Abstract
In order to provide for a linguistically and cognitively sound theory of negation, we argue for the introduction of a dyadic negation predicate lack and a force dynamic account of affirmation and negation in general.
1 Introduction
Arnauld observes (p. 208; PRL 160) that the sentence There are few pastors nowadays ready to give their lives for their sheep, though superficially affirmative in form, actually contains implicitly the negative sentence ‘Many pastors nowadays are not ready to give their lives for their sheep.’ In general, he points out repeatedly that what is affirmative or negative ‘in appearance’ may or may not be in meaning, that is, in deep structure. In short, the real ‘logical form’ of a sentence may be quite different from its surface grammatical form. The identity of deep structure underlying a variety of surface forms in different languages is frequently stressed. (Chomsky 1966, 87)
While the transformational analysis gradually came to include a variety of non-sentential negatives (see Jackendoff 1969 for an intermediate, and den Dikken 2019 for a well worked out modern version), the primary focus of this theory is still on logical negation at the sentence level, at the expense of key pragmatic aspects such as presuppositions and conversational/conventional implicatures.
In this paper, we retain the transformational generative grammarian's identification of semantic representation with underlying form, a method most popular in Generative Semantics (Goldsmith & Huck 2013), but one that actually goes back to Pāṇini. However, we avoid calling it Logical Form (LF), and speak simply of semantic representation for two reasons. First, because the term ‘LF’ is increasingly used in a narrow sense to mean formulas of Montague's Intensional Logic (Montague 1973) and related calculi, whereas the work presented here relies on a different system of Relevant Logic (Meyer & Martin 1986). Second, because the representation format we use, the Resource Description Framework (RDF, Beckett 2004) has its roots in a different tradition, that of Knowledge Representation (King 1979; Sowa 2000) and conceptual graphs (Quillian 1967; Minsky 1975; Sowa 2008) that computational linguistics uses extensively (see Ch. 1 of Kornai 2023).
logical structure accounts for no more than 12–16% of the information conveyed by a sentence, a number that actually goes down with increased sentence length. (Kornai 2019, 6)
(…) accounts of truth that treat mathematical and nonmathematical discourse in relevantly similar ways do so at the cost of leaving it unintelligible how we can have any mathematical knowledge whatsoever; whereas those which attribute to mathematical propositions the kinds of truth conditions we can clearly know to obtain, do so at the expense of failing to connect these conditions with any analysis of the sentences which shows how the assigned conditions are conditions of their truth.
(…) the form and function of negative statements in ordinary language are far from simple and transparent. In particular, the absolute symmetry definable between affirmative and negative propositions in logic is not reflected by a comparable symmetry in language structure and language use. Much of the speculative, theoretical, and empirical work on negation over the last twenty-three centuries has focused on the relatively marked or complex nature of the negative statement vis-a-vis its affirmative counterpart. (p. xiii)
Although it is not a particle of English, it should be. (p. 309, emphasis in the original)
Remarkably, it is not only English that lacks a * morpheme but, to the best of our knowledge, all natural languages do, an empirical fact just as striking as the marked/unmarked asymmetry emphasized by Horn. The solution proposed here is to use an actual morpheme, lack, as the primitive element, and derive the linguistic asymmetry from the fact that it is dyadic. As we shall see in Section 2, the analogy between logical and arithmetical negatives is quite clear as long as we restrict ourselves to the ancient Greek understanding of arithmetic, where 7 − 5 could be easily computed as 2, but 5 − 7 simply made no sense, as there was no concept of −2 to begin with. lack subtracts something that is by default present, and makes no sense (is infelicitous) otherwise. If blind is defined as ‘lack sight’, blind person makes eminent sense as ‘person lack sight’ whereas #blind stone runs into the problem of subtracting a property that only animals and image-making devices enjoy by default.
We lay out a theory of negation built on the information-theoretic insight that positives, the unmarked case, are not just more frequent but, as befits a communication system, have less information content (require fewer bits). While there is no strict quantitative correspondence between frequency and the size of the code of the kind we find in artificially constructed codes (Huffman 1952), the tendency is unmistakable in natural language and has been noted as early as Zipf (1949). From this perspective, monadic negation is overanalysis in the morphological case, and in Section 3 we argue that the unary negation operator
Our discussion of compositional constructions in Section 3 also aims at exhaustiveness, including many forms that involve negation only in an indirect fashion. We offer a simple, finite state formalization that embodies a more nuanced understanding of affirmation and negation, seeing these as opposing forces in the force dynamic setting (Talmy 1988). The machinery is put to use in Section 4, where we describe how some puzzles generally considered central to the semantics of negation such as double negation, compositional quantifiers, disjunction, and scope ambiguities can be handled with the dyadic system presented here.
There are several aspects of the system presented here that cannot be justified within the bounds of the paper, such as the lack of underlying ternary operators (ditransitives, see Kornai 2012); no probabilistic or other semiring weighting in the metalanguage (Gyenis & Kornai 2019); using grammatical functions as linkers (Kiparsky 1987; Butt 2006); hypernode (as opposed to hyperedge) graphs as the basic data structure (Woods 1975); and no doubt many smaller design decisions that sometimes go against the mainstream choices. For the reader interested in a more extensive defense of these we offer a ‘hook’ to Kornai (2023), where many of these issues are discussed in greater detail, and the same machinery is used not just for negation, but also for spatial and temporal semantics (Ch. 3); probabilistic reasoning (Ch. 5); modals and counterfactuality (Ch. 6); implicature and gradient adjectives (Ch. 7); proper names and the integration of real-world knowledge (Ch. 8); and some computational linguistic applications (Ch. 9). Readers who prioritize syntax over morphology, and in general syntactic phenomena over lexical ones, are particularly urged to take a look at Ch. 2.4 on linking, and readers with a more lexical/morphological set of concerns are advised to look at Ch. 2.5.
2 Negation in the lexicon
Another difference between the generative and the algebraic approach is that only the former implies commitment to a specific set of primitives. To the extent that work on lexical semantics often gets bogged down in a quest for the ultimate primitives, this point is worth a small illustrative example. Consider the cyclic group Z3 on three points given by the elements e, a, b and the following multiplication table (Table 1).
Table 1.Multiplication in Z3
e a b e e a b a a b e b b e a
The unit element e is unique (being the one and only y satisfying yx = xy = x for all x) but not necessarily irreducible in that if a and b are given, both ab and ba could be used to define it. Furthermore, if a is given, there is no need for b in that aa already defines this element, so the group can be presented simply as a, aa, aaa = e i.e., a is the ‘generator’ and a3 = e is the ‘defining relation’ (as these terms are used in group theory). Note, however, that the exact same group is equally well presented by using b as the generator and b3 = e as the defining relation – there is no unique/distinguished primitive as such. This non-uniqueness is worth keeping in mind when we discuss possible defining vocabularies (Kornai 2010a)
About 12% of the defining set (144 items altogether) involve some form of negation: accept accident acid arrive atom bad bar behind bend black block building burn calm catch chance child clean close coal continue continuous cover curve dark dead destroy different dry eager easy elephant end fail finish firm first flat free full gas gradual green hang hard hide ill instead jump laugh leave light limit long lose mean middle must narrow natural necessary need negative new night no nothing object off offensive one only open opinion oppose out park permanent plant police practice preserve prison private protect public quiet reach remove rest right romantic rough rubber rude sad safe same send separate serious sharp short simple sincere single sleep slope smoke smooth soft solid sometimes special steady steal stiff stop straight strange stupid success sudden sure surprise take tent thick thin tie tight together twist unless waste water weak without wrong. This list is actually a bit shorter (139 elements), because in the 144 we count with multiplicity elements that are homophonic in English, such as thin ‘liquidus’ as in thin paint versus thin ‘tenuis’ as in thin reed. The technical means of disambiguating such lexical entries are irrelevant for this paper, but we note that we avoid spurious duplication of entries for metaphorical senses, treating, e.g., acid in vinegar is an acid and in an unnecessarily acid remark by one and the same lexical item, so that disambiguation is rarely called for.
The list has many elements such as water which seem to lack any negative aspect. But a closer look at the definition ‘liquid, life NEED, has no color, has no smell, has no taste’ shows how negative statements enter the picture. Many of these can be handled by our central innovation, in our case replacing the above definitions by
In many adjectival oppositions, normally handled by some version of scalar semantics such as (Kennedy 2007), it is very easy to pinpoint the asymmetry that Horn talks about, and assign negative value to one side of the scale unambiguously – for a summary of standard marked/unmarked diagnostic tests see Lehrer (1985). For example, invisible carries overt negative marking relative to visible, so we conclude that conceptually it is invisible things that have no visibility, rather than visible things that lack invisibility. Yet other oppositions, such as between full and empty, offer no overt morphological cues, but are nevertheless trivial to classify, because their definition hinges on words (in this case presence v. absence of filling material) one of which is broadly synonymous to overt negatives: in this case, absence to lack or want (Merriam-Webster).
In many cases like dirty or blind the lexical entry carries a negative (prejudicial) sentiment, but not all of these are amenable to an analysis that contains a negative. Every analysis of blindness invokes a logical negative: ‘sightless’ (Merriam-Webster) ‘unable to see’ (Longman), etc. Within the bounds of our defining vocabulary, we can write this as
The same treatment can be effortlessly extended to many antonym pairs, e.g., defining good as the object of WANT, and bad as
There is of course an entire class of lexical items whose primary function is to negate: the words no, not, the clitic n't, the prefixes un-, im-, de-, non-, anti- and the like. Ideally, we wish to represent these by a unary negation operator, provisionally written as
Establishmentarianism is the ‘movement or ideology advocating the principle of an established Church with special rights, status, and support granted by the state’, an issue most people never heard of and most likely stand neutral on. Disestablishmentarianism is the directly opposed ‘movement or ideology advocating the withdrawal of special rights, status, and support granted an established church by a state’, and antidisestablishmentarianism is of course the movement or ideology directly opposed to this. Conservative people who prefer the status quo will likely be antidisestablishmentarian, but not establishmentarian, since neither of these movements/ideologies would be content to leave things as they are.
(…) based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well.
As our survey demonstrates, the dyadic negation operator lack, even though it is limited to cases of incompatibility with lexical prespecification, is already sufficient for capturing the lexical semantics by means of RDF-style meaning postulates – the traditional neg, which has no such limitation, is plainly overkill for this. The modal aspect is also clear, not just for the epistemic case used for ⊚ above, but also for the deontic cases that will be central to our discussion of compositional negation in Section 3. Consider up and down. Let's say we are at a construction site, perhaps standing on a ladder, and receive the instruction move up! which we want to defy. This can be achieved not just by moving down, but also by moving sideways, or by not moving at all. All three of these acts will conform to the negated command don't move up. Don't move or rest are contrary to move, and move down is contrary to move up, but these simply don't exhaust the entire space of possibilities, which also contains moving sideways, an action contrary to rest, move up, and move down alike. Thus, the classical Boole/De Morgan picture where negation satisfies the involution law is simply not tenable for natural language – we present our own solution in Section 3, and return to double negation in Section 4.1.
Several variants of quantum logic have resources to express the idea that the door is in some quantum superposition state. However, neither the logic used here nor natural language have such resources: what ⊚ describes is the common situation when we cannot (or just do not wish to) make a committment to ⊕ or ⊖, rather than some exotic ‘Schrödinger's Cat’ situation.
2.1 Quantifiers
Ever sice Aristotle's De Interpretatione, and particularly in the hands of Frege (1879), Russell (1905), the treatment of a restricted class of lexical elements, quantifiers, has become virtually inseparable from the treatment of negation. In this regard, our treatment is a considered return from Montague (1973) and subsequent work to the earlier tradition, whose last significant exponent was Peirce (Böttner 2001). While Montague Grammar eventually treated nominals as generalized quantifiers (Gärdenfors 1987; Badia 2009), we move in the other direction, and treat quantifiers as nominals whose compositional behavior, which we defer to Section 3, is largely dictated by their semantic content, rather than as special term-binding operators. In doing this “we make purposely very little distinction between an individual fox, the species Vulpes vulpes, the set of foxes in the world, or the class of potential foxes in all possible worlds” (Kornai 2018).
That some kind of quantificational ur-element is needed is already clear from a closer look at our definition of good as the object of WANT. To write out the definiens in infix (SVO) order, it is not enough to write
Lexicalized quantifiers either in their base form some, any, no, … or in a subtyped form someone, somebody, something, somewhere, somehow, anyone, anybody, anything, anywhere, anyhow, noone/no-one, nobody, nothing, nowhere, … will be treated on a par with pronouns, including interrogatives, as members of a new lexical category proquant, whose crosslinguistic coherence (but not the name proquant) is argued for by Szabolcsi (2015). Quantifiers of a clearly compositional nature, like at most seven, no more than ten, are deferred to Section 4.2, but we note here that two is not defined as one plus one: ‘being two’ is an inherent perceptual property just like ‘being blue’. This works well up to the limits of human numeracy, the magic number seven plus or minus two (Miller 1956), and means exactly two (as opposed to the ‘at least two’ readings sometimes proposed as default (Horn 1972) from which the ‘exactly two’ reading is derived by exhaustification, see Haida & Trinh 2020, fn. 1). Many, if not most, of the proquants are either lexical primitives, or have a compositional analysis that directly relies on abstract primitives such as the
At this point, the question may be asked: rather than using gen and lack as primitives, and deriving unary no from these as gen lack, why not use binary has and unary ¬ as primitives, and derive lack as ¬ has? To the extent ¬ and has are independently motivated, this appears considerably simpler (even though gen is also well motivated, see Carlson & Pelletier 1995), and certainly more in keeping with tradition. Our answer is threefold. First, such an analysis of lack misses the key lexical insight, that it negates something that is ordinarily present. It is true that blind means ‘not have sight’, but if this were the entire story then #blind stone should be just as felicitous as blind person. Second, the contrast in felicity smoothly extends from the lexical to the compositional domain: consider the priest at the end of a marriage ceremony uttering
- (1)The bride and the groom shall HAVE each other (for the rest of their lives, in sickness and health, …).
If the paraphrase of lack as ‘not have’ were reasonable, by negating (1) (e.g., as a response to the standard callout Speak now or forever hold your peace) we should obtain
- (1a)The bride and the groom shall not have each other (for the rest of their lives …).
or, by the suggested analysis
- (1b)#The bride and the groom shall lack each other (for the rest of their lives …).
which is clearly infelicitous. A third point against this suggestion is that it leaves Horn's observation about the linguistic asymmetry between positive and negative statements entirely unexplained, indeed, mysterious. This will be particularly clear for double negation, a matter we will return to in 4.1.
3 Negation in compositional constructions
From our perspective, the traditional Square of Opposition (Parsons 2017) is inhomogeneous. ‘A’ statements of the form every s is p are simply written
For example, when we say in naive physics (Hayes 1978) that atoms are small particles that have nuclear energy (never mind how well this definition fits modern physics, our target is ordinary language), the definiens is formulated as
Of particular interest here is the style of default inference supported: if energy is provided by atoms, that energy is nuclear, if a cane is owned by a blind person, that cane is white, and so forth. This is indeed in opposition to ‘E’ statements no s is p whose central goal is to block similar inferences: persons have organs, these organs are typically functioning, so persons can walk, talk, see, etc. – this all goes without saying. The inferences are highly automatic/preconscious, yet we rely on such inferences in the process of making sense of natural language utterances all the time.
Clearly, the raison d’être of the word blind is to guarantee that some of these inferences are blocked, hence our definition
Finally, ‘O’ statements, some s is not p mean lack of implication from s to p, a view equally compatible with Aristotle's original formulation not every s is p, which need not carry the existential implicature that many take for granted in the analysis of some. This becomes a bit clearer if we take into account the Aristotelian view that the predicate inheres in the subject: there is no difference, other than surface form, between Joe is fat and Joe has fatness or Joe fat(ten)ed. Whether the predicate is expressed adjectivally, nominally, or verbally has no bearing on its relation to the subject, which is one of subsumption. On this view, O forms are simply
The outstanding issue is explaining why unary
Forces in negation and affirmation
Citation: Acta Linguistica Academica 71, 1-2; 10.1556/2062.2024.00656
A word of caution is in order: while finite state automata of the sort depicted here are capable of counting modulo the number of states, the iteration could go to any depth. For example,
To see how the state transitions actually work, and to refine the picture to include not just negation but also affirmation, we analyze some ordinary language expressions here. We start with imperatives, both because they are the dominant source of negation in primary linguistic data (kids are urged don't touch and don't put it in your mouth long before they encounter any other negatives), and because they make clear how key features of our model fit with the Australian Plan. An imperative X calls for some action that results in a state of affairs (situation, possible world) where X is fulfilled. Using the
Now, to negate (defy) an imperative means doing some Y that is incompatible with the X targeted by it. Any Y ⋏ X, including the important special case where X calls for some direct action we refuse to perform, is a good way for negating the force of the command, as in the construction site example above. Since phrasal verbs like move up are often considered lexical (semicompositional) here we will consider the negatives Don't smoke! or No smoking and their paired affirmative Smoke! where the negation is clearly compositional. Frege already noted that the deontic element takes wide scope over the negation. In the semantic representation, written here with unary ¬ to make the point in standard terms, we have
- (2)Diamonds are valuable to John (but Mary considers them stupid, overpriced trinkets).
In the deep structure most analyses would posit a matrix element with an experiencer subject and the embedded proposition as object:
- (2a)
John CONSIDER (diamond HAS value) - (2b)
Mary CONSIDER (diamond LACK value)
The work to be done by the syntax, relating the matrix subject John/Mary to the experiencer expressed by to on the surface; and getting the subject diamond of the predication expressed by the small clause in surface subject position is the same work for the positive HAS and the negative LACK case. There are many syntactic theories capable of doing this, from old-style transformational grammar to minimalism, and it is not clear how on the strength of examples like (2) we could choose among them. The picture is further complicated by the interaction of tense with modals and negation: as Han (2001) notes, in many languages (e.g., Italian, Modern Greek, and Spanish), imperatives cannot be negated, whereas in English, German, etc. they can be.
Normally, locations are unspecified for smoking/nonsmoking, though there are many places where the default is nonsmoking and some where the default is still smoking. A sign that simply says No smoking has the same force as one with an overt deontic operator Smoking prohibited. The opposite of this is a sign smoking (permitted), and not #smoking mandatory which would carry a much stronger affirmation of smoking. This is not because we don't find obligatory rules, there are many from seatbelts mandatory to you must agree to our privacy policy first, but rather because we find smoking increasingly restricted to special settings like dedicated smoking rooms at airports.
Returning to a moment to our earlier example, it is clear, even if we don't take overt morphological marking into consideration, that the normal (default) state of things is to be visible, and invisibility, to the extent it exists, is the marked case. The primary goal of prohibitions is to designate their object as abnormal. Consider You shall not murder. Biblical Hebrew (and English at the time of King James) made no distinction between imperative and imperfect, the normative effect (of an ideally kept command) is that in the future there is simply no murder (retzach). In our formal language of semantic definitions we can write this as
We often see antonyms that fit well with the tripartite picture of Figure 1: heavy really means ‘has weight greater than gen’ and light means ‘has weight less than gen’. Since the generic will unify with the subject, the effect that (Parsons 1970) illustrates with the example of enormous flea, that such a flea is still rather small, is easily explained: such a flea has size much larger than
Returning to our theory of You shall not murder,
4 Putting it all together
The picture of negation that emerges from our considerations is somewhat nontraditional: instead of the standard, unary negation operation
How the (primitive) dyadic negation operator
The semantics of any expression must be fully accounted for by the lexicon and the grammar taken together.
The Principle of Responsibility is only slightly stronger than the standard Principle of Compositionality which takes the semantics of any expression to be determined by the semantics of its lexical components and by the grammatical way those are combined. The additional requirement it imposes is that the ‘pragmatic wastebasket’ remain empty at all times: it doesn't matter whether we call ordinary inferences grammatical, lexical, or pragmatic (and perhaps extragrammatical), the overall system needs to account for these, either in one specific component, or by means of tracing the inference process through several components.
Here the entire work is performed in the semantics, using one and the same implicational mechanism for pragmatic and ‘purely semantic’ inference, a mechanism that includes substitution of the definiens for the definiendum salva veritate (There are other parts, such as kal va-chomer, see Kornai (2019, Ch. 19.4), and spreading activation (Nemeskey et al. 2013) that are not discussed here as they don't play a constitutive role in the following.) We analyze our X constituent, trades take effect as
4.1 Double negation
‘Natural’ negation only involves objects or elements a speaker or listener is attending to … It makes no sense to instruct a listener to suppress a thought he is not considering or an idea he is not having.
The only standard case of double negation is when the first negative is syntactic and the second morphological: a not unhappy person, a not unfriendly letter, … (see Horn 1989 5.1.3). What is remarkable about such cases is that they are no longer about the negation of some default: there is no assumption that people are generically happy or letters are friendly. It is the unhappiness of a person that is being negated here, an idea that we couldn't reasonably assume to have been already present in the listener's mind as a default assumption. Rather, it is the compositional meaning
In this case, the negation of the predicate is easy: both ¬
Since standard tests of constituency (Wells 1947) support the second analysis, we start with not unhappy and substitute, salva veritate, the definition of unhappy, to obtain
In the other analysis, we start with unhappy person with the semantics
When both
4.2 Compositional quantifiers
One area where the standard theory appears vastly superior to the one presented here is assigning semantics to obviously compositional quantifier structures such as at most seven, no more than ten. But this is accomplished at the price of sweeping under the rug the fundamental problem we started out with, assigning semantics to the atomic units. What is the semantics of seven? The dictionary suggests ‘the number 7’, but this is not exactly helpful, since ‘7’ is left undefined.
Could we actually use here the standard mathematical semantics that rests on the Peano axioms? The requisite formulas
Altogether, the standard logical approach is inappropriate for handling what little overlap there is between the semantics of logical and natural language expressions. It offers spurious precision, not just in the handling of ‘fuzzy’ quantifiers but also for any number above the magical number 7 ± 2 (Miller 1956). Since the standard theory was developed in order to overcome the well-known limits of human numerosity (Dehaene 1997), it is incapable, by design, of accounting for these limits. A fuller discussion would go beyond the scope of this paper, but a step in the right direction is already taken in Gordon & Hobbs (2017), who restrict Peano arithmetic to the metatheory, and concentrate on the cognitively relevant structures like ‘half orders of magnitude’.
Using this notion, we can assign meaning to lexically complex quantifiers such as somewhat in constructions such as It will be somewhat warm(er) which we take to mean ‘it will be perceptibly warm(er)’ where perceptibly means ‘by half order of magnitude’. Since this is arguably an adverbial meaning, we will concentrate here more on the proquants, where some- has a pure existential import. Deriving the lexical meaning of quantifiers is made easier by the fact that in most languages they share a sortal type with pronouns, so we will have interrogatives who, what, where, when, …and follow the same typing everyone/anyone/someone/noone, everything/anything/something/nothing, everywhere/anywhere/somewhere/nowhere, everytime/anytime/sometime/never.
The sortal types are quite transparent: who requires a
As standard (Katz & Postal 1964; Langacker 2001), we analyze who as
In systems of Knowledge Representation (KR) such as Cyc (Lenat & Guha 1990) it is common to distinguish individuals, e.g., some particular poet, say Allan Ginsberg, from the class Poet, of which Ginsberg is an InstanceOf. The semantics of any-, however conceived, will have to express the choosing of one particular instance from a class, the central element of the meaning being that it doesn't matter which instance (Kadmon & Landman (1993) call this the ‘free choice’ reading of any). Here we take advantage of the thematic role mechanism that we have at our disposal independent of negation and quantification (Dowty 1989) and the fact that we already have a fundamental
With the other proquantal roots out of the way, we can turn to our central subject matter here, the semantics of noone, nothing, nowhere, …. This requires no special effort, in that no- is already defined as
4.3 Disjunction
In BAs, De Morgan's Laws connect conjunction to disjunction in a perfectly symmetrical fashion. But in natural language semantics conjunction is the default operation: unless some other particle is present we interpret phrases and clauses conjunctively. In case of proper nouns, we treat the conjunct as a collective (Scha 1981). Given that negation is a marked operation, there is no way to follow the BA technique and reduce disjunction to conjunction by means of De Morgan's laws. In fact,
An alternative (or or) proposition contains two statements, the acceptance of one of which involves the rejection of the other … either may be agreed to, but not both. (Lakoff 1971, 142)
This makes
We note that
4.4 Scope ambiguities
Compare Everyone on Cormorant Island speaks two languages to Two languages are spoken by everyone on Cormorant Island. There is a sense that the active sentence does not require these to be the same two languages for everyone, whereas the passive sentence does. But how strong is this sense? Early generative theory (Katz & Postal 1964) assumed that both readings are available for both sentences. This left explaining which reading is preferred in which context to factors that go beyond syntax and semantics such as communicative dynamism (Firbas 1971), as there is a similarly strong sense that the active sentence is about the inhabitants of Cormorant Island while the passive is about two languages. Also, it is worth keeping in mind that the entire phenomenon is somewhat marginal. The ratio of passives to actives is somewhere between 4% and 18% depending on genre (Givón 1979), e.g., the BNC has 662 instances of killed by compared to 4407 instances of kill. Quantifier phrases (nearly 70k examples in the BNC) will appear in the by-phrase only in about 1.5% of the cases.
In the semantic representation system we rely on (see Ch. 1.3 of Kornai 2023), the active sentence means
Here we consider, very briefly, the other proquants. Anyone on Cormorant Island speaks two languages versus Two languages are spoken by anyone on Cormorant Island has the same level of uncertainty in regards to judgments of grammaticality and readings as the everyone examples we started out with. To avoid bracketing, we will write
Finally, let us consider the examples most relevant to our subject matter, negated universals or ‘E’ statements. Clearly, Noone on Cormorant Island speaks two languages means
As for ‘E’ passives, we get
4.5 Open problems
One area of notorious complexity that we left untouched is prosody, especially contrastive stress, what Manaster Ramer (1995) called ‘the last refuge of the formal grammarian’. Comparing It was not ‘Peter and ‘Kate but only ‘Kate who made this mistake to It was not Peter ‘and Kate but (‘)only ‘Kate who made this mistake is hard. The task of the grammarian is made even harder by a key design decision of the mainstream syntactic framework, starting with the ‘T model’ of Chomsky's Pisa Lectures, to separate semantic and phonological interpretation early on in the derivation. In contemporary such as the ‘new Minimalist Program’ of Chomsky et al. (2023) this decision is preserved, so the underlying representation must contain some contrastive stress morpheme “’ to track effects that impact both the phonology and the semantics. Following (Harley 2014), the morphemes are classified as roots or as features, but any contrastive stress element “’ seems to display properties of both.
5 Conclusions
There is no question that the proposal made here sacrifices quite a bit on the mathematics side: conjunction is not commutative, Boolean duality is gone, and there are many ripple effects through the entire system we haven't even discussed, e.g., that existential quantification no longer amounts to infinite disjunction. But the gains on the linguistic side are considerable: we have a formal theory of word meaning whereby we can assign semantics to morphological operations in a manner that smoothly extends to compositional semantics.
In regards to negation, the semantic theory proposed here and in related work (Kornai 2010a; Kornai et al. 2015) captures well the key observation that negation is not an involution, and in general offers translations whose processing difficulty (Xiang, Grove & Giannakidou 2016) correlates inversely with their frequency. Clearly, the theory is a better fit with the classical Knowledge Representation tradition (Brachman & Levesque 1985, 2004) and with database logic than with the first- and higher-order (intensional) calculi familiar from Montague Grammar and related theories. We do not see this as a loss, especially not from the learnability perspective (Gyenis & Kornai 2019).
We started with Benacerraf's observation that sentences in natural language and in mathematics are different enough to merit separate semantic frameworks. Were this not so, it would actually be hard to explain why Boolean Algebra, and modern logical calculi in general, took so long to develop from Aristotle's logic. Our work, in many ways a considered return to a more Aristotelian perspective, is not an attempt to ‘reform’ standard mathematical logic, which we take to be the correct theory of the domain. Rather, our goal is to build, with the same care, a formal theory of natural language semantics, even at the price of finding this theory insufficient in the mathematical domain.
Acknowledgments
We thank Anna Szabolcsi (NYU), András Máté (ELTE), and Hans-Martin Gärtner (RIL) for incisive comments on an earlier version of this paper. We are grateful to Dávid Nemeskey (ELTE), Gábor Recski (TU Wien), Márton Makrai (Institute of Cognitive Neuroscience and Psychology), Attila Zséder (Lensa), and all other contributors to the ongoing development effort at GitHub.
We are grateful to the anonymous reviewers for Acta Linguistica. The author was particularly pleased by the tough report by Referee ‘A’, which was tremendously helpful for making this a better paper. To quote the timely Bonnici & Simske (2023) “A truly excellent review, whether recommending acceptance or non-acceptance, motivates the authors to do more in the area their submission addresses”, and this is precisely what happened here. We are also grateful to Márta Abrusán (CNRS) and Geoff Pullum (Edinburgh) for pointing the author to very relevant literature.
Kornai was supported in part by 2018-1.2.1-NKP-00008: Exploring the Mathematical Foundations of Artificial Intelligence, by the Hungarian Scientific Research Found (OTKA), contract number 120145, and by the European Union project RRF-2.3.1-21-2022-00004 within the framework of the Artificial Intelligence National Laboratory.
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