A Semantic Theory of Adverbs

A Semantic Theory of AdverbsAuthor(s): Richmond H. Thomason and Robert C. StalnakerSource: Linguistic Inquiry, Vol. 4, No. 2 (Spring, 1973), pp. 195-220Published by: The MIT PressStable URL: http://www.jstor.org/stable/4177764Accessed: 23/03/2010 20:25

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Linguistic Inquiry Volume IV Number 2 (Spring, 1973), 195-220.

Richmond H. Thomason A Semantic Theory of Adverbs* Robert C. Stalnaker

i. Introduction

In recent years adverbial constructions have attracted the attention of many logicians, philosophers of language, and linguists. Most of this work is relevant to the content of the present article, but that of Richard Montague and his associates deserves special discussion since our formal semantic theory can be regarded as a special case of Montague's.'

Montague's semantics for adverbs was a component of a general program for developing a formal semiotic theory of natural languages such as English. The task of the syntactic part of the program is to develop a recursive definition of the sets of phrases of various syntactic categories of expressions (e.g. noun phrases, intransitive verb phrases, adverb phrases, sentences) of a fragment of English. The semantic part of the program furnishes an interpretation of this fragment in terms of intensional model theory, and the pragmatic part deals with the interpretation of context- dependent or "indexical" expressions such as I, here, and now. In Montague's linguistic writings2 the general semiotic program is the center of attention, and detail is not lavished on specifics such as the treatment of adverbs. One function of this paper is to remedy this defect.

Also, though we accept the main features of Montague's semiotic program, we have not adopted its methodology here, but instead have followed a more conservative approach emerging from recent work in intensional logic. According to this approach, artificial logi-al languages are used as the objects of formal semantic theory, and are

* Each of the authors would like to thank the other for pointing out numerous errors in earlier drafts of this article and making many helpful suggestions; but he assures the reader that his coauthor is not responsible for any errors that remain.

We are indebted to the Council for Philosophical Studies for their sponsorship of a summer institute in the philosophy of language, where many of the ideas of this article took shape, and to George Lakoff for an extensive and provocative correspondence. This research was supported by the National Science Foundation under Grants GS-25I 7 and GS-2574.

We owe special thanks to two referees of this paper who took pains to provide us with extensive comments and criticisms. This version of the paper owes much to their care and helpfulness.

1 Of Montague's papers, Montague (I 970a) devotes most attention to adverbs. See especially pp. 212-214 of this work.

2 All of these writings are listed below in our bibliography. Some of these are difficult to obtain, but this should be corrected by the end of I973, when the Yale University Press plans to publish an anthology of Mon- tague's papers.

i96 RICHMOND THOMASON AND ROBERT STALNAKER

linked to natural languages by an informal procedure of formalization.3 The application of semantic theories to English is therefore indirect, rather than direct as in Montague's approach, since it is mediated by formalization.

This makes our account of adverbs in natural language less rigorous than Mon- tague's, but at the same time it enables us to give a freer and more open-ended discussion of semantic problems arising in connection with adverbs. By being less direct in relation to English syntax we can consider problematic and puzzling phenomena, as well as those that are more perspicuous.

2. Logical Form and Adverbial Constructions

Adverbs are notoriously resistant to perspicuous formalization in first-order logic, which apparently requires them to be petrified components of the predicate.4 For instance, if (I)

(i) John walks

is formalized by Pa, where P stands for walks and a for john, then we must, it seems, formalize (2)

(2) John walks slowly

by Qa, choosing a new predicate parameter, say Q, to stand for walks slowly. Perhaps this is not as bad as rendering (i) by Pa and John doesn't walk by Ra, but still one feels that Qa is unfaithful as a formalization of (2). The complex structure of the verb phrase walks slowly, with its explicit relation to the verb phrase walks, has been erased. Logicians, who are interested in accounting for valid inferences, may make the stronger objection that this formalization will not make (i) a logical consequence of (2). We will argue below, in Section 8, that this objection is questionable. But whether or not it stands up, the need for a more faithful logical translation is clear.

An option explored by Donald Davidson is to seek a more sophisticated formalization in first-order logic itself. But Davidson's theory is explicitly narrow in its sCope, adverbs such as slowly, greedily, carefully, and intentionally being excluded; and we suspect it may prove to be even more narrow than he anticipated. There remains, then, a need for a general semantic theory of adverbs. It is in response to this need that we

3 Formalization is the procedure of translating statements of a natural language into formulas of an artificial language for the purpose of evaluating arguments using the statements, exposing ambiguities in them, or revealing their "true logical form". The procedure is informal, since the rules for carrying it out are never made fully explicit. One must use his intuitive understanding of the content and structure of the given statements. An adequate formalization must yield a formula that has the same truth conditions as the given statement, but beyond this, the standards of adequacy are unclear. All one can say is that the "relevant structure" of the given statement should be reflected in the formalized equivalent. Although this standard is unsatisfactory as an abstract account of what formalization is, in concrete cases the procedures for formalizing and evaluating formalizations are often unproblematic.

4 For discussions of the problem, see Reichenbach (1947), Davidson (I968), Clark (1970), Quine (1970), and Parsons (1970).

A SEMANTIC THEORY OF ADVERBS

propose to construct our extension of first-order logic. Research in other areas, e.g. in the logic of tenses, has shown that it can be rewarding to build such extensions, adding new notation to the underlying formal language as it is needed to formalize a particular kind of discourse. The methodological rule governing such projects is that they must be carried out with the same degree of rigor associated with familiar logical theories. In particular, all new notation must be provided with an acceptable semantic interpretation, or model theory.5

We now return to (2). The obvious way to handle this sentence is simply to add a new piece of logical notation to stand for slowly. Then the formalization of (2) will be obtained from that of (i) by adding a new symbol, say e, to Pa. But how to add it ? Since slowly modifies walks, not John or Joohn walks, something like (3) seems good.

(3) ePa The task of formalizing (2) is then just a matter of transposing word order and replacing English with logical vocabulary. John walks slowly becomes Slowly walks John, which then becomes EPa.

This is nice, but not yet quite right. The trouble is that it gives ePa the same syntactic structure as Pa, standing for the English sentence John does not walk. But our grammatical intuitions suggest that these should be distinguished from one another. The formula -Pa represents the negation of a sentence formed by attaching a predicate to a subject. But ePa is the result of modifying a predicate with an adverb, and then applying the modified predicate to a subject. So Pa and ePa have the following two structures.

Pa {Pa

Pa ep a

P / \ a %/ P

In a well-designed logical language such distinctions are built into formulas themselves. What we need, then, is something like (6P)a. This notation is more perspicuous as to syntactic structure, but is too restrictive because it provides no way of applying adverbs to complex predicates. There is no way, for instance, to formalize examples such as the following two.

(4) With his pencil, John doodled and took notes. (5) John walked or was carried to the hospital.

At first glance this may not seem a serious limitation, since (4) could be simply assigned the logical form (6P)a A (Q )a and (5) the form (eP)a v ( Q)a. Here the

5 As a branch of mathematical logic, model theory originates in the work of Tarski. Kalish (i 967) contains an introduction to model theory suitable for nonspecialists.

i98 RICHMOND THOMASON AND ROBERT STALNAKER

fact is exploited that (4) can be paraphrased as (6) and (5) as (7):

(6) John doodled with his pencil and John took notes with his pencil. (7) John walked to the hospital or John was carried to the hospital.

Some use of paraphrase in the application of a logical calculus to English is probably inevitable, but there is a general methodological objection to this strategy. When a sentence has to be paraphrased before being analysed, the task of interpreting natural language becomes less formal, as stress is shifted from formal semantics, where problems can be dealt with explicitly at the theoretical level, to our pretheoretic and unarticulated understanding of spoken language. Paraphrase sweeps semantic problems under the rug, and whenever we can see how to avoid using it we should do so.

In this case, however, there is a more specific objection: the paraphrases do not always work. For example, consider (8) and (9).

(8) Reluctantly, John bought gas and had the oil changed. (g) John intentionally kissed Mary or kissed Susan.

Sentence (8) can be true because it was doing both that John disliked-say, because it cost too much-while he was not reluctant to do either separately. Similarly, John could have intentionally kissed Mary or Susan by kissing the girl in the mask, knowing that the girl in the mask was Mary or Susan. But in this case he neither intentionally kissed Mary nor intentionally kissed Susan.

The structure of (g) is something like this.

(Io) John intentionally kissed Mary or kissed Susan

John intentionally kissed Mary or kissed Susan

intentionally kissed Mary or kissed Susan

kissed Mary or kissed Susan

To capture this structure we need some formal way of building up complex predicates, such as kissed Mary or kissed Susan. Diagram (i o) suggests that this predicate is obtained from kissed Mary and kissed Susan by an operation of disjunction. But in our logical language the symbol v, corresponding to or, is a sentence connective. It links formulas, not predicates.

It would be possible to solve this particular problem by creating another kind of disjunction acting on predicates. But there is a far more general and deeper solution. This consists in regarding the predicate kissed Mary or kissed Susan as derived from the sentence He kissed Mary or he kissed Susan by a process whereby predicates are formed from sentences. The relevant part of (io) will then appear as follows.


kissed Mary or kissed Susan

he kissed Mary or he kissed Susan

/Z I \ he kissed Mary or he kissed Susan

There is a familiar logical device, known as abstraction, which performs precisely this task. Abstraction has been used by logicians primarily in formalizing set theory (and especially in higher-order logics), but recent work suggests that it may also have a direct bearing on problems having to do with natural language.6 Abstraction can be added to our formal language by stipulating that if A is any formula then (2Ax/u) is a predicate,7 and that if X is a predicate and t an individual term then X(t) is a formula.8

Abstraction yields a direct and natural formalization of sentence (9): (6x(Pxb v Pxc)) (a), where 6 stands for intentionally, Puv for ... kissed , a for Joohn, b for Mary, and c for Susan. This formula has the following syntactic structure.

(ex(Pxb v Pxc))(a)

(6x(Pxb v Pxc)) a

7 K - x(PxbIv Pxc)

Pub v Puc

Pub v Puc

3. Scope and Adverbs

The difference between (7) and (i i)

( II) John intentionally kissed Mary or John intentionally kissed Susan.

emerged in our logical language as a difference in scope, in the order in which formation rules are applied in constructing the formulas (6X(Pxb V Pxc)) (a) and (6XPxb) (a) V

(6Pxc) (a). Such formal differences of scope frequently correspond to distinctions with 6 We refer here to work of Barbara Partee's, as yet unpublished, concerning extensions of Montague

grammar. Also see Thomason and Stalnaker (I968) and McCawley (1970).

7 Ax/u is the result of replacing all occurrences of the individual parameter u in the formula A by occurrences of the individual variable x; for example, Puxfu is Px. When the notation 'Axlu' is used, it is presumed that u is free for x in A, i.e. that no occurrences of u in A are within the scope of an abstraction operator x or a quantifier (x). See Thomason (1970, I49-I52, 175-178) for further discussion of the syntactic terminology used here. We will sometimes drop parentheses where doing so does not result in ambiguity, and occasionally will add them where this makes a display easier to read.

8 This introduces some of the apparatus of second-order logic, but even so our amended language remains essentially first-order. The distinctive element of second-order logic, second-order quantification, has not been added.

200 RICHMOND THOMASON AND ROBERT STALNAKER

which speakers of English are familiar, and which are expressed in English in a variety of ways. For example, the sentences (I2) and (I3) will most naturally be formalized by ( I4) and (I 5) respectively.9

(I 2) He slowly tested all the bulbs. (I 3) He tested each bulb slowly.

(I 4) (6X(y)Pxy) (a) (I 5) (y) (6XPxy) (a)

It is important to realize that the truth conditions of (I 2) and (I 3) differ. Sentence (I 2) would be true if he took a long coffee break between each testing, even though he tested each single bulb quickly. His testing of all the bulbs would then be slow. The difference between (i 6) and (I7) is represented similarly.

(i 6) He slowly tested some bulbs. (I 7) He tested some bulbs slowly.

In many cases like these there seems to be no difference in meaning between the two scope readings of a single English sentence, even though the strategy of formalization predicts the existence of two such readings. An example is (i 8), which can be rendered as (I9) or (20)

(i8) John took all his friends to the ballgame.

( I 9) (X) 6jPyx (a) (20) (6j(x)Pyx) (a)

where 6 stands for to the ballgame, Puv for . .. takes , and, for simplicity, the quantifier ranges over John's friends. In such cases, the absence of a difference in meaning may be ascribed to particular semantic properties of the lexical items appearing in the example: in this instance, to properties of to the ballgame.

In other cases, different scope readings of a single sentence will correspond to significant ambiguities. An example due to Lakoff is the sentence (2I):

(2I) Harry was willingly sacrificed by the tribe.

Let Puv stand for . . . sacrificed , a for Harry, and b for the tribe. Then the sense of (20) in which the tribe was willing appears as (22) and the sense in which Harry was willing as (23).

(22) ( eXPxa) (b) (23) ( 6XPbx) (a)

4. Predicate Modifiers and Sentence Modifiers

In Section 2 we took it as intuitively evident that the proper parsing of John walks slowly is john (walks slowly), not (John walks) slowly. That is, we assumed that the

9 For simplicity here, the quantifier has been taken to range over the bulbs.


adverb modifies the verb rather than the sentence. This may seem an innocent assumption; slowly is, after all, an ad-verb and not an ad-sentence. But there are many so-called adverbs, like possibly, probably, usually, and unfortunately, for which the alterna- tive parsing seems more appropriate. The distinction between the two parsings is no mere matter of convention since it influences logical relationships among sentences containing adverbs. We will therefore try to back up our intuition with arguments and general criteria.

We shall argue that there are two kinds of adverbs in English; some modify predicates, while others modify sentences. Since a unified theory of adverbs would be simpler than ours, the burden of proof lies on us to make this claim good. We shall first give some general reasons why it is important to analyze the scope of a modifier correctly and second give some specific arguments for the claim that there are modifiers of both kinds in English. These will in turn yield criteria for classifying various adverbs either as sentence modifiers or as predicate modifiers.

It is obvious that if the scope of a modifier is analyzed as narrower than it should be one will get a defective account of the job done by the modifier in some sentences. If, for example, negation were treated as a predicate modifier, one would have no way of representing the distinction between the two readings of (24)

(24) Everybody didn't come,

one of which can be paraphrased by (25)

(25) Nobody came,

and the other by (26)

(26) Not everybody came.

In order to get the second reading, one must be able to include the subject of (24),

everybody, within the scope of the negation, and this is possible only if negation can operate on sentences.

It is less obvious, but equally true, that if the scope of a modifier is analyzed as wider than it should be, one will get an incorrect account of the logical relationships among sentences containing the modifier and of the syntactic ambiguities in such sentences. For example, if slowly is a predicate modifier, then John walks slowly exhibits the form X(a), the general form of a subject-predicate sentence. Hence, since (27) is a valid form, the inference in (28) is valid.

(27) X(a) a=b

X(b)

(28) John walks slowly. John is the mayor of New York.

Therefore the mayor of New York walks slowly.


But if slowly were analyzed as a sentence modifier, then John walks slowly would not be an instance of X(a), and so the argument would not be valid in virtue of (27). The analogous argument form for sentence modifiers, (29)

(29) o(Fa) a =b

o(Fb)

is notoriously invalid, as is shown by examples from the philosophical literature on opacity such as the following.

(30) Necessarily nine is odd. Nine is the number of the planets.

Therefore, necessarily the number of the planets is odd.

On the relevant reading of the conclusion, it is false, even though the premisses are true.10

We could, of course, treat adverbs like slowly as sentence modifiers and account for the phenomena by introducing special semantic principles, or meaning postulates, to validate inferences that seem valid for such adverbs. But as a general methodological policy it is best, all else being equal, to explain semantic relationships in terms of structure rather than in terms of the unanalyzed content of specific words. Thus the apparent validity of the substitution principle when applied to contexts containing a particular adverb will be treated as prima facie evidence that such adverbs are predicate modifiers. This evidence, together with the absence of any counterevidence, is the principal justification for treating slowly and other manner adverbs as predicate modifiers.

What kinds of counterevidence will show other adverbs to be sentence modifiers ? We will give four semantically based criteria for identifying sentence modifiers, and show why these criteria establish that a great many adverbs and adverbial phrases are in fact sentence modifiers. These criteria involve semantic notions such as scope, ambiguity, and paraphrase, and they will depend for their application on only the the most general and elementary assumptions about English syntax.

First, if an adverb precipitates counterexamples to the substitution principle, this will establish it to be a sentence modifier.

10 In classical first-order logic, any sentence containing a name a is treated as having the form X(a). Even though negation is a sentence operator, Pa is treated as an instance of the form X(a) and the inference of Pb from -Pa and a = b is regarded as valid in virtue of the general substitution principle. Generalizing from this model, one might treat any sentence of the form F(Pa) as having the form X(a), and regard the counterexample as showing that the substitution principle is not valid. But this, we think, would be a mistake. The only reason that it is harmless in first-order extensional logic to treat any sentence containing a name as an instance of X(a), is that for the restricted language Aalu is always logically equivalent to (xEAxIu)(a), which explicitly has the form X(a). But that equivalence does not hold generally, as we have argued in Thomason and Stalnaker (i 968).


Criterion i Only if an adverb is a sentence modifier can it give rise to opaque contexts everywhere in a sentence in which it occurs.

Even though Richard Nixon is the President of the United States, the sentence On a number of occasions Richard Nixon has died in office is false while On a number of occasions the President of the United States has died in office is true. This shows that on a number of occasions modifies the sentence The President of the United States has died in office and not just the predicate has died in office. Modal adverbs like necessarily and probably and adverbs of attitude like unfortunately are shown by this test to be clear cases of sentence modifica- tion. Some locative adverbs like in several restaurants also seem to create opaque contexts. In several restaurants, the maitre d' wears a tuxedo may be true, even if there is no one who wears a tuxedo in several restaurants.

One must use a certain amount of care in applying this criterion. First, in order to be sure that it is the adverb being tested that is responsible for substitution failure, one should find a sentence that would have no opaque contexts if the adverb were removed. Second, in order to be sure that the adverb can give rise to opaque contexts everywhere in the sentence, one should find a sentence with all its singular terms in an opaque context created by the adverb. An adverb may give rise to opacity within a logical predicate and still be a predicate modifier."1 For example, John willingly trusted Harry may be true while john willingly trusted his worst enemy is false (on one reading of the latter sentence), even if Harry is John's worst enemy. From this one can conclude that the object of trusted must be within the scope of the adverb willingly in those sentences, but one cannot conclude that John must be within its scope. And unless john is within its scope it is not modifying the whole sentence. Hence the example fails to show that the adverb is a sentence modifier.

If the opacity criterion fails to apply, this is no proof that the adverb is not a sentence adverb, for it is possible for an adverb to be, like negation, a referentially transparent sentence modifier. The adverb actually is a paradigm of a transparent sentence modifier.

The second criterion succeeds in showing negation to be a sentence modifier.

Criterion 2

Only if an adverb is a sentence modifier can it give rise to quantifier scope ambiguities in simple universal or existential sentences.

In particular, if there can be a semantic contrast between Q-ly someone F's and Someone F's Q-ly then Q-ly is a sentence modifiier. So, for example, Frequently, someone got drunk contrasts with Someone got drunk frequently (on one reading of these sentences). The

11 By a logical predicate, we mean a part of a sentence that corresponds semantically to a propositional function. From the sentence Dick beats Hubert one may abstract two singulary logical predicates: being one who beats Hubert and being one who is beaten by Dick. Or, using the abstraction operator, x(x beats Hubert) and ?(Dick beats x).


semantic difference can be explained only on the assumption that frequently is capable of modifying the sentence Someone got drunk, and hence is a sentence adverb.12

By a simple universal or existential sentence, we mean one containing only one quantifier and no singular terms. The claim is restricted to sentences which are simple in this sense because it is not meant to cover scope ambiguities which result from quantifiers within a logical predicate. For example, the contrast between Sam carefully sliced all the bagels and Sam sliced all the bagels carefully does not show carefully to be a sentence adverb. For this contrast can be explained by the difference between (3I)

and (32)

(32) (x)(Pyx)(a) (32) (X) ((ejPyx) (a))

where Puv stands for sliced, a for Sam, and 6 for carefully. These two formalizations capture the required distinction, and in neither of them is carefully represented as a sentence modifier.13

But again, this test is not decisive. It is possible for an adverb to be a sentence adverb, and yet operate transparently through quantifiers. There is, for instance, apparently no semantic contrast between Actually, someone got drunk and Someone actually got drunk.

Criterion 3 If an adverb includes within its scope an adverb or adverbial phrase that has already been shown to be a sentence modifier, and if the whole rest of the sentence is within the scope of that sentence modifier, then the original adverb is also a sentence modifier.

The truth of this claim is obvious, even trivial. To apply it, it is useful to use conditional sentences, since if clauses are clearly shown by Criterion i to be sentence modifiers.'4 If a conditional sentence begins with an adverb, and if one cannot paraphrase the sentence by putting the adverb in the consequent, then there is reason to conclude that the initial adverb modifies the sentence following it.

12 There is no need to classify frequently as a predicate adverb as well as a sentence adverb in order to account for the ambiguity between Frequently someone got drunk and Someone got drunk frequently. These can be formalized by F(3x)Px and (3x)FPx, respectively. In the first formula F modifies a closed formula, (3x)Px; in the second, it modifies an expression, Px, derived by substituting the variable x for u in the open formula Pu. But in both cases F acts as a sentence adverb.

13 G. Lakoff, in Lakoff (1970, 233-234) uses the contrast between Sam carefully sliced all the bagels and Sam sliced all the bagels carefully to argue that manner adverbs like carefullv must be represented as sentence operators, not as operators mapping predicates into predicates. His argument, however, fails to take account of the pos- sibility that the contrast between the two sentences can be explained in terms of the contrast between two predicates in the way we suggest.

14 To illustrate this application of Criterion i, suppose it is in fact true that John Connally is the man who will be elected President in 1976. On one reading of If Daniel Berrigan will be elected President in 1976, then the man who will be elected President in I976 will be a priest this sentence is true. But it does not follow that John Connally will be a priest if Daniel Berrigan will be elected President in 1976.


For example, the two sentences (33) and (34) differ in meaning.

(33) Frequently, if John walked to school Mary walked to school with him. (34) If John walked to school, Mary frequently walked to school with him.

On the other hand, (35) is unacceptable.

(35) Slowly, if John walked Mary walked with him.

If one forces a reading on it, it will be the same as that given to (36):

(36) If John walked, Mary slowly walked with him.

Locatives and temporal modifiers show themselves to be sentence modifiers by this criterion, even though many of them fail the first two tests. For example, (37) and (38) differ in meaning, as do (39) and (40).

(37) In the morning, if John is told to walk then he walks. (38) If John is told to walk, then he walks in the morning. (39) In that restaurant, if John is asked to wear a necktie, he wears a necktie. (40) If John is asked to wear a necktie, he wears a necktie in that restaurant.

The fourth criterion involves paraphrase.

Criterion 4 Only if Q-ly occurs as a sentence modifier can one paraphrase the sentence by deleting the adverb and prefacing the resulting sentence by It is Q-ly true that.

Since the form of the paraphrase is an adjectival phrase that modifies a nominalized sentence, the acceptability of the paraphrase would seem to be strong evidence that the adverb modifies a sentence.

Again, this test works well on our paradigm cases. The sentences (4I) and (42) are alike in meaning.

(40) Sam frequently sucks lemons. (42) It is frequently true that Sam sucks lemons.

But (43) is clearly deviant, and not an acceptable paraphrase of (44).

(43) It is slowly true that Sam sucks lemons. (44) Sam slowly sucks lemons.

To take a slightly different example, although (45) is not a deviant sentence, it differs in meaning from (46).

(45) It is happily true that Sam sucks lemons. (46) Sam sucks lemons happily.

According to our fourth criterion, this is evidence that happily is ambiguous, having one sense (roughly equivalent to fortunately) that is a sentence modifier and another (roughly equivalent to gladly) that is a predicate modifier.

2o6 RICHMOND THOMASON AND ROBERT STALNAKER

Even though (47) is somewhat awkward, it is certainly grammatical, and means the same thing as (48).

(47) It is true in the morning that Mary beats her dog. (48) Mary beats her dog in the morning.

But like (43), (49) is deviant and does not qualify as an acceptable paraphrase of anything.

(49) It is true with a stick that Mary beats her dog.

Locative adverbs seem to be borderline cases for this criterion. Sentence (50) is at best awkward as a paraphrase of (5I):

(50) It was true in the kitchen that Henri dropped the souffle. (5I) Henri dropped the souffle in the kitchen.

But it is not as manifestly unacceptable as (52).

(52) It was true at his assistant that Henri threw the souffl.

And some sentences similar to (50), such as (53), seem perfectly acceptable.

(53) It is true in several restaurants that women in trouser suits will not be admitted.

All our criteria have been presented as sufficient conditions of being a sentence modifier. Though it is strong prima facie evidence that an adverb is a predicate modifier if it fails all four tests, we have no conclusive criteria that will prove it is not a sentence modifier. Criterion 4, however, comes close to being a necessary and sufficient condition. If the operation of this criterion results in utter nonsense, such as (43), (49), and (52), we believe that this is good evidence that the adverb is a predicate modifier.

All four criteria are semantic and are justified by semantic arguments. While we feel these arguments are decisive in many cases, in others they are not entirely conclusive. It may be possible to use syntactic evidence to supplement our arguments and criteria. If syntactic features can be found that distinguish the clear cases of predicate modifiers from the clear cases of sentence modifiers (as distinguished, in part, by the above criteria), then these features might help to decide cases on which the semantic evidence is indecisive.15

15 Another test for separating predicate from sentence modifiers is suggested in Geach (I970). Geach points out that passionately and presumably is "syntactic nonsense" and suggests that this will apply to any conjunction of a predicate with a sentence modifier. But this test is unreliable. On the one hand there is nothing peculiar about passionately and often, as in He spoke passionatety and often against the bill, or willingly but seldom, or intentionally and because popular opinion was against the war. And on the other hand, carelessly and to Mary and twice and if it is foggy sound just as peculiar as passionately and presumably. Without refinements of the responses to evidence such as this, Geach's test does not seem workable.


5. Syntax of the Formal Language

The syntax of a formal language is specified by giving formation rules for generating complex members of the syntactic categories of the language from simpler members of any of these categories.16 Given a vocabulary for the language, which lists the primitive expressions of each syntactic category of the language, the syntax will determine the membership of each syntactic category.

Like the formal languages discussed in Thomason (I 970, Chapter 9), our language will have the following syntactic categories: individual constants, individual variables, individual parameters, and formulas. In addition, there is for each n (n > o) a category of n-ary predicates, for each n (n > o) a category of n-ary predicate adverbs, and a category of sentence adverbs.

Many of the formation rules of the language are copied from first-order logic. If the language is to possess connectives v and for disjunction and negation, and the universal quantifier-other truth-functional connectives and the existential quantifier can be defined in terms of these-then we will have the following formation rules.

(54) If A, B are formulas then (A v B) is a formula. (55) If A is a formula then (A) is a formula. (56) If A is a formula, u is an individual parameter, and x an individual vari-

able, then (x)Ax/u is a formula.'7

A fourth formation rule serves to define the category of individual terms, whose membership consists of individual constants and individual parameters.

(57) If t is an individual constant or an individual parameter, t is an individual term.

The formation rules having to do with predication are not to be found in the standard syntax of first-order logic, which does not deal with the abstraction operator. These rules are as follows.

(58) If A is a formula, u is an individual parameter, and x is an individual variable, then (SAxIu) is a singulary predicate.

(59) If X is an n-ary predicate (n > i) and t, .. . , t,, are individual terms then X(tl, ... , tn) is a formula.

Before adding a clause allowing predicates to be constructed by means of adverbial expressions, an analogy is needed. Consider for a moment the generalization that leads from individual constants to operators. An n-ary operator corresponds to an n-ary function on individuals; iff is a singulary operator, for instance, thenf(t) is an individual term if t is an individual term. More generally, iff is an n-ary operator and tl, . . ., tn are individual terms thenf(t1 ... tn) is an individual term. Individual terms

16 In other words, the categories are built up by simultaneous recursion. See Montague (0970b, 377) and Montague (I97oa, 202).

17 See Note 7, above, for an explanation of this notation.


can themselves be thought of as o-ary operators since they require no individual terms in order to produce an individual term.

We will not add operators to our formal language, but instead will perform a similar generalization on predicate adverbs. A singulary predicate adverb f would yield formulas of the kind {(a) (SPx) (b). Constructions corresponding to this formula are common enough in English; typically a singulary predicate adverb will be realized by a preposition. For instance, if Pu stands for walks, e for to, a for the store, and b for John then {(a) (XPx) (b) will formalize the sentence John walks to the store.

The formation rule governing n-ary predicate adverbs (n > o) is as follows.

(6o) If X is a singulary predicate, e an n-ary predicate adverb, and tl, ... , individual terms, then (f(tl, . . ., tn)X) is a singulary predicate.

The last formation rule of our language deals with o-ary sentence adverbs. As one would expect, this resembles the rule for negation.

(6i) If A is a formula and F is a sentence adverb then F (A) is a formula.

Suppose for the moment that the vocabulary of our language is arranged so that x, y, and z are individual variables, a and b are individual constants, P is a singulary and Q a binary predicate, e and 4 are o-ary predicate adverbs, s1 is a singulary predicate adverb, and F a sentence adverb. Then among the formulas generated by the above syntactic rules will be the following expressions.18

({P)a F(Pa) (4eP) a {(XPx) (a)

'6(SQPx) (y)) (a) (x) (rlx (Y Qxy) (x))

b (X ,Px) (a) {(2(y) (Q( Qz.y) (x))) (a) eX(Qxa v -qaPx) (b)

6. Formal Semantics

In setting up our semantic theory, we will make use of possible situations.'9 We will think of propositions as functions from possible situations into the truth values T and F,

18 Some of the formulas listed below do not seem to have counterparts in English. It is a commonplace feature of logicians' formal languages that they permit syntactic constructions not corresponding to ones found in natural languages, but this is not thought to reduce their logical interest or even their applicability to argu- mentation in natural language. Many syntactic discrepancies between English and the formal language developed in this paper could be explained by supposing that English has no syntactic mechanism capable of doing in full generality what abstraction does in logical languages.

19 Or possible worlds, if you prefer more familiar terminology with a more robust, metaphysical ring; or points of reference if you want a more neutral term that emphasizes the abstract character of the notion. Although these terminological differences suggest different applications and philosophical interpretations, from a formal point of view they are equivalent.


and of propositional functions as functions from individuals into propositions.20 Our present task of providing a semantic interpretation of predicate adverbs shows up clearly the advantages of this decision. The framework of possible situations makes a routine exercise of this task, and the resulting theory is at once general and explicit, applying uniformly to all predicate adverbs of English and minimizing the need for paraphrase in formalization.

Propositions are the intensions of sentences and singulary propositional functions the intensions of predicates. Sentence adverbs will therefore denote functions taking propositions into propositions, and predicate adverbs will denote functions taking singulary propositional functions into singulary propositional functions. For example, walks will have as its intension the propositional function that attaches the value T in a situation a to those and only those individuals who are walking in a. Slowly will denote a function applying to singulary propositional functions. When applied to the intension of walks, for instance, the value of this function will be the intension of walks slowly, which in turn will be the propositional function that attaches the value T in a situation a to those and only those individuals that are walking slowly in a.

This can be formulated precisely by treating the semantic interpretation of our language like that of Q3 in Thomason (969)).21 Suppose we are given a model structure <X, 9>. *' is the set of possible situations of the structure and 9 the set of possible individuals. Let 90 be the set of functions from X to {T, F}. According to what we said at the beginning of this section, 90 is to be thought of as the set of propositions associated with the model structure. Let 91 be the set of functions from .9 to g?o; gA1 is the set of singulary propositional functions associated with the model structure. Let VO/ be the set of functions from 90 to 90 and -/2j be the set of functions from 91 to g1. WO is the set of denotations appropriate for sentence modifiers and VI the set of denotations appropriate for predicate modifiers. For n > i, let Q11 be the set of functions from the Cartesian power 9n to d11. Thus, if f E c1 and dl, . . . , dnl E 9 then f(dl, . .. , dn) will be a function from 91 to '1.

A valuation V on a model structure <k, 9> assigns values to individual constants, individual parameters, and primitive predicates as described in Thomason (I969). If a is an individual constant, then V(a) is a function from M to 9; Va(a) is V(a) (a). If u is an individual parameter, V(u) is a member of 9. If P is a primitive singulary predicate, V(P) is a member of 21. If e is a o-ary predicate adverb, V(t) E s/l and, in general, if e is an n-ary predicate adverb, V(f) E Q1. Finally, if F is a sentence adverb then V(F) E,40.

This suffices to characterize the lexical aspect of valuations, which consists in the assignment of semantic values of the appropriate type to the primitive expressions of

20 For general discussions of this technique see Montague (I 969), Scott (I970), Lewis (I970), and Stalnaker (1970). For a methodological account see Thomason ( I972a).

21 The semantic theory of Thomason (I969) has been modified here to make our presentation more simple. We do not allow the domain to vary from situation to situation, nor do we introduce the relations between situations that are often employed in the interpretation of sentence modifiers.


the language. To complete the semantic interpretation of the language we lay down the following semantic rules showing how an arbitrary valuation V must assign values to complex expressions. In these rules, Va(A) is the truth value given by V to the formula A in the situation a. And where t is a term (i.e. an individual constant or individual parameter), Va(t) is the member of -9 given by V to t in the situation a. Where X is a singulary predicate, V,r(X) is the member of 91 given by V to X in the situation a.22 Vd/u iS the valuation like V in its assignment of values to all primitive expressions except the individual parameter u; Vd/u gives u the value d.

(62) If X is an n-ary predicate and t1, . . ., tn are individual terms then Va(X(ti, ... , tn)) = Tif and onlyifV (X)(<Va(tl), * * ,Va(tn)>)(ae) = T.

(63) V(xEAx/u) is the function f in g1 such that where d e 9 and i e , f (d) (fi) = T if and only if Vd/u86 (A) = T.

(64) If X is a singulary predicate, f and n-ary predicate adverb and t1, ...,tn

are terms then Va(s(ti1 . .. , tn)X) = f(Va (X)), where f = V(e)(<V(tl), * * a )Var(tn)>)-

(65) If F is a sentence adverb and A a formula then Va(F(A)) = V(F) (f), where f is the function in d40 such that for all fi E X, f() = V8 (A).

7. Some Applications

The semantic theory developed in the previous section is highly abstract and leaves unexplored many questions that would have to be settled in the process of applying the theory in detail to natural languages. For example, it is not possible to specify, for any of the predicate adverbs of English we have used in our examples, the function that is to count as its intension.23 Applications of this sort, if they are possible at all, will have to await further theoretical developments.

But if we count logical consequence as a feature of natural languages, semantic theories such as ours will be applicable at least to this feature. A relation 1 of logical consequence can be defined abstractly by referring to the class of all model structures

22 IfXis a primitive predicate or a predicate formed by means ofthe abstraction operator, Va(X) = V(X) for all situations a and ,B; so, dropping the subscript, we may speak simply of V(x). But this generalization fails to hold for predicates formed by means of predicate modifiers. For instance, Va(-qaP) is the result of applying the function Va(qa) to the propositional function V(X). But Va(qa) will vary according to the denotation Va(a) of a, and need not be a constant function of a. For this reason, neither need Va(,qaP) be a constant function of a.

23 All that the abstract theory tells us is that, given a model structure <X., 9>, the intension of walks will be some function taking members of 9 into functions from .Y* to {T, F}. It does not tell us how to go about characterizing particular sets of possible situations and of possible individuals, such that there will be such a function that can plausibly be called the intension of walks. To do this would presuppose a metaphysical theory of what is to count as possible, as well as an analysis of the conditions under which an individual can be said to walk. By saying that it is not possible to specify the intensions of certain English expressions, we mean that a theory and analysis of this kind do not at present exist.


and the class of all valuations on these model structures, without having to designate any particular model structure and valuation as the ones that are intended.

(66) Let r be a set of formulas and A be a formula. r IF A if and only if for all model structures <X, 9>, all a E M' and all valuations V on <M, 9>, if Va(B) = T for all B E r, then Va(A) = T.

An inference of a conclusion A from a set r of premisses is said to be valid (or, strictly speaking, to be logically valid) if and only if r IF A.

We can test and explain the definitions of Section 6 by showing how they account for the logical character of certain inferences. As an example, we will show why the inference (67) is logically valid while (68) is not.

(67) Oedipus intentionally married Jocasta. Oedipus is the son of Laius. Therefore, the son of Laius intentionally married Jocasta.

(68) Oedipus intentionally married Jocasta. Jocasta is Oedipus' mother. Therefore, Oedipus intentionally married his mother.

As a first step, note that the formalizations of (67) and (68) are (69) and (70), respectively.

(69) (XPxb) (a) a = a.,

6 (XPxb) (a,)

(70) (.Pxb) (a) b =b

e(fPxb1) (a)

Here Puv stands for ... married , a for Oedipus, b for Jocasta, a, for the son of Laius, b1 for Oedipus' mother, and e for intentionally.

Now, it follows from the definitions of Section 6 that any predicate X constitutes a referentially transparent context for X in the formula X(t). That is, any inference having the form (7I) is valid.

(7I) X(t) s=t

X(s)

To see this, note that if Va (X(t)) = T and Va(S = t) = T then Va(X) (Va,(t)) (a) = T, so that Va(X(s)) = T. Hence, as a special case of (7I), the inference (69) is valid.


On the other hand, inference (70) is not valid according to our theory. This inference fails because Va(exPxb) and V,(exPxbj) depend on the propositional functions Va,(SPxb) and VE(XPxbi). Even though Va(b) may coincide with Va(bl) for some situations a the two need not coincide for all a and, if they do not, the propositional functions can differ, and so Va(Q(XPxb) (a)) and VaC(eQ(xPXbbi) (a)) need not be the same truth value.

We noted above, in Section 4, that the analogue of (67) fails for sentence adverbs. And according to our theory the inference (72)

(72) Nixon never met Charles Dickens Nixon is the President of the United States

Therefore, the President of the United States never met Charles Dickens

though it resembles (67), is invalid because never is classified as a sentence adverb. The formalization of (72), that is (73)

(73) N(Pcd) c=c,

N(Pcl d)C

does not have the form (7I), and in fact in N(Pcd) the term c occurs within the scope of the operator N. It is a routine matter to show formally that (73) is invalid.

In making this point, however, it is important to remember that our formalized language is syntactically disambiguated, in that each grammatical expression of the language is assigned a unique syntactic structure.24 The semantics of the language is then coordinated with the syntax in such a way that each grammatical expression is assigned a unique semantic structure. But in view of examples such as John and Bill or Mar will go in the car and John met a man who kissed a woman who loved him, English cannot be regarded as disambiguated. Thus, it may happen that one expression of English may be represented by more than one expression of our formalized language. There is much evidence to suggest that the formal operation of abstraction is hardly ever represented explicitly in English, and so is a plentiful source of such ambiguities.25

We may infer from this that the conclusion and first premiss of (72) are ambiguous. And among the other formalizations of the inference is one that has the form (7I),

and so is valid.

(74) N (Pxd) (c) C=-C,

XN (Pxd (cl)

24 See Montague (197ob) for information concerning disambiguated languages. 25 The evidence is discussed in Thomason and Stalnaker (I968), where the corresponding ambiguity is

identified with the traditional distinction between de re and de dicto uses of terms.


This reading of (72) can be captured in English by means of the locution is true of.

(75) It is true of Nixon that he never met Charles Dickens. Nixon is the President of the United States. Therefore, it is true of the President of the United States that he never met Charles Dickens.

Two important conclusions can be drawn from the ambiguity of (72). First, it is an oversimplification to characterize the difference between (67) and (72) by saying that the first is valid while the second is not. More accurately, we should say that whereas (72) admits a reading on which it is invalid, (67) does not admit such a reading.26 Second, it is possible to regard predicate modifiers in English as "special cases" of sentence modifiers, in the following sense: sentences in which a sentence modifier is used will admit a reading in which the sentence modifier is used, so to speak, as a predicate modifier. Thus, The President frequently visits California has a formalization (76)

(76) (.fF(Pxb)) (a)

where F stands for frequently, Puv for . . . visits , a for the President, and b for California, as well as a formalization

(77) F(Pab).

In sentence (76) the predicate is constructed by abstraction on F(Pub), in which F modifies the formula Pub. But it would be equivalent to regard this reading of The President frequently visits California as having the form (GF(2Pxb))(a), where {F is a predicate modifier derived from F and satisfying a semantic postulate to the effect that (G('xAxlu))(t) is to be synonymous with (HF(Ax/u))(t), for all formulas A and terms t. In this way, each sentence adverb can be thought of as giving rise to a derived predicate adverb.27

26 According to Montague's semiotic theory these readings would be identified with assignments of syntactic analyses to the components of the inferences, and these syntactic analyses would consist of trees showing the order in which syntactic rules have been applied in generating these sentences from basic expressions. See Montague (1970a) and Montague (1972).

27 We cannot, however, treat predicate modifiers in general as derived in this way from sentence modifiers without imposing some semantic constraints on predicate adverbs that would not be required by a direct theory according to which predicate adverbs are not derived from sentence adverbs. To see this, consider the formulas (i) (x((yPxy(b)) (a) and (2) (50(xPxy) (a)) (b). They are not logically equivalent; there are interpretations on which they express different propositions. Yet if the predicate modifier e were derived from a sentence modifier F in the way suggested above, then (i) would be synonymous with (i a) xFPxy(b) (a) and (2) would be synonymous with (2a) yxFPxy(a) (b). The formulas (ia) and (2a) are logically equivalent, and so on any interpretations they must express the same proposition. This in itself does not show that the required semantic constraint is not a plausible one, but there is reason to think that the constraint is not satisfied by all predicate adverbs of English. For example, suppose that Puv stands for . . . examines , a for Mary, b for the doctor, and e for reluctantly. Then (2) corresponds to The doctor reluctantly examines Mary, and (i) to the reading of Mary is reluctantly examined by the doctor in which the reluctance is on Mary's part. Since we are clearly dealing here with two different propositions, reluctantly must not be considered to be definable by means of sentence modifiers.


Though this train of thought makes it possible to assimilate certain predicate adverbs to sentence adverbs, it does not make the distinction between the two types of adverbs trivial. For while a sentence adverb will also admit "de dicto" readings having the form (77), predicate adverbs will not.

Before turning to other matters we will consider one more example, due to Donald Davidson, who in Davidson (I968) points out that (78) is a correct inference.

(78) I fly the spaceship to the morning star. The morning star is the evening star. Therefore I fly the spaceship to the evening star.

This is confirmed by our semantic theory, according to which the inference (79) is valid.

(7 9) ea (iPxb) (c) a = al

ea, (.fPxb) (c)

On the other hand, the more complicated inference (8o) must be regarded as an'- biguous.

(8o) I intentionally fly the spaceship to the morning star. The morning star is the evening star. Therefore I intentionally fly the spaceship to the evening star.

On one reading it yields an invalid inference resembling (68), namely (8i).

(8 I) (y 'ea (XPxb) (y)) (c) a=al

(y'ea, (-fPxb) (y) ) (c)

Abstraction can be used in the way discussed above, however, to produce a reading of (8o) that is valid.

(82) (A( Aez(XPxb) (y)) (c)) (a) a=al

(zg(y 'z(x~Pxb) (y)) (c)) (a,)

And still another reading of (8o), namely (83)

(83) ea(jA(QPxb) (y)) (c) a=al

6an (

Av (XPxb) (y) v(c )


results from reversing the order in which the adverbs of (8o) are regarded as applying. This reading of (8o), in which only the flying is intentional and not its destination, is harder to associate with the English sentence. But such a reading is readily associated with slightly different examples, such as He intentionally flew his plane within a few feet of the rooftops, over a town called Hadleyville.

8. Adverbial Constructions and Validity

Logical validity is not a palpable, overt property of inferences couched in natural language. For one thing, questions of validity depend on a syntactic analysis of the language: if adverbs that are sentence modifiers are not distinguished from those that are predicate modifiers, then inference (67) must be considered invalid, since it has the same form as (72). Logical validity also depends on the extent to which the semantic theory of the language has been developed. A theory that does not treat few and most as having fixed semantic interpretations will be unable to account for the difference in logical validity between the following two inferences.

(84) Most doctors are men. Most doctors are nonsmokers.

Therefore, some men are nonsmokers.

(85) Few doctors are women. Few doctors are smokers.

Therefore, some women are smokers.

If the theory puts no restrictions whatever on the quantifiers that interpretations may assign to few and most, then an interpretation in which the customary meanings of these expressions are exchanged is not excluded. In fact, the semantic theory of logical validity will givefew and most only those inferential characteristics that can be ascribed to all expressions that are construed as denoting quantifiers.28

Moreover, any syntactic-and-semantic theory of a language will determine a relation of sameness of form among inferences such that all inferences sharing the same form are alike valid or invalid. This means that imaginative talent is required in testing inferences for validity, since an inference that strikes the ear as appropriate and correct

28 Inadequacies such as the one exemplified by (84) and (85) can be alleviated by adding certain meaning postulates that serve to rule out unwanted interpretations, thus defining a nonlogical type of validity that renders more inferences valid. As a referee of this paper pointed out, for instance, one could introduce a meaning postulate for most to the effect that P(most X) entails P(more than half X). But though meaning postulates render a semantic theory less abstract, in no nontrivial case will they make it so concrete that there will fail to be many significantly different interpretations, each of which satisfies all the meaning postulates. And there are methodological reasons for seeking to avoid the use of meaning postulates. They detract from the coherence and systematic character of the resulting theory.

2I6 RICHMOND THOMASON AND ROBERT STALNAKER

may well have the same form as one that does not. Consider, for instance, the pairs

(86)-(87) and (88)-(89):

(86) This is a tiger.

Therefore, this is an animal.

(87) This is a cigar.

Therefore, this is an animal.

(88) John realizes it is raining.

Therefore, it is raining.

(89) John believes it is raining.

Therefore, it is raining.

A theory that does not distinguish between the ways in which tiger and cigar, or knows and believes, interact with interpretations cannot distinguish between the validity of (86) and (87), or that of (88) and (89).

These cautions are appropriate because many inferences in which adverbs figure seem correct but nevertheless are invalid relative to our theory. We have put no restrictions whatever on the set _s10 of functions that can be assigned to predicate adverbs. In particular, these functions need not commute with one another: where f,g E a? f(g(p)) may differ from g(f(p)) for various p E bY1. Consequently, it is not the case that (UeP) (a) IF (U4P) (a); therefore inferences such as (go)

(go) John carried the eggs quickly to the wrong house

Therefore, John quickly carried the eggs to the wrong house

are classified as invalid, a result that clashes with the strong feeling we all have that this inference is correct. But in the absence of a theory that enables us to distinguish between quickly and carefully we must assign (go) the same form as (gi).

(gi) John carried the eggs carefully to the wrong house.

Therefore, John carefully carried the eggs to the wrong house.

This inference does not appear valid. To be fair, however, we should add that it does not appear invalid, either; the first impression it presents is one of puzzlement. But the following account of (9I) explains why this should be so. In one of its senses, the premiss means (92):

(92) that John was careful in his manner of carrying the eggs, but not necessarily in carrying them to the wrong house.

And in one of its senses, the conclusion means (93):

(93) that John was careful in carrying the eggs to the wrong house, but not necessarily careful in his manner of carrying them.


Word order in English is not in general a very reliable clue to adverbial scope; an adverb can usually be inserted in a variety of positions in a sentence without obtaining results that differ in meaning. Thus, though the word order of (9I) is slightly more natural in association with the senses we chose above, the premiss of (9i) can mean (93) and the conclusion of (9 I) can mean (92). Since (93) clearly does not follow from (92), this explains why (9i) should not appear valid; it has a sense in which its conclusion can be false while its premiss is true. On the other hand, since (9i) also has senses in which its conclusion and premiss say the same thing, we can explain why it does not appear invalid.

Notice that it is natural to explain the ambiguity of John carefully carried the eggs to the wrong house as one of scope.29 On the reading associated with (92) the scope of carefully is carried the eggs; on that associated with (93) it is carried the eggs to the wrong house. If this is so, then commutativity must fail.

Another principle that holds for many adverbs and at first appears plausible is that for all f, g e Q1, f(p, A P2) = f(pl) A f(P2)230 This corresponds to the joint validity of inferences such as (94) and (95)

(94) E(XPx) (a) A t(fQx) (a) 5e((Px A Qx))(a)

(95) e(X(Px A Qx))(a) Q(fPx) (a) A e( Qx) (a)

But there are counterexamples to this principle, at least insofar as it implies (95): (8) is such a counterexample. It is more difficult to find evidence against (94), and it may be that this inference embodies a meaning postulate that could reasonably be imposed on all predicate adverbs of English.

Perhaps the best known and most controversial inference pattern involving adverbs is exemplified by John walks slowly; therefore, John walks. Formally, this is (96).

(96) 6(SXPx) (a) Pa

This inference is endorsed in Davidson (I968), but Montague in Montague (I970a, I972) rejects it because of examples such as allegedly. T. Parsons in Parsons (I970),

likewise rejects it because of in a dream. But our tests for distinguishing predicate and sentence adverbs show both allegedly and in a dream to be obvious sentence modifiers; for instance, the premisses In a dream, the President of the U.S. is Abbie Hoffman and The President of the U.S. is Richard Nixon do not imply In a dream, Richard Nixon is Abbie Hoffman.

29 What is more, there seems to be no other way of explaining the ambiguity in question. For instance, it cannot be due to a lexical ambiguity in carefully, since John carefully carried the eggs in his left hand to the wrong house is three ways ambiguous. Referring the ambiguity to scope predicts this phenomenon, while lexical ambiguity is incompatible with it.

30 Here the symbol " A " represents conjunction of propositional functions.

2I8 RICHMOND THOMASON AND ROBERT STALNAKER

Once all adverbs are excluded that show themselves to be sentence adverbs according to our tests, it becomes very difficult to find counterexamples to (96). There is at least one class of such counterexamples, however, that is general enough to deserve

mention. It often happens that a verb expressing the completion of some prolonged activity will take adverbs such as halfway, indicating that the activity is completed to a certain extent. These adverbs fail our tests for sentence modifiers and presumably are predicate adverbs. But clearly, inferences such as (97)-(99) all fail.

(97) He filled the tank halfway.

Therefore, he filled the tank.

(98) He scaled the cliff to the first ledge.

Therefore, he scaled the cliff.

(99) He sang the aria from the first cadenza.

Therefore, he sang the aria.

If enough examples such as these are forthcoming, the most natural approach to (96), and the one to which we ourselves are most attracted, will be to refuse it logical validity but to establish meaning postulates that guarantee the validity of many of its instances.3' Since they must be imposed on a piecemeal basis, meaning postulates are to be avoided. But it is difficult to see how to dispense with them entirely in any systematic account of inference.

9. Some Open Problems

The syntax of our artificial language permits the formation of certain expressions corresponding to nothing that can be said in English. One case of this phenomenon originates in the fact that many predicate adverbs of English can be applied sensibly only to certain kinds of predicates. The following examples illustrate cases where this restriction has been violated.

(i oo) John is slowly tall.

(Ioi) John slowly ignored the music. (I02) John slowly is captain of the football team. (103) John slowly does not walk. ( I04) John slowly if he walks, walks barefoot. (I05) John slowly walks frequently. (i o6) John carefully walks allegedly.

The peculiarity of these sentences need not be explained by classifying them as syntactically ill-formed. It may be, in particular, that at least some of these examples

31 See Montague (1972) for examples of the use of meaning postulates.


are to be regarded as resulting from the violation of selectional constraints imposed at the semantic level.32 If this is so, then this phenomenon should be explained by setting forth various kinds of propositional functions, and regarding adverbs like slowly as denoting functions that apply only to certain of these kinds of propositional functions. For instance, if the propositional functions are divided into states on the one hand and events or actions on the other, then the function denoted by slowly will be defined only for events or actions. If the predicates is tall, ignored the music, is the captain of the football team, does not walk, and if he walks, walks barefoot are then shown to express states, this will account for the pecularity of (IOO)-(IO5).

This, of course, is highly informal and indefinite. As it stands, our semantic theory provides us only with certain functions as the semantic values of predicates, with no means of determining which such functions are states and which are not. The distinction between propositional functions that are "negative", for instance, and those that are not cannot be made in terms of our present semantic theory. We are envisaging here an extension of the theory, not a series of definitions carried out within the theory itself.33

References

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32 Here we assume that at least some deviant sentences, for instance The fact that he is late regrets itself, are to be explained in this way. For a semantic treatment of such examples see Thomason (0972b).

33 It is at this point that research in theories such as the one we have presented begins to make contact with the work of Davidson and his associates. Davidson has restricted himself all along to adverbs that are to be conceived of as modifying actions or events, and his program has involved philosophic inquiry into the nature of these entities.


Quine, W. (I970) Philosophy of Logic, Prentice-Hall, Englewood Cliffs, New Jersey. Reichenbach, H. (I947) Elements of Symbolic Logic, Macmillan, New York. Scott, D. (1970) "Advice on Modal Logic," in K. Lambert, ed., Philosophical Problems in

Logic, D. Reidel, Dordrecht, Holland. Stalnaker, R. (I970) "Pragmatics," Synthese 22, 272-289.

Thomason, R. and R. Stalnaker (I968) "Modality and Reference," Nous 2, 359-372.

Thomason, R. (I969) "Modal Logic and Metaphysics," in K. Lambert, ed., The Logical Way of Doing Things, Yale University Press, New Haven, Conn.

Thomason, R. (1970) Symbolic Logic: An Introduction, Macmillan, New York. Thomason, R. (0972a) "Philosophy and Formal Semantics," in H. Leblanc, ed., Truth,

Syntax and Modality, North Holland, Amsterdam. Thomason, R. (I972b) "A Semantic Theory of Sortal Incorrectness," Journal of Philosophical

Logic I, 209-258.

Thomason Department of Philosophy University of Pittsburgh Pittsburgh, Pennsylvania 152I3

Stalnaker Department of Philosophy Cornell University Ithaca, New York 14850

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A Semantic Theory of Adverbs