Deictic Implicatures

Speranza

In philosophy of language, that Grice mastered, an indexical behavior or utterance points to (or indicates) some state of affairs.

For example, I refers to whoever is speaking; now refers to the time at which that word is uttered; and here refers to the place of utterance.

For Charles Sanders Peirce, indexicality is one of three sign modalities (see further down), and is a phenomenon far broader than language; that which, independently of interpretation, points to something — such as smoke (an index of fire) or a pointing finger — works indexically for interpretation. Social indexicality in the human realm has been regarded as including any sign (clothing, speech variety, table manners) that points to, and helps create, social identity.

 

Indexicality is often treated as part of the study of language called pragmatics – in contrast to such fields as phonology, syntax, and semantics – in that it concerns the use and effects of language. Indexicality is sometimes seen as an alternative way of understanding reference (a concept of semantics) since it allows for an expansion of the way we understand language, and communication in general, to work.

Scholars in linguistic anthropology, Elinor Ochs for example, note how gender can be indexed by the stances one adopts, whether physical or linguistic. This can be accomplished by the way one stands (e.g., the conventionally feminine: "hand on hip with body bent"; in contrast to the conventionally masculine: "thumb in pocket, standing straight with legs apart"). Gender can also be indexed by the language styles one uses (e.g., the conventionally feminine: "large variable range in speaking tones, favoring higher pitches" or "lisping, soft tones"; in contrast to the conventionally masculine: "deep tones within a narrow range of low pitches"). Indexicality is closely related to deixis, which denotes a behavior or an utterance whose meaning varies according to certain features of the context in which it is uttered. Now, here, and I are also typical examples of deictic terms, as well as examples of indexical terms.

The related term "index" comes from Charles Peirce's trichotomy of signs: icon, index, and symbol.^[1]

Indexicals are closely related to demonstratives (this, that), in that both vary in meaning depending on context. Demonstratives may be thought of as forming a subset of indexicals: they are often accompanied, in ordinary usage, by pointing gestures or other non-verbal expressions of their sense. Many but not all indexicals are also egocentric, which means that in order to successfully interpret them the hearer must have knowledge of the respective speaker, time, and place of utterance.

C.S. Peirce elaborated three central trichotomies of sign. The first depends on whether the sign itself is a quality or an actual thing or a habit (tone, token, type, also called qualisign, sinsign, legisign). The second (icon, index, symbol) depends on the kind of reference to the denoted object. The third depends on the kind of reference which the sign will be interpreted as making.^[2] Most famous is the second trichotomy:

Icon, also called a likeness or semblance: a sign that is linked to its represented object by some shared quality (which may vary from physical appearances, common actions, distinct sounds, etc.). An example of this would be the stick-figure pictorial representations of men and women on the door of a public restroom. This is iconic because it is meant to signify a man or woman through a simplified visual representation. An icon does not depend on an actual connection to its object (which may fail to exist) or on a habit of interpretation.^[2]

Index: a sign that is linked to its object by an actual connection or real relation (irrespectively of interpretation), for instance, by a reaction, so as to compel attention, in a definite place and time. A simple example is an "Exit" sign which has an arrow pointing towards the exit. Smoke billowing from a house is an index for a fire inside.^[2]

Symbol: A symbol represents its denoted object by virtue of an interpretive habit or rule that is independent of any shared physical quality, contextual contiguity, or lack thereof, with that which it denotes. A symbol consists in that rule. A word such as "horse" is an example of a symbol which, additionally, is specific to a particular language and prescribes the qualities of its instances, which, then, are noticeably arbitrary with respect to iconic qualities and indexical connections. Most spoken language (with the exception of instances of onomatopoeia like 'hiccup' and 'roar') is symbolic because it is arbitrary in those senses. For example, the English word "window" has no relation to any actual physical window. Peirce usually considered personal names and demonstratives like "this" to be indices, not symbols.^[2]

 

It is possible for signs to have two kinds of meaning, referred to as indexical and referential. Indexical meaning is meaning that is context-dependent. For examples, consider the traditional deictic categories of person, place, and time. Some frequently-used English examples are pronouns, demonstratives, and tense markings. Referential meaning, also called 'semantico-referential function', is when a word functions to describe events or states of affairs in the world independent of the context of the utterance. An example of this could be

A cat is on a mat

because the meaning that it conveys is independent of who says it, when they say it, etc.
A referential indexical, also called a 'shifter', is a sign which contains both referential and indexical meaning. So for example, the word 'I', as in

"I went to the store." (vide Grice, "Personal Identity" for the meaning of "I" and check any good dictionary for the meaning of "store").

 

is a referential indexical. It has referential content, in that it refers to the singular first person, and indexical content, in that its meaning depends on who uttered the word.^[3]

 

Indexical sign types are defined by rules of use that state that there exists a relationship between mutually implied existence of sign vehicle token (i.e. icon, index or symbol) and certain aspects of the context of discourse. The indexical sign token presupposes the aspect of the speech situation and is referentially uninterpretable without some knowledge of context. In other words, some aspect of the context is spelled out in the rules of use, fixed and presupposed, and must be understood for the referential contribution to be made.In the use of pure indexical tokens the sign can also have a creative or performative aspect in that rather than change the context, it creates boundaries to the structure of the event. For example in the case of English indexical pronouns, I and we (as opposed to he/she/it/they) create parameters that specify the parties to whom one is referring. Indexes, both referential and non-referential, therefore exist on a sliding scale, some more presupposing, some more creative, and some containing clear aspects of both.^[3]

 

Non-referential indices or "Pure" indices do not contribute to the semantico-referential value of a speech event yet "signal some particular value of one or more contextual variables."^[3] Non-referential indices encode certain metapragmatic elements of a speech event's context through linguistic variations. The degree of variation in non-referential indices is considerable and serves to infuse the speech event with, at times, multiple levels of pragmatic "meaning."^[4] Of particular note are: sex/gender indices, deference indices (including the affinal taboo index), affect indices, as well as the phenomena of phonological hypercorrection and social identity indexicality.

 

In much of the research currently conducted upon various phenomena of non-referential indexicality, there is an increased interest in not only what is called first-order indexicality, but subsequent second-order as well as "higher-order" levels of indexical meaning. First-order indexicality can be defined as the first level of pragmatic meaning that is drawn from an utterance. For example, instances of deference indexicality such as the variation between informal "Tu" and the more formal "Vous" in French (See T/V deference indexes) indicate a speaker/addressee communicative relationship built upon the values of 'power' and 'solidarity' possessed by the interlocutors.^[5] When a speaker addresses somebody using the V form instead of the T form, they index (via first-order indexicality) their understanding of the need for deference to the addressee. In other words, they perceive/ recognize an incongruence between their level of 'power' and/or 'solidarity', and that of their interlocutor and employ a more formal way of addressing that person to suit the contextual constraints of the speech event.

Second-Order Indexicality is concerned with the connection between linguistic variables and the metapragmatic meanings that they encode. For example, a woman is walking down the street in Manhattan and she stops to ask somebody where a McDonalds is. He responds to her talking in a heavy "Brooklyn" accent. She notices this accent and considers a set of possible personal characteristics that might be indexed by it (such as the man's intelligence, economic situation, and other non-linguistic aspects of his life). The power of language to encode these preconceived "stereotypes" based solely on accent is an example of second-order indexicality (representative of a more complex and subtle system of indexical form than that of first-order indexicality).
Michael Silverstein has also argued that indexical order can transcend levels such as second-order indexicality and discusses higher-order indexicality in terms of what he calls "oinoglossia" or "wine talk".^[4](For discussion see below)


Examples of non-referential forms of indexicality include sex/gender, affect, deference, social class, and social identity indices. Many scholars, notably Silverstein, argue that occurrences of non-referential indexicality entail not only the context-dependent variability of the speech event, but also increasingly subtle forms of indexical meaning (first, second, and higher-orders)as well.^[4]

 

One common system of non-referential indexicality is sex/gender indices. These indices index the gender or "female/male" social status of the interlocutor. There are a multitude of linguistic variants that act to index sex and gender such as:

word-final or sentence-final particles:many languages employ the suffixation of word-final particles to index the gender of the speaker. These particles vary from phonological alterations such as the one explored by William Labov in his work on postvocalic /r/ employment in words that had no word final "r" (which is claimed, among other things, to index the "female" social sex status by virtue of the statistical fact that women tend to hypercorrect their speech more often than men);^[6] suffixation of single phonemes, such as /-s/ in Muskogean languages of the southeastern United States;^[3] or particle suffixation (such as the Japanese sentence-final use of -wa with rising intonation to indicate increasing affect and, via second-order indexicality, the gender of the speaker (in this case, female))^[6]

morphological and phonological mechanisms: such as in Yana, a language where one form of all major words are spoken by sociological male to sociological male, and another form (which is constructed around phonological changes in word forms) is used for all other combination of interlocutors; or the Japanese prefix-affixation of o- to indicate politeness and, consequently, feminine social identity.^[7]

Many instances of sex/gender indices incorporate multiple levels of indexicality (also referred to as indexical order).^[4] In fact, some, such as the prefix-affixation of o- in Japanese, demonstrate complex higher-order indexical forms. In this example, the first order indexes politeness and the second order indexes affiliation with a certain gender class. It is argued that there is an even higher level of indexical order evidenced by the fact that many jobs use the o- prefix to attract female applicants.^[7]

This notion of higher-order indexicality is similar to Silverstein's discussion of "wine talk" (see below) in that it indexes "an identity-by-visible-consumption^[4] [here, employment]" that is an inherent of a certain social register, (i.e. social gender indexicality).

 

Affective meaning is seen as "the encoding, or indexing of speakers emotions into speech events."^[8] The interlocutor of the event "decodes" these verbal messages of affect by giving "precedence to intentionality";^[8] that is, by assuming that the affective form intentionally indexes emotional meaning.
Some examples of affective forms are: diminutives (for example, diminutive affixes in Indo-European and Amerindian languages indicate sympathy, endearment, emotional closeness, or antipathy, condescension, and emotional distance); ideophones and onomatopoeias; expletives, exclamations, interjections, curses, insults, and imprecations (said to be "dramatizations of actions or states"); intonation change (common in tone languages such as Japanese); address terms, kinship terms, and pronouns which often display clear affective dimensions (ranging from the complex address-form systems found languages such a Javanese to inversions of vocative kin terms found in Rural Italy);^[8] lexical processes such as synecdoche and metonymy involved in affect meaning manipulation; certain categories of meaning like evidentiality; reduplication, quantifiers, and comparative structures; as well as inflectional morphology.
Affective forms are a means by which a speaker indexes emotional states through different linguistic mechanisms. These indices become important when applied to other forms of non-referential indexicality, such as sex indices and social identity indices, because of the innate relationship between first-order indexicality and subsequent second-order (or higher) indexical forms. (See multiple indices section for Japanese example).

 

Deference indices encode deference from one interlocutor to another (usually representing inequalities of status, rank, age, sex, etc.).^[3] Some examples of deference indices are:

 

The T/V deference entitlement system of European languages was famously detailed by linguists Brown and Gilman.^[5] As previously mentioned, T/V deference entitlement is a system by which a speaker/addressee speech event is determined by perceived disparities of 'power' and 'solidarity' between interlocutors. Brown and Gilman organized the possible relationships between the speaker and the addressee into six categories:

1. Superior and solidary
2. Superior and not solidary
3. Equal and solidary
4. Equal and not solidary
5. Inferior and solidary
6. Inferior and not solidary

The 'power semantic' indicates that the speaker in a superior position uses T and the speaker in an inferior position uses V. The 'solidarity semantic' indicates that speakers use T for close relationships and V for more formal relationships. These two principles conflict in categories 2 and 5, allowing either T or V in those cases:

1. Superior and solidary: T
2. Superior and not solidary: T/V
3. Equal and solidary: T
4. Equal and not solidary: V
5. Inferior and solidary: T/V
6. Inferior and not solidary: V

Brown and Gilman observed that as the solidarity semantic becomes more important than the power semantic in various cultures, the proportion of T to V use in the two ambiguous categories changes accordingly.
Silverstein comments that while exhibiting a basic level of first-order indexicality, the T/V system also employs second-order indexicality vis-à-vis 'enregistered honorification'.^[4] He cites that the V form can also function as an index of valued "public" register and the standards of good behavior that are entailed by use of V forms over T forms in public contexts. Therefore, people will use T/V deference entailment in 1) a first-order indexical sense that distinguishes between speaker/addressee interpersonal values of 'power' and 'solidarity' and 2) a second-order indexical sense that indexes an interlocutor's inherent "honor" or social merit in employing V forms over T forms in public contexts.

 

Japanese provides an excellent case study of honorifics. Honorifics in Japanese can be divided into two categories: addressee honorifics, which index deference to the addressee of the utterance; and referent honorifics, which index deference to the referent of the utterance. Cynthia Dunn claims that "almost every utterance in Japanese requires a choice between direct and distal forms of the predicate."^[9] The direct form indexes intimacy and "spontaneous self-expression" in contexts involving family and close friends. Contrarily, distal form index social contexts of a more formal, public nature such as distant acquaintances, business settings, or other formal settings.
Japanese also contains a set of humble forms (Japanese kenjyoogo 謙譲語) which are employed by the speaker to index their deference to someone else. There are also suppletive forms that can be used in lieu of regular honorific endings (for example, the subject honorific form of taberu (食べる^?, to eat): meshiagaru 召し上がる). Verbs that involve human subjects must choose between distal or direct forms (towards the addressee) as well as a distinguish between either no use of referent honorifics, use of subject honorific (for others), or use of humble form (for self). The Japanese model for non-referential indexicality demonstrates a very subtle and complicated system that encodes social context into almost every utterance.

[

Dyirbal, a language of the Cairns rain forest in Northern Queensland, employs a system known as the affinal taboo index. Speakers of the language maintain two sets of lexical items: 1) an "everyday" or common interaction set of lexical items and 2) a "mother-in-law" set that is employed when the speaker is in the very distinct context of interaction with their mother-in-law. In this particular system of deference indices, speakers have developed an entirely separate lexicon (there are roughly four "everyday" lexical entries for every one "mother-in-law" lexical entry; 4:1) to index deference exigent of contexts inclusive of the mother-in-law.

 

Hypercorrection is defined by Wolfram as "the use of speech form on the basis of false analogy."^[10] DeCamp defines hypercorrection in a more precise fashion claiming that "hypercorrection is an incorrect analogy with a form in a prestige dialect which the speaker has imperfectly mastered."^[11] Many scholars argue that hypercorrection provides both an index of "social class" and an "Index of Linguistic Insecurity". The latter index can be defined as a speaker's attempts at self-correction in areas of perceived linguistic insufficiencies which denote their lower social standing and minimal social mobility.^[12]
Donald Winford conducted a study that measured the phonological hypercorrection in creolization of English speakers in Trinidad. He claims that the ability to use prestigious norms goes "hand-in-hand" with knowledge of stigmatization afforded to use of "lesser" phonological variants.^[12] He concluded that sociologically "lesser" individuals would try to increase the frequency of certain vowels that were frequent in the high prestige dialect, but they ended up using those vowels even more than their target dialect. This hypercorrection of vowels is an example of non-referential indexicality that indexes, by virtue of innate urges forcing lower class civilians to hypercorrect phonological variants, the actual social class of the speaker. As Silverstein claims, this also conveys an "Index of Linguistic Insecurity" in which a speaker not only indexes their actual social class (via first-order indexicality) but also the insecurities about class constraints and subsequent linguistic effects the encourage hypercorrection in the first place (an incidence of second-order indexicality).^[4]
William Labov and many others have also studied how hypercorrection in African American Vernacular English demonstrates similar social class non-referential indexicality.

 

Multiple non-referential indices can be employed to index the social identity of a speaker. An example of how multiple indexes can constitute social identity is exemplified by Ochs discussion of copula deletion: "That Bad" in American English can index a speaker to be a child, foreigner, medical patient, or elderly person. Use of multiple non-referential indices at once (for example copula deletion and raising intonation), helps further index the social identity of the speaker as that of a child.^[13]
Linguistic and non-linguistic indices are also an important ways of indexing social identity. For example, the Japanese utterance -wa in conjunction with raising intonation (indexical of increasing affect) by one person who "looks like a woman" and another who looks "like a man" may index different affective dispositions which, in turn, can index gender difference.^[6] Ochs and Schieffilen also claim that facial features, gestures, as well as other non-linguistic indices may actually help specify the general information provided by the linguistic features and augment the pragmatic meaning of the utterance.^[14]

 

For demonstrations of higher (or rarefied) indexical orders, Michael Silverstein discusses the particularities of "life-style emblematization" or "convention-dependent-indexical iconicity" which, as he claims, is prototypical of a phenomenon he dubs "wine talk." Professional wine critics use a certain "technical vocabulary" that are "metaphorical of prestige realms of traditional English gentlemanly horticulture."^[4] Thus, a certain "lingo" is created for this wine that indexically entails certain notions of prestigious social classes or genres. When "yuppies" use the lingo for wine flavors created by these critics in the actual context of drinking wine, Silverstein argues that they become the "well-bred, interesting (subtle, balanced, intriguing, winning, etc.) person" that is iconic of the metaphorical "fashion of speaking" employed by people of higher social registers, demanding notoriety as a result of this high level of connoisseurship.^[4] In other words, the wine drinker becomes a refined, gentlemanly critic and, in doing so, adopts a similar level of connoisseurship and social refinement. Silverstein defines this as an example of higher-order indexical "authorization" in which the indexical order of this "wine talk" exists in a "complex, interlocking set of institutionally formed macro-sociological interests."^[4] A speaker of English metaphorically transfers him- or herself into the social structure of the "wine world" that is encoded by the oinoglossia of elite critics using a very particular "technical" terminology.
The use of "wine talk" or similar "fine-cheeses talk", "perfume talk","Hegelian-dialectics talk", "particle-physics talk", "DNA-sequencing talk", "semiotics talk" etc. confers upon an individual an identity-by-visible-consumption indexical of a certain macro-sociological elite identity^[4] and is, as such, an instance of higher-order indexicality.

 

The terms deixis and indexicality are frequently used near-interchangeably, and both concern essentially the same idea; contextually-dependant references. However, each has a different history and tradition associated with it. In the past, deixis was associated specifically with spatio-temporal reference, while indexicality was used more broadly.^[15] More importantly, each is associated with a different field of study; deixis is associated with linguistics, while indexicality is associated with philosophy.^[16]

 

There are various extensions of the basic idea of indexicality, some of which arise outside of linguistics and philosophy of language. One notorious example is David Lewis's indexicality of actuality, according to which actual is itself an indexical term, and the ontological distinction between merely possible worlds and the actual world is just that the actual world is this world (see Modal realism, Modal logic).

[edit] See also

• Deixis
• Semeiotic
• Semiotics
• Charles Sanders Peirce#Classes of signs
• Semiotic elements and classes of signs (Peirce)
• Yehoshua Bar-Hillel
• Speech act theory

References

1. ^ Peirce, C.S., "Division of Signs" in Collected Papers, 1932 [1897]. OCLC 783138
2. ^ ^{a b c d} Commens Dictionary of Peirce's Terms, see especially under "Icon", "Index", and "Symbol", Eprint.
3. ^ ^{a b c d e} Silverstein, Michael. "Shifters, Linguistic Categories, and Cultural Description." In Meaning in Anthropology. K. Basso and H.A. Selby, eds. Albuquerque: School of American Research, University of New Mexico Press, 1976.
4. ^ ^{a b c d e f g h i j k} Silverstein, Michael. "Indexical order and the dialectics of sociolinguistic life". Elsevier Ltd., 2003.
5. ^ ^{a b} Brown, R., Gilman, A. "The pronouns of power and solidarity, IN: Sebeok, T.A. (ed.) Style in Language. Cambridge: MIT Press, 1960.
6. ^ ^{a b c} Wake, Naoko. Indexicality, Gender, and Social Identity.
7. ^ ^{a b} Kamei, Takashi.Covering and Covered Forms of women's language in Japanese.'Hitotsubashi JOurnal of Arts of Sciences' 19:1-7.
8. ^ ^{a b c} Besnier, Niko. Language and Affect. Annual Reviews, Inc., 1990.
9. ^ Dunn, Cynthia. "Pragmatic Functions of Humble Forms in Japanese Ceremonial Discourse. 'Journal of Linguistic Anthropology', Vol. 15, Issue 2, pp. 218–238, 2005
10. ^ Wolfram, W. Phonological Variation and change in Trinidadian English-the evolution of the vowel system. Washington: Center for Applied Linguistics, 1969.
11. ^ DeCamp, D. 'Hypercorrection and Rule Generalization. 1972
12. ^ ^{a b} Winford, Donald. 'Hypercorrection in the Process of Decreolization: The Case of Trinidadian English. Cambridge, England: Cambridge Univserity Press, 1978.
13. ^ Ochs, Elinor. "Indexicality and Socialization". In J. Stigler, R. Shweder & G. Herdt (eds.) 'Cultural Psychology: Essays on Comparative Human Development'. Cambridge: Cambridge University Press, 1990.
14. ^ Ochs, Elinor and Shieffelin, Banbi. "Language has a heart". 'Text 9': 7-25.
15. ^ Silverstein, Michael. (1976) "Shifters, linguistic categories, and cultural description". In K. Basso and H. Selby (eds.), Meaning in Anthropology. SAR pp.25
16. ^ Levinson, Stephen C. (2006) "Deixis". In Laurence R. Horn, Gregory L. Ward (eds.) The Handbook of Pragmatics, pp. 978–120. Blackwell Publishing.

[edit] External links

• Arché Bibliography of Indexicals
• Commens Dictionary of Peirce's Terms, consisting in Peirce's own definitions and characterizations. See "Index".

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Categories:

• Semantics

A Shaggy Dog

Speranza

The publishing story of Grice's "William James Lectures" is a bit of a  
funny piece. His WoW:vi -- which contains "The shaggy dog story" was published  
in a rather obscure journal, "Foundations of Language". I was able to find 
it  and always studied it with affection. It was repr. in Searle, Philosophy 
of  Language (Oxford UP) which made it a bit of a classic -- and it's now 
safely in  chronological order as WoW:6. I was recently, elsewhere, 
discussing bits of  this, and it may do here to re-transcribe the Grice quotes 
(selected) as it  applies to The S is P  00 "the dog is shaggy", or That dog -- 
i.e. Smith's  dog -- is shaggy -- where "The S" is a nominal phrase, and "P" 
is an adjectival  phrase. THIS IS THE SIMPLE TYPE of syntax Grice wants to 
restrict the lecture  to. "Smith's dog is shaggy", say. 

He uses 

z ∈ y

and 

a ∈ b

embedded in intentional contexts. 

Grice wants to be able to say how such  a conversational move may shed 
light on intentional aspects of the U's behaviour  vis a vis basic postulates of the 

the theory of action. In particular,
"What  d'you mean, 'shaggy'?". "Hairy-coated." So Grice proposes to 
restrict this --  his stage 6 of his project, or grand plan, to "definite 
descriptor" ("Smith's  dog") and an "adjectival phrase" ("is shaggy").

Grice writes: "We need to  be able to apply some such notion of a 
PREDICATION", or indicatation "of beta  (adjectival) on alpha (nominal)". We 
have thus reached the stage where we have  "two species" of co-relation:
i. R-co-relation, where "R" for REFER,  and ii. D-co-relation (for 
DENOTATE). "We want to be able to speak of some  particular [thing] as an 
R-correlate of alpha, and of "each member of some  class" as being a 
D-correlate of 
beta." (WoW:130).

Grice then goes on to  provide an 'intentional' (basic, rather than 
resultant) procedure: that will  co-relate a belief, say, or a desire, with an 
assertion, or an imperative move  ("Bring that shaggy dog over here!"). There 
is 
a P1, then which corresponds to  the R-correlate. This he just formulates 
as an imperative, "To utter s if U  means the S to be P."  A second, P2, 
focuses on the D-correlate: "To utter  a psi-cross-correlated (cfr. P1 and P1' 
predication of beta on alpha)", and here  again he produces conditions which 
do not claim to be necessary and sufficient  jointly, to the effect that U 
intends psi-cross a particular R-correlate of  alpha to be one of a 
particular set of D-correlates of beta" (WoW:  131).

At this point Grice wants to extend BEYOND a merely disquotational  Truth 
scheme (vide M. K. Davies for an extended approach -- his book with RKP on  
Meaning). It's not just 'The dog is shaggy' is true iff the dog is shaggy.  
Rather he wants to say, "Smith's dog (his example, p. 131), called Fido, is  
shaggy iff the thing is hairy coated. So he needs to work on an equivalence, 
in  an intentional context for 'the dog' to mean, first, 'THAT dog that 
Smith owns'  and second that it is shaggy, with a sort of explication for the 
'meaning' of  'shaggy'. Next, Grice applies basic procedures to create a 
'resultant' one: "to  utter "p", a PREDICATION of beta on alpha ... if U 
intends 
to express a  particular R-correlate of alpha to be one of a particular set 
of D-correlates of  beta". Referring to Smith's dog, unimaginatively, as 
"Fido", Grice proceeds with  a more expansive resultant procedure:

"to utter ... a predication of  'shaggy' on 'Fido'" if U intends to express 
the belief that Jones's dog is "ONE  OF THE SET OF" hairy-coated things 
(i.e. is hairy-coated)". "U has the procedure  of uttering a 
psi-cross-corelated predication of 'shaggy' on alpha if ... [he is  expressing 
the belief re 
the psi-cross "a particular R-correlate of alpha to be  one of the set of 
hairy-coated things." At this point, Grice displays an  interest in something 
like intensional isomorphism when in footnote to p. 133 he  notes the caveat 
that reads as a very fine distinction indeed, "To the  definiens, then, we 
should add, within the scope of the initial quantifer, the  following clause: 
'& U's purpose in effecting that (Ax) (......) is that  (ER') (Az) (R' 
shaggy' x iff x [belongs] to y" --, where he uses the  set-theoretical sign for 
'belongs' (as per below). Grice goes on to refer to  'ostending' here (p. 
134) which may relate to the idea of explicit or implicit  definitions. 

An act of ostension makes explicit what is implicit. We are  providing a 
definition of what a correlation is: under what circumstances we  hold the 
'shaggy' = df. 'hairy-coated'? And in doing thus he goes into a  problem. Does 
'shaggy' mean, simpliciter, as it does, 'hairy-coated'? But then  this 
intentional programme seems to yield, rather, and we do not want that, that  
'shaggy' means, "in U's view unmistakably hairy-coated", so we need a tweak  
there (p. 135). So he opts for "non-explicit" correlations.

Grice  concludes the lecture with a nod to what he will later have as the 
PERE, or  principle of economy of rational effort (in "Reply to Richards"): 
The rule --  IMPLICIT (meaning postulate, say) -- is it subterranean?. Grice 
writes: "in some  sense", "implicitly" we DO accept these rules" (p. 136).  
His P.E.R.E makes  sense of that in terms of potential explicitation of what 
we are _deemed_ to  follow or accept implicitly. No subterranean, thanks! 
(This was later the  polemic of, say, Gricean M. K. Davies in the sequel to 
his book with RKP in the  pages of Mind and elsewhere on 'tacit' knowledge of 
a language and what the  thing is supposed NOT to mean!).

OF COURSE GRICE IS RIGHT IN providing  some charming illustration with 
Jones's dog being hairy-coated (colloquially  'shaggy'). For 'shaggy' is, after 
all, a predicate. And what we are dealing with  here is what I think R. Dale 
in his essay on Grice calls 'a first-order  language', i.e. a predicate 
calculus. Dale indeed plays with the idea of  J-English. English, as it 
comprises only one sentence: "June loves to dance".  Similarly one can imagine 
S-English. S-English only contains: "Fido is shaggy".  Grice provides an 
ostensive definition of 'shaggy' -- too, in the remaining bits  of WoW:VI. Dale 
touches on that fascinating point in the theory of Fodor that  the meaning of 
"Mentalese" is 'circumscribed' as it were, by grasping the  non-logical terms 
(i.e. the predicates) involved. Dale plays with 'cows'  (property of being 
a cow -- cfr. Grice, "the property of being hairy-coated",  _sic_ in WoW:VI 
for those who think he is only a committed  extensionalist). Dale also plays 
with 'dog' and Schiffer's 'schmdog'. One  point to consider here may have a 
historical side to it. I recall having to pass  a seminar -- using Greek 
Loeb -- on ancient scepticism. So I read all the  Sextus, and in looking for 
contemporary literature, came across a review by  Dummett on "The language of 
appearance". The idea that there are noumenal- and  phenomenal-predicates 
as it were.

It would seem that 'shaggy' belongs to  the sort of physicalist (or 
physical, or naturalistic) predicates. A asks B:  "What kind of dog are you 
buying?" "A shaggy one" 'Shaggy' does not seem a  _primitive_ predicate. Grice 
I 
think would hold that 'RED' is a primitive  predicate (discussed extensively 
in his "Remarks about the senses", in WoW).  Then there's 'sofa' that Dale 
also mentions! I tend to think that had it not  been for Strawson's 'mistake' 
in "Introduction to Logical Theory" in finding  formal logic otiose, Grice 
would have explored areas that perhaps interested him  more intrinsically, 
like the philosophy of perception.

Why is it that a  PIROT may need to tell another, "That pillar-box isn't 
red", "It SEEMS red"  ("Causal theory of perception" -- unfortunately the 
section II on 'implication'  not repr. in WoW). Talking of 'red', it was good 
to 
find, online, a reply by  Fodor to Schiffer indeed on 'simple 
compositionality', as it were. The concept  of a 'red flag' I think it is -- 
with Fodor 
arguing how this cannot mean but a  'pirot' being equipped with the concept 
'red' AND the concept  'flag'.

Grice seems to have been charmingly obsessed with things like:  "The 
pillar-box seems red"/"The pillar-box looks red". Why is it that '... looks  
...' 
carries this (what Grice calls) 'doubt-or-denial' implicature? Surely  
cancellable. What else can a red pillar box do but LOOK red? (the philosopher 
of  
perception -- Grice, and, why not? I -- wonders). One little bit about the  
politics behind Grice may be in order before too long. 
A beautiful section  in that ch. iv of Dale's PhD dissertation, online, 
"The theory of  meaning"

Recall Grice: "Fido is shaggy" --- R-correlate: By uttering  "Fido" U meant 
Jones' dog. --- D-correlate. By uttering 'shaggy', U meant  'hairy-coated.'

Dale: "[O]n the sort of theory that Fodor argues for, a  predicate like 
"dog" (pretend that's Mentalese) will have as its meaning the  property of 
being a dog." "But, for well known reasons that property will not  suffice as 
the thing assigned to "dog" by a C[ompositional] M[eaning] T[heory]."  "A 
story by Schiffer is helpful here."
"Ralph came upon a race of creatures  which he thought comprised a 
previously unencountered biological species, and he  introduced the word  
'shmog'  
to designate members of that species."  ""A thing shall be called a 'shmog'," 
Ralph said, "just in case it belongs to  the species of those creatures."  
"Unbeknown to him, however, shmoghood IS  doghood." "Ralph had stumbled not 
upon a new species but a new race of dogs, and  thus the property that 
'shmog' has been introduced as standing for is none other  than doghood."
"But 'shmog' and 'dog' will have to be synonymous for Fodor's  theory since 
they will both stand in the relation that Fodor offers to the same  
property, the property of being a dog." It is sad that Grice focussed on the  
'shaggy', in retrospect -- i.e. on the D-correlate, rather than the R-correlate 
 
[Jones's shmog?], in retrospect, that is. In his sixth (almost there!) 
William  James Memorial Lecture, Grice educated his audience:

"Suppose that for U  (utterer), the following two correlations hold: i. 
Grice's dog is an  R-correlate of "Plato"
ii. Any hairy-coated thing is a D-correlate of  "shaggy". "Given that U has 
the initial procedures that he has, we can infer  that U has the following 
resultantprocedure, to wit:
RP: To utter the  indicative version of a predication of Beta on "Fido" if 
U wants A (Addressee)  to think U to think Grice's dog to be one of a 
particular set of D-correlates of  Beta. Given RP and (ii) we can infer that U 
has: RP2: to utter the indicative  version of a predication of "shaggy" on 
"Fido" if U wants A to think U to think  Grice's dog is one of the set of 
hairy-coated things (i.e. is hairy-coated). And  given the information from the 
linguist that "Plato is shaggy" is the indicative  version of a predication of 
"shaggy" on "Fido" (assumed),
we can infer U to  have:
RP3: To utter "Fido is shaggy" if U wants A to think U to think that  
Grice's dog is hairy coated. And RP4 is an interpretant of "For U, "Fido is  
shaggy" means 'Grice's dog is hairy-coated'". I now provide a definiens which  
may be adequate for adjectival X (e.g."shaggy"): For U, X (adjectival) means  
'...' iff U has this procedure: to utter a psi-correlated predication of X 
on  ALPHA if (for some Addressee) U wants A to believe a particular  
Referentially-correlate of Alpha to be ..." (where the two lacunae represented  
by 
dots are identically completed). Any specific procedure of the form 
mentioned  in the defininens can be shown to be a resultant procedure: if U has 
(2) 
it is  inferable that he has the procedure of uttering a psi-correlated 
predication of  "shaggy" on Alpha if for some A U wants A to belief a 
particular 
 referential-correlate of Alpha to be one of the set of hairy-coated 
things, that  is, that for U, "shaggy" means "hairy-coated"". More formally: By 
uttering  V, U has correlated "shaggy" with (and only with) each hairy-coated 
thing iff  There is a Reference such that U effected by V that there is an x 
such that  R(shaggy, x) iff x belongs to y (y is a hairy-coated thing) andU 
uttered V in  order that U effect by V that there is an x..." And this is 
insufficient as it  stands".
(p.133).

For certainly, why wouldn't "one want to know that  my dog is shaggy unless 
she wants to beautify it?" INTERLUDE: A shaggy dog story  is defined by the 
OED as "a lengthy tediously detailed story of an  inconsequential series of 
events, more amusing to the teller than to his  audience, or amusing only 
by its pointlessness; also shaggy dog yarn, etc  --  1945 D. Low in N.Y. 
Times Mag. 4 Feb. 40/1 -- The logical lunacy of  `Shaggy Dog'. -- 1946 Coll. 
Shaggy Dog Stories facing p. 1 -- Stories of the  Shaggy Dog variety are 
essentially tales to be told rather than read. -- 1947  Beat Apr. 6/3 -- Here's 
one of my favourite `shaggy dog' stories. -- 1952 A. R.  K. Barnard in A. Red
man Somewhat `Shaggy' 4 -- The comparatively recent type of  story-the 
`Shaggy Dog' yarn. -- 1952 Koestler Arrow in Blue i. viii. 68 -- The  people of 
Budapest have a peculiar shaggy-dog kind of humour. -- 1958 Listener  16 Oct. 
623/1 -- It was a shaggy-dog story about a small-town worthy who shams  
madness to avoid paying bills. -- 1972 P. Ruell Red Christmas xi. 102 -- He  
seemed to be in the middle of an autobiographical shaggy-dog story.

I  have elsewhere, indeed in most fora I have been able to push Grice (as 
if he  needed my pushing, but I enjoy doing it -- I push other things too, 
like  wheelbarrows, if that´s the word, not when I´m selling cockles in Dublin 
(which  I don´t, but when I garden at the Villa Speranza). Anyway, a recent 
query  elsewhere was looking for experimental evidence of the 
´procedural´thing about  things. The interesting thing, if I may be redundant, 
is that 
the query appeared  to be from someone who is well aware of the literature of 
what is called L2 --  not your mother tongue, but not your father tongue, 
either. Instead, the querier  was looking for evidence _in_ the mother tongue. 
This is confusing. The mother  tongue while etymologically IS my mother´s 
tongue, I guess she (my mother) will  have NO idea. So I take ¨mother 
tongue¨to mean, by metonymy, my tongue. And what  does "procedure" mean in my 
tongue? It was fun to have the most active of my  sexual years -- the ten of 
years -- involved in the Gricean analysis of  propositional attitudes and 
communicative reasoning alla whatever. Not only was  great at bars, A: Do you 
come 
here often? B: Only in the mating season.
--  and cfr. my The content of content -- but I was able to _think_. For 
ten years I  was looking -- true I was writing my PhD thesis too, and needed a 
LOT of field  work, for the right word for various things, and discovered 
it all, one dark  night, in Grice´s

A Shaggy Dog Story Grice tells, then, in WoW:vi.  It involves a dog which 
is shaggy. For Grice, the shaggy dog story is a way of  illuminating us 
against the evil (he then thought, and vice versa) influence of  Chomsky. For 
Grice, there are two types of procedures: basic and resultant. It  is _both_ 
basic procedures, as he calls them, which are involved in the Meaning  of 
¨Words¨. This is not to say that this is "procedural words". For the basic  
procedures, for Grice, involve the incorporation of content.
Notably, what is  the content of "dog"? What is the content of "shaggy", 
when we say "The dog is  shaggy."
We need a basic procedure for ¨dog¨, and we need a basic procedure  for 
"shaggy". Both fall within the same nominal category, while Grice takes  
provisions to deal with "shaggy" as not "the shaggy one", but "shaggy" qua  
adjective. However, the procedures are similar. Here Grice applies work by  
Strawson and himself on reference, in "Individuals" by Strawson and elsewhere.  
¨The dog¨involves the referential bit of the utterance; "is shaggy" involves 
the  predicational bit. It´s these functional categories that tell us then 
what a  noun versus an adjective is. I won´t inverse the terms here, since 
"The sh*g is  doggy" makes so much sense, and according to my friend, J. M. 
Geary, much MORE  sense than the original sentence used by Grice. WoW indeed.

So this  leaves us what another "use" of "procedural" which I find easy to 
digest, or  understand. Grice seemed to have problems with some "procedural" 
procedures. I  would think that what is elsewhere, or in some elsewheres, 
called "procedural"  is merely esultant procedure in Grice. For consider, 
"¨The dog is shaggy and the  sh*g is doggy". Here the resultant procedure 
involves "and". And, for some  reason, "and" was the first resultant procedure, 
as analysed by Grice that  struck Chomsky. Anyone familiar with his 1966 
Theory of Syntax will find a  reference in the name index which is bound to 
amuse him: Grice, A. P. That _is_  our Grice. The "A" possibly meaning, 
"Aristotle" -- and "P", Plato? Anyway, for  Grice, "and" involves various 
procedures, all resultant. The first is the  "&" thing. "The dog is shaggy & 
the sh*g 
is doggy", where "&" is  defined in terms of the truth-value table. The 
second resultant procedure is  more complicated, for it involves a FLOUT of the 
maxim, Be orderly. So, the  following dialogue may ensue:
A: The dog was shaggy and the sh*g was  doggy.
B: Are you saying you found it out in that order?
A: No. Matter of  fact, I did feel the sh*g was being doggy
well before I percieved that it  _was_ a shaggy dog.
B: Still, I don´t see what kind of sh*g you were  expecting
from a dog other than the one that struck you so
deeply.
A:  Dunno. Is this the Griceian in me?
Now, experiments, made on oneself are --  painful. I am going to experiment 
with myself, whose mother tongue involves  "shaggy" and "dog" and proceed 
to see if I see a distinction between "shaggy",  "dog", and "and". And I 
don´t! I mean, "and" can become a noun. ¨His ands bore  me, especially as 
follwing "try", "try and"¨". Here "and" is "mentioned", not  used, but in 
metalogic, "p & q" may become the metadiscourse at some higher  level, where 
"and" 
is the focus of our attention, and hence it achieves  "content". After all, 
Grice WILL talk of something like a Fregean SENSE [this  should interest 
McEvoy, since he (McEvoy) senses sense senses] iin Reply to  Richards, for 
things like "not", "and", and in WoW:ii, he refers to the  "meaning" of things 
like "to" and "of" as being just as tricky as "or". But then  I am not an 
innocent informant. Chapman has a good one on this when she  mentions, in her 
bio of Grice (Macmillan, 2006) that Grice would use Tim´s and  Karen´s (his 
children) playmates as naive informants, for things like "Nothing  can be 
green and red all over". What kind of procedures, basic and resultant  does 
this 
"judgement" of synthesis a priori involve? Years, later, and in  Lancaster, 
too, Nigel Morley-Bunker reported a same experiment -- cited by G. R.  
Sampson in his book in Experimental Linguistics. People don´t _know_ what they  
are talking, is mainly Sampson´s conservative point. T. Wharton who teaches 
at  Sussex should approach Sampson and get the real answer about this! I 
first  learned of Morley Bunker via Sampson´s more philosophical "Making Sense" 
 (Clarendon), which amused me, since deals with the complete protocol for 
the  experiment. Something like that I assume is the way to go with the 
original  query. And in any case, going through it, has allowed the Grice Club 
to 
expand  on distintions made by this mastermind the club is dedicated to.

Grice on  shaggy (five-step semi-inferential sequence) après WoW:364  ---- 
Since  Grice (WoW: 364) is careful to use 'feature' (which he distinguishes 
in WoW:vi  from 'item') it's best to state his semi-inferential sequence 
thus (his  shaggy-dog story): STEP 1: It is, speaking extensionally, general 
practice  (merely) -- which can be with myself --, to treat 'shaggy' as 
signifying  hairy-coated. Or 'runt' to mean 'undersized person'.  STEP 2: It 
is, 
now  speaking, with Carnap, INTENSIONALLY too, general boring practice to 
treat  'shaggy' to signify (or mean) 'hairy coated' (or 'hairy-coated' to mean 
shaggy).  STEP 3: It is generally, as a matter of fact, rather 'de iure', 
acceptED that it  is LEGITIMATE (sc.) acceptABLE) to treat 'shaggy' as 
signfiing or to signify  'hairy-coated'. STEP 4: It is legitimate (ceteris 
paribus 
acceptable) to deem  and treat 'shaggy' to signify or as signifying 
'hairy-coated'. STEP 5: "'shaggy'  signifies 'hairy-coated'". Then there's 
Grice on "
α ∈ β" (WoW:VI:133n1) -- repr.  in Searle and scaring Chomsky
One may wonder: how are the concepts to  which  the uppercase words refer 
actually represented?
I pointed out  that Grice will NOT engage in this silly practice, and use 
rather, if he must  ('must' understood in the preterite, here -- no such 
trick in English), by  variables, using the more distinguished phi and khi, or 
alpha, alpha', beta, and  beta', as in the footnote referred to in title 
which scared Chomsky when he read  it in Searle, "The philosophy of language" 
(Oxford, 1971) and which him fodder  for his boring 3rd lecture against Grice 
at Oxford (The Locke  Lectures).

α ∈ β

So Grice wants to just stick with any 'feature'  that belongs to an 'item'. 
We have four features in his long shaggy-dog  story:
the dog that Jones calls 'Fido' -- alpha
the dog that Jones owns  ----------- alpha'
shaggy ---------------------------- beta
hairy-coated  ---------------------- beta'
When we think of co-extensionality of features,  we mean that all items 
which have feature beta, also have feature beta' (All  shaggy things are 
hairy-coated things). We are concerned with Grice on 'beta',  here because for 
the 
sub-mechanism of referring the logic is slightly different  from the more 
basic sub-mechanism of 'predicating'. So what that 'infamous'  footnote that 
does display Grice as the extensionalist he once was reads: "The  definiens 
suggested for explicit correlations is, I think, insufficient as it  
stands." To see that Grice managed Harvard University Press to have this as a  
footnote is miraculous. He goes on in same self footnote:
"I would not wish  to say that if A deliberately detaches B from a party, 
he has thereby  correlated
himself with B, nor that a lecturer who ensures that just ONE  blackboard 
is visible to EACH member of his audience (and to no one else) has  thereby 
explicitly correlated the blackboard with EACH member of the audience,  even 
though in each case the analogue of the suggested definiens is satisfied."  
His ability to bring in the most disparate illustrations is just genial. It  
brings the whole abstract field he is plowing into something that even the  
dullest student or reader should understand. It's like he is saying: 'No 
way you  can defend yourself by saying that I did use convoluted examples.' He 
goes on:  "To have explicitly correlated X
with EACH MEMBER [i.e. each ITEM  that is a member. JLS] of a set K, not 
only must I have intentionally  effected that a particular relation R holds 
between X and all those (and   only those) items which belong to
K [feature. JLS], but also my purpose or  end in setting up this  
relationship must have been to perform
an act."  Imagine if you wanted to say this -- of such an importance -- in 
just a  footnote! Grice goes on:
"to perform an act as a result of which there will  be some relation or 
other which holds between X and 
all those (and only  those) things [or items. JLS] which belong to K." And 
here is the important  bit, where he plays Zermelo-Fraenkel: "To the 
definiens, then, we would ADD,  within the scope of the initial quantifier, the 
following clause." And what does  the clause look like? It looks LIKE this. In 
fact it IS this: & U's purpose  in effecting that (x) (......) [six dots. 
JLS] is that
 
 
 (ER') (z)(R' 'shaggy'z <-> z ∈ y (sc. y is  hairy-coated)).

Crystal clear, right? To understand him fully we need to expand on the NEED 
 of this footnote, but I bring it to the forum because it's the ONLY place  
(surely for abbreviatiory purposes) that Grice cares to use that rather 
infamous  concept, ∈, that he will later criticise when he sees 
"Extensionalism" as a bête  noire. Of course he KNEW he wasn't an 
extensionalist (enemy of 
intenSions) even  then, because he IS using at least intenTional terms (like 
'end' or 'purpose'),  and he is well aware of the problems of quantifying 
in: "I [did say] at one  point that intenSionality seems to be embedded in 
the very foundations of  the  theory of language". Grice allows, though, that 
"it may be possible to  derive ... the intenSional concepts which I have 
been using from more primitive  EXTENSIONAL concepts." (WoW:137). So, the 
"R-correlates" (rather than  "D-correlates") he has in mind is for the 
'subject' 
line of the canonical: Fido  is shaggy, rather than Whiskers is shaggy -- 
where "Fido" has as a R-correlate,  what Grice has as "Jones's dog", rather 
than "Smith's cat" (Grice uses "Smith's  cat" elsewhere in this essay, 
WoW:VI). He does not spend much time at all on  alpha-correlates, since he is 
more 
notoriously going with Frege in D-correlates  then, or beta-type correlates. 
I.e. extension of the predicate 'shaggy' itself.  Here he is treading the 
ground of compositional semantics. If Tarski was into  disquotational: "Snow 
is white" iff snow is white. Grice wants to go inside the  'propositional 
complex' and want to say that "shaggy" means 'hairy-coated'. It  is via 
D-correlates that the thing is achieved. He allows for an extensionalist  
reading 
in set-theoretical terms, and an intensionalist reading in terms of the  
'property of hairy-coatedness', and so on. Now, it is interesting that for 
Grice  those D-correlates feature WITHIN a 'phrastic' as it were. "Tactful" is 
another  predicate he considers (along with "shaggy" -- if not applied to  
'dog'). Suppose that we define 'tactful' as considerate.
So, he wants to  say that:
"Smith is tactful", "Smith, be factful!", "that Smith be tactful!",  etc. 
These utterancesintroduce the same D-correlate. But obviously there is a  
variancy there. In the 'assertion' (for which he uses Frege's symbol,  
judgement-cum-content, /-), 'tactful' is predicated of Smith. Also in "Smith, 
be  
tactful" but not within the scope of an 'assertion' operator, but what Grice  
would symbolise, naively, as "!", and where, notably, the psychological 
attitude  involved will be a 'desire' rather than a 'belief'. In the analysis 
of 
 D-correlata themselves, Grice uses variables for psychological attidudes 
(psi),  variables for moods (*) and the basic notion of a neutral 
'intention',  rather.

Interestingly, Grice was concerned with Positivism as it was then  known in 
Oxford back in the 1930s. Recall that Ayer was more or less of Grice's  
generation (Ayer born 1911, Grice, born 1913). So, while Grice will be of 
course  identified, post-war, 1945, with Austin), back in the 1930s, when Grice 
was a  student at Corpus Christi, he knew what was going on. There was a lot 
of  activity (positivist and other) with Ayer, Hart, Austin, McNabb, and 
others,  meeting at Hampshire's rooms in All Souls, as I recall -- or perpahs 
Hart's  rooms. In any case, Grice always reminded that he never was invited 
to those  'seminal' advances of Oxford philosophy (vide Berlin, "Austin and 
the early  origins of Oxford philosophy") because he had been born on the 
wrong side of the  tracks. Corpus Christi catered for "Midlands" types like 
Grice. Whereas All  Souls, and Hart in particular gathered around him a more 
'pretentious' group.  Ah, in any case, as I think M. Arnold once said, and he 
knew: "Only the poor  learn in Oxford". Or something. It is good to review 
Grice's earliest  publications for traces of 'positivism', or 'empiricism' of 
a radical type. His  analysis of "I"-sentences ("I was hit by a cricket 
ball") look empiricist enough  (in terms of mnemic states, _Mind_, 1941), and 
he has unpublished contemporary  stuff on "Negation and privation" (negation 
in terms of 'ignorance') and  "Intention and disposition". But soon enough 
Grice took all his introspections  for valid, and never questioned again 
their validity. Positivism as a creed  ceased to be a dogma, as it were.

And so on.

Grice on numerical quantifiers: (∃x), "at least ONE" (WoW: 22)

Speranza

We are considering Grice's claim that "(∃x)" reads as "at least ONE", with emphasis on "ONE" qua allegedly numerical quantifier (WoW, 22).

In English, number-words (numerals) appear to be used as quantifiers, as in the following examples:

"Three students came to the party."

"There are four dogs in the yard."

On the other hand, numerals also figure in the following sorts of constructions.

"the three dogs"

"my four dogs"

"all five dogs"

"No two dogs are exactly alike."

"we three kings"

If numerals are quantifiers, then the first four phrases violate the prohibition against double-determiners.

In the last phrase, the appositive ‘three kings’ cannot be replaced by QPs such as ‘some kings’, ‘most kings’, or ‘all kings’. Given the latter data, we propose that numerals are fundamentally, not quantifiers, but are rather adjectives, as illustrated in the following tree.

"the three dogs"
DP
Det CNP
the
Adj CNP
three dogs

Here the numeral ‘three’ serves as a CNP-modifier, which is the principal role of adjectives.
Numerals can also be used as bare adjectives, although it is less common, as in the following poetic description of the arrival of one's first child.

"Now we are three."

S
S-Adv S
now
NP VP
we
Cop Adj
are three

Numerals are sometimes regarded as a species of number words – being logograms like ‘2’ rather than phonograms like‘two’.

Since this is not a semantically relevant distinction, we will simply use the terms interchangeably.

Also note this is in exact agreement with traditional lexicography.

There is an important difference, however, between numerical adjectives and most other

adjectives. In particular, most adjectives are conjunctive, but numerical adjectives are not. The latter concept may be defined as follows.
´
is conjunctive ü for any x1, …, xk
x
1 and … and xk are ´
if and only if
x
1 is ´, and … , and xk is ´

So, for example ‘happy’ is conjunctive, since for example
Jay and Kay are happy if and only if Jay is happy and Kay is happy.

On the other hand, ‘two’ is not conjunctive, since for example
Jay and Kay are two, but neither Jay nor Kay is two.

Notice carefully that the previous sentence introduces a novel use of ‘and’, which we refer to as
mereological ‘and’, which differs from the more common logical ‘and’. Recall that logical ‘and’ is
a two-place S-operator, which is illustrated in the following.

Jay and Kay are happy

.
(
1õ.)õ. 1õ.
are happy

1 .2õ. 1
Jay
1 and Kay1
H
[J] & H[K]
l
P1 { P(J) & P(K) } lx1 H[x]
.
are happy.
J
1 & K1
.
Jay1. .and. .Kay1.

We can apply a completely parallel treatment to

"Jay and Kay are two."

which accordingly analyzes this sentence as saying that Jay is two and Kay is two. This is an admissible reading of the sentence, to be sure, but it is not the most plausible reading.

Rather, the most plausible reading of this sentence treats  ‘and’, not as logical, but as mereological, the latter being categorially rendered as follows.

type(
and) = 2õ
.
and. = lxy{x+y}
Here,

+ is the mereological-sum operator, which is explained in the next section. In the meantime, the
following is an example grammatical analysis.

This is closely related to another notion – distributivity.

"Jay and Kay are two."

.
 õ
1 1õ.
[+1] are two
 
2õ 
Jay and Kay
2
[J+K]
J
+K lx{x1} lx1 2[x]
.
+1. .are two.
J
+ K
.
Jay. .and. .Kay.
Here,

2[.] is the two-predicate, in the metalanguage, which is explained in the next section.

We propose to expand the domain of entities to include the usual singular-objects (individuals)
as well as plural-objects (pluralities), the union of which is the class of count-objects.

On the one hand, singular-objects are the usual suspects of first-order logic, and include particular individual persons, places, and things.

On the other hand, plural-objects are entities that first-order logic studiously sidesteps, and include groups of individuals of various sorts and functions.

For example, whereas Nomar Garciapara is an individual, the Boston Red Sox infielders are a plurality.

Even the grammar of plurals sounds strange, as we strain to treat each plurality simultaneously as a "one" and as a "many".

Set theory presents a formal account of collections, which reduces every "many" to a "one", and in particular reduces all plural-talk about collections to singular-talk about sets, which are treated as singular-objects (individuals) completely on a par with "ordinary" singular-objects like numbers and
space-time points.

Although we don't officially propose to identify pluralities with (mathematical) sets, we do propose to model/explicate pluralities in terms of sets.

In this connection, we offer the following official definitions.

Where  Æ is a set of singular-entities (individuals), the associated set à of plural-entities
is defined as follows,

Ã
ü { X : X Ú Æ & X¹Æ & ;$y[X={y}] }

and the associated set
¶ of count-entities is defined as follows.
ü Æ È Ã

In other words, plural-entities correspond to non-empty non-singleton subsets of singular-entities.

We can now define  + as follows.

a
+b ü a+ È b+
where:
e+ ü {e} if eÎÆ
ü
e if eÎÃ

Count-objects, the counterparts of count noun phrases, are distinguished from mass-objects, the counterparts of mass noun phrases.

Some groups are mere collections; other groups take on a quasi-autonomous existence, and in particular can serve as agents of actions. For example, a jury can hand down a verdict.

To say that sets model pluralities is to say that many properties of pluralities correspond in a specified way to properties of sets.)

So, for example, granting that
J and K are individuals (i.e., elements of Æ),
J
+K = J+ È K+
=
{J} È {K}
=
{J, K}

This means, in particular, that the interpretation of the compound
‘Jay and Kay’ is (modeled by) the
set {Jay, Kay}, which consists of Jay and Kay and nothing else.

Granting that pluralities are (modeled by) sets, we can offer the following simple definitions of
the numerical adjectives.
1
[a] ü a+ has exactly one member
2
[a] ü a+ has exactly two members
3
[a] ü a+ has exactly three members
etc.

These in turn have strictly first-order rewrites, as follows.
ü
$x "y{yÎa+«y=x}
ü
$x1x2 { x1¹x2 & "y{yÎa+«. y=x1 Ú y=x2} }
ü
$x1x2x3 { x1¹x2 & x1¹x3 & x2¹x3 & "y{yÎa+ «. y=x1 Ú y=x2 Ú y=x3} }
etc.

We propose that plural-predication is a primitive notion on a par with singular-predication. So,
for example, our lexicon might include the following entries.
.
student. = lx0 S[x]
.
meeting. = lx0M[x]
Here, the variable ‘

x’ ranges over count-entities.

So, although we use singular-terms in our mathematical description, we understand them as modeling count-entities, including singular-entities and plural-entities.

So if x is a plural-entity, then S[x] means that x are students, and M[x] means that x are meeting.

We further propose that the lexicon will additionally include the following "logical" information.

(
L1) distributive[S]
(
L2) plural[M]
These in turn are expanded in accordance with the following definitions.
(d1) distributive[P]
ü "x { P[x] « "y{yáx ² P[y]} }
(d2) plural[P]
ü "x { P[x] ² Ã[x] }

Here, . is the mereological part-whole relation on the class ¶ of count-entities, which is officially
defined as follows.

a
.b ü a+Úb+
a
áb ü a.b & b./a
30

In other words, to say that P is (fully) distributive is to say that a count-entity
x is P if and only if every
count-entity that
x properly contains is P. And to say that P is (inherently) plural is to say that only plural-entities are P.

Notice that no (inherently) plural predicate is (fully) distributive, although many plural predicates are
plural-distributive, which may be defined as follows.

(d3) plural-distributive[P]
ü "x { P[x] «. Ã[x] & "y{Ã[y] & yáx .² P[y]} }

For example, it is fairly plausible to propose that  ‘meeting’ is plural-distributive.

For example, if Jay,
Kay, and Elle are meeting, then Jay and Kay are meeting, but we would say that Jay (or Kay, or Elle) is
meeting.

The following is an example that uses
‘and’ both logically and mereologically.
Jay and Kay, and Ray and Fay, are married
.
(
õ.)õ. õ1 1õ.
[+1] are married
 .
2õ. 
and
 
2õ   2õ 
Jay and Kay Ray and Fay
M
[J+K] & M[R+F]
l
P { P(J+K) & P(R+F) } lx{x1} lx1 M[x]
.
+1. .are married.
J
+K & R+F
.
and.
J
+ K R + F
.
Jay. .and. .Kay. .Ray. .and. .Fay.

First-order logic studiously avoids plural quantifiers, paraphrasing what it can, and ignoring the
rest, as illustrated in the following.

all dogs (are)
à every dog (is)
some dogs (are)
à some dog (is)
no dogs (are)
à no dog (is)
most dogs (are)
à Æ

Notwithstanding the clarity and precision afforded by singular-only speech, the overall
translation method is not entirely successful, as the following examples illustrate.

(1)

some students are waiting in the lounge

(2)

some students are gathering in the lounge

(3)

some theatre critics only listen to each other

For example, given the plural-inflection, (1) conveys the information that the students involved
are a plurality, which is grammatically analyzed as follows.

Some students are waiting in the lounge
.
(
1õ.)õ. 1õ.
are waiting…
Ïõ
[(1õ.)õ.] Ï
some
1 students
Ï ÏõÏ
student [+plural]
$
x { S[x] & Ã[x] & W[x] }
l
Q1 $x { S[x] & Ã[x] & Q(x) } lx1 W[x]
.
are waiting….
l
P0lQ1$x{…} lx0 { S[x] & Ã[x] }
.
some1 . .students.
l
x0 S[x] lP0 lx0 { P(x) & Ã[x] }
.
students. .+plural.

This reads the sentence as saying that there is a plural-set of students whose members are waiting in the lounge.

We are now in position to grammatically analyze numerical quantifiers, as in the following example.

Three dogs are barking.

In particular, we postulate that this sentence contains a covert existential quantifier, as in the following.

[some] three dogs are barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
[ some
1 ]
ÏõÏ Ï
three dogs
$
x { D[x] & 3[x] & B[x] }
l
Q1 $x { D[x] & 3[x] & Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1$x{…} lx0 { D[x] & 3[x] }
.
some1 .
l
P0 lx0 { P(x) & 3[x] } lx0{D[x] & Ã[x]}
.
three. .dogs.

Note that the Ã-predicate becomes redundant after we add the 3-predicate, so it is dropped.

This analysis reads the original sentence as saying that the members of at least one 3-membered set of dogs are barking.

Notice that, granting distributivity, this is tantamount to saying that  at least three dogs are
barking.

In order to convey that exactly three dogs are barking, we need additional semantic
information, not postulated in the above analysis.

Compare this sentence to the following.

The three dogs are barking.

1 1õ.
are barking
Ïõ
1 Ï
the
1
ÏõÏ Ï
three dogs
B
[ ·x{D[x] & 3[x]} ]
·
x { D[x] & 3[x] }1 lx1 B[x]
.
are barking.
l
P0·xP(x) lx0 { D[x] & 3[x] }
.
the1 .
l
P0 lx0 { P(x) & 3[x] } lx0{D[x] & Ã[x]}
.
three. .dogs.
the dog is barking
.

1 1õ.
is barking
Ïõ
1 Ï
the
1 dog
ÏõÏ Ï
one dog
B
[ ·x{D[x] & 1[x]} ]
·
x { D[x] & 1[x] }1 lx1 B[x]
.
is barking.
l
P0·xP(x)1 lx0 { D[x] & 1[x] }
.
the. .dog.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one. .dog.

This first sentence is read as saying, in effect, that there is exactly one 3-membered set of dogs (in the
relevant domain), and its members are barking.

The second sentence is read as saying that there is exactly one 1-membered set of dogs (in the relevant domain), and its unique member is barking, which is to say there is exactly one dog, and it is barking.

Note that we employ the covert morpheme ‘one’ (as Grice does, WoW: 22).

We can alternatively employ singular-number inflection

[+singular].

Traditionally, definite-determiner phrases are logically mapped to definite descriptions involving
the iota-operator as we have done in previous examples (and also cited by Grice, p. 22).

The following example, however,
demonstrates the shortcomings of this approach.
the dogs are barking
.

1 1õ.
are barking
Ïõ
1 Ï
the
1 dogs
B
[ ·xD+[x] ]
·
x D+[x]1 lx1 B[x]
.
are barking.
l
P0 ·xP(x)1 lx0 D+[x]
.
the1. .dogs.

Here, we introduce the following abbreviation.

P
+[a] ü P[a] & Ã[a]
30

According to this analysis, the sentence says, in effect, that there is exactly one plural-set of dogs, and its members are barking.

The problem is immediate and fatal.

Suppose there are in fact three dogs barking; then how many plural-sets of dogs are there whose members are barking?

Well, since ‘barking’ is distributive, there are four such sets!

But the above analysis claims that there is exactly one such set.

What is needed is an alternative account of ‘the’.

The best-known alternative is due to Godehard Link, who proposes the following.

.
the. = lP0 mxP(x)
Here, ‘
m’ (mu) is short for ‘maximum’, which is defined relative to the mereological part-whole relation
.
among count-objects.8 In particular,
mnF
ü ·n{ F & "u{F[u/n] ² n.u} } [n free for u in F]

In other words,

mnF is the unique count-object that is F and furthermore contains every count-object
that is
F.

Let's see how this works in our earlier case.

Given our notation, we don't have to change much – just replace ‘

·’ with ‘m’.

the dogs are barking
.

1 1õ.
are barking
Ïõ
1 Ï
the
1 dogs
B
[ mxD+[x] ]
m
x D+[x]1 lx1 B[x]
.
are barking.
l
P0 mxP(x)1 lx0 D+[x]
.
the1. .dogs.

According to the analysis, the sentence says that there is a maximal plural-set of dogs (in the relevant
domain), and its members are barking.

Link's account of  ‘the’ works on problematic examples, but does it work on examples for
which we already have a satisfactory solution?

If not, then we must postulate two different meanings of ‘the’, which is, as anybody who loves Grice knows, theoretically inelegant.

Fortunately, our earlier examples work just as well with the new account of ‘the’, as seen in the following examples.

7
reference+++
8

We note that the Link account of ‘the’ also applies to mass nouns like ‘water’ as in ‘the water in the bathtub’, although it requires postulating an expanded domain of entities that includes mass-entities, which include all manner of amorphous "quantities of matter".

For example, these entities figure in explaining what I mean when I say that the gold in
this ring was once scattered across the Galaxy [as currently suggested by cosmologists].

the three dogs are barking the dog is barking
B
[ mx{D[x] & 3[x]} ]
m
x { D[x] & 3[x] }1 lx1 B[x]
.
are barking.
l
P0 mxP(x)1 lx0 { D[x] & 3[x] }
.
the1.
l
P0 lx0 { P(x) & 3[x] } lx0 D+[x]
.
three. .dogs.
B
[ mx{D[x] & 1[x]} ]
m
x { D[x] & 1[x] }1 lx1 B[x]
.
is barking.
l
P0 mxP(x)1 lx0 { D[x] & 1[x] }
.
the1. .dog.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one. .dog.

The first sentence is read as saying that there is a unique maximal 3-membered set of dogs, and its
members are barking, and the second sentence is read as saying that there is a unique maximal 1-
membered set of dogs, and its members are barking.

Note that, as a matter of set theory, there is a maximal 3-membered set of dogs precisely if there is exactly one 3-membered set of dogs, and similarly, there is a maximal 1-membered set of dogs precisely if there is exactly one dog.

Accordingly, we can replace ‘m’ by ‘·’ in the above examples.

According to Link's account,
.
the. = lP0 mxP(x)
where
mxP(x) is the maximal set of count-entities that are P.

Consider the following analysis in accordance with this account.

the people I met-with this week

Ïõ Ï
the
Ï ÏõÏ
people [that] I met
with this week
m
x { P[x] & M[I,x] }
l
P0 mxP(x) lx0 { P[x] & M[I,x] }
.
the.
l
x0P[x] lP0 lx0 { P(x) & M[I,x] }
.
people. .that I met
with this week
.
Here, we take ‘meet with’ as a phrasal verb. Now, suppose that the following summarizes the meetwith events in which I was involved in this week.

I met with Jay on Monday at 10:00 a.m.;

I met with Kay and Elle on Tuesday at 11:00 a.m.;

In particular, on no occasion did I meet with Jay

+Kay+Elle, so there is no maximal set of individuals that I met with this week.

Yet it seems reasonable to claim that the people I met with this week include Jay, Kay, and Elle (and no one else).

To account for this reading, we propose the following cumulative reading of ‘the’.
.
the+. = lP0 ÅxP(x)
where:
Å
nF ü lub {n : F} least upper bound
ub
(S) ü {x : "y(yÎS² y.x} upper bound
l
(S) ü ·x{xÎS & "y{yÎS² x.y}} least

For example, the following is the set of all count-entities that I met with.

{Jay, Kay
+Elle}

So, the least upper bound of this set in the class of count-entities is:

Jay
+Kay+Elle [= {Jay, Kay, Elle}]

We next consider the use of universal quantifiers in plural domains, as illustrated in the
following examples.

(1)

All dogs are barking

(2)

All the dogs are barking

(3)

All three dogs are barking

Item (1) is fairly straightforward, being analyzed as follows.

All dogs are pets
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1 dogs
"
x { D+[x] ²B[x] }
l
Q1 "x { D+[x] ² Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0 D+[x]
.
all1. .dogs.

This reads the sentence as saying that every plural-set of dogs has the following property – its members are barking.

Granting that barking is distributive, this amounts to saying that every dog is barking.

Item (2) is not so straightforward, since it appears to have a double-determiner, involving in
particular a type-mismatch between ‘all’ and ‘the’.

In order to resolve this problem, we propose an optionally-pronounced partitive ‘of’ interposed between ‘all’ and ‘the’, as in the following grammatical analysis.

all [of] the dogs are
barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1
õÏ 
[of]
Ïõ Ï
the dogs
"
x { D[x] ²B[x] }
l
Q1 "x { D[x] ² Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0 D[x]
.
all1.
l
y lx0 {x.y} mx D+[x]
.
of.
l
P0 mxP(x) lx0 D+[x]
.
the. .dogs.

Thus, according to this analysis, the sentence says that every dog-entity "is" barking, which granting
distributivity is the same as every individual dog (in the relevant domain) is barking.

Note the partitive use of ‘of’, which is categorially rendered as follows.

type(
of) = õÏ
.
of. = ly lx0 {x.y}

Thus, ‘of’ converts a proper-noun phrase into a common-noun phrase, pretty much reversing the effect of
‘the’. 9

Item (3) is even less straightforward.

First, a naïve analysis goes as follows.

all three dogs are barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1
ÏõÏ Ï
three dogs
"
x { D[x] & 3[x] .²B[x] }
l
Q1 "x { D[x] & 3[x] .² Q(x) } lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0 { D[x] & 3[x] }
.
all1 .
l
P0 lx0 { P(x) & 3[x] } lx0 D+[x]
.
three. .dogs.

This reads the sentence as saying that the members of every 3-membered set of dogs are barking.

This is not a very plausible reading, unless we bring a lot of IMPLICATURE to save it (as Stephen Yablo, a former student of Grice's once said, "Implicatures happen").

A more plausible reading posits unpronounced material as in the following.

Not perfectly, however, since .dogs. includes only pluralities, whereas .of the dogs. includes individuals as well as pluralities.

all [of the] three dogs are barking
.
(
1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] Ï
all
1
õÏ 
[of]
Ïõ Ï
[the]
ÏõÏ Ï
three dogs
"
x { x.mx{D[x]&3[x]} ² B[x] }
l
Q1"x{x.mx{D[x]&3[x]}²Q(x)} lx1 B[x]
.
are barking.
l
P0lQ1"x{…} lx0{x.mx{D[x]&3[x]}}
.
all1.
l
ylx0{x.y} mx{D[x]&3[x]}
.
of.
l
P0 mxP(x) lx0{D[x]&3[x]}
.
the.
l
P0lx0{P(x)&3[x]} lx0D+[x]
.
three. .dogs.

This reads the sentence as saying, in effect, that there are exactly three dogs, and they are
all barking.

For the sake of comparison, we conclude this section with counterpart examples that employ the
singular-quantifier

‘every’.

Every dog is barking.

.
(
1õ.)õ. 1õ.
is barking
Ïõ
[(1õ.)õ.] Ï
every
1 dog
ÏõÏ 
one dog
"
x { D[x] & 1[x] .²B[x] }
l
Q1"x{D[x] & 1[x] .² Q(x)} lx1 B[x]
.
is barking.
l
P0lQ1"x{…} lx0 { D[x] & 1[x] }
.
every1. .dog.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one. .dog.

Every one of the dogs is barking
.
(
1õ.)õ. 1õ.
is barking
Ïõ
[(1õ.)õ.] Ï
every
1
ÏõÏ 
one
Ïõ Ï
of the dogs
"
x { D[x] & 1[x] .²B[x] }
l
Q1"x{D[x] & 1[x] .² Q(x)} lx1 B[x]
.
is barking.
l
P0lQ1"x{…} lx0 { D[x] & 1[x] }
.
every1.
l
P0 lx0 { P(x) & 1[x] } lx0 D[x]
.
one.
l
y lx0 {x.y} mx D+[x]
.
of. .the dogs.

Earlier, we proposed that the fundamental meaning of numerical QPs involves the phrase that Grice explicitly uses on p. 22 of WoW: "at least" ("at least one" for the meaning of "(∃x)"), rather
than "exactly".

In order to convey "exactly", the sentence either needs appropriate contextual factors, or it needs explicit modifiers such as ‘exactly’, ‘precisely’, and ‘only’.

The later are examples of exclusive adverbs, which also include the following, among others:

"just two" "just one"

"merely" -- "merely one", "merely two"

"simply"

"solely",

"alone",

"uniquely",

"exclusively",

"specifically",

"particularly",

"barely",

"scarcely"

The general idea is that an exclusive adverb focuses attention on a phrase, and conveys that other
possibilities are excluded.

What makes exclusive adverbs so interesting and perplexing is that the very same surface form
can receive many different interpretations.

Our favorite example comes from a popular song from the 1950's, whose title and key lyric is:

"I only have eyes for you."

We think we all sort of understand the sentiment of this song.

But imagine a considerably more gruesome scenario in which the village butcher, who saves body parts for the infamous surgeon Dr. Frankenstein, one day declares:

"Sorry, Dr. Frankenstein, but today I only have eyes for you."

It is also not hard to imagine a less flattering reading of the song in which the speaker (let's say, a boy) tells his girlfriend that only he has eyes for her.

So, depending upon how it is intonated, the sentence can be paraphrased in the following three different manners.

(1) I have eyes for you, but for no one else.

(2) I have eyes for you, but I have nothing else for you.

(3) I have eyes for you, but no one else does.

Given its focus-sensitivity, and given the variety of phrase types that ‘only’ can modify, the
semantics of ‘only’ is subtle and difficult.

We propose that ‘only’ is a multi-categorial adverb with the following multi-type.

type(
only) = (Kõ.)õ(Kõ.) [one for each type K]
Here,

K is the type of the focused phrase, and Kõ. is the type of the matrix that contains the focused
phrase.

For example, in

"I only have eyes for you"

the focused phrase is ‘you’, which has type 2, and the matrix is ‘I have-eyes-for…’, which has
type
2õ..

The semantics of  ‘only’ is a bit complicated.

Our first approximation goes as follows.

(We can also concoct a reading in which ‘have’ is focused, and one in which ‘for’ is focused, but these are grammatically far-fetched).

only. = lF ln { F(n) & ;$n¢ { F(n¢) & n¢¹n }
º lF ln "n¢
{ F(n¢) «n¢=n }
where
F Î .Kõ..
n
,n¢ Î .K.

For example, in our current example,
K=2, and
.
only. = lP2 ly2 { P(y) & ;$z{P(z) & z¹y} }
º l
P2 ly2 "z { P(z) «z=y }

Thus, we have the following grammatical analysis, in which we treat ‘have eyes for’ as an
idiomatic unit (lexical item).

"I only have-eyes-for you"
.

1 1õ.
I
1

2õ(1õ.) U2
you
2
(
2õ.)õ(2õ.) 2õ(1õ.)
only have-eyes-for
"
z { E[I,z] «z=U }
I
1 lx1 "z { E[x,z] «z=U }
.
I1.
l
y2 lx1 "z { E[x,z] «z=y } U2
.
you2.
l
P2 ly2"z{ P(z) «z=y } ly2 lx1 E[x,y]
.
only. .have-eyes-for.

The top node in the semantic tree says that the speaker of the sentence (I) has eyes for the audience of
the sentence (U), but for no one else.

The key composition is underwritten by the following derivation.

(1) (N
2²S)²(N2²S) 1 Pr lP2 ly2 "z { P(z) «z=y }
(2) N
2²(N1²S) 2 Pr ly2 lx1 E[x,y]
(3) N
2 3 As y2
(4) N
1 4 As x1
(5) N
2²S 24 2,4,MP2 ly2 E[x,y]
(6) N
2²S 124 1,5,²O ly2 "z { E[x,z] «z=y }
(7) S 1234 3,6,
²O "z { E[x,z] «z=y }
(8) N
1²S 123 4-7,²I lx1 "z { E[x,z] «z=y }
(9) N
2²(N1²S) 12 3-8,²I ly2 lx1 "z { E[x,z] «z=y }

Compare the above to the following alternative reading in which ‘I’ is focused.

In this case,
K
=1, and
.
only. = lP1 lx1 "z { P(z) «z=x }
30
I
only have-eyes-for you
.

1 1õ.
I
1

2õ(1õ.) U2
you
2
(
1õ.)õ(1õ.) 2õ(1õ.)
only have-eyes-for
"
z { E[z,U] «z=I }
I
1 lx1 "z { E[z,U] «z=x }
.
I1.
l
y2 lx1 "z { E[z,y] «z=x } U2
.
you2.
l
P1lx1"z{ P(z) «z=x } ly2 lx1 E[x,y]
.
only. .have-eyes-for.

The top node in the semantic tree says that the speaker of the sentence (I) has eyes for the audience of
the sentence (U), but no one else does.

The key composition is underwritten by the following derivation.
(1) (N
1²S)²(N1²S) 1 Pr lP1 lx1 "z { P(z) «z=x }
(2) N
2²(N1²S) 2 Pr ly2 lx1 E[x,y]
(3) N
2 3 As y2
(4) N
1 4 As x1
(5) (N
1²S)²S 14 1,4,MP2 lP1 "z { P(z) «z=x }
(6) N
2²S 124 2,5,TR ly2 "z { E[z,y] «z=x }
(7) S 1234 3,6,
²O "z { E[z,y] «z=x }
(8) N
1²S 123 4-8,²I lx1 "z "z { E[z,y] «z=x }
(9) N
2²(N1²S) 12 3-9,²I ly2 lx1 "z { E[z,y] «z=x }

Our account of  ‘only’ works great for the two previous examples, but consider the following
example,

"Jay only has eyes for Kay and Elle"

where we presume that ‘Kay and Elle’ is the focused phrase.

First, we must face the issue of whether ‘and’ is logical-conjunction or mereological conjunction.

If ‘and’ is mereological, then ‘Kay and Elle’ is a proper-noun phrase (), and Kayand-
Elle is a plural entity, in which case we obtain the following analysis.

Jay only has eyes for Kay and Elle
.

1 1õ.
Jay
1

2õ(1õ.) 2
Kay and
2 Elle
(
2õ.)õ(2õ.) 2õ(1õ.)
only has-eyes-for
"
z { E[J,z] «z=K+L }
J
1 lx1 "z { E[x,z] «z=K+L }
.
Jay1.
l
y2 lx1 "z { E[x,z] «z=y } (K+L)2
.
Kay and2 Elle.
l
P2ly2"z{ P(z) «z=y } ly2 lx1 E[x,y]
.
only. .has-eyes-for.

According to this analysis, the sentence says that Jay has eyes for an entity if and only if that entity is
(identical to) the plurality Kay +Elle.

Therefore, since Kay ¹ Kay+Elle, Jay does not have eyes for Kay, and similarly he does not have eyes for Elle.

Jay only has eyes for Kay-and-Elle (as a unit so to speak).

Perhaps the envisaged circumstances are odd (or even kinky!), but it is nevertheless an admissible reading of the sentence.

On the other hand, if  ‘and’ is logical-conjunction, then ‘Kay and Elle’ is a quantifier phrase [
(õ.)õ.], and Kay-and-Elle is a QP-object, in which case we categorially render ‘only’ as
follows.

type(
only) = [(2õ.)õ.]õ[(2õ.)õ.]
.
only. = lÄ2 lP2 "Q2 { Ä(Q) « Q=P }
º l
Ä2 lP2 { Ä(P) & ;$Q { Ä(Q) & Q¹P } }
where
Ä2 Î .(2õ.)õ..
P
2,Q2 Î .2õ..

Unfortunately, this does not yield appropriate truth-conditions for the above sentence, since it implies
that the above sentence says that Jay has eyes for neither Kay nor Elle (exercise).

In light of the difficulties faced by our original account of .only., we now consider the
following  second approximation account of .only..
.
only. = lF ln { F(n) & ;$n¢ { F(n¢) & n¢^n } }
where
F Î .Kõ..
n
,n¢ Î .K.

Note that this account is obtained from the first approximation by replacing ‘

By ‘^’, where ‘^’ refers to the disjointness relation, the exact definition of which varies from sort/type to sort/type.

For example, when applied exclusively to singular-entities, disjointness (^) coincides with nonidentity, and the revised account subsumes our earlier account.

This is to be expected, since the earlier account makes correct predictions when applied to singular-entities.

On the other hand, the revised account does not coincide with the original account when applied to plural-entities.

For example, when applied to our standing example, we obtain the following analysis.

"Jay only has eyes for Kay and Elle."

E
[J,K+L] & ;$z{E[J,z] & z^K+L}
J
1 lx1 { E[x,K+L] & ;$z{E[x,z] & z^K+L} }
.
Jay1.
l
y2 lx1 { E[x,y] & ;$z{E[x,z] & z^y} } (K+L)2
.
Kay and2 Elle.
l
P2 ly2 { P(z) & ;$z{P(z) & z^y} } ly2 lx1 E[x,y]
.
only. .has-eyes-for.

This analysis reads the sentence as saying that Jay has eyes for the plural-entity Kay-and-Elle, but for no entity disjoint from Kay-and-Elle.

So it does not say that Jay does not have eyes for Kay, or for Elle.

Unfortunately, it does not say that Jay  does have eyes for Kay or Elle, unless we further hypothesize that the E-relation is distributive.

According to the most plausible reading , the sentence under scrutiny says that Jay has eyes for Kay,
and Jay has eyes for Elle, but Jay has eyes for no one else.

This suggests that the appropriate reading of ‘and’ is the logical reading, in which case the focus of
‘only’ is a QP, but the alternatives considered are not all QP-objects, but only those QP-objects that
are "like" Kay, Elle, and Kay-and-Elle.

Putting all this together, we obtain the following categorial rendering of
‘only’.
.
only. = lt2 lÄ2 {x(Ä)} { t(Ä) & ;$Ã{x(Ã) & t(Ã) & Ã^Ä} }
where
t Î .[(õ.)õ.]õ..
Ã
,Ä Î .(õ.)õ..
x
(Ã) ü $P [ $xP(x) & Ã = lQ"x{P(x)²Q(x)} ]
Ã
^Ä ü ;$P{ Ã(P) & Ä(P) }

Note the introduction of the additional restriction on the domain of QP-objects, which in particular
restricts the domain to universal-QP-objects.

The following is the associated analysis, where

E(a) ülnE[a,n] [a free for n].

Jay only has eyes for Kay and Elle
.

1 1õ.
Jay
1
[(
2õ.)õ.]õ(1õ.) (2õ.)õ.
Kay and
2 Elle
{ [(
2õ.)õ.]õ. } õ { [(2õ.)õ.]õ.} 2õ(1õ.)
only has-eyes-for
E
[J,K] & E[J,L] & ;$Ã{ x(Ã) & Ã(lxE[J,x]) & Ã^lP{P(K) & P(L)} }
J
1 lx1 { E[x,K] & E[x,L] & ;$Ã{ x(Ã) & Ã(lyE[x,y]) & Ã^lP{P(K) & P(L)} } }
.
Jay1.
l
Ä2 {x(Ä)} { Ä(lyE[x,y]) & ;$Ã{x(Ã) & Ã(lyE[x,y]) & Ã^Ä} } lP2 { P(K) & P(L) }
.
Kay and2 Elle.
l
t2 lÄ2 {x(Ä)} { t(Ä) & ;$Ã{x(Ã) & t(Ã) & Ã^Ä} } ly2 lx1 E[x,y]
.
only. .has-eyes-for.

First, note that, in the second composition, the input is of the appropriate sort for the functor.

Next,
since the top node quantifies over second-order predicates, it is not obvious what it says.

As it turns out, it is equivalent to the following, which is exactly what we want.
"
x { E[J,x] «. x=K Ú x=L }

The equivalence is demonstrated in the following type-theory derivations, where ‘
E’ is short for
‘
lxE[J,x]’.
(1)
"x { E(x) «. x=K Ú x=L } Pr
(2)
­: E(K) & E(L) & ;$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } 3,4,SL
(3)
E(K) & E(L) 1,IL
(4)
­: ;$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } ID
(5)
$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } As
(6)
x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } $O
(7)
­: þ 16,19,SL
(8)
$P { $xP(x) & Ã = lQ"x{P(x)²Q(x)} } 6a,Def x
(9)
$xP(x) $&O
(10) P(
a) $O
(11)
à = lQ "x { P(x) ² Q(x) } 10, $&O
(12)
lQ "x { P(x) ² Q(x) } ^ lP{P(K) & P(L)} } 6c,11,IL
(13)
lP{P(K) & P(L)} = lQ"x { x=K Ú x=L .² Q(x) } lC
(14)
lQ "x { P(x) ² Q(x) } ^ lQ "x { x=K Ú x=L .² Q(x) } 12,13,IL
(15)
;$x { P(x) &. x=K Ú x=L } 14,Def ^
(16)
a¹K & a¹L 10,15,QL
(17)
"x { P(x) ² E(x) } 6b,8b,IL,lC
(18)
E(a) 10,17,QL
(19)
a=K Ú a=L 1,18,QL
30
(1)
E(K) & E(L) & ;$Ã{ x(Ã) & Ã(E) & Ã^lP{P(K) & P(L)} } Pr
(2)
­: "x { E(x) «. x=K Ú x=L } 1a,1b,3,QL
(3)
­: "x { E(x) ². x=K Ú x=L } UCD
(4)
E(a) As
(5)
­: a=K Ú a=L DD
(6) [
lP{P(a)}](E) 4,lC
(7) P(
a) «"x{x=a² P(x)} IL
(8)
lP{P(a)} = lP"x{ [lx{x=a}](x) ² P(x) } 7,lC
(9)
x( lP{P(a)} ) 8,QL,Def x
(10)
lP{P(K) & P(L)} ^/ lP{P(a)} 1c,6,9,QL
(11)
lP{P(K) & P(L)} = lQ"x { x=K Ú x=L .² Q(x) } lC
(12)
lP{P(a)} = lQ"x { x=a² Q(x) } lC
(13)
lQ"x { x=K Ú x=L .² Q(x) } ^/ lQ"x { x=a² Q(x) } 10-12,IL
(14)
$x { x=K Ú x=L & x=a } 13,Def ^
(15)
a=K Ú a=L 14,IL

What happens when the focus is a CNP?

Consider the following example.

Only poisonous snakes are dangerous

Depending on the focus of ‘only’, this is ambiguous among the following.


Snakes are dangerous, but non-poisonous snakes are not dangerous
poisonous

Snakes are dangerous, but other poisonous things are not dangerous
poisonous.

Snakes are dangerous, but other things are not dangerous

We first consider the last one, since its overall structure is the simplest.

The first thing to notice is that the paraphrase is not well-formed by simple categorial formation rules.

poisonous snakes are dangerous…
?
CNP VP
are dangerous
Adj CNP
poisonous snakes

We need an NP to serve as the subject of the VP, but what we have is a CNP.

There seems to be a missing determiner – but which one?

When we say that poisonous snakes are dangerous, do we mean "all" poisonous snakes, "most" poisonous snakes, "some" poisonous snakes, or *what*?

There does not seem to be a generally agreed upon answer to this question. Grice says that "it depends on context", which hardly illuminates (His analysis of "Englishmen are brave" -- in Jill's reasoning, "Jack is an Englishman; he is, therefore, brave", "Aspects of Reasoning" -- Grice further explores the basis for such a generalisation: empirical, tautological, deductive?)

However, we propose that, at least in the presence of ‘only’, the missing determiner is ‘some’.

So, for example, the sentence

Only men play NFL football

says that some men play NFL football, but no non-men play NFL football.

It most certainly does not say that all, or most, or generally, men play NFL football.

How does this fit into our account of  ‘only’?

The focus phrase is ‘men’ which is a CNP, so the applicable sub-category of ‘only’ is:

(Ïõ.)õ(Ïõ.)

and the associated interpretation of  ‘only’ is:

l
Ä0 lQ0 { Ä(Q) & ;$P{Ä(P) & P^Q} }
where
P
^Q ü ;$x { P(x) & Q(x) }

With this account of .only. in hand, we offer the following analysis.

Note the covert determiner ‘some’.

Also note that, for the sake of brevity, we occasionally write ‘M’ in place of ‘lxM[x]’.

only men play NFL football
.
(
1õ.)õ. 1õ.
play NFL
Ïõ
[(1õ.)õ.] (Ïõ.)õ.
[some
1]
(
Ïõ.)õ(Ïõ.) Ï
only men
$
x{ M[x] & F[x] } & ;$P{ $x{P(x) & F[x]} & P^M}
l
Q1{$x{M[x]&Q(x)} & ;$P{$x{P(x)&Q(x)} & P^M}} lx1 F[x]
.
play NFL.
l
P0lQ1{…} lÄ0 { Ä(M) & ;$P{Ä(P) & P^M} }
.
some1.
l
Ä0 lQ0 { Ä(Q) & ;$P{Ä(P) & P^Q} } lx0 M[x]
.
only. .men.

The key composition is underwritten by the following derivation.

(1) (
¤²S)²S 1 Pr lÄ0 { Ä(M) & ;$P{Ä(P) & P^M} }
(2)
¤²[(N1²S)²S] 2 Pr lP0 lQ1 $x{ P(x) & Q(x) }
(3) N
1²S 3 As Q1
(4)
¤²S 23 2,3,MP2 lP0 $x{ P(x) & Q(x) }
(5) S 123 1,4,
²O $x{M[x]&Q(x)} & ;$P{ $x{P(x)&Q(x)} & P^M}
(6) (N
1²S)²S 12 3-5,²I lQ1 $x{M[x]&Q(x)} & ;$P{ $x{P(x)&Q(x)} & P^M}

Now, let us examine the top node more carefully. It clearly contains the information that some men play NFL football.

The question is whether it also says (or merely "implicates", to use Grice's insidious parlance) that no one else does.

In other words, is the top node equivalent to the following.

Note carefully, however, that there is also a weak sense of ‘only’.

$
x{ M[x] & F[x] } & ;$x { ;M[a] & F[a] }

By way of answering this question, we offer the following type-theory derivations.
(1)
;$P{ $x{P(x) & F[x]} & P^M } Pr
(2)
­: ;$x { ;M[a] & F[a] } ID
(3)
$x { ;M[a] & F[a] } As
(4)
;M[a] & F[a] 2,$O
(5)
­: þ 4a,12,SL
(6) [
lx(x=a)](a) IL,lC
(7)
$x{ [lx(x=a)](x) & F[x] } 4b,6,QL
(8)
lx(x=a) ^/ M 1b,7,QL
(9)
$x{ [lx(x=a)](x) & M[x] } 8,Def ^
(10) [
lx(x=a)](b) & M[b] 9,$O
(11)
b=a 9a,lC
(12)
M[a] 9b,10,IL
(1)
;$x { ;M[x] & F[x] } Pr
(2)
­: ;$P{ $x{P(x) & F[x]} & P^M } ID
(3)
$P{ $x{P(x) & F[x]} & P^M } As
(4)
­: þ 6b,8,SL
(5)
$x{P(x) & F[x]} & P^M 3,$O
(6) P(
a) & F[a] 4a,$O
(7)
;$x { P(x) & M[x] } 3b,Def ^
(8)
;M[a] 6a,7,QL
(8)
;F[a] 1,7,QL

We next consider an example in which the focus is an adjective, as in the following example.

Only poisonous snakes are dangerous.

In this case the focus-type is

ÏõÏ, and the matrix-type is (ÏõÏ)õ., and the associated interpretation of
‘only’
is as follows.
.
only. = l» lh { »(h) & ;$h¢{»(h¢) & h¢^h} }
where
» Î .(ÏõÏ)õ..
h Î
ÏõÏ
h¢^h
ü ; $P0 $x0 { h¢(P0)(x0) & h(P0)(x0) }
Hardegree,
Numerical Quantifiers page 23 of 30 30

Note carefully that the construction only makes sense for subsective adjectives,12 so in particular, the h-variables range over subsective adjectives.

Compare, if you have the time, the following two pieces of nonsense involving nonsubsective
adjectives:

Only alleged snakes are dangerous.

Only former snakes are dangerous

The following is the associated grammatical analysis.

Only poisonous snakes are dangerous
.
(1õ.)õ. 1õ.
are dangerous
Ïõ
[(1õ.)õ.] (Ïõ.)õ.
[some
1]
[(
ÏõÏ)õ.]õ. Ï
snakes
[(
ÏõÏ)õ.]õ[(ÏõÏ)õ.] ÏõÏ
only poisonous
$
x{ P(S)(x) & D[x] } & ;$h{ $x{h(S)(x) & D[x]} & h^P }
l
Q1{ $x{P(S)(x) & Q(x)} & ;$h{ $x{h(S)(x) & Q(x)} & h^P} } lx1 D[x]
.
are dangerous.
l
P0 lQ1 $x { P(x) & Q(x) } lÄ0 { Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P} }
.
some1.
l
» { »(P) & ;$h{»(h) & h^P} } lx0 S[x]
.
snakes.
l
» lh { »(h) & ;$h¢{»(h¢) & h¢^h} } P
.
only. .poisonous.

The key compositions are underwritten by the following derivations.

Basically, h is subsective if h(S) Ú S, for every set S of entities.

For example, ‘poisonous’ is subsective since every poisonous N is an N.)

Note that these two alleged pieces of nonsense make sense if ‘alleged snakes’ and ‘former snakes’ are focused, although they describe rather odd worlds, even for Grice.

(1) [(
¤²¤)²S]²S 1 Pr l» { »(P) & ;$h{»(h) & h^P} }
(2)
¤ 2 Pr lx0 S[x]
(3)
¤²S 3 As Ä0
(4)
¤²¤ 4 As h
(5)
¤ 24 2,4,²O h(S)
(6) S 234 3,5,
²O Ä0( h(S) )
(7) (
¤²¤)²S 23 4-6,²I lh Ä0( h(S) )
(8) S 123 1,7,
²O Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P}
(9) (
¤²S)²S 12 3-8,²I lÄ0 { Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P} }
(1) (
¤²S)²S 1 Pr lÄ0 { Ä0( P(S) ) & ;$h{ Ä0( h(S) ) & h^P} }
(2)
¤²[(N1²S)²S] 2 Pr lP0 lQ1 $x { P(x) & Q(x) }
(3) N
1²S 3 As Q1
(4)
¤²S 23 2,3,MP2 lP0 $x { P(x) & Q(x) }
(5) S 123 1,4,
²O $x{P(S)(x) & Q(x)} & ;$h{$x{h(S)(x) & Q(x)} & h^P}
(6) (N
1²S)²S 12 3-5,²I lQ1 $x{P(S)(x) & Q(x)} & ;$h{$x{h(S)(x) & Q(x)} & h^P}

Let us now examine the top node which is
$
x{ P(S)(x) & D[x] } & ;$h{ $x{h(S)(x) & D[x]} & h^P }

This clearly says that some poisonous snakes are dangerous. The question is whether it denies that any non-poisonous snakes are dangerous, which is to say whether it is equivalent to the following.
$
x{ P(S)(x) & D[x] } & ;$x { S[x] & ;P(S)(x) & D[x] }

This is settled in the following type-theory derivations.

(1)
;$h { $x{h(S)(x) & D[x]} & h^P } Pr
(2)
­: ;$x { S[x] & ;P(S)(x) & D[x] } ID
(3)
$x { S[x] & ;P(S)(x) & D[x] } As
(4)
­: þ 13b,13c,SL
(5)
S[a] & ;P(S)(a) & D[a] 3,$O
(6) [let]
h¢ = lQ0 lx0 { Q(x) & ;P(Q0)(x0) } lC
(7)
h¢(S) = lx0 { S[x] & ;P(S)(x0) } 6,IL
(8)
h¢(S)(a) 5a,b,7,IL,lC
(9)
$x{h¢(S)(x) & D[x] 5c,8,QL
(10)
h ^/ P 1,9,QL
(11)
$Q0 $x0 { h¢(Q0)(x0) & P(Q0)(x0) } 10,Def ^
(12)
h¢(Q0)(b0) & P(Q0)(b0) 11,$O
(13)
Q(b) & ;P(Q0)(b0) & P(Q0)(b0) 6,12,IL,lC
(Note 14

Note carefully the difference between a non-poisonous snake and a non(poisonous snake). Here, it is critical to the semantics that the  h-variables range over subsective adjectives.

(1)
;$x { S[x] & ;P(S)(x) & D[x] } Pr
(2)
­: ;$h { $x{h(S)(x) & D[x]} & h^P } ID
(3)
$h{ $x{h(S)(x) & D[x]} & h^P } As
(4)
­: þ 5b,9,SL
(5)
$x{h(S)(x) & D[x]} & h^P 3,$O
(6)
h(S)(a) & D[a] 5a,$O
(7)
;P(S)(a) 6b,5a,Def ^
(8)
S[a] 6a,h is subsective
(9)
;D[a] 1,7,8,QL

One of the more perplexing combinations involves ‘the’ and ‘only’.

First, note that the following are not equivalent.

"The only people I respect are kind."

"Only the people I respect are kind."

So, clearly ‘the only’ =/= ‘only the’, so we have to be careful in constructing our categorial
analysis.

We propose that, when it appears in the phrase ‘the only’, the role of ‘only’ is largely
emphatic, similar to how ‘unique’ operates inside ‘the unique’.

In other words, ‘the’ does the real work, and ‘only’ simply provides EXPLICATURAL emphasis (as we may call it, to use a noun that Grice avoided like the plague, 'explicature').

Let us apply this proposal to a few examples.

First consider the following:

The only people I respect are Jay and Kay.

which is equivalent to

"Jay and Kay are the only people I respect"

which suggests that ‘be’ is transitive.

This is further clarified in the following.
.

1 1õ.
Ïõ
1 Ï 1õ(1õ.) 1
the
1 are Jay and1 Kay
Æ
Ï
only people I respect
30
m
x R[x] = J+K
m
x R[x]1 lx1 { x = J+K }
l
P0 mx P(x)1 lx0R[x] ly1 lx1 { x=y } (J+K)1
.
the1. .are. .Jay and1 Kay.
Æ l
x0R[x]
.
only. .people I respect.

According to this analysis, the sentence says that the maximal count entity containing all the people I
respect is identical to the count-entity Jay+Kay.

By standard mereological reasoning, granting that "respect" is distributive, this is equivalent to saying that I respect an individual if and only if that individual is Jay or Kay.

Now, back to our original example.

"The only people I respect are kind"

Note that this is not equivalent to

"Kind are the only people I respect"

since the latter is, well, ill-formed (even if Speranza can utter it -- and worse, get understood!).

This suggests that ‘be’ is a copula.

This is further clarified in the following.
.

1 1õ.
are kind
Ïõ
1 Ï
the
1
Æ
Ï
only people I respect
K
[ mx R(x) ]
m
x R[x]1 lx1K[x]
.
are kind.
l
P0 mx P(x)1 lx0R[x]
.
the1.
Æ l
x0R[x]
.
only. .people I respect.

According to this analysis, the sentence says that the maximal count-entity containing all the people I
respect is counted among the entities that are kind.

This entails that everyone Speranza respect is kind, provided we read the predicate
‘is kind’ as distributive.

This is strongly encouraged by the presence of the word ‘only’ in Speranza's original phrase.

We next consider how  ‘only’ interacts with ‘if’ in phrases such as:

"I will get into Cal Tech only if I ace all my exams"

"A number is even only if it is divisible by two."

We propose that, in ‘only if’ clauses, the focus of ‘only’ is the antecedent (i.e., the complement
of ‘if’).

Thus, applying our general analysis of ‘only’, we have the following.

type(
only) = (.õ.)õ(.õ.)
.
only. = lh lp { h(p) & ;$q{h(q) & q^p} }
where
h Î ..õ..
q
^p ü q&;p

In other words,

The following is an example analysis.

"I will get into Cal Tech only if I ace all my exams."

.
.õ. .
I ace all my exams
.
0 .0õ(.õ.)
I will get into Cal Tech
(
.õ.)õ(.õ.) .õ(.0õ.)
only if
{
A²C} & ;$r{ (r²C) & r^A}
l
p{ {p²C} & ;$r{ (r²C) & r^p} } A
.
I ace all my exams.
C
0 lq0 lp { {p²q} & ;$r{ (r²q) & r^p} }
.
I will get into Cal Tech.
lh l
p { h(p) & ;$q{h(q) & q^p} } lp lq0 {p²q}
.
only. .if.
(1) (S
²S)²(S²S) 1 Pr lh lp { h(p) & ;$r{h(r) & r^p} }
(2) S
²(S0²S) 2 Pr lp lq0 {p²q}
(3) S
0 3 As q0
(4) S
²S 23 2,3,MP2 lp {p²q}
(5) S
²S 123 1,4,²O lp { {p²q} & ;$r{ (r²q) & r^p} }
(6) S
0²(S²S) 12 3-5,²I lq0 lp { {p²q} & ;$r{ (r²q) & r^p} }
30

The following type-theory derivations demonstrate that the top node is equivalent to the following.
A
«C
(1) {
A²C} & ;$r{ (r²C) & r^A} Pr
(2)
­: A«C 1a,3,SL
(3)
­: C²A 6,SL
(4)
C²C SL
(5)
C ^/ A 1b,4,QL
(6)
;(C & ;A) 5, Def ^
(1)
A«C Pr
(2)
­: {A²C} & ;$r{ (r²C) & r^A} 1,SL,3,SL
(3)
­: ;$r{ (r²C) & r^A} ID
(4)
$r{ (r²C) & r^A} As
(5)
­: þ 6a,7,SL
(6) B
²C & B^A 4,$O
(7) B &
;A 6b,Def ^

Now, here is the problem.

It does not seem that ‘only if’ is equivalent to ‘if and only if’ (cfr. Pears in Berlin et al, essays on Austin), so we propose that ‘only’ has both a strong sense and a weak sense, the latter being the negative half of the former.
.
onlyw. = lF ln ;$n¢ { F(n¢) & n¢^n }
where
F Î .Kõ..
n
,n¢ Î .K.

This gives us the following truth conditions for
‘C only if A’.15
C only if A
º if C then A

The weak sense of  ‘only’ also applies to CNP applications, according to which
only A's are B's does not logically entail

some A's are B's
but only (the weak half):
no non-A's are B's

This is left as an exercise for the Grice Club.

Note carefully that this equivalence, which is counterintuitive, is largely a product of the oddity (so well explained by Grice) of our truth conditions for ‘if’, according to which ‘if’ is truth-functional.

We should not automatically expect this equivalence to obtain for other (more robust) versions of
‘if…then’.

Finally, we consider how  ‘exactly’ works in phrases such as

Exactly three dogs are barking.

We propose that, in this context, ‘exactly’ behaves semantically like ‘only’ where the focus is the
numerical adjective ‘three’.

In particular, for numerical adjectives, we propose the following typeanalysis of
‘only’.

type(
only) = [(ÏõÏ)õ.]õ[(ÏõÏ)õ.]
.
only. = l» lh { »(h) & ;$h¢ { »(h¢) & h¢^h } }
where
» Î .(ÏõÏ)õ..
h
,h¢ Î Á [the class of numeral objects, a subclass of .ÏõÏ.]
where
h¢^h ü h¢ > h
With this proposal in hand, we now offer the following analysis.
exactly three dogs are barking
S
l
(1õ.)õ. 1õ.
are barking
Ïõ
[(1õ.)õ.] (Ïõ.)õ.
[ some
1 ]
[
(ÏõÏ)õ.]õ. Ï
dogs
[
(ÏõÏ)õ.]õ[(ÏõÏ)õ.] ÏõÏ
exactly three
$
x { D[x] & 3[x] & B[x] } & ;$h { $x{ h(D)(x) & B[x] } & h>3 } }
l
Q1 $x{D[x] & 3[x] & Q(x)} & ;$h{ $x{h(D)(x) & Q(x)} & h>3 } } lx1 B[x]
.
are barking.
l
P0 lQ1 $x { P(x) & Q(x) } lÃ0 { Ã0(lx{D[x]&3[x]}) & ;$h{ Ã0(h(D)) & h¢>3 } }
.
some1 .
l
» { »(lP0lx{P(x) & 3[x]}) & ;$h¢ { »(h¢) & h¢>3 } } lx0 D[x] [üD]
.
dogs.
l
» lh { »(h) & ;$h¢ { »(h¢) & h¢>h } } lP0lx{P(x) & 3[x]} [ü3]
.
exactly. .three.

According to this analysis, the sentence says that there is a three-membered set of dogs whose members are barking and there is no  larger set of dogs whose members are barking. The following derivations underwrite the key compositions. Note the abbreviation:
3D[a] üD[a]&3[a]

(1) [(

¤²¤)²S]²S 1 Pr l» { »(lP0lx{P(x) & 3[x]}) & ;$h¢ { »(h¢) & h¢>3 } }

(2)

¤ 2 Pr lx0 D[x]

(3)

¤²S 3 As Ã0

(4)

¤²¤ 4 As h

(5)

¤ 24 2,4,²O h ( lx0 D[x] )

(6) S 234 3,5,

²O Ã0 ( h ( lx0 D[x] ) )

(7) (

¤²¤)²S 23 4-6,²I lh { Ã0 ( h ( lx0 D[x] ) ) }

(8) S 123 1,7,

²O Ã0(lx{3D[x]}) & ;$h¢{Ã0(h¢(lx0 D[x])) & h¢>3 }

(9) (

¤²S)²S 12 3-8,²I lÃ0 {Ã0(lx{3D[x]}) & ;$h{Ã0(h(lx0D[x])) & h¢>3}}

(1) (

¤²S)²S 1 Pr lÃ0 {Ã0(lx{3D[x]}) & ;$h{Ã0(h(lx0D[x])) & h>3}}

(2)

¤²[(N1²S)²S] 2 Pr lP0 lQ1 $x { P(x) & Q(x) }

(3) N

1²S 3 As Q1

(4)

¤²S 23 2,3,MP2 lP0 $x { P(x) & Q(x) }

(5) S 123 1,4,

²O $x{3D[x]&Q(x)} & ;$h{$x{h(D)(x) & Q(x)} & h>3 }}

(6) (N1²S)²S 12 3-5,²O lQ1 $x{3D[x]&Q(x)} & ;$h{$x{h(D)(x) & Q(x)} & h>3 }}

There are possibly other problems left out in this account, but as Grice says, "leave it to implicature".

--- By courtesy of G. Hardegree.