unsatisfactory, for the same reasons that render Russell's theory of descriptions unwelcome. Thus, this attempt to define abstraction fails, and in fact it can be proved that any such attempt will fail; abstraction, as we have characterized it semantically, is not definable in the modal logic 43. 9. Before concluding our discussion of abstracts and their use in modal logic, there is a further point which merits consideration. In devising a notation to resolve certain ambiguities which we found in natural language, we found we needed a syntactical device which would distinguish between applying U P to a, and applying to P ( a ) , and we used ? u P ( x ) ( a ) and n P ( a ) for this purpose. This notation is familiar, as is the term "abstract", which we have also used. However, we have not justified our use of this familiar terminology; do formulas of the sort f n P ( x )( a ) , as we have interpreted them, really involve abstracts? This is a legitimate question, since according to our theory the familiar law of abstraction, fails.12 To answer it, we must show that our theory is embeddable within a plausible second-order modal logic. In such a logic, can abstracts be regarded as referring to properties? A detailed discussion of the semantics of second-order modal logic is beyond the scope of this paper; however, it is not difficult to see, informally, how we can interpret formulas of the sort 2A(t) as expressing the result of applying a property, or propositional function, referred to by EA to an object referred to by t. The modal analogue of the classical notion of a propositional function (of one argument) is a function taking possible worlds and substances into truth values. Thus, if f is a propositional function, a a possible world and d an object, then f ( a , d ) = T or f ( a , d ) = F. Singular

free

IqIlere, At/x is the result of replacing all occurrences of x in A by occurrences of t (relettering bound variables, if necessary, to avoid rendering cation to make the distinction we formalize using, e.g., u P ( a ) and x n P ( x ) ( a ) . any of the variables occurring in the new occurrences of t bound).

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terms, on the other hand, will correspond with what are called world-lines in 191: functions taking possible worlds into objects. If h is such a function and a a possible world, then h ( a ) = d for some object d. (Note that since individual variables range over substances, an individual variable x must always be assigned a constant world-line: a world-line h such that for all worlds a and @ h ( a ) = h ( @ ). ) Now, suppose that 2A is correlated with such a propositional funtion, f, and t with a world-line h. Then, f applies to h in a possible world a if f ( a , h ( a ) ) = T, and ?A(t) is true in a if f applies to h in a. This understanding of things generates precisely the semantical rules we developed in 6, above. And this shows, we believe, that the similarity of our notation to that of higher-order logic is not merely accidental. What we have developed here is a correct fragment of higher-order logic. This fragment is, nevertheless, entirely first-order in character, since abstracts appear only in contexts of the sort 2A( t ) . In view of the modal failure of (28), however, such formulas are required in first-order modal logic for expressive completeness. 10. Logicians call classical logic "classical" not because of any affinities with antiquity, but because it serves as a locus classicus, or paradigm for research. In developing a novel logical theory, one loolzs for the right analogues of classical results, and then tries to establish these; failure to do this is often a sign that something has gone wrong. But, of course, if the material is really new, it will differ in some respects from the classical case; one cannot expect all classical results to carry over. Thus, one of the most important prerequisites of successful research is a good feel for what is "important" and "natural" in classical logic, a sense for distinguishing those features which should be expected in new material from those which may be discarded. We can now see that the major impediment to the development of a successful theory of quantified modal logic has been lack of success in making distinctions of this sort: in particular, in realizing which features of classical logic are mere special cases, and need not hold generally. In (91, it was argued that it was failure to dispense with the principle of instantiation, (x)A > At/x, which hindered a successful generalization of classical logic; here, we have shown that this may equally well be regarded as failure to discard the principle (28) of abstraction.

1. Goodman, N. Fact, Fiction, and Forecast. (Cambridge, Mass.: Harvard University Press, 1955). 2. Ilintikka, J. "Towards a Theory of Dofinite Descriptions," Analysis, XIX ( 1959) : 79-85. 3. Hintikka, J. "Knowledge and Belief." (Ithaca, New York: Cornell University Press, 1962). 4. Hintikka, J. "Individuals, Possible Worlds, and Epistemic Logic," h'ous, 1 ( 1 9 6 7 ): 33-62. 5. Scott, D. "Existence and Description in Formal Logic," Bertrand Russell: Philosopher o f t h e Century, ed. R. Schoenman. (Boston: 1967): 181-200. 6. Stalnaker, R. "A Theory of Conditionals," Studies in Logical Theory (American Philosoplzical Quarterly supplementary monograph series), forthcoming. 7. Stalnaker, R. and R. Thomason. "A Semantic Analysis of Conditional Logic." Mimeo., 1967. 8. Stalnaker, R. and R. Thomason. "Abstraction in First Order Modal Logic," Theoria, forthcoming. 9. Thomason, R. "Modal Logic and Metaphysics," T h e Logical W a y of Doing Things, ed. K . Lambert. (New Haven, Conn., 1968). 10. Thomason, R. "Some Completeness Results for Modal Predicate Calculi," Philosophical Deaelopments in hTon-standard Logic, ed. K. Lambert, forthcoming. 11. Van Fraassen, B, and K. Lambert. "On Free Description Theory," Zeitschrift fur nzatheinatische Logik und Grundlagen der Mathematik, 13 ( 1 9 6 7 ) : 225-240.