LOGICAL FORMS IN CONTEXT

LOGICAL FORMS IN CONTEXT: PRESUPPOSITIONS A N D OTHER PRECONDITIONS* I. Introductory Natural Logic is the science of valid argumentation in natural languages. Every native speaker of any natural language has logical intuitions, pronounceable by his assent or dissent to arguments presented to him in his natural language, like: (1) Dubito ergo Cogito or He promised you yesterday to send you a postcard from Greece; therefore, he believed then that you preferred his sending you a postcard from Greece to his not doing it. The logical intuitions reflect the logical competence that governs the logical behaviour, the ability to judge and provide alleged relations of certain types between sentences of a natural language. A theory of Natural Logic might provide adequate descriptions and explanations of the logical competence of a particular speaker, but more interesting than these "idiological" theories are those concerned with the families of idiolects called natural languages. A further step of abstraction leads us to general Natural Logic, being the theory of logical competence of speakers of actual natural languages. Finally, there should be universal Natural Logic, concerned with the logical competence of speakers of natural languages, whether actual or possible.1 The natural logician is not interested in creating catalogs of valid, * The research summarized in this paper was conducted under Rome Air Development Center Contract F30602-70-C-0118 (University of Texas at Austin). I am in debt to many with whom I discussed problems related to this paper, particularly to Emmon Bach, John Bacon, Leroy Baker, Lauri Karttunen, Stanley Peters and my students Orvoki Heinamaki, Thomas Hester and Susan Prather. 1. A possible natural language is roughly a language that an actual human being can learn.

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natural arguments; neither is he engaged in producing assent- or dissent-protocols. He is concerned with explanations of logical intuitions by the formulation of rules that govern them. Every generalization determines a class—a "tribe"—of valid arguments, and logical intuitions pertaining to some arguments are explained by demonstrating that a given argument belongs to an appropriate tribe of arguments. A logical form of an argument is a clear and distinct tribal emblem. The arguments are divided into tribes according to their emblems. However, there is no unique division of arguments into tribes: abstractions of different extents result in different tribal limitations. Consequently, formal validity is not an abstractness invariant. (2), for example, is formally invalid in the framework of a propositional calculus, but formally valid in the framework of (nonfree) predicate calculus. (2) If there is a black tulip then all tulips are black. All tulips are black. Therefore, there is a black tulip. Our tribes are linguistic ones: Sentences are linguistic entities and natural arguments are linguistic molecules, the atoms of which are sentences. Logical forms, being tribal emblems of linguistic entities, have at least some linguistic ribs, because they are products of manipulating linguistic data. If a deep structure is an integral part of an adequate description of a sentence, then that structure is one component of the compound which produces logical forms. At least this connection between depth grammar and logical analysis seems to exist. Establishing closer connections would require a closer look at the "problem". The linguistic entities under consideration, whether atoms or molecules are products of certain abstractions. Our input and output are physical, and in a sense even concrete entities, which are contextual representations of sentences: in context we consider an utterance as a possible instance of a sentence. In a different context we might consider it as related to another sentence or to no sentence at all. Now, not only the identity of the sentence involved in an utterance depends on the context. The intuitive judgments about the grammaticality of a sentence or the validity of an argument do also

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depend on it. But the context-dependency of validity amounts to the context-dependency of logical form. How is that dependency expressed in the language of logical forms? We shall try answering this question by investigating appropriate ways for representing presuppositions and other preconditions in Natural Logic. II. Preconditions We shall introduce now the pragmatical concept of precondition. There will be apparent relations between this concept, as denned here, and various definitions of presupposition,2 and we shall outline some of them later. The relation of precondition has four arguments: a speaker, A, a sentence, S, a context, C, and a proposition, P. The speaker A, uttering (an instance of) the sentence S in the context C, is committed to granting in C that the proposition P is true, if and only if A's belief that P is true is a necessary condition for A's performing a happy speech act by uttering (an instance of) S in C. When A is committed to granting in C that P is true, upon uttering S in C, we shall call P 'a precondition' of S, for A, in C. Trying to simplify the presentation of precondition-statements, we observe, first, that propositions are representable as ordered pairs consisting of sentences and contexts.3 The proposition that the present president of the United States is Republican is expressed in the present context by the sentence "The present president of the United States is Republican". An adequate description of the sentence and the context amounts to an adequate description of the proposition. Hence, a precondition-statement involves speaker A, sentence S, context C and the pair < sentence 5", context C > representing the proposition P. Assuming that P is expressible in every context, we can take C to be C using an appropriate 5". Consequently, the precondition-statements involve speaker A, context C and the sen2. Cf. Richard Garner, " 'Presupposition' in Philosophy and Linguistics," in Studies in Linguistic Semantics, ed. by Charles J. Fillmore and D. Terence Langendoen (New York: Holt, Rinehart and Winston, 1971), pp. 22-42 for references to Frege's, Strawson's and W. Sellars's explications and a discussion thereof. 3. I developed this way of representing propositions in my thesis "The Logical Status of Indexical Sentences" (The Hebrew University of Jerusalem, 1970).

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tences S and 5". Naturally, when the identity of the speaker and the context of utterance are fixed, or when they can be treated as parameters, precondition-relations are representable as relations between sentences. Our concept of precondition shares some parts of the classical explications of presupposition, suggested by Frege, Strawson and W. Sellars, but it also differs from each of them substantially. What is it that has preconditions? We follow Frege, but not Strawson and Sellars, taking purported speech acts to be the type of objects having preconditions. With Sellars, we relate the preconditions to the beliefs of the speaker in the truth of statements, rather than to those statements themselves. Finally, we consider the result of a precondition failure to be an unhappy speech act, which is similar to what Frege and Strawson take it to be. Later we shall show how Strawson's semantic concept of presupposition—"a precondition of the truth-or-falsity"—is related to our pragmatic concept of precondition. We would like to consider now some types of preconditions. For that purpose we introduce the term 'preconditional function', to mean a function the values of which are preconditions, where the arguments are the other members of the precondition-relation, viz. a speaker, a sentence and a context. Types of preconditions can be characterized by the form of some preconditional functions generating their members. Among the types of preconditions there is the type of universal preconditions, which are characterized by the fact that the preconditional function involved, f(A,S,C), is, in a sense, constant.4 One example is a condition mentioned by Searle5 among those required for the performance of a "sincere and non-defective illocutionary act": "Normal input conditions obtain" (p. 57). This includes, according to Searle, the exclusion of impediments and imperfections which are not relevant to the study of the linguistic competence, as opposed to the linguistic actual performance.6 Consequently, the speaker under consideration is committed to 4. The formulation of a precondition in a certain context might include reference to the speaker—"the speaker is ideal". 5. John Searle, Speech Acts (Cambridge: Cambridge University Press, 1970), pp. 57 ff. 6. Searle does not use this Chomskyan distinction, but it seems only natural to use it here.

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granting, for example, that he shares with the hearer7 at least a substantial, appropriate part of the language used in the context of utterance. Indeed, the speaker under consideration is the famous ideal speaker; he not only knows his language perfectly, but he is also committed to granting that all the participants in the discourse know their own languages perfectly, all sharing at least an appropriate part of it. Some nontrivial restrictions are imposed on the way the ideal speaker uses his knowledge of his language. For example, the ideal speaker means exactly what he says; when he says "I did not like Ike" he believes that he did not like Ike; when he says to his wife "Did you like Ike?" he believes that he has not been aware of the answer and that his wife does know it. Lying and playing, asking rhetorical questions or faking, in short, playing foul is impossible in a community of ideal speakers; they are all possible only in the realm of actual linguistic performance. Indeed, the hearer under consideration is also an ideal one, who understands perfectly what is said to him. Again, misunderstanding and misinterpretation, naivete and mistrustfulness are impossible for an ideal hearer to share with an actual performer. Moreover, the ideal speaker knows that he is ideal and that his addressee is an ideal hearer, and the ideal hearer in turn also knows that all the participants in the conversation under consideration are ideal ones. Another universal precondition is exemplified by another felicity condition mentioned by Searle with regard to promises to do A: "It is not obvious to both speaker and hearer that the speaker will do A in the normal course of events" (p. 59). The ideal hearer knows that the ideal speaker has a point; the speaker's words are supposed to carry some point in the context of utterance. The ideal speaker will not try to order the hearer to do something immediately if the hearer does it in the context of utterance anyway. Generally, the ideal speaker believes that what he says is not self-evident and completely foreseeable by the ideal hearer; for him to achieve his general objects, keeping quiet during C would be harmful, saying what he said, profitable. Obviously, the universal preconditions should be represented 7. If there is any. For a discussion of some cases, cf. O. H. Green, "Intentions and Speech Acts," Analysts, 29, No. 3 (1969), 109-12.

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somewhere in an adequate linguistic theory. However, they should appear as universal preconditions exactly once, not being an explicit part of the representation of each sentence separately. Consequently, these universal preconditions are not recognizable in logical forms, the later being distinction-oriented rather than universal. We shall consider now some particular preconditions—where the preconditional functions are not constant—and discuss ways of representing appropriately logical forms that are involved in speech acts in which those preconditions also play a role. Under consideration will be mainly three types of preconditions: those pertaining to the context of utterance, e.g. to the speaker, the hearer and their mutual relations; those pertaining to the radical of the sentence and those related to the mood of the sentence.8 TYPE I:

CONTEXT-PRECONDITIONS

1. The Speakers In many cases one can recognize speech subcommunities that are characterizable by some nonlinguistic features. In some languages there are, for example, pairs of synonyms, instances of one of which occur only in the speech of men, whereas instances of the second occur only in the speech of women. O. Jespersen9 mentions the Caucasian language Tshetshensian in which " 'I am' is suo wu when spoken by a man, suo ju by a woman, suo du by a child" (p. 227), and L. Bloomfield10 brings similar examples from the Indian language Yana, in which, for example, "fire" is 'auna in the speech of men, and 'auh in the speech of women. Even better examples can be drawn from Hebrew in which the system of verb-inflections includes a distinction between forms used by a speaker only if he (or she) believes that he (or she, respectively) is a man, and forms used by those who believe that they are women. Gender-deviation reflects misbeliefs the ideal speaker has about himself (or herself). 8. For the distinction between the radical and the mood of a sentence, cf. David Lewis, "General Semantics," in Semantics of Natural Languages, ed. by Donald Davidson and Gilbert Harman (Dordrecht: D. Reidel Publishing Co., 1972), pp. 169-218. 9. O. Jespersen, The Philosophy of Grammar (London: Allen and Unwin, 1924), p. 227. 10. L. Bloomfield, Language (London: Allen and Unwin, 1935), p. 45.

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2. The Addressees Speakers have similar preconditions pertaining to their addresses. Again, in Hebrew, for example, second person pronouns and verbforms are strictly divided according to gender and number. Therefore there is no unique Hebrew translation of the English "you wrote"; actually it has four different translations according to what the speaker is committed to granting about the number and sexes of his addressees.11 3. Speaker-Addressee Relations Particularly interesting are the preconditions as to the differences between the speaker and addressees with respect to their membership in different speech subcommunities. The system of Japanese honorifics is well known,12 but similar phenomena appear in other languages as well: consider the use of "sir" and "ma'm" in English, the German third person plural pronoun functioning as a second person singular pronoun, and the Samoan high-chief words.13 They all involve a belief, on the part of the speaker, concerning his relative status as compared to that of the addressee, in terms of social rank or protocol. TYPE II: RADICAL-PRECONDITIONS

Now we would like to mention some radical-preconditions. Many of them can be described using Strawson's concept of presupposition, and the move from that description to a precondition-type description can be performed naturally and quite easily. 1. Restrictive Relative Clauses Relative clauses are, roughly, embedded sentences which modify noun phrases in larger structures; the restrictive relative clauses are those that we do not set off in writing by commas. Now, restrictive relative clauses introduce interesting existential presuppositions (and 11. When the speaker believes that some of the addressees are women and some are men, he uses the same form he uses when he believes that all are men. 12. Cf. Gary D. Prideaux, The Syntax of Japanese Honorifics (The Hague: Mouton, 1970). 13. Dr. Augustin Riska (private communication).

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hence involve preconditions). For example, consider the following advertisement: (3) "There are two cars built in Sweden. This is the one with bumpers that do what bumpers are supposed to do".1* Indeed, (3) suggests that not every bumper does what bumpers are supposed to do. Actually, the advertisement goes on saying: Bumpers were originally put on cars to protect the body. Somewhere along the line, however, the stylists forgot what bumpers were all about and instead of bumpers that protected the car, people had to buy bumper-guards to protect the bumpers.15 Restrictive relative clauses may include other restrictive relative clauses, thus introducing presuppositions, which in turn involve further presuppositions (4) The man whom the boy whom the students pointed out is a friend of mine 16

recognized

presupposes: (5) There is exactly one man that the boy whom the students recognized pointed out and (6) There is at least one man that the boy whom the students recognize did not point out which in turn presupposes: (7) There is exactly one boy that the students recognized and (8) There is at least one boy that the students did not recognize. For some native speakers of English not only embedding but also stacking of restrictive relative clauses is possible:17 14. Time, April 24, 1972. 15. Ibid. 16. Notice that this sentence is grammatical, being considered from the point of view of the ideal speaker. 17. Cf. Robert Stockwell et al., Integration of Transformational Theories on English Syntax (Los Angeles: University of California, 1969).

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(9) I want to buy a watch that keeps good time that is cheap which presupposes that (10) There are watches that do not keep good time and (11) There are watches that keep good time and are not cheap. Notice the difference between (9) and the conjunction (12): (12) I want to buy a watch that keeps good time and that is cheap18 which presupposes that (13) There are watches which do not keep good time and there are watches that are not cheap which is indeed different from the conjunction of (10) and (11). 2. Adjectives and Adverbs Related to restrictive relative clauses are many of the adjectives: (14) a. a new deal b. a deal which is new. Notice that many but not all the adjectives stand in similar relations: (15) a. an alleged murderer b. a murderer who is alleged. Adjectives of the first type introduce presuppositions like those introduced by their relative clause-paraphrases. The (ideal) speaker who uses (16) The emperor's new clothes are green is committed to granting that (17) There is at least one garment which is not the emperor's and 18. This should not be confused with sentence (12') I want to buy a watch that keeps good time and is cheap.

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(18) There is at least one garment which is the emperor's but not new. Adjacent adjectives are related to relative clauses, but they are ambiguous in a way that makes it possible for them to be paraphrased as conjunctions or embed dings: (19) Busing is a fine old Southern tradition19 has at least the following readings: (20) a. b. c. d.

Busing Busing Busing Busing

is is is is

a a a a

(fine (old (Southern (tradition)))) (fine and old (Southern (tradition))) (fine, old and Southern (tradition)) (fine and (old (Southern)) (tradition)).

While each of these readings involves the presupposition that there are traditions which are not Southern, readings a and b but not c and d involve the presupposition that there are Southern traditions which are not fine. Adverbs present similar relations: using the following sentences: (21) After tackling a passage of his speech in atrociously bad French, he apologized for his pronounciation of it20 involves the existential presupposition concerning French which is bad but not atrociously so. A specially interesting case appears in the following sentence: (22) If the Gambino family literally buried its opposition in New York, then Carlo Gambino could, if he wished, control the entire national rackets-combine of La Cosa Nostra.21 In this case the precondition is metalinguistic: the word "buried" could have been interpreted in the context (23) If the Gambino family in a nonliteral sense.22 19. 20. 21. 22. similar

its opposition etc.

Time, April 24, 1972. Ibid. Ibid. S. Kuno mentioned in a lecture (at the University of Texas, Austin) metalinguistic phrases, e.g. "He is yellow in both senses of the word."

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3. Cleft Sentences Another syntactical structure which is used for introducing presuppositions is that of cleft sentences:23 (24) (25) (26) (27) (28)

It was the car that John was in. It is a handkerchief that Mary wants of Sue's. It was yesterday that he decided to quit. It was suddenly that the ghost appeared. ?It was that Bill was prejudiced that I ignored.

which involve commitment to granting that the following sentences are true in contexts of uttering ( 2 4 ) - ( 2 8 ) , respectively: (29) There was something, different from the car, that John could be in but wasn't. (30) There is something, different from a handkerchief, that belongs to Sue, which Mary could want but doesn't. (31) There is or was a day, different from yesterday, in which he could decide to quit, but didn't. (32) There is a nonsudden manner in which the ghost could appear, but didn't. (33) There was a proposition, different from the proposition that Bill was prejudiced, that I could ignore, but didn't.

*•

*

According to one of the universal preconditions there should be a point in stressing, for example, that it was yesterday, rather than today or yesterweek that he decided to quit. Usually the point is to reject a suggestion that is believed to be naturally acceptable in the context of utterance, were the cleft sentence not to have been uttered. 4.

Complements Several linguists have recently observed24 that there are some clear

23. The following examples are taken from Stockwell, Transformational Theories, the question mark included. 24. Paul and Carol Kiparsky, "Fact," in Recent Advances in Linguistics, ed. by Manfred Bierwisch and Karl-Heinz Heidolph (The Hague: Mouton, 1971); and Lauri Karttunen, "The Semantics of Complement Sentences," in Papers from the Sixth CLS Meeting (Chicago: Chicago Linguistic Society, 1970), pp. 328-40.

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relations between certain sentences and the complements of their main verbs. Consider, for example, the sentences: (34) Bill saw fit to (force ) 1 persuade I Mary to prevent Sheila from having an opportunity to avoid trying the new detergent. Now, "saw fit" is an implicative verb,25 i.e. in sentences like (34), in which it is the main verb and not negated, its deep complement is presupposed (and in case it is negated, the negation of the complement is presupposed): (35) Bill (forced [ 1 persuaded I Mary to prevent Sheila from having an opportunity to avoid trying the new detergent. The verb "force" in turn is an "if-verb", i.e. in sentences like (35) in which it is the main verb and not negated, the complement is presupposed (while a similar relation does not hold when it is negated): (36) Mary prevented Sheila from having an opportunity to avoid trying the new detergent. Now, "prevent" is a "minus if-verb", i.e. in sentences like (36), but not if it were negated, the negation of the complement is presupposed: (37) Sheila did not have an opportunity to avoid trying the new detergent. The presupposition of (37) and its own presupposition are determined similarly, noticing that "have an opportunity" is an "only if-verb" and "avoid" is a "minus implicative" one. Other kinds of verbs are "minus only if-verbs" (e.g. "hesitate") and "factive verbs" (e.g. "regret"). Verbs like "persuade" are different from all those mentioned above. What is presupposed when using the sentence 25. Cf. Lauri Karttunen, "Implicative Verbs," Language, 47, No. 2 (1971), 340-58.

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(38) Bill persuaded Mary to prevent Sheila etc. is (39) Mary intended to prevent Sheila etc. Since (39) is an amplification of the complement of ( 3 8 ) , these verbs—"amplifiers"—seem to be substantially different from all the rest. We shall see later that this is not the case with regard to the way the related presuppositions are reflected in logical forms. T Y P E III:

MOOD-PRECONDITIONS

1. One of the other conditions Searle presents for sincere and nondefective promises is that the hearer would prefer the speaker's doing A to his not doing it and that the speaker believes that this is so.26 More accurately, the hearer prefers one possible course of events over another if they differ from each other exactly to the extent that in the first possible course of events the speaker does A while in the second course of events he does not do it. A further precondition is that the speaker believes that that is the hearer's preference with respect to the speaker's doing A. As I argued elsewhere,27 similar preconditions play a similar role in other types of speech acts. For example, using (40) I ask you to listen to her right away commits its (ideal) utterer to granting that he prefers his addressee's listening to her right away over his not doing it. A pair of such preference-relations characterizes threats like: (41) I hereby threaten you: I will do A unless you do B, in which case the speaker believes that the hearer prefers a possible course of events in which the speaker does not do A over another possible course of events, in which the speaker does A, everything else being the same in the two compared possible courses of events. Furthermore, the speaker prefers one possible course of events over 26. Speech Acts (cf. n. 27. In "Worlds, Games Acts," in Logic, Language, Niinoiluoto (Dordrecht: D.

5), p. 58. and Pragmemes: A Unified Theory of Speech and Probability, ed. by R. J. Bogdan and I. Reidel Publishing Co., forthcoming).

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the other if they differ from each other exactly to the extent that in the first one the hearer does not do B and the speaker does A, while in the other one the hearer does not do B and the speaker does not do A A different type of preference appears in a precondition of advising: (42) I advise you to strike. The speaker is committed to granting that he prefers one possible state of affairs over a second one, if the only differences between them are due to the fact that in first the hearer prefers to strike rather than not (everything else being the same), while in the second possible state of affairs the hearer prefers not to strike rather than to do it (again, everything else being the same). 2. There is an interesting interaction between radical-preconditions and mood-preconditions, which might be reflected in their basic representations. Consider again the case of (40). If the addressee is deaf, or at least not in a position to listen to her right away, then the speaker's use of (40) will be taken to be bizarre. If the ideal speaker asks you to listen to her right away he believes that you can listen to her right away. Similarly, when the speaker promises to visit you tomorrow morning at home, he will not be taken to violate his promise if he does not come because he happens to be injured, arrested or dead tomorrow morning, or even if he gets hold of the information, unknown to him earlier, that your apartment has been taken over by some relatives of yours who are dangerously hostile to him. It seems that every promiser has some presuppositions with respect to those items of occurrence which are at issue in his promise. Consequently, instead of describing the preference-relations involved in mood-preconditions in terms of "possible courses of events" or "possible states of affairs", we have to consider courses of events and state of affairs in which all the presuppositions the speaker has at the context of utterance, concerning those possible worlds and histories, hold. How should all these facts be described and explained in a Natural Logic theory? How are the preconditions reflected in logical forms, if at all? These will be our topics in the next sections.

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III. New

Representations

The general framework we adopt for representing logical forms and meaning relations will be that of a predicate calculus; by this we mean that we accept the basic distinctions between predicate terms and individual terms and between constants and variables. We shall also distinguish between "logical" particles like connectives, quantifiers and abstractors, and the "descriptive" particles, but we shall not stick to connectives that are definable by finite truth-tables or to classical quantifiers. Although "classical quantification theory enjoys an extraordinary combination of depth and simplicity, beauty and utility" 28 we shall not hesitate crossing its borders to whatever direction that will seem fruitful for the development of Natural Logic. In particular we shall not take our general framework to be a calculus of first order. We shall present now the basic part of the grammar of our calculus NLo, first formally, as a set of rewrite rules and ruleschemes, and then informally. Basic Rules: 1.

a -* Con(m> ( « ! , . . . , « m )

2.

« -» P r e d , < * i , . . . , V

3.

a -» Quant'a

4.

t -*D]

»

5.

Arg" -»

a

6.

Arg'P

7.

Arg'i,

m = 2, 3 , . . , Arg* ) n = 1, 2, i = l,2,.

Const 3 Vari p Abst' Indiv 0 Pred

Abst* -»

(

(

ALL ALMOST ALL MOST

Vari' Vari' : Restr "Vari*

THE

e Vari' : Restr 10.

Restr -» (

11.

s -»

12.

Con'2'

13.

Con(m)

a : Restr )

HUMAN CONCRETE NONHUMAN ANIMATE

BUT AND THEN

AND(m> OR(n,>

m = 2, 3 , .

\ 14.

Const'

15.

Vari' -

i = 1, 2,

Ci

xf

i = 1, 2,.

•P Strings of symbols generated by applying some or all of rules 1-15 (and no other rule) starting from the string 'a', will be called 'sentential logical forms'. These should not be confused with the well-formed formulas of the calculus, which will be generated from the natural logical forms when we add to the basic rules and logical dictionary a nonlogical dictionary. Before we do this we shall explain rules 1-15. The sentential logical forms are logical forms of sentences. Rules

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1-3 determine the ways these logical molecules may be built from smaller molecules or atoms. According to rule 1, a sentential logical form may result from combining other sentential logical forms by means of an appropriate connective. The logical dictionary (rules 9-15) includes a list of NLo connectives, to be discussed later. Rule 2 provides another possible molecular structure of sentential logical forms: a predicate of a certain type is attached to an appropriate number of arguments, each of an appropriate type. We impose restrictions on the attachments of strings to one another by dividing them into types and only strings of certain types may be attached to a given predicate to form a sentential logical form. Thus, no logical form will be provided in NLo for the English strings 'ideas are not green' and 'let the prime minister be a prime number'.30 Types are denoted by superscripts. When a 2-or-more place predicate is under consideration, the series of required types of arguments is taken to be the type of the argument. Notice that predicates are logical atoms, i.e. they have no internal structure as far as the grammar of NLo is concerned. (The typesuperscript of a predicate does have a structure, but this is not taken as a component of the predicate, but as a restriction on cooccurrence relations in which it appears.) Rule 3 is similar to rule 1, showing that a sentential logical form may consist of a quantifier of an appropriate type and another sentential logical form. Rule 4 provides the basic division into types: a-strings are the sentential ones, all the rest being ^-strings. Now, strings of type a are simply sentential logical forms (rule 5). The nonsentential strings are either of the type "individuals of sort s" or "predicates of type " (rule 7). Hence the three basic types of NLo-strings are sentences, individual-terms and predicates, the latter two being further subdivided according to sort and co-occurrence-type respectively. Nonsentential arguments of any type are either constants, or variables or abstractors of that type (rule 6). Hence, we have among our ^-strings individual constants, predicate variables and both individual and predicate descriptors. 30. Some counterparts of these strings in the metalanguage of English may be meaningful.

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Rules 8-15 provide the specification of all the logical particles of NLo by identity and structure. A quantifier includes a quantity-sign, e.g., ALL, MOST, etc., a variable which determines which (if any) of the variables occurring in the attached a-string (see rule 3) becomes bound by the attachment, and optionally a restrictor, which limits the variable to range over a certain subset of its general domain (rule 8). 31 Similarly, every abstractor consists of a sign and a possibly restricted variable (rule 9). The restrictors themselves consist of either an a-string, or an a-string aattached to another embedded restrictor (rule 10), Rule 11 is the list of sorts of individuals. The distinction between different sorts of individuals, partially based on Chomsky's system of distinctive features,32 is an effective part of the type-system which makes it possible for us to reject syntactically ungrammatical and semantically anomalous strings. Rules 12 and 13 are lists of connectives33 and rules 14 and 15 specify the constants and variables of the calculus. Before presenting and discussing some examples we introduce a third part of NLo—a nonlogical dictionary, which includes all the descriptive particles required. We shall bring only a few examples: 16. a. Vtcd