Not Only Barbara
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Abstract
With this paper I aim to demonstrate that a look beyond the Aristotelian square of opposition, and a related nonconservative view on logical determiners, contributes to both the understanding of Aristotelian syllogistics as well as to the study of quantificational structures in natural language.
Keywords
Categorial logic Square of opposition Cube of oppositions Generalized quantifiers Conservativity Topical restriction Focus sensitivity Existential there Only Many1 Introduction
Aristotelian syllogistics is confined to a square of oppositions that has fascinated linguists and logicians for centuries. It has overshadowed a cube of oppositions, discovered by Reichenbach some decades ago, of which the square constitutes just one side. In this paper I restage this cube and uncover its syllogistics, polishing and slightly improving Reichenbach’s observations (Sect. 2). The onesided focus on the square has aligned with a bias for conservative determiners in the theory of generalized quantifiers. However, certain structural features of quantificational structures in natural language can be properly delineated only if nonconservative determiners are not excluded from the outset (Sect. 3). I conclude with an outline of some natural and interesting subsections of the cube other than the square (Sect. 4). In each of the three sections the trivial conclusion is that a restricted focus hampers seeing things.
2 Categorial Logic
2.1 The Square of Opposition
A syllogism counts as valid if and only if the conclusion logically follows from the premises.^{1} That is to say, if and only if the relations between predicate and middle term and between middle term and subject that are stated by the premises, logically enforce a specific relation between the predicate and subject themselves, as stated by the conclusion. A clear, also called ‘perfect’, example is one in which the premises predicate \(P\) of all that is \(M\), and \(M\) of all that is \(S\), in which case we can conclude that, if the premises hold, then \(P\) applies to all that is \(S\). Formally, \(MaP\!,SaM / SaP\). This is a valid syllogism of the first figure, also known as ‘Barbara’.
Aristotle’s syllogistics can be, and has been, substantiated and supplemented in various ways. The terms \(A\), \(B\), ..., can be supposed to make a distinction in some ontological (e.g., settheoretical) space, to the effect that \(A\) and not\(A\) (indicated by \(A'\)) satisfy at least two conditions: noncontradiction (nothing is both \(A\) and \(A'\)) and excluded middle (everything is either \(A\) or \(A'\)). (It is natural to also assume nontriviality, that there actually is or are \(A\) and \(A'\).)
We see that the conclusion \(AoB\) of Ferio, indicated by the cross in \(A\) but not in \(B\), has already been established through the premises. The inference, thus, is valid. The (in)validity of the other syllogistic schemes can be demonstrated likewise.
The two vertical lines in this square connect propositions which contradict each other. If we assume nontriviality, the diagonals indicate that the propositions on the top each presuppose the diagonally connected ones on the bottom. The two top propositions, on this assumption, can therefore not both be true, and are called ‘contraries’; the two bottom propositions cannot both be false then, and are called ‘subcontraries’. Thus, as Aristotle held, “All men are just.” and “No man is just.” are contraries, since the predicate “just” is both ascribed to and withheld from the substance “man”. Assuming, as Aristotle did, that there are instances of “man”, individual men, these sentences cannot both be true. Likewise, “Some men are just.” and “Not all men are just.” cannot both be false then. Upon the stated assumption there must be some man, and he must be either just (\(AiB\)) or unjust (\(AoB\)).
Standard categorial logic has been confined to the (propositions or combinators from this) square. It is no exaggeration to say that for some it constitutes a wholly logical space. Logically speaking, however, the square fills only part of logical space, as, among few others, Hans Reichenbach observed.
2.2 Cubing the Square
 (7)
A syllogistic scheme \(uvw\)\(x\) is valid iff its inverse \(u^iv^iw^i\)\(x\) is.^{5}
2.3 Cubistic Syllogistics
Since the cube is converse and negationcomplete, its logic is quite transparent. Conversions help us get around some practical limitations imposed by the standard presentation of the syllogisms. As said, upon its standard presentation the first, major, premise of a syllogism should contain the major term \(P\) of the conclusion, and the minor should contain the subject term \(S\). But, of course, it is immaterial, logically speaking, in what order the arguments of a valid inference are presented. If a proposition \(r\) follows from propositions \(p\) and \(q\), it follows from \(q\) and \(p\). The slightly annoying point is that if we reorder the premises of an inference of the first figure, the subject and predicate terms switch roles, and the middle term switches places in both of the premises. But since we found the cube conversion complete, we can rescue the situation. The role switch of the subject and the predicate term can be made undone by a, meaning preserving, conversion on the conclusion, and the double switch of the middle term can be made undone by a conversion of the two premises.
Observation 1

A syllogistic scheme \(uvw\)1 is valid iff \(v^cu^cw^c\)1 is valid.
(If one scheme is valid if and only if another scheme is, we henceforth call the two schemes ‘equivalid’.) Since all combinators in the cube have their converses, reordering of premises is always possible.
Conversions also help crossing the borders between the four figures. By a conversion of its premises a syllogism in one figure can be translated into an equivalid one in another figure. If we have a syllogism \(BuC, AvB / AwC\) of the first figure, and we use \((BuC)^c\), which is of the form \(Cu^cB\), instead of \(BuC\), we get a syllogism of the second figure, and since \(BuC\) and \(Cu^cB\) are equivalent, the latter syllogism is valid if and only if the first is. This insight gives us a recipe to quadruply any valid syllogism.
Observation 2

The syllogistic schemes \(uvw\)1, \(u^cvw\)2, \(uv^cw\)3 and \(u^cv^cw\)4 are equivalid.
The two observations also tell us how to reorder premises in the figures other than the first one. For, also, \(uvw\)2 and \(vuw^c\)2 are equivalid, as are \(uvw\)3 and \(vuw^c\)3, as well as \(uvw\)4 and \(v^cu^cw^c\)4.
Since the cube is negationcomplete, and the syllogistic validities are schematic, we can obtain valid syllogistic schemes from other valid schemes by systematically negating the predicate term, the subject term, the middle term, or any number of them. This is laid down in the following, fairly obvious, observation about syllogisms of the first figure.
Observation 3
(Negation) A syllogistic scheme \(uvw\)1 is valid iff \(u^{P'}vw^{P'}\)1 is valid (\(P\)Negation) iff \(uv^{S'}w^{S'}\)1 is valid (\(S\)Negation) iff \(u^{S'}v^{P'}w\)1 is valid (\(M\)Negation).
Thus, given the validity of one of the eight schemes, they are all valid.
The syllogistic schemes dealt with thus far all have general conclusions. Valid schemes with particular conclusions can be obtained by the principle of noncontradiction. If a conclusion follows from two premises then a negation of the conclusion entails that the premises cannot both be true. There are, of course, two options. We can take the negation of the major premise to follow from the negation of the conclusion and the sustained (but converted) minor premise, and also the negation of the minor can be seen to follow from the (converted) major and the negation of the conclusion.
Observation 4

The syllogistic schemes \(uvw\)1, \(w'vu'\)3 and \(w'v^cu'\)1 are equivalid (MCP).

The syllogistic schemes \(uvw\)1, \(uw'v'\)2 and \(u^cw'v'\)1 are equivalid (mCP).
(The two new cubes are the CPimages of the original one. The legend explicating the lines therefore changes, but in a systematic fashion of course. For instance, in the cube in the middle the vertical lines indicate \(S\)negation, and in the cube on the right they indicate \(P\)negation.) We have by now derived 24 syllogistic schemes of the first figure, and, by conversion, four times that number in total. Actually, they are all the valid schemes. A more general observation may serve to substantiate this claim.^{8}
Observation 5
 (1.)
the middle term \(M\) occurs both negatively and positively in the premises,
 (2.)
the polarities of the major term \(P\) and the minor term \(S\) in the conclusion align with those of the two terms in the premises, and
 (3.)
there is at most one particular premise, and then, and only then, also a particular conclusion.^{9}
In the outline of the proof we have already employed the fact that the syllogistic schemes displayed in the three cubes, and their conversions into the other figures, are all valid schemes. It also shows they are the only valid ones.
The generalization stated here may serve to adjust a common and old conviction about syllogistic validity. According to received wisdom, first, a scheme can be valid only if it has at least one positive premise, and, second, a valid scheme with a negative premise can only yield a negative conclusion. Both generalizations are correct if restricted to syllogisms obtained from the square of opposition, but they do not hold in general, as we see from the syllogisms obtained from the cube. A valid scheme can after all employ two negative premises: \(\ddot{a}\ddot{a}\ddot{a}\)1 (Bärbärä), and \(\ddot{a}ee\)1 (Cälerent) (this was also observed by Reichenbach 1952). Moreover, we can draw a positive conclusion from a negative premise. An example is the valid scheme \(x\ddot{a}x\)1 (Xäx), and we also have Xena (xea), again. Xena has an \(a\)conclusion which is positive, or affirmative, without any doubt, and the minor \(e\)premise is undisputably negative.
The present findings also lead up to the following conclusion. The validity of a syllogistic scheme in any figure can be traced back to that of one in any other figure. Moreover, the valid schemes of the first figure can be derived from only one. Thus:
Observation 6
(One for All) All valid syllogistic schemes from the cube can be derived from any one of them by means of conversion, negation and contraposition.^{10}
This is a conceptually welcome result about the syllogistics of the cube. Notice that a completely analogous result can be stated for the restricted set of validities emanating from the square. The slightly disturbing point is that the structural principles involved have only a limited application within the confines of the square.
3 Generalized Quantifiers
In this section the combinators from the cube are taken up in the wider theory of generalized quantifiers. In the first Sect. 2.1 give a characterization of the combinators as determiners featuring characteristic semantic properties. The features characterizing the cube are next shown to be more widely relevant for the analysis of quantified constructions in natural language. It will be seen that some of the phenomena described cannot be properly described, or even perceived, if it is assumed that the combinators belong to the square, or that determiners more in general ought to be ‘conservative’. The last subsection discusses somewhat more extensively this common, but upon reflection too limiting, assumption that natural language determiners actually are conservative.
3.1 A Generalized Quantifier Characterization of the Cube
The theory of generalized quantifiers is concerned with the study of quantifiers and determiners in natural and formal languages. The primary interest is in the semantic conception of a determiner, settheoretically understood as a relation between sets of individuals.
Definition 1
(Determiners) Given a universe of individuals \(E\), a determiner \(D_E\) on \(E\) is a relation on \(\mathcal{P}(E)\), i.e., \(D_E \subseteq (\mathcal{P}(E) \times \mathcal{P}(E))\).^{11}
(I write \(D_E(A,B)\) iff \(\langle A,B \rangle \in D_E\).) Our primary interest is in the semantic conception of determiners, and their syntactic realization, as lexical determiners or otherwise, is of secondary interest. Even so, syntactically, a determiner can be taken to combine with two expressions, understood as denoting sets, so as to produce a sentence. The resulting sentence is true if and only if the determiner, semantically conceived, holds of the pair of associated sets. These sets are often, but not exclusively, denoted by a nominal and a verbal phrase, respectively.
The categorial combinators we have seen in the previous section are essentially determiners. The corresponding settheoretical objects are called by distinguished names, the first six of them fairly standard. That is, we have \({ {NO}}\) (\(e\)), \({ {SOME}}\) (\(i\)), \({ {ALL}}\) (\(a\)), \({ {NOT\_ALL}}\) (\(o\)), \({ {ONLY}}\) (\(\ddot{a}\)), and \({ {NOT\_ONLY}}\) (\(\ddot{o}\)); the less common ones are phrased here as \({ {UNLESS}}\) (\(x\)) and \({ {ELSE}}\) (\(y\)). The theory of generalized quantifiers enables a straightforward and general characterization of the cube, as well as of its sides.
The eight categorial combinators can be characterized as logical determiners that are minimally sensitive to extension of the universe. If an element is added to a universe \(E\) it is added to either one of the four subsets relevant for the valuation of a determiner. A determiner counts as minimally sensitive to addition if such an addition affects the valuation of the determiner only once.
Observation 7
(Characterization of the Cube) The cube hosts the logical determiners that are minimally sensitive to extension of the universe.
The Venndiagrammatic representations in Sect. 2 show that \(D_E(A,B)\), when true, may get falsified when exactly one individual is added to one of the distinguished areas—or, when false, may get verified by one and only one such an addition. These are exactly the eight possible casualties. I will call these eight determiners the minimal logical determiners in what follows.
The possible impact of an extension of the universe enables us to distinguish the combinators from the top and the bottom plane of the cube.
Observation 8
(Top and Bottom Plane) The top plane of the cube hosts the minimal logical determiners which can be falsified through an extension of the universe; the bottom plane of the cube hosts those that can be verified.
We can identify the four other sides of the cube by means of some other general, characteristic, semantic properties of determiners, as we will see presently.
The domains or universes of quantification employed in natural language are normally contextually determined, or constrained, and the first argument of a determiner can be taken to establish that domain. If a determiner \(D\) is used that way I will refer to it as yielding the nominally restricted reading \(D^n\) of \(D\). (Such a reading is, slightly misleadingly, called ‘nominally restricted’, because in paradigmatic uses of determiners the first argument is supplied by a nominal phrase.) It is defined as follows.
Definition 2

\(D^n_E(A,B)\) iff \(D_{E \cap A}(A,B)\) (that is, iff \(D_A(A,A \cap B)\)).
A nominally restricted reading of a determiner makes us look at the individuals in the extension of the determiner’s first argument only—as if \(E=A\), and if the determiner is logical its interpretation only depends on the cardinalities of \(A\setminus B\) and \(A \cap B\).
If we look at the cube of oppositions, we find that the quantified propositions at its front are immune to nominal restriction, and those at the back are allergic to it. A determiner \(D\) is here said to be immune for a certain operation if and only if the operation does not change its interpretation: \(D=D_n\); the determiner is said to be allergic to it if and only if the operation trivializes it, i.e., if and only if it renders one or both of its arguments redundant.
Observation 9
(Front and Back Plane) The front plane of the cube hosts the minimal logical determiners that are immune to nominal restriction; the back plane host those that are allergic to it.
Of course, the domain of quantification can also be restricted, systematically, by the second argument of a determiner, thus yielding socalled verbally restricted readings of determiners. (These readings are, also slightly misleadingly, called ‘verbally restricted’, because in paradigmatic uses of determiners the second argument is supplied by a verbal phrase.)
Definition 3

\(D^v_E(A,B)\) iff \(D_{E \cap B}(A,B)\) (that is, iff \(D_B(A \cap B,B)\)).
Verbal restriction makes one look at the individuals in the extension of the determiner’s second argument only—as if \(E=B\). If \(D\) is a logical determiner, \(D^v(A,B)\) only depends on the cardinalities of \(A \cap B\) and \(B \setminus A\). Verbal restriction serves to distinguish the left and right sides of the cube.
Observation 10
(Left and Right Plane) The left plane of the cube hosts the minimal logical determiners that are immune to verbal restriction; the right plane hosts those that are allergic to it.
3.2 Linguistic Applications
Observation 11
(\(^*\)NounDeletion) Determiners allergic to nominal restriction resist deletion of their nominal argument.
The existential contexts we are considering here in a sense annul, or trivialize, the second argument of the embedded determiners. If this is correct it is to be expected that determiners allergic to verbal restriction do not figure well there, whence the following tentative generalization.
Observation 12
(\(^*\)Ethere and \(^*\)Rhave) Determiners allergic to verbal restriction resist existentialthere and relationalhave configurations.
It appears that not even Keenan’s generalization has caught on in the literature, and this may arguably be because of the persistent intuition that natural language determiners are conservative. (Cf., discussion in Peters and Westerståhl 2006; the issue is taken up in Sect. 3.5.) For upon this assumption the initial generalization is provably correct. If a determiner is (as is generally assumed) conservative, and if it has to be regressive (as Keenan requires for the configurations at issue), then the quantifier must be intersective and hence symmetric, as is easily proved. Thus we find that symmetry shows up again as a characteristic feature allowing determiners to figure in existentialthere constructions in, e.g., (Peters and Westerståhl 2006; Westerståhl 2014). However, note that the last generalization is tenable only upon a theoretical assumption—about conservativity of natural language determiners—and that it renders potentially relevant facts out of scope.
It is not the purpose of this paper to supply an analysis of these constructions. (But, cf., e.g., Moltmann 2013, Ch. 5.) They are mentioned here only to point out that the very same semantic property serves to identify the noun phrases that do, or that do not, allow for such a nonextensional interpretation. The ones that do not are precisely the noun phrases headed by a determiner allergic to verbal restriction. Notice that, again, the proposed distinctive feature would be left unidentified if the determiners involved are assumed to be conservative.
3.3 Many Readings
The observations in the preceding subsection can be ignored by simply denying the existence of nonconservative determiners. “Only” is not a determiner falsifying the symmetry generalization about existentialthere configurations simply because it is not a determiner, or so it can be said. Things are not that simple when we turn to the English lexical item “many”. The item “many” can combine with a nominal and a verbal predicate so as to yield a sentence, like other determiners in natural language do. Semantically it can also be taken to behave like a determiner, i.e., as relating two sets. It is famously focus sensitive, but it is also topic sensitive, and it is not obviously immune to either nominal or verbal restriction. This fact makes it interesting for the present discussion.
Observation 13
Upon a nominally restricted interpretation, associated with (41), the number of Belgian cyclists is compared to an average among the Belgians; upon a verbally restricted interpretation, associated with (42), the number of Belgian cyclists is compared to an average among the cyclists. The truthconditions are obviously, and intuitively, quite different.
A nagging question of course is what this ‘\({ {AVR}}_X\)’ eventually is, or how it should be determined. A substantial part of a reply to this question in specific circumstances of use will surely draw from highly contextual features such as (ir)relevance, attention, expectation, and so on. But it seems to be generally agreed upon that focal emphasis on linguistic items in such constructions induces alternatives for the focused phrases, which can be used to set a standard for evaluation (cf., e.g., Cohen 2001; Greer 2014).
It may hardly need comment that the determination of the average or standard in actual cases of use may depend on more than just the extensions of the terms employed here, and it has been argued that, of course, subjective, intensional features, expectations, play a role as well. (Fernando and Kamp 1996; Bastiaanse 2014) There can be overt disagreement on the choice of alternatives that one on occasion takes into account, which is not a (syntactically) given. However, it is important to note that all such qualifications, also depend on the particular choice for either a nominally or a verbally restricted interpretation of “many”, a choice which, hence, cannot be neglected in any analysis.
There is a considerable literature on the generalized quantifier interpretation of “many”, that aims to show that it is essentially intensional, and trying to rescue the idea that it is conservative (cf., the recent Bastiaanse 2014; Greer 2014 for discussion). The present discussion may serve to show that many if not most of the specific findings in the literature can actually be obtained from a very rude and simple, plainly extensional, interpretation, as in (39). The readings discussed here are derived from principles, assumed generally available, of nominal and verbal restriction. Notice that these results would not obtain if it were assumed from the very start that “many” is conservative, that is, immune to nominal restriction.
3.4 Only a Determiner
It must be beyond doubt that the lexical item “only” can be used in combination with a common noun phrase \(N\) and a verb phrase \(V\), to yield something like a sentence, one that can be used to assert that, indeed, only \(N\) are \(V\), i.e., that all those who \(V\) are \(N\). “Only students sleep.” simply says that all sleepers are students. As we have seen above, such an interpretation of “only” as \({ {ONLY}}\) (or \(\ddot{o}\)) may serve to explain why, on the one hand, it does not allow its nominal argument to be deleted, or elided, and also why, on the other, it does figure well in existentialthere contexts.^{16}
It has been observed that, while “only” is immune to verbal restriction, it is focus sensitive. As we have seen with the determiner “many”, nonfocal material can be assumed to be backgrounded, in that the material is projected into the domain of quantification. Even though we have also seen that “only” is allergic to nominal restriction, we do find partial instances of nominal restriction. Partial restriction can be defined, for all determiners, along the following lines. In the following definition I use \(\langle A\rangle B\) to indicate a structured nominal argument with a backgrounded condition \(A\), and focal material \(B\).
Definition 4

\(D^b_E(\langle A\rangle B, C)\) iff \(D_{E \cap A}(B,C)\) (that is, iff \(D_{E \cap A}(B \cap A,C \cap A)\)).
Backgrounded (normally deemphasized) material \(A\) is ‘projected’ (as some would say) into the domain of quantification, leaving the material that is also \(B\) as the first argument of the determiner. We may subtly, but significantly, observe that this kind of projection, or backgrounding, can be assumed to be generally available in all compound constructions, but without this having any effect if the construction hosts a determiner immune to nominal restriction. (This is easily seen.) However, in construction with a determiner which, like “only”, is not immune to nominal restriction, this backgrounding has a substantial and interesting effect.
Observation 14

\({ {ONLY}}^b_E(\langle A\rangle B, C)\) iff \(B \supseteq (C \cap A)\).
We see that truly interesting semantic features of “only” can be stated in terms of the generalized quantifier \({ {ONLY}}\), that is, in terms of its Aristotelian interpretation \(\ddot{a}\). The reader may wonder why all this has not been established before. The answer seems to be that research on focussensitive “only” has left the nominal uses of “only” out of its scope, and, with the positive exception of (de Mey 1991), the generalized quantifier approach has exclusively focused on determiners that are ‘conservative’, and “only” is obviously not that.
3.5 Determiners, Conservative and Regressive
The preceding findings—viz., on allergy to verbal restriction, a proportional reading of “many” and restricted readings of “only”—still don’t make much sense if one maintains the assumption that all natural language determiners are conservative. As I have already indicated nonconservative items like “only” are often denied the status of a determiner in the literature. The phenomena observed could thus be taken to not even exist. In this Sect. 2.1 want to argue that the support for such claims is in no way conclusive, if not simply begging the question.
It is useful to first clarify the terminology employed. A determiner \(D\) is said to be conservative iff always \(D_E(A,B)\) iff \(D_E(A,A\cap B)\), and it is said to be regressive iff always \(D_E(A,B)\) iff \(D_E(A\cap B,B)\). Now for any determiner \(D\) that satisfies extension (\(D_E(A,B)\) iff \(D_{E\cap (A\cup B)}(A,B)\), cf., below), we find that \(D\) is conservative iff \(D\) is immune to nominal restriction, and \(D\) is regressive iff \(D\) is immune to verbal restriction.
An initially appealing, but upon reflection hardly convincing, argument for taking conservativity as a universal property of natural language determiners, is that it serves to severely restrict the enormously large class of possible determiner denotations. On a universe of \(n\) individuals the limitation to conservative determiners reduces the number of \(2^{4^n}\) possible determiners to \(2^{3^n}\), which is a substantial reduction indeed, and which has even been appealed to to account for the fact that we can actually learn the right denotations of determiner expressions in the acquisition of natural language. This argument loses much of its original appeal, though, once we realize that a child, in a universe with 3 individuals only (its mother, its father, and itself, for instance), is supposed to find himself faced with almost 135 million possible conservative determiner denotations. If we assume, perhaps more realistically, that a child would be designed to learn, for instance, minimal logical determiners, then the limitation to conservative determiners would reduce the number of possibilities from 8 to 4. Useful, of course, but not significant.^{19}
The expression “only”, like the expression “many”, has also been denied the status of a determiner, because, unlike other determiners, it is also used as an adverb, so it is an adverb, not a determiner, or so it is suggested. This is a curious argument, because, by the same reasoning, we may say that while “only” behaves like an adverb, yet, since it is also used as a determiner, it is a determiner, and hence not an adverb. Also, by the same type of reasoning we might say that “and”, as in “Harry opened the door and Milly came in”, is not a sentential connective because “and” can be used to conjoin expressions other than sentences. Independent motivation for such a claim thus should be provided, and the fact that the terms “only” and “many” can be used and interpreted like other determiners is prima facie evidence against it.^{20}
There is no reason to doubt that most, if not almost all, lexically realized determiners are conservative, and that conservative determiners are more ‘natural’, and more ‘easy to learn’. While this does not supply any evidence for conservativity as a universal semantic property of (lexical) determiners, it may help explain why many researchers are eager to suppose so. Van Benthem (1991, p. 30) observes “This universal already illustrates the earliermentioned interaction between semantic observation and semantic theory. (...) nowadays one finds that this failure of conservativity rather becomes one in an already existing list of reasons for not counting “only” as a determiner (but, rather, say, an adverb). Thus, successful pieces of theory exert pressure on previous empirical schemes of classification.” We can indeed witness such interactions at work in syntax and semantics, but the reliance on conservativity is not, I think, as safe as it seems. A semantic universal will always be a welcome substantial discovery, but a theoretical convenience may turn into an inconvenience or even handicap if not carefully handled.
The interaction alluded to by van Benthem can be observed, for instance, in work on dynamic semantics and discourse representation that deals, among others, with ‘donkey phenomena’ in quantified and conditional constructions. Chierchia (1992, p. 151) argues that “weak readings [of quantified donkeysentences, PD] are linked to conservativity and since the latter is universal, the former must be universal too.” Even more outspoken are van Eijck and Kamp (1997, p. 225): “It is one of the central claims of DRT that this kind of anaphoric connection [in quantified donkeysentences, PD] is possible because the material of the quantifying sentences that makes up the restrictor is also, implicitly, part of the quantifier’s body. This principle also explains why natural language quantifiers are always conservative (...).”
Such claims, however appealing, as a matter of fact render the theories for which they are stated vulnerable. For if a proposed theory or analysis entails (predicts, explains, ...) conservativity as a universal, then the eventual detection of nonconservativity would actually falsify the theory or analysis. This worry is not just imaginary. Next to the socalled donkey phenomena in constructions headed by (universal) quantifiers and in ifconditionals, we witness analogous phenomena in constructions headed by “only” and in only_ifconditionals. (von Fintel 1994; Dekker 2001) The first are successfully dealt with in the mentioned theories, but by the claimed inalienable bond with conservativity the treatments are inapplicable to the second, by principle.
Fortunately, the situation is not that bad, because a successful, and uniform, dynamic or discourse representation theoretic treatment of both types of phenomena is feasible. (Dekker 2001) However, it appears that such can only be achieved once the claimed link between anaphora relationships and conservativity is given up. Thus, while there may be a theoretical urge or inkling to embrace a (quasi)empirical generalization like that of the conservativity of natural language determiners, such urges, if not resisted, better be handled with caution.
Conservativity is a generally shared assumption that is often taken for granted. But it is mostly only taken for granted and it is almost never sanctioned, or actually or empirically verified. On the contrary, it seems, since leading advocates of conservativity themselves question its status as a semantic universal. Quite a few authors (von Fintel and Matthewson 2008; Francez and BenAvi 2014; Greer 2014; Herburger 1997; Keenan 1996; de Mey 1996; Zuber 2004) come to the conclusion that if determiners are not conservative, in their first argument it is said, they are assumed to be ‘conservative in their second argument’, that is, they are what I call ‘regressive’.^{21} Obviously, if determiners are generally assumed to be ‘conservative in their first or second argument’, they are no longer required to be conservative in the original sense.
The relevant observations may lead to the formulation of a ‘conservativity’constraint, or universal, that we can very well live with, but notice that it involves a terminological move that ought to be handled with care. If the term ‘conservative’ comes to mean what I take ‘conservative or regressive’ to mean, all previous claims about conservativity will have to be qualified or trashed. For, for instance, the statement that the quantifiers from the square are precisely those minimal logical determiners that satisfy conservativity would no longer be true. For, likewise, the claim that “only” is not conservative would no longer be true. And a claim to the effect that conservative determiners are easier to learn than nonconservative ones, would suddenly become an entirely different claim.
I prefer to stick to the use of the term ‘conservative’ in the way it was originally intended. But even so, independent of my preferences, the argument that “only” is not a determiner because it is not conservative will have to be qualified. Either the argument fails, because natural language determiners are no longer required to be conservative in the original sense. Or the argument fails because the premise is no longer true, and this is because “only” is deemed conservative after all, in its second argument, that is.
4 Natural Se(le)ctions
In the first section we have left the twodimensional square behind us and studied the syllogic of the threedimensional cube of oppositions. But it is not for no good reason, of course, that the square has ruled the logicolinguistic landscape. The four corners of the square make up a very natural selection from the cube, as has been observed in various linguistic domains. However, recent interest in Aristotelian ‘hexagons’ and ‘dodecahedrons’ may be indicative of the fact that the square might be too much of a straightjacket for the classification of natural language combinators. In this Sect. 2 briefly indicate some alternative subsections of the cube that can be distinguished.^{22}
The Square of Opposition The four corners of the square show up distinctly in various guises. Germanic languages host some elementary quartets that are naturally accommodated in the square. In English we find “nothing”, “everything”, “something” and “not everything”, and, similarly, “nobody”, “everybody”, “somebody” and “not everybody”, and, likewise, “never”, “always”, “sometimes” and “not always”.^{23} It has been observed that, typically, English does not contain noun phrases like “onlything”, or “onlybody”, or an adverb “onlytimes”, neither do related natural languages. The reason, however, is not, as seems to be appealing to assume, because ONLY and NOT_ONLY are not determiners, which we deem a farfetched and unwarranted conclusion, but simply that they are determiners with the specific semantic characteristic property of being allergic to nominal restriction. As we have seen in Sect. 3.2, they resist emptying their focal first argument. Likewise for the temporal counterpart of “only”. Trivially, we can only characterize these items as determiners with a distinctive semantic property, if we characterize them as determiners in the first place.
A Prism of Oppositions In the previous section we witnessed some attempts to characterize ‘natural’ or ‘naturally occurring’ logical combinators or determiners, in order to restrict logical space and to settle upon a ‘tractable’ portion of it. (Keenan 1996, fn. 1) seems to have been the first to argue that if determiners are not ‘conservative’, they are what we call ‘regressive’ and (Greer 2014) aptly implemented this idea in an arguably syntactic manner. We can cash out the idea semantically using a notion of extensional restriction. On the extensionally restricted reading of a determiner it is assumed that its first and second argument jointly constitute the universe, i.e., as if \(E = A \cup B\).
Definition 5

\(D^e_E(A,B)\) iff \(D_{E \cap (A \cup B)}(A,B)\).
We can observe that the minimal logical determiners UNLESS and ELSE are allergic to extensional restriction, and that the other six are immune to it.
Observation 15
(Extension) A minimal logical determiner \(D\) is immune to extensional restriction iff \(D\) is NO, ALL, ONLY, SOME, NOT_ALL, or NOT_ONLY (i.e., \(e\), \(a\), \(\ddot{a}\), \(i\), \(o\), \(\ddot{o}\)).
Immunity to extensional restriction implies that if \(A,B \subseteq E\subseteq E'\), then \(D_E(A,B)\) if and only if \(D_{E'}(A,B)\). (Actually this is the standard formulation of the condition for a determiner to ‘satisfy Extension’, whence the name.) The six mentioned logical determiners that are immune to extensional restriction hold of two sets \(A\) and \(B\) independent of the wider universe ‘surrounding’ these two sets.
The combinators in the prism are of course mutually related like they are in the cube; more interestingly, like the cube, the prism is contradiction and conversecomplete. This is to say that the validity of syllogistic schemes in all figures can be derived (by conversion) from the validity of those in one figure only, and that the validity of particular ones can be derived (by contraposition) from the validity of general ones, and vice versa. The syllogistics of the prism thus is logically wellbehaved, as one may also conclude from the table at the end of this paper.
The possibility to present logical determiners this way indicates that computational results obtained in terms of semantic automata running over trees of numbers, can be preserved, in so far as the corresponding automata can be run over pyramids of numbers. (van Benthem 1987)
It does not look like these four combinators share a characteristic semantic property, but they are obviously intrinsically related. The whole can be seen to emerge by the three possible types of negation from the junction \(AeB\). (As can be seen from the picture, the fork extends in the three dimension of the cube.) The set of four combinators has less expressive power than the cube or the square, but it is conversecomplete. The combinators \(a\) and \(\ddot{a}\) are each others converses, and \(e\) and \(i\) are selfconverse. The syllogistics of the fork, hence, is minimal but transparent. (An overview of the first figure validities in the various systems is given at the end of this paper.)
This is all very tentative and sketchy, of course. Still the range of uses of the modalities can be matched with these interpretations. That is, “may (not) \(P\)” expresses (in)consistency, and hence indicates or instigates permission (prohibition). “Must \(P\)” expresses inevitability of \(P\) and is, thus, imperative. (It is taken to say that only if \(P\) will things be alright; \(P\) is mandatory.) “Should \(P\)” proposes that \(P\) brings about the goal, and with the qualification “all else being equal”, may properly count as advice.
This subsection is, also surprisingly perhaps, converse complete, but negation free. The syllogistics of the diamond is, thus, transparent again, but very minimal (See the table at the end of the paper). Notice that with (propositional) negation added, the full cube gets restored.^{30}
In the figure above conjunction and disjunction appear in the two corners of the cube which are positive in both of their arguments, which is surely appropriate. Conjunction (“and”) however, is rendered particular, and disjunction (“or”) general. It should be noticed that this does by no means conflict with the ‘universal’ character of “and”, and the ‘existential’ character of “or”, because that aspect is of a different nature. It may also be noticed that the choice between particular or general is immaterial if we consider propositions as zeroplace predicates. (See footnote above.) The domain that sentences are predicates of has only one member, the true, so a sentence has some truth if and only if it has every truth. Once we start making distinctions among the true, for instance, when we introduce proofs or states of affairs as truth makers, then the choice between particular and general does start making a difference. It seems to me that it is more appropriate, then, to have a conjunction \((P \wedge Q)\) require some proof to be a proof and \(P\) and \(Q\), rather then every. However, I must submit that I have no substantial opinion on that matter.
5 Conclusion
In this paper I have first considered the square of opposition as constituting just one side of a logical cube of opposition. In line with Reichenbach’s observations of the cube I have shown its syllogistics to be totally transparent, if not trivial.
By extending our restricted focus on the square, or by more generally lifting a conservativity constraint, we can also, so I claim, open up our eyes to a wider range of structural properties of natural language determiners. More particularly the concept of what is called allergy to nominal restriction has shown relevant for a characterization of the determiners that oppose nominal deletion, and allergy to verbal restriction characterizes those that oppose existentialthere and relationalhave contexts, and allow nonextensional readings as arguments of intensional transitive verbs. Certain proportional readings of “many” become almost automatically available. These readings have been acknowledged before, but have not been that easily accounted for. Also various contextually readings of “only” become available, on the basis of the simple and I claim intuitive interpretation of it as ONLY (\(\ddot{a}\)). In the final Sect. 2 have demonstrated that the square need not figure as the only interesting subsection of the cube, and that others can be detected.
The observations in this paper could not have been stated, not even made, if we were to were to use the square as the sole paradigm, or conservativity as a linguistic universal. If the set of combinators or of determiners is not restricted to begin with, you see more of them.
Footnotes
 1.
In the terminology adopted here the term ‘syllogism’ is used for inferences of a particular form, including invalid ones. This is perhaps not the standard locution, but convenient.
 2.
The square is here presented in a slightly unusual order. The presentation is logically speaking isomorphic to the usual one, but more convenient for what follows.
 3.
This is why I have chosen to use the ‘umlaut’ed vowels, like (Kraszewski 1956).
 4.
Running out of vowels, I have chosen ‘\(y\)’, because it most looks like one, and ‘\(x\)’, mnemonic for exhaust.
 5.
Obviously, \(u^iv^iw^i\) is schematically the same as \(uvw\) except for the fact that each of the terms \(P\), \(S\) and \(M\) have been systematically replaced by \(P'\), \(S'\) and \(M'\), respectively.
 6.
Janusz Ciuciura (p.c.) pointed out to me that the Polish logician Tadeusz Czežowski has identified this cube already in 1931. His work has however been inaccessible to me.
 7.
By performing majorminormajor contraposition on \(uvw\)1 we obtain \(v{^c}''u''^cw'{^c}'\)1, which is \(v^cu^cw^c\), as is easily checked; likewise, minormajorminor contraposition, mMmCP, gives \(v''^cu{^c}''w'{^c}'\)1, which is \(v^cu^cw^c\) as well.
 8.
It is hard to believe that others have not made this generalization, so, in lack of having seen it stated thus before, I hereby attribute it to common sense.
 9.
Outline of the proof. If: By a mere inspection of the numbers of types of possibilities one can conclude that out of the \(8 \times 8 \times 8 \times 4\) syllogistic schemes precisely \(24 \times 4\) satisfy the conditions mentioned. (Condition 1 eliminates \(1/2\), condition 2 eliminates \(1/2\) twice, and \(3/8\) of the possibilities satisfy condition 3.) In the first figure these are seen to be the 24 schemes displayed in the three cubes above, and since these are equivalid, and at least one of them is valid, they are all valid. Their conversions to the other three figures preserve both the mentioned conditions and validity.
Only if: By contraposition. If one of the first two conditions is not satisfied, we can present a case (with \(M\), \(S\), or \(P\) universal or empty) in which the conclusion is trivially false while the premises are contingent, or in which the premises are trivially true, and the conclusion is contingent. In either case the conclusion can be false while the premises are true. If both premises are particular no independent conclusion follows. If both premises are general, no particular conclusion follows.
 10.
Proof. The observations in this section are all stated as equivalences or equivalidities, whence they are convertible. Now suppose the validity of some scheme \(u^*v^*w^*\) \(i\) is not derivable by means of the three structural observation. Then it can be turned into one of the form \(uvw\) \(1\) which is also not derivable, and which does not satisfy the three conditions stated in the previous observation. It is, hence, invalid, and \(u^*v^*w^*\) \(i\) is likewise.
 11.
It may, and will, happen, that the universe \(E\) gets contingently, contextually, restricted to a subset \(C \subseteq E\), so that \(A\) and \(B\) are no longer guaranteed to be a subset of the universe. In that case I write \(D_{E\cap C}(A,B)\) iff \(D_{E\cap C}(A\cap C,B\cap C)\).
 12.
Surely the restricted rephrasings in the examples here—rephrasings which are rendered equivalent—carry additional meaningful overtones, due to induced information structure and assumed intonation. I will come back to this in the next section.
 13.
With the proviso perhaps that “The ␣ failed” be, obligatorily, rendered as “They failed.”
 14.
There appear to be conflicting intuitions about a determinerlike compound “none of the”, which combined with a noun phrase and a verb phrase makes up a sentence, like “None of the cats meowed.” Interestingly, precisely the discord can be explained by the present generalization. For observe that there are two ways of understanding the embedded determiner “the”, yielding two different analyses of the compound. The definite article can be taken anaphorically, yielding a presupposition of there being cats, in the universe or under discussion. A verbally restricted reading of the determiner in “None of the cats meowed.” could thus be taken to express the proposition that, among those given cats none in the domain of meowers meowed, i.e., that none of them meowed. This is a nontrivial proposition, the compound thus is not allergic to verbal restriction, and hence “There are none of the cats.” is rendered felicitous upon the anaphoric reading of “the cats.” Alternatively, the embedded article can be read as yielding a quantificational reading, to the effect that “None of the cats meowed” yields the conjunctive proposition that there are cats and that none of them meowed. The verbally restricted reading then reads “There are cats who meowed, and none of them meowed” which is inconsistent. Attempting this reading, the construction would be infelicitous in an existentialthere context. For as far as both readings are possible, both judgments can thus be argued for.
 15.
Barwise and Cooper (1981) have proposed roughly onethird of the cardinality of \(E\) as an initial, default, standard of \({ {AVR}}_E\)—a number that of course is supposed to be variable.
 16.
Thus, the observation “There are only students.”, can be rendered by “Everybody is a student.” and the observation “There is only fish (on the menu).” amounts to the observation that “There is nothing else but fish.” Let me emphasize again that it is not the purpose of the present paper to supply a compositional semantic analysis of existentialthere constructions. It suffices to observe the relevant equivalences.
 17.
 18.
I only give a schematic reconstruction of the crucial parts of the schemes; a compositional analysis of the sentences actually hosting “only” is much more involved of course.
 19.
(Hunter and Lidz 2013) recently tested the relative learnability of two new lexical determiners with conservative meaning \({ {NOT\_ALL}}\) and nonconservative meaning \({ {NOT\_ONLY}}\), respectively. The results indicated “a (marginally) significant dependency between conservativity of the determiner and success in learning” (p. 326). Interestingly, despite the authors’ conviction that “[a] striking crosslinguistic generalization about the semantics of determiners is that they never express nonconservative relations” (p. 315), they had it that “ ‘only’ refers to the interpretation (...) that reverses the inclusion expressed by ‘all’ (i.e. ‘only girls are on the beach’).” (p. 327). “Only” was said to be “not representable by a conservative determiner,” (p. 327), though, apparently not because it is not conservative, but because ‘only’ is not a determiner (p. 332).
 20.
Even Peters and Westerståhl apparently agree that “only” has a determiner use, acknowledging that it is “an expression with much wider and more varied use.” (Peters and Westerståhl 2006, p. 219,emphasismine)
 21.
 22.
I thus remain constrained myself, by the cube, and do not consider, e.g., Boolean meets or joins of its corners. Notice, however, that any Boolean meet (conjunction) of two general corners would violate differential import. “All and only”, like “if and only if”, is used to state an identity, which is trivial if not conceived of intensionally; likewise, a construction with “All and no” implies that the first argument argument is empty, and that the second, hence, is vacuous.
 23.
Here and above, I deliberately include the apparently compound expressions “not everything”, “not everybody” and “not always”. The reason is that the constituent “not” does not itself seem to figure as a productive lexical item or morpheme in these constructions. We can of course understand, for instance, “not nobody” and “not someone” and “not sometimes,” but not as smoothly as we do understand the ‘quasicompounds’ just listed.
 24.
Independent from linguistic motivations Kraszewski 1956 has singled out precisely this set of categorial combinators. I thank Janusz Ciuciura for pointing out the reference to me.
 25.
Sentences like these notoriously allow for a variety of alternative interpretations, partly correlating with the choice of material that can be taken to be presupposed, that goes in the restrictor, and that goes in the nuclear scope of the quantifying adverb (see, e.g., Hendriks and de Hoop 2001).
 26.
The apparently compound combinators “not always” and “not only” are again taken as a unit, since here as well “not never” and “not sometimes” are indeed awkward. I take this to provide motivation for taking the two compounds as quasicompounds.
 27.
(Keenan and Paperno 2012) reports that a lexical item corresponding to ONLY (\(\ddot{a}\)) shows up in all languages in a recent crosslinguistic survey.
 28.
I have taken “may not” as a quasicompound expression for exclusion or prohibition, that is, as the denial or negation of “may”. I believe the default interpretation of “You may not enter” is that it is not allowed or permitted to enter; not that it is allowed to not enter.
 29.
We can conceive of sentences as zeroplace predicates, which either have the empty tuple (the true) in their denotation, or not, in which case they are false. A conjunction \((P \wedge Q)\), rendered as \(PiQ\), then comes out true if and only if the intersection of \(P\) and \(Q\) is not empty, i.e., if and only if the true is in both \(P\) and \(Q\), as desired. The three forms of negation can then be uniformly conceived of as complementation, and the cube unfolds as shown.
 30.
In the last two figures conjunction, and “and”, are associated with the \(i\)corner of the cube. As a reviewer points out conjunction has a close and natural tie with the universal quantifier, and disjunction with the existential one. For this reason the two have been traditionally associated with the \(a\) and \(i\)corners of the cube, respectively. The two connectives plus “nand” (\(\uparrow \)) and “nor” (\(\downarrow \)), can be seen to form their own (Boolean) square, which also can be cubed, as recently expanded upon in (Westerståhl 2012). However, Westerståhl’s Boolean square of dualities does not match the Aristotelean one, in which the \(a\)corner is notoriously not symmetric, and \(\rightarrow \) comes much closer to \(a\) than \(\wedge \) does. Westerståhl’s findings are therefore orthogonal to the ones discussed in this paper.
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