Double privation and
multiply modified artefact properties
ABSTRACT
This paper presents and justifies a logical framework that will enable us to calculate whether an individual a having a multiply modified property F is an F, whenever this can be calculated. The logic is encapsulated in ten rules of inference involving subsective and privative, first-order and second-order modifiers. The predicate ‘is anything but a false friend’ exemplifies one variant of multiple modification, namely double privation. Double privation is the juxtaposition of two privative modifiers. Our main thesis is that the logic of double privation is a logic of contraries. One of two major results we present is that the received rule of single privative modification proves too strong when extended to first-order double privation and inoperative when extended to higher-order double privation. We revise the rule by replacing boolean negation by property negation in order to operate on the contraries of properties. The other major result is that first-order and higher-order double privation validate
1 the elimination of only one of the two privatives. However, when semantic information is added to an inference, there will be cases where it can be calculated whether a is an F. For instance, a former fallen angel is (again) an angel while almost half a pound is not a pound.
KEYWORDS
Nested modification privative modification higher-order modification property negation contraries artefact
2
INTRODUCTION
This paper explores nested modifiers, especially clusters of so-called privative modifiers like fake and almost. The paper covers themes from philosophy of language, logic, semantics, metaphysics, and philosophy of artefacts. The main question we are addressing in this paper is one of (extensional) set-theoretic classification rather than (intensional or hyperintensional) semantic characterization or description of multiply modified properties:
If a property F has been multiply modified in such-and-such a manner, is
an individual a that has the so modified property an F?
For instance, if F is the property of being a Samurai sword, does an artefact with the property of being a previously malfunctioning theatre prop Samurai sword have the property of being a Samurai sword? Calculemus! Assign a semantic interpretation to the predicate ‘is a previously malfunctioning theatre prop
Samurai sword’, according to one’s construal of the two adjectives
(‘malfunctioning’, ‘theatre prop’) and the adverb (‘previously’) involved, and apply the relevant inference schemas to calculate whether an object with that property emerges as a Samurai sword. This is not to imply, though, that there is necessarily going to be a logical fact of the matter as to whether an individual instantiating an arbitrarily modified property F emerges as an F. In fact, there are interesting cases where logic alone cannot decide. We will be discussing some of these cases below.
3
To enable a wide array of calculations, we advance and investigate ten different inference schemas whose respective premises involve subsective or privative, first- order or second-order property modifiers. This paper will focus on subsective, privative, and modal property modifiers, at the expense of the intersective ones, since whatever applies to subsectives applies equally well to intersectives in all the regards relevant to this paper.
To fix ideas, let Ms be a subsective modifier, Mp a privative modifier, F a property with respect to which the modifier is subsective and privative, respectively, and a an individual. Then these two rules will, preliminarily, characterize subsection and privation, respectively:
(Ms F) a (Mp F) a
Fa Fa
A rough-and-ready criterion for distinguishing between subsective and privative modifiers is that the former permit rephrasing by means of relative clauses (thus inverting the word order), as the latter do not. E.g. a round peg is a peg that is round; a fake banknote is not a banknote that is fake, for it is not a banknote to begin with. Subsection states what something is; privation, what something is not; and modal modification, what something may be. A modal modifier, preliminarily speaking, is one that is sometimes subsective and sometimes
privative with respect to one and the same property. Where Mm is a modifier
4 modal with respect to F, an object with the modally modified property (Mm F) may, but need not, be an F. Thus an alleged assassin is maybe an assassin, while an expected outcome may, or may not, be the actual outcome. Below we provide a definition of modal modification that captures its unique features and incorporates it into a general theory of property modification. The fact that modal modifiers lend themselves to being defined at all runs counter to the received view that they defy definition for want of entailments uniquely characterizing them.1
The first of two key findings we present here is that the received rule of single privative modification stated above proves too strong when extended to first- order multiple privation, and turns out to be inoperative when extended to higher-order multiple privation. We proceed to amend the problem. We weaken the rule sufficiently, as well as make it operative, by replacing boolean negation by property negation in order to operate on the contraries rather than contradictories of properties. This seems more intuitive, anyway, since something that operates on properties (a modifier) is replaced by something else that also operates on properties (property negation). Our main thesis is that the logic of multiple privation is a logic of contraries.2
1 See, e.g., Partee (2001, p. 9) and Chatzikyriakidis and Luo (2013, §4).
2 It has been put to us that it is being tacitly assumed by everyone working on the semantics of adjectives that pairs of privatives yield contraries. If this is so, it would go a long way toward explaining why we, at least, have not come across
5
The second key finding we present here applies equally both to a pair of first- order privative modifiers, as well as an ordered pair whose first element is a higher-order privative modifier and whose second element is an arbitrary lower- order modifier. The finding is that only the outermost, and not also the innermost, modifier can be eliminated. There is sometimes a way, though, to infer whether object a has, or else lacks, the root (i.e. unmodified) property F.
The premise set must be augmented by semantic information. The meaning of some, but emphatically not all, predicates involving one of the two above pairs of modifiers is such that it is determinate whether an individual instantiating the property is an F. The meanings of the remaining predicates carry too little information for a determinate answer to which way the cookie crumbles, so to speak.
Our overall philosophical objective is to avail ourselves of the means to calculate the logical behaviour of nested or iterated modifiers, such as these are encoded in natural language. The more specific philosophical motivation for taking a closer look at multiple modification stems from the philosophy of artefacts, both technical, artistic and institutional. Such artefacts abound and permeate our lives. They run the gamut from bulky doorstops, point-and-shoot cameras and cellular phones through elaborate social hierarchies, advanced social institutions
any explicit discussion of the logic of multiple privation. What this paper has to offer is, minimally, a discussion and substantiation of this allegedly tacit assumption.
6 like money, cutting-edge hybrid cars, paintings in the league of Caravaggio’s
Judith and Oloferne or Vermeer’s The Milkmaid, nanorobots going up your bloodstream, vital limbs and organs being replaced by state-of-the-art surrogates, to space aircraft landing on comets or exploring Pluto and its vicinity.3
Our planet, alas, is also replete with such low-caste artefacts as fake banknotes, forged paintings, and sham jewellery. These are all artificial artefacts, as it were, by being parasitic on original artefacts.4 But they are no less artefacts for that, are as amenable as the next artefact to the vicissitudes pertaining to artefacts in general and must not be missing from any metaphysician’s inventory of man- made reality. Typical examples of such vicissitudes would include being subjected to improper use, malfunctioning, or being forged or otherwise imitated.
A brief interlude on improper use; malfunction and forgery will briefly receive attention below and have been discussed at length elsewhere.5 An obvious
3 In this paper artistic, technical and institutional artefacts are treated as being on a par. We remain neutral as to whether anything bearing on any of these three kinds of artefacts carries over to biology, where a phenomenon such as mimicry might conceivably be a counterpart of forgery.
4 It is not a foregone conclusion, however, that fake Fs depend conceptually on
Fs. For discussion, see Ryle (1954, pp. 94-5) and YZ* (2010).
5 See XY (2011), (2013) on malfunction; see Wreen (2002), Hick (2010), YZ*
(2010) on forgery.
7 example of improper use of, say, a toy hammer would be to use it to pound nails. But improper use of a counterfeit banknote is also an option. Its proper use is to deceive the receiver, so if the bearer uses a counterfeit banknote to settle a debt with a receiver who knows that the slip of paper is a counterfeit banknote and attaches value to it qua collector of counterfeit banknotes, then the receiver is not being deceived and the counterfeit banknote is being used in violation of its proper function. The payment is not monetary in kind, but rather comparable to barter. What about Monopoly money then? It forms a subgenre of fake money
(like counterfeit money does) consisting in make-believe money where everybody involved knows they are pretending to be buying and selling by means of money. Hence common knowledge, in the technical sense known from epistemic logic, is required of the players to get a game of Monopoly going. The proper use of Monopoly money does not consist in deceiving the receiver, but in pretending collectively that money is shifting hands. Improper use will consist in breaking the pretence, e.g. by attempting to pass Monopoly money off as money, as though Monopoly money were counterfeit money instead of fantasy money.
Improper use of Monopoly money may also coincide with improper use of counterfeit money, as described above, if encountering a collector of fantasy money. Note that there need be nothing dismissive about ‘improper use’. In fact, incompetent use aside, improper use may well betray resourcefulness and originality in the user who detects ways of using an artefact that deviate in a fruitful manner from the artefact’s standard or designed or intended use.
8
A counterfeit banknote, say, may appear to be a lowly object, since there is a straightforward intuitive sense in which a counterfeit banknote is a lesser thing than a banknote, professing to be something it is not. A counterfeit banknote, a forged painting, etc., may have the feel of something like an empty shell, the intuition being that a counterfeit banknote and all the rest have (more or less) the right ‘exterior’ but the wrong ‘interior’. Not only that, but some artefacts are, for instance, fake malfunctioning artefacts (think of a theatre prop sword with a broken ‘blade’), others are malfunctioning fake artefacts (think of a forged banknote failing to deceive the receiver), and still others are fake fakes (see footnote 11 for the example of forged counterfeit banknotes). From a conceptual point of view, artefacts that are fake or malfunctioning are highly sophisticated.
Malfunctioning fake, fake malfunctioning, and fake fake artefacts are even more conceptually complex, since their respective concepts come with an additional layer of qualification. Such doubly qualified artefact properties form the main topic of this paper, where we work the dark side of modification, i.e. privation.
The rest of the paper is organized as follows. Section 1 provides further philosophical motivation for investigating multiply modified artefact properties and lays out the basic notions. Section 2 sets out the relevant logical and semantic portions of the foundations of property modification, together with a definition of subsective modifiers. Section 3 defines a general rule governing privative modifiers that is suited to accommodate multiple privation and also includes single privation. Section 4 defines a rule governing modal modifiers.
Section 5 provides the advertised ten schemas accompanied by examples.
9
1. PHILOSOPHICAL FOUNDATIONS
Two of the ten inference schemas in Section 5 have premises exhibiting double privation. The logic of double privation would appear to be murky. Iterated subsective modification, by contrast, is always smooth sailing, as we simply peel off modifiers until we are left with the root property F. The source of the apparent murkiness might arguably be the failure to keep sufficiently separate three different kinds of negation:
privation, which applies to properties
the complement function, which applies to sets
the boolean not, which applies to propositions-in-extension, i.e. truth-values
The rule for privation works just fine in the case of single privation, but wreaks havoc when two privative modifiers are replaced by two instances of boolean negation. The problem, in rough outline, is that, e.g., an imitated rhinestone diamond would be not not a diamond, hence a diamond (!) by double negation elimination. So by forging a fake diamond you would have created a diamond.
This is a whisker away from alchemy. It is also illogical. We show in this paper how to steer clear of ‘the paradox of double privation’ of two privatives necessarily returning a positive property.
Our approach to modifiers is continuous with the model-theoretic one that
Montague developed formally and which Kamp, Partee and a host of others
10 have since contributed to.6 In the interest of a uniform treatment of modifiers, we construe a modifier as an extensional entity associating one intensional entity with another intensional entity. Accordingly, a property modifier is a function-in-extension that generates one property from another property.7 The property-to-property construal creates first a new property from an existing property and only then descends to an extension to pick up those individuals that instantiate the modified property at the empirical index of evaluation.
We are opposed to construing modifiers as functions from sets to sets, basically because the set-to-set construal works only for subsective and intersective modifiers and fails to generalize to non-subsective and non-intersective modifiers. The property-to-property construal of modifiers allows modifiers to be unaffected by this or that particular extension (set) of a property. This is a critical feature for the non-subsectives and non-intersectives to have, because it is undesirable for any theory of modifiers that a set of, say, fake banknotes would have to be extracted, by comprehension, from a set of banknotes, or a set of alleged assassins from a set of assassins. No fake banknote is a banknote; some, not all, alleged assassins are assassins.
6 See Abdullah and Frost (2005), Clarke (1970), Cresswell (1978), X*XY* (2010,
§4.4), XY (2013), Kamp (1975), Kamp and Partee (1995), Montague (1970),
Parsons (1970), Partee (2001), (2007), Rotstein and Winter (2004).
7 The locus classicus of the generalization to property-to-property mappings is
Montague (1970).
11
As for subsective modifiers, an individual with the property of being a short
Dutchman is a Dutchman, so short is subsective. But a short Dutchman is not short, pure and simple. A short Dutchman is not an element of the (presumed) intersection of the respective extensions of the property of being a Dutchman and the (presumed) property of being short, for there is no such property as being (absolutely) short, hence there is no such intersection. If it is asserted that a is short, a perfectly legitimate question would be, ‘a short what?’ Felicitous uses of ‘short’ are elliptic, for both speaker and hearer know which suppressed, yet contextually salient, property is being modified by short. That is, “a is short” is elliptic for “a is a short F”, where F is the salient property. This also goes to show that ‘short’ does not, in the final analysis, denote a property but a modifier.
Formally phrased, the domain of properties is closed under modification, and so is the domain of modifiers. Thus, if large is a first-order modifier and elephant a property then large elephant is also a property. Large is a first-order modifier whenever its arguments are properties. Very is at least a second-order modifier because its arguments are at least first-order modifiers. Thus very applies to large and not to elephant. Hence, using brackets as scope indicators, the correct scope distribution is ((very large) elephant) rather than (very (large elephant)). In our formal notation, we shall append an asterisk, ‘*’, to words (adverbs) denoting higher-order modifiers, like ‘very*’.
12
Assume that African is subsective with respect to the property of being an elephant, very* is subsective with respect to any lower-order modifier, and large is subsective with respect to any property. Then a very large African elephant is a large African elephant, a large African elephant is an African elephant, and an
African elephant is an elephant. This is because, necessarily, any given set of very large African elephants is a subset within a set of large African elephants, any given set of large African elephants is a subset within a set of African elephants, and any given set of African elephants is a subset within a set of elephants.
Let almost* be a higher-order modifier privative with respect to the modifier recaptured, and let recaptured be a first-order privative modifier of the property of being a fugitive. Then a recaptured fugitive is not a fugitive (though they were a fugitive prior to being caught again), so a recaptured fugitive is located in the
(relative) complement of a given set of fugitives. However, an almost recaptured fugitive is still a fugitive (though on the verge of not remaining one much longer). The correct scope is ((almost* recaptured) fugitive).8
The modified property ((almost* recaptured) fugitive) exemplifies double privation, since two privative modifiers are juxtaposed. Double privation can also be generated by means of two first-order privative modifiers, as in the doubly dynamic former heir apparent, whose scope distribution may equivalently be
8 Cf. Enç (1986) for the example of recaptured fugitive.
13
(former (apparent heir)) or ((former* apparent) heir). In (former (apparent heir)), former operates on the property formed by applying apparent to heir. Still two privatives, former and apparent, are juxtaposed in the first-order manner of generating the property of being a former heir apparent. The dual dynamics is due to the backward-looking aspect of former/former* and the forward-looking aspect of apparent, in the special sense of ‘apparent’ as ‘designated to become’. A designated F is currently not yet an F, though they are supposed to become one.
A former heir apparent is not an heir apparent, for one of two reasons: either they succeeded in succeeding the previous monarch, or they failed to. We are deploying the strict interpretation of former/former* as a privative rather than modal modifier to get the example of ‘is a former heir apparent’ off the ground.
From “a is a former F” we are to infer that a is no longer an F, hence is not an F.
On the more permissive modal interpretation, a former F may also be a current
F, if reinstated.9
9 Other dynamic examples of ‘stages of loss in the privative process’ and
‘incomplete realizations of possible privational histories’ (Martin 2003, p. 439, p. 441, resp.) would include going bald, i.e. progressing (or perhaps regressing) toward being almost or entirely without hair (where the privative feature of being bald is not lexicalized by, e.g., ‘-un’ as in the made-up adjective ‘unhairy’). Note that ‘is going bald’ and ‘is a former heir apparent’ are dynamic for two different reasons. The former is related to an interval due to the processual character of going bald; the latter, to two instants of time due to the point-like character of being an heir apparent and being a former heir apparent.
14
A datum that any theory of modifiers must respect is the fact that far from all adjectives found in natural language denote invariantly modifiers of one order only. Many natural-language adjectives shift routinely between orders. For instance, when ‘New York’ is used as an adjective, what ‘New York’ denotes in
“I am in a New York state of mind” is a first-order modifier whereas ‘New
York’ denotes a second-order modifier in “Dominique is a New York hotel maid” if New York qualifies hotel. Dominique is predicated to be a ((New York* hotel) maid), the sort of maid who works in a hotel that is of the New York kind
(i.e. one situated in New York City), and not a (New York (hotel maid)), a hotel maid of the New York kind, i.e. one hailing from New York City. Thus ‘New
York’ (as an adjective) is ambiguous between denoting a first-order and a second-order modifier in these examples. Upon disambiguation, we have ‘New
10 York1’ denoting New York and ‘New York2’ denoting New York*.
Another datum that a theory of modification must respect is that, say, a wooden table is a table and a wooden horse is not a horse. Likewise, whereas a North
Korean statue is a statue, a ((North Korean) (US banknote)) is not a US banknote,
10 A topic we will not be delving further into here is how to decide for (a given token of) a given adjective whether it denotes a property or a modifier. See, however, Siegel (1976), Kamp (1975), Montague (1970), Beesley (1981).
Schematically put, Montague pairs all adjectives off with modifiers, Beesley pairs all adjectives off with properties, and Kamp pairs some adjectives off with modifiers and the rest with properties.
15 or a banknote altogether, so North Korean is privative when its argument property is (US banknote).11 Similarly, Nordic gold is not gold, but an alloy; fides punica is not trust, but treachery; and Rocky Mountain oysters are not oysters.12
Only an entity having obtained the octroy to print US currency is capable of issuing US banknotes, and the Democratic People’s Republic of Korea is no such entity. That is, there is the meaning of the predicate ‘is a US banknote’ specifying when a piece of paper passes muster as a US banknote, and there is the empirical fact that the DPRK fails to meet one or more requirements inherent in that meaning. Absent this meaning and that empirical fact, we are barred from detecting the linguistic fact that this particular occurrence of ‘North
Korean’ denotes a privative modifier.
11 Cf. http://en.wikipedia.org/wiki/Superdollar. For an example of a fake fake, if you design and manufacture a piece of paper that is a forgery of a North Korean
US banknote you have forged a forgery. So what you are trying to pass off as a
North Korean US banknote is a fake fake, whereas the real McCoy is a North
Korean US banknote.
12 Makinson (1973, pp. 64-65) adduces the example of white: a white pen
(extrapolating from the original example of blue pen) is both white and a pen, whereas white wine is ‘not a white liquid, but rather a light yellow one’.
Therefore white ‘may function as a qualifier in some of its uses, and as a proper modifier in others’, where a qualifier is an intersective or subsective modifier and a proper modifier is modal or privative.
16
The fact that the status as subsective, privative, etc. of a given modifier depends on the given argument property does not saddle us with semantic contextualism.
An analogy may be helpful. The meaning of the functor ‘’ (‘minus’) is the same in all contexts. But when the subtraction function is applied to one argument pair of numbers, its value is positive, but negative if applied to some other pair.
Nor are we saddled with some sort of logical relativism. We do index a modifier’s status to its particular argument property. But, first of all, while we do believe that this specific kind of relativization is on the right track, we wish to leave conceptual space for modifiers that are absolutely subsective, absolutely privative, etc., which means that the modifier’s status is not a function of its argument property. Candidates for absolute modifiers would arguably include modifiers representing degrees, such as large, little, very* and almost*. Secondly, however, the distinction (if viable) between absolutely and relatively subsective, etc. modifiers is a linguistic one without logical significance. All privatives (etc.) obey the same logic, whether they are absolutely or relatively privative (etc.).
Still it might be interesting for a linguist to chart, for a given language, the various ways in which the same adjective (including nouns used as adjectives) oscillates between denoting subsective and privative, etc. modifiers.
On a methodological note, set theory is an excellent tool for classification and fully sufficient for the extensionalist approach pursued in this paper. We are recording only the effects that various combinations of first-order and second- order subsective and privative modifiers have on properties and modifiers, respectively, paying little heed to the internal semantic mechanisms underlying
17 the very formation of these iterated or compound modifiers. Given a domain of artefacts, a property, whether plain or modified, induces a dichotomy over that domain. We want to drop individual a on the correct side of the fence. The logical machinery deployed in this paper is on purpose not-all-too fine-grained, since we are, for now, only interested in addressing the coarse-grained question stated at the outset: if property F has been modified in such-and-such a manner, is object a bearing the so modified property an F?
Set theory, of course, is a poor tool for characterization or description, which calls for a study of properties and not just of their extensions, which for empirical properties are world-and-time-indexed satisfaction classes. A set-theoretic answer to the question what an alpha++ city is would be that it is either element of the doubleton London, New York City, which leaves it unexplained why
London and New York City are the members of the set of alpha++ cities. The question is not which cities are actually and currently alpha++ cities, which does require a set as an answer. It is left for a future study to look further into the semantic features shared by, for instance, Fs and fake Fs, like banknotes and fake banknotes. A fake banknote is namely not just any old non-banknote, but a non- banknote being a closer approximation to banknotes than are any other non- banknotes by being, in some intuitive sense, ‘almost’ a banknote. Counterfeit banknotes share more of the properties defining banknotes than do any other non-banknotes. Something similar holds for hammers and those non-hammers that are toy hammers, although the latter have no need for the sort of institutional embedding that banknotes require and which counterfeit banknotes
18 are being parasitic on. ZX (2010) makes the point that individuals with the
property (Mp F) end up in the particular subset of the complement of an
extension of F which contains all and only (Mp F)-objects. For instance, a fake banknote will belong to that subset of a given complement of the extension of the property of being a banknote that contains only those non-banknotes that are fake banknotes and not any other kind of non-banknotes.13 Incidentally, this goes to show that privative modifiers are, indirectly, via the notion of complement set, a variant of subsective modifiers, in that they identify a subset of a set, in casu a set of non-Fs. This fact vindicates, in a roundabout manner, Partee’s project of ‘reducing’ privative modifiers to subsective ones.14
2. LOGICAL FOUNDATIONS
We offer in this section a formal framework in which to study modified properties. The framework in question is the fragment of Tichý´s Transparent
Intensional Logic that conceptualizes extensional and intensional entities. The
13 Uidhir (2010) also zooms in on a subset in the complement. What is ‘almost’, or failed, art (as opposed to poor art) forms ‘a meaningful subclass of non-art’.
Similarly, of all the things in the universe incapable of housing a dog, only some are failed doghouses (ibid., §3.1). These are products of unsuccessful attempts to produce doghouses using the same methods that also eventuate in doghouses.
14 Cf. Partee (2001, p. 7). Note we need not invoke coercion, unlike Partee (ibid.) and Chatzikyriakidis and Luo (2013, §4).
19 remaining (and much larger) fragment of TIL conceptualizes structured hyperintensional entities organized in a ramified type hierarchy. The semantics of TIL is a procedural one, but a model-theoretic (Montague-style) semantics can be provided for the simple-type fragment of TIL in which modifiers and properties are located. We rest content, in this paper, with a model-theoretic account of the semantics of terms for such mappings as are modifiers because our investigation is restricted to a set-theoretic one, for which model-theoretic semantics is tailor- made. That is, the meaning of a term for a modifier will be that modifier itself (a function), rather than a hyperintensional procedure producing the modifier. 15
15 Programmatically put, whereas a procedural semantics identifies meaning with the very procedure for obtaining an output object from one or more input objects, a model-theoretic semantics identifies meaning with the output object itself. For instance, where a procedural semantics construes ‘2 + 3’ as a name of the arithmetic procedure of applying the function plus to the numbers 2 and 3, a model-theoretic semantics construes the term as a name of the number 5.
Phrased in terms that may be helpful for those who distinguish between object language and metalanguage, in a procedural semantics procedures are denoted already in the object language rather than being relegated to the metalanguage.
See Tichý (1986a), Tichý (1988, §2), X*XY* (2010, §3.2.1).
20
The model-theoretic fragment of TIL shares obviously various similarities with
Montague’s Intensional Logic.16 In fact, Montagovians can readily translate our results back into IL. However, TIL differs in at least two respects important for our present purposes. First, TIL encodes abstraction over worlds and times directly in the logical syntax, allowing us to trace in a notationally and conceptually transparent manner how modified properties are predicated of individuals relative to empirical circumstances such as worlds and times.
Second, TIL deploys the operation of functional application as the logic both of the extensionalization of intensions and of the predication of properties of individuals. TIL has no special operations earmarked for extensionalization or predication; Montague uses his cup operator, denoted ‘ˇ’, for extensionalization, and application for predication. Thus TIL makes do with one operation less.
The framework relevant to modifiers and properties is a typed universe whose types are organized in a simple (as opposed to ramified) type theory. We begin by defining the simple type theory.
DEFINITION 1 (types of order 1 over B) Let B be an ontological base, i.e. a collection of pair-wise disjoint, non-empty sets. Then:
(i) Every member of B is an elementary type of order 1 over B.
16 See X*XY* (2010, §2.4.3) for a close comparison between TIL, IL and Ty2- style type theories, TIL being by far the most elaborate type theory of the three.
21
(ii) Let , 1, …, n (n 0) be types of order 1 over B. Then the collection (1…n)
of all n-ary partial mappings from 1…n into is a functional type of order 1
over B.
(iii) Nothing is a type of order 1 over B unless it so follows from (i), (ii).
The ground types currently encompass these four:
= the set of truth-values T,
= the universe of discourse (a constant domain of individuals)
= the set of reals, doubling as times (time being a continuum)
= the logical space of logically possible worlds
Some logical objects, like truth-functions, quantifiers and singularizers, are extensional entities: (conjunction), (disjunction), (material implication) are all of type (). Importantly, (boolean negation) is of type (). TIL types boolean negation as a partial function taking T to , to T, and a truth- value gap to a truth-value gap. However, nothing in this paper hinges on partiality, so for present purposes boolean negation may be taken to simply invert truth-values. Quantifiers and singularizers will be needed below and are defined thus:
22
DEFINITION 2 (quantifiers, singularizer).
(i) The quantifiers , are type-theoretically polymorphous, total functions of type (()), for an arbitrary type , defined as follows. The universal quantifier
is the function that associates a set S of -elements with T if S contains all the elements of type , otherwise with . The existential quantifier is the function that associates a set S of -elements with T if S is a non-empty set, otherwise with .
(ii) The singularizer ɿ is the type-theoretically polymorphous, partial function of type (()) that associates a set S of -elements with their unique element whenever S is a singleton and is otherwise undefined.
We introduce the following notational conventions:
‘X/’ means that entity X is of type
‘x ’ means that variable x ranges over type
Quantifiers and singularizers will not be dressed up in type assignments,
in order not to clutter the notation
‘ɿx’, ‘x’, ‘y’ abbreviate ‘ɿx’, ‘x’, ‘y’: variable binding is always -
binding
In the full type theory of TIL all the variables occurring in this paper would be of
type *1, which is the lowest of the types in the ramified hierarchy. However, we
23 are not defining the ramified type hierarchy here, because we are only concerned with intensional and extensional entities. Hence only the values of variables, and not the variables themselves, can and will be typed here.17
DEFINITION 3 (intensional entity). An intensional entity (or ‘intension’ for short) is a function with domain in and range in an arbitrary type .
Typically, intensions are both modally and temporally sensitive. TIL models such intensions as functions from worlds to functions from times to , i.e. as functions of type (()), abbreviated ‘’. In particular, a property is a function of type ((())), abbreviated ‘()’, from worlds to a function from times to sets of individuals.18 Relative to a world/time pair, there is a set (perhaps empty) of those individuals that have the relevant property at that dual index.19 When understood in a formally precise manner in this paper, a proposition is a function of type (()), abbreviated ‘’, from worlds to a partial function from times to
17 For details, see X*XY* (2010, pp. 42-43), (ibid., p. 45, def. 1.2), (ibid., p. 52, def. 1.7). See also X* (2013).
18 See X*XY* (2010, §1.4.2.1) for a classification of empirical properties.
19 Speaking of a ‘world/time pair w, t’ is a slight simplification, because our intensions are not defined on such pairs. The modal and the temporal dimension are deliberately kept separate in TIL: see X*XY* (2010, pp. 205-07). However, the simplification is innocuous in this paper, because we have no need for this separation.
24 truth-values. That is, propositions are empirical truth-conditions and modelled as temporally sensitive sets of possible worlds, as in possible-world semantics enriched with temporal indices (in which a proposition is identified with its satisfaction class, or equivalently, with the characteristic function of the class).20
Predication, extensionalization and modification are all instances of functional application, denoted by square brackets ‘ … ’. Thus when a property F is predicated of an individual a, the logical procedure is this. Fwt, abbreviated
21 ‘Fwt’, is the set/() of individuals instantiating the property F at w, t. The
characteristic function Fwt is applied to a, resulting in a truth-value, according as
a is a member of Fwt. Accordingly, predication is logically a matter of applying an extensionalized property to an individual. The resulting truth-value is
abstracted over by means of w, t to obtain the truth-condition wt Fwt a, which is satisfied by all and only those w, t pairs at which a has property F.22
20 See X*XY* (2010, pp. 276-78) for a philosophical justification of partiality, which is a feature that makes TIL non-classical. E.g., for an arbitrary
proposition P, the proposition wt Pwt Pwt does not come out a tautology,
for either of P and wt Pwt may be undefined at w, t. See also Materna (2014, pp. 49-51).
21 See X (2008) or X*XY* (2010, §2.4.2).
22 0 0 In canonical TIL, wt Fwt a, or rather wt Fwt a, is a structured hyperproposition, itself an instance of a procedure called Closure, producing or constructing a possible-world proposition / empirical truth-condition. The
25
DEFINITION 4 (first- and second-order modifier). A property modifier is of type
(() ()), forming a property from a property, and is thus a first-order modifier. Abbreviate ‘(() ())’ as ‘()’. Then a modifier of property modifiers is of type (() ()), i.e. a second-order modifier.
Modifiers are not intensional, but extensional, entities, to revisit a topic broached in Section 1. They are typed as functions-in-extension from one intension to another and as such impervious to modal, temporal and whatever other empirical vicissitudes. A modifier M/() takes property F/ to the property distinction between hyperintensions and intensions is essential to TIL, but may be ignored here, since it is not relevant for present purposes how intensions are presented. Here we are interested merely in the intensions themselves. In the idiom of procedural semantics, structured hyperintensions are procedures and some of their products are intensions. See X*XY* (2010, Ch. 1) for the foundations and (ibid., Ch. 5) for various applications of what TIL calls constructions, i.e. structured hyperintensions. We allow ourselves, in this paper, the liberty of identifying constructions with what they construct. Thus M F will be considered the result of applying modifier M to property F and not the structured procedure detailing that M is to be applied to F. In canonical TIL, 0M
0F is an instance of the construction (i.e. procedure) called Composition which in this case constructs a property, where the constructions 0M, 0F are so-called
Trivializations, which are atomic procedures producing the modifier M and the property F, respectively. See X (2014) on atomic constructions.
26
MF/ formed by way of functional application of M to F. A function typed as
() is not the sort of function that trades arguments for values relative to w, t.
For this to happen the type would have to be ().
The reason why we want the type () rather than () is this. If the type of M
were () then it would be possible that at some w, t the resulting property
[Mwt F] would be G whereas at another w’, t’ the result [Mw’t’ F] would be a different property G’. In other words, it would be an empirical fact which property was produced, and we would know the result only a posteriori. Yet we do know a priori which property is the result. For instance, happy applied to child results in the property [happy child], and fake applied to banknote results in [fake banknote]. What is modally and temporally sensitive is instead which particular individuals (if any) are in the extension of some modified property M F at a given w, t of evaluation. Which property [M F] is depends only on the semantics of the modifier term ‘M’ and the predicate ‘F’, not on empirical facts.
Hence it can be nothing but a strictly analytical, as opposed to empirical, matter which modified property M takes F to. For instance, if M is privative with respect to F then at no w, t are the properties [MF] and F co-instantiated. If there is a pair w, t at which [MF] and F are co-instantiated and another w, t at which they are not then M must be modal with respect to F. And if the set of pairs w, t at which [MF] is instantiated is a subset of the w, t at which F is instantiated then M is subsective with respect to F. In all three cases the
27 modified property that M takes F to is a function exclusively of the argument property F and not also of some w, t. That is, the nature of the relation between
[MF] and F depends on whether M is subsective or privative or modal with respect to the property F.
The insensitivity to empirical vicissitudes explains why vagueness is no complicating factor. Vagueness would indeed be an issue if we were
investigating the relationship between the sets Fwt, Gwt, for arbitrary w, t,
wondering where to draw the line between membership of Fwt and of Gwt, e.g. between a set of orange surfaces and a set of yellow surfaces. Likewise, fuzziness
would be an issue if we were investigating the relationship between a and Fwt,
wondering to what degree a was a member of Fwt. But we are studying exclusively the relationship between M and F, and between M* and M. It does not matter whether F is vague or fuzzy, for if M is applied to F the result will still be a property, MF, that obeys the definition of M as this or that sort of modifier. For instance, if chubby is a subsective modifier then a chubby child must be a child, however vague or fuzzy the properties of being a child or a chubby child may be. The point is that any fluctuations of the extension of F have no bearing on how F behaves when modified, for it is an intension (namely, F)
and none of its extensions (namely, Fwt) that M operates on.
28
When a modified property MF is predicated of an individual a, the logical
procedure, as with F, consists in applying the extensionalized property MFwt to a thus:
wt MFwt a
Where M*/(() ()), the modification of M by M* and the resulting application to F looks like this: M* M F. Hence the predication of M* M F of a looks like this:
wt M* M Fwt a
The three kinds of modifiers deployed in this paper will be defined in three separate definitions. Subsection is straightforward and will be defined first. The general rule of privation we will be putting forward requires some explanation and justification, and will be defined only in the next section. The definition of modal modification presupposes the definitions of subsective and privative modification, and so will be presented last.
DEFINITION 5 (modifier subsective with respect to a property). Let M be a first- order modifier/(); let x ; f, g ; ɿ/(()): the function from a property singleton to its element; =/(): identity between properties; =/(()()):
29 identity between property modifiers. (We are using infix notation for ‘=’ for better readability.) Then M is subsective with respect to property f iff
M = f ɿg [[g = [Mf]] wt x [[gwt x] [fwt x]]].
Def. 5 defines M as a function from f to Mf. From Def. 5 we obtain the
following elimination rule for the arbitrary first-order modifier Ms subsective with respect to the arbitrary property F:
[Ms Fwt x]
Fwt x
To put this rule into a wider perspective, it is a rule of right subsectivity, because
the right-hand F in Ms F is detached and carried through to the conclusion. A rule of left subsectivity is more intricate, because no M/() can be carried
through to the conclusion by simply eliminating F from M F: M x, and Mwt x for that matter, is type-theoretically inappropriate. TIL has formulated and proved the validity of its own rule of left subsectivity: X*XY* (2010, §4.4), XY
(2013, §2.5) provide the details. In a framework in the vein of Montague’s, a rule of left subsectivity is required to validate the first conjunct of this inference,
which defines intersective modifiers: [Mi Fwt x]Mwt x Fwt x, where M is
the property obtained from the intersective modifier Mi in the manner made
30 valid by the rule of left subsectivity. For instance, if Mi is round then the rule validates the conclusion that x is round.
Because intersective modification is the conjunction of left and right subsection, intersection turns out to be a special case of subsection. Note that the above rule,
[Mi Fwt x]Mwt x Fwt x, defines intersectives and exceeds the definition of subsectives, where subsectives are understood to be right-hand subsectives. This is important in order to distinguish between intersectives and subsectives. For
from [Ms Fwt x] the conclusion Mwt x Fwt x follows: the second conjunct follows from the above definition of right-hand subsectives, and the first conjunct follows from the rule of left subsectivity, which applies to both intersective, subsective, modal, and privative modifiers. That the rule of left subsectivity applies across the board is little surprise, for the intersectives and the privatives (see the end of Section 1) are special cases of subsection, and the modals flit between being subsective and being privative.
Subsectives have an interesting limiting case. Trivial subsectives, like genuine, are such that subset and superset coincide:
wt x [genuine Fwt x] Fwt x
Genuine is the identity function defined over . The above elimination rule
applies also when Ms is trivial, but drawing an inference would be an exercise in
31 futility, for the inference in the opposite direction is also valid. Hence a trivial Ms makes no logical difference, as indeed it should not. A trivial modifier returns the modified property unmodified, as it were. See X*XY* (2010, pp. 189-90) for the argument that terms for identity functions may have rhetoric (e.g. perlocutionary) though no logical or semantic import. There is but one identity function over , but natural language has many names for it, such as ‘genuine’, which emerges as a trivial functor. A term for a trivial modifier may serve the pragmatic purpose of highlighting contrasts, as between owning a Vermeer and a fake Vermeer (“No, that painting is not a fake Vermeer, as you claim; it is a real
Vermeer!”), or paying literally a thousand euro as opposed to slightly less (“No,
I didn’t pay close to a thousand euro for that philosophy book; I paid a full thousand euro”). The time-honoured drinking toast “Champagne for my real friends and real pain for my sham friends” exemplifies similarly the contrast between reality and pretence, but also highlights the additional phenomenon that an adjective like ‘real’ may quiver between denoting a trivial subsective modifier and an intensifying subsective modifier. When a token of ‘real’ is construed as denoting an intensifier, ‘is real pain’ denotes only a special kind of pain, namely severe pain.23
23 See also X*XY* (2010, p. 402, n. 56) for a non-trivial reading of the predicate
‘is a true man of the desert’. See X*XY* (ibid., p. 400, n. 52) for a critique of
Partee (2001, §4) on how to analyze ‘fake fur’ and ‘real fur’.
32
To put the elimination rule for first-order subsectives into a still wider perspective, the perhaps most fundamental distinction among modifiers is the one between what Kamp (1975) calls extensional and non-extensional
(‘intensional’) modifiers. (Kamp’s use of ‘extensional’ should not be confused with ours, according to which every single modifier is extensional, because it takes a property to a property or a modifier to a modifier independently of
empirical indices.) If Fwt and Gwt are the same set, then by the definition of
extensional modifier (see the rule of inference below) Me Fwt and Me Gwt are the
same subset within the set Fwt (i.e. Gwt).
Kamp (ibid., pp. 125-26) notes that if a modifier is intersective then it is extensional, speculating whether the converse also holds. An (admittedly superficial) counterexample to the converse holding would be that trivial subsectives are extensional, but not intersective: a genuine banknote is not genuine simpliciter, but only with respect to a property, in casu banknote. It is obvious that subsective, privative and modal modifiers are not extensional in
Kamp’s sense. This fact is actually what makes them philosophically, logically and semantically intriguing. It requires explanation in each individual case why a given non-extensional modifier fails to distribute, i.e. why it fails as an instance of this rule of inference:
[Me Fwt x]; Fwt = Gwt
[Me Gwt x]
33
For a valid instance of the rule, if at w, t some object is a round peg and the set of pegs is identical to the set of toys, then the object is a round toy. 24
3. PRIVATION GENERALIZED
The basic question is: what sense can be made of the privation of a privation?
Here is what would not work. Apply twice over the rule, broached in the
Introduction, that Mp is to be replaced by . Then:
[[Mp [Mp F]]wt a]
[[Mp F]wt a]
[Fwt a]
[F wt a]
24 For comparison, Makinson (1973, pp. 64ff) presents the rule of transference, which is similar to, but still different from, the rule governing Kamp’s
extensional modifiers. The rule of transference is this: [Gwt x][MGwt x]. For instance, “if something is a blue tie and is also a museum exhibit then it is a blue museum exhibit” (ibid., p. 64).
34
What a powerful logical rule this would be! The thing is, of course, that it is not a logical rule, for there are counterexamples. Just recall the spectre of a fake fake diamond being a diamond, of logical necessity. The essential feature of privation is that privation tells you what something fails to be, because the thing in question is deprived of the property in question. So it is already cause for suspicion if the conclusion is the predication of a positive property.
Let us take a closer look at the derivation above, beginning in reverse with the
step from [Fwt a] to [F wt a]. Classically, where P is a truth-condition, e.g. in
the form of a possible-world proposition, P and wt Pwt are one and the
same truth-condition, and co-entailment, P wt Pwt, reduces to self- entailment, where , /(()) are relations-in-extension between propositions. But in the case of double privation, double negation elimination
ought not to get the chance to kick in, for fake fake Fwt a, say, ought
obviously not to go into Fwt a. This suggests to us that the original rule of single privation has not been stated generally enough. The original rule trades privation, which is tantamount to the deprivation of a property, for , which is the inversion of a truth-value.25 Trading privation for boolean negation works fine for single privation, as we noted above, but not for multiple privation, where it amounts to an overstatement. The reason why it works for single privation is because it is required that property F not be true of a in the conclusion, and
Fwt a achieves at least this much. One way to block the transition from
25 Or else if Pwt is a truth-value gap then so is Pwt, and vice versa. See Section 1.
35
[Fwt a] to [Fwt a] would be to dismiss double negation elimination as a valid rule of inference. This option is not open to us, however, on pain of adhockery since TIL assigns a classical logic to the truth functions. Another way is to
prevent [Fwt a] from arising in the first place. This is the approach we will be exploring below.26
The place to start looking is the step from [[Mp F]wt a] to [Fwt a]. The rule of
single privation lays down what to do if [[Mp F]wt a] is true. The rule does not
state what to do if [[Mp’ F] a] is negated. So the dialectics is this. The rule stated at the outset of this Section is invalid, which is a good thing, and it fails to tell us
what does follow from [[Mp’ F]wt a]. The way to fill this gap is obviously not to
define Mp the way we go on to describe right now. Instead the way to fill the gap
is to define Mp in terms of contraries, as we go on to do afterwards.
The step from [[Mp F]wt a] to [[Fwt a]] would be correct if Mp were defined as a function assigning to a given property f the unique property contradictory to f:
Mp =df f wt [x [fwt x]]
Necessarily, [[Mp F]wt a] = [Fwt a], and [[Mp F]wt a] = [Fwt a]. From this definition we would obtain the equivalence between the privatively modified
26 It is beyond us to explore to what extent Hegel’s dialectical law of the negation of negation can shed light on double privation as this notion is understood here.
36 property [Mp F] and its contradictory property not being an F: wt [x [Fwt x]].
By running the following equivalences, we would be able to start out with
[[Mp [Mp’ F]]wt a] and end up with [Fwt a], where =/():
[Mp [Mp’ F]] =
[Mp wt [x [Fwt x]]] =
w’t’ [y [wt [x [Fwt x]]w’t’ y]] =
w’t’ [y [x [Fw’t’ x] y]] =
w’t’ [y [Fw’t’ y]] =
w’t’ [y [Fw’t’ y]] =
F
Obviously, this is not plausible. For each property f there is just one
contradictory property wt [x [fwt x]]. Hence each modifier privative with respect to a given property f would take f to one and the same contradictory
property wt [x [fwt x]]. This cannot be right. For instance, wooden and iron
(when ‘iron’ is used as an adjective) are privative modifiers with respect to the property of being a horse. But being a wooden horse and being an iron horse are two different properties.
37
3.1 SINGLE PRIVATION
The lesson is that we must define the first-order modifier Mp in another way.
Our proposal is this. We submit that the logic of privation, whether single or multiple, is a logic of contraries. Contraries contrast with contradictions:
a is an F a is a non-F versus
a is an F not (a is an F)
The former is not a classical tautology; the latter is. “a is a non-F” expresses the contrary of a being an F; “not (a is an F)”, its contradictory. It has been known at least since Aristotle that contrariety and contradiction differ as for their logical behaviour:
The sentences “It is a not-white log” and “It is not a white log” do not
imply one another’s truth. For if “It is a not-white log”, it must be a log:
but that which is not a white log need not be a log at all.
(Prior Analytics I, 46, 1)
That is, in modern parlance, a set of logs divides into those that are white and those that are non-white, whereas a set of non-(white logs) divides into those elements that are non-white logs and those that are not even logs (though perhaps white).
38
And directly relevant for our present purposes:
From the fact that John is not dishonest we cannot conclude that John is
honest, but only that he is possibly so. (La Palma Reyes et al. 1999, p.
255.)
The alternative is namely that John is neither dishonest, nor honest, so “John is not dishonest”, if true, tells us what John fails to be and what the alternatives are: either honest, or neither honest nor dishonest.27 The contradiction is that John is not not honest, which is logically equivalent to him being honest.28
Let us see how this proposal plays out formally. We begin by defining how a property modifier qualifies as being privative with respect to a property f.
DEFINITION 6 (modifier privative with respect to a property). Let M be a first-order modifier/(); let x ; f, g ; ɿ/(()): the function from a property
27 See Horn (1989, pp. 296-308), (1991) for discussion of the logic and rhetoric of double negatives, e.g. as expressed by ‘not un-F’ (‘not unhappy’, ‘not impolite’, etc.). Is a not impolite remark a polite remark (perhaps even a very polite remark
(litotes); cf. 1991, pp. 86ff); or a remark that is neither polite nor impolite, ending up in the neutral mid-interval? See (ibid., pp. 38-41) for a historical survey of various takes on contrariety and predicate term negation.
28 See also Kaneiwa (2007).
39 singleton to its only element; and let w w, t t. Then M is privative with respect to a property f iff
M = f ɿg [[g = [M f]]
x [wt [[fwt x] [gwt x]] wt [[fwt x] [gwt x]]]].
The definition states that a privative modifier associates a given property f with a property g contrary to f. It is not possible that x be both an f and a g, and possibly
x is neither an f nor a g. The conjunct wt [[fwt x] [gwt x]] is the clause that f
and g are mutually exclusive. The conjunct wt [[fwt x] [gwt x]] is the
contrariety clause that the negation of one of the conjuncts fw’t x and gw’t x does not entail the truth of the other. Hence any property [M F] formed from a property F by a modifier M privative with respect to F is a property contrary to
F. Contrariety provides the weaker form of negation that is suitable for privative
M. We use the notation ‘[Mp F]’ to indicate that Mp is privative with respect to F, as in the Introduction.
From Def. 6 we obtain the elimination rule that characterizes the first-order
modifier Mp privative with respect to a property F presented at the outset of this paper:
40
[Mp Fwt x]
Fwt x
Necessarily, if x is an [Mp F] then x is not an F. This follows from the second
clause wt [[fwt x] [gwt x]], which is equivalent to wt [[gwt x] [fwt x]].
Since by definition g = [Mp f], for all x and a definite property F the following necessary implication holds:
wt [[[Mp F]wt x] [Fwt x]]
This in turn means that for all x the proposition wt [[Mp F]wt x] entails the
proposition wt [[Fwt x]]:
wt [[Mp F]wt x] wt [[Fwt x]]
Note, however, that wt [[Mp F]wt x] only entails wt [[Fwt x]] rather than
being necessarily equivalent to it, because by Def. 6 [Mp F] is contrary rather than contradictory to F.
For each property F there may be more than one modifier privative with respect to it, and thus more than one such property contrary to F, so we can generalize.
41
First we define the function Cp/((())) that associates a property f with the class of those modifiers that are privative with respect to f, as follows.
DEFINITION 7 (function from a property f to the set of modifiers privative with respect to f). Let non () be a variable ranging over first-order modifiers; x ; f
; w w, t t. Then:
Cp = f non [x [wt [[fwt x] [[non f]wt x]]
wt [[fwt x] [[non f]wt x]]]].
Now we are in a position to define the general modifier Non, which is needed for
the elimination rule pertaining to Mp. Non is privative with respect to a property f in such a way that [Non f] is the general property contrary to f. Figuratively speaking, [Non f] is the union of all the properties contrary to f. Literally
i speaking, let {Mp } = [Cp f] be the set of all the modifiers privative with respect to
f. Then in any world w at any time t the extension [Non f]wt of the property [Non f]
i i is the set-theoretical union of the extensions [Mp f]wt of the properties [Mp f] contrary to f.
DEFINITION 8 (general modifier privative with respect to a property f). Let non
() be a variable ranging over first-order modifiers; and let =/(()()) be the identity relation defined over first-order modifiers. Then
42
Non = f wt x non [[[non f]wt x [[Cp f] non]]]
is the general first-order modifier privative with respect to f.
Non is the unique general privative modifier, which takes a property f to the general contrary property Non f. For instance, [Non banknote] is the general property contrary to the property of being a banknote. Necessarily, the extension
of [Non banknote]wt includes forged banknotes, banknotes dissolved in acid,
Monopoly banknotes, etc. One might worry that it would be too strong to maintain that, necessarily, for all and any w, t, the extension contains the full panoply of non-banknotes. But it follows from Definition 8 that the full panoply is indeed involved. In any w, t, for any individual x it holds that
[[[Non F]wt x] = non [[[non F]wt x [[Cp F] non]]]
Hence individual a has the property [Non F] iff a has any property [non F] for
some non. Thus the set [Non F]wt is almost as large as the complement of the set
Fwt. In some, but not all, w, t it can even be the case that [Non F]wt = \Fwt. Or, if
Fwt happens to be the entire type , then [Non F]wt happens to be the empty -set,
i.e. the union of all empty sets [non F]wt. Definitions 6 through 8 do not exclude such modifier functions as do not even have a name in our vernacular.
43
We should not forget, however, the limiting case where F is a trivial, non- contingent property with a constant extension like being self-identical.29 In this
case, necessarily, Fwt is the entire type and [Non F]wt is the empty -set, because at no w, t is there an individual that would be neither identical with itself nor non-identical with itself. Another example of a non-contingent property is the property of being identical to a or b. At all w, t the extension of this property is the set {a, b}, and at no w, t is there an individual that would be neither identical with a or b, nor non-identical with a or b. The upshot is that non- contingent properties do not lend themselves to being modified by privative modifiers on pain of falsity. Hence if T is such a trivial non-contingent property then the extension of [Non T] is necessarily, in all w, t, the empty -set.
From Def. 8 we obtain the following elimination rule for modifiers Mp privative with respect to property F:
[Mp Fwt x]
[Non Fwt x]
The conclusion of this rule states that predication of F eludes x: F does not get to be predicated of x. For instance, if the premise is that a is a fake banknote then the conclusion is that a is a Non-banknote, therefore the property banknote is not
29 For classification of empirical properties see X*XY* (2010, § 1.4.2.1).
44 predicated of a. Or if b is a wooden horse then b is a Non-horse. But if c is a wooden porcupine then it follows neither that c is a Non-horse, nor that c is a horse.
The elimination rule for Mp is a cornerstone of this framework, so it may be in order to provide a proof of the rule.
PROOF.
1. [Mp Fwt x]
2. y [wt [[Fwt y] [[Mp F]wt y]]
wt [[Fwt y] [[Mp F]wt y]]] 1, Def. 6
3. [[Mp Fwt x]
y [wt [[Fwt y] [[Mp F]wt y]]
wt [[Fwt y] [[Mp F]wt y]]]] 1, 2, I
4. [[Mp Fwt x] [[Cp F] Mp]] 3, Def. 7
5. [non [[[non F]wt x] [[Cp F] non]] Mp] 4, -exp
6. non [[[non F]wt x] [[Cp F] non]] 5, EG
7. [[f [wt x non [[[non f]wt x] [[Cp f] non]]] F]wt x] 6, -exp
8. [[Non F]wt x] 7, Leibniz’s Law
The elimination rule above makes it explicit what the set-theoretic relationship is
between Mp Fwt and Non Fwt. Where /(()()) is the subset relation, the following necessary relation holds:
45
wt [Mp Fwt Non Fwt]
Due to the first clause wt [[Fwt x] [[Non F]wt x]] of the definition of
contrariety, which is equivalent to wt [[[Non F]wt x] [Fwt x]], the following necessary relation also holds:
wt [Non Fwt x Fwt x]
Hence, by transitivity, wt [Mp Fwt x Fwt x], thus validating the original rule of single privation presented in the Introduction.
Again, it is crucial not to confuse Non/(), which operates on properties, with the complement function \, which operates on sets of individuals and is of type
(()()). The complement of a complement is the original set, thereby reinstalling the problematic boolean negation, whether with or without the
misguided definition of Mp as a function from being an f to not being an f. The assignment of elements to sets is only a reflection at the extensional level of what effect Non has on properties. Therefore, what Martin (2003, p. 449) says about ‘infinite negation’ (e.g. non-human, to use Martin’s example) does not carry over to our Non: “Semantically infinite negation converts a term into one that stands for its non-empty complement ...”. Our Non comes with no such ontological restriction as non-emptiness; however, more importantly, ‘Non-F’ does not denote a complement set, but a contrary property. Nonetheless, since
46 necessarily, for every w and t, Non Fwt \Fwt, there is a logical link between complement sets and contrary properties, namely material implication, if restricted to single privation:
wt [[Non Fwt a] [\Fwt a]]
But the reverse implication is not valid. From the extensional fact that a is an
element of \Fwt it cannot be reconstructed whether a is in the set Non Fwt because a is a Non-F or because a is not an F.
3.2 DOUBLE PRIVATION
First and foremost, due to the contrariety of Non there can be no entailment like this involving double privation:
wt Non Non Fwt a wt \ \ Fwt a
And since \ \ Fwt is equal to Fwt, the invalid, ‘powerful’ rule from the beginning of Section 3 is blocked. The reason is this. First, Non takes F to its generalized contrary property G = [Non F], the extension of which is not
necessarily equal to \Fwt; rather Gwt is only a subset of \Fwt. When privation strikes again, Non takes G to another generalized contrary property H = [Non G],
for which it again holds that Hwt is only a subset of \Gwt. Hence a is an element
47 of \Gwt, but may fail to be an element of \ \ Fwt, i.e. Fwt. This demonstrates the important fact that double privation may, but need not, return a to F. To illustrate, the distribution of extensions (sets) at a particular w, t may, for instance, look like this:
[Mp[Mp’ F]]wt [Non [Mp F]]wt
[Mp’’ F]wt F wt
[Non [Non F]]wt
[M F] p wt
FIGURE 1.
Having availed ourselves of Non, let us revisit the first two steps of the problematic inference discussed at the beginning of Section 3:
[[Mp [Mp F]]wt a]
[[Non [Mp F]]wt a]
?
48
The first step is the application of the elimination rule. But we cannot apply this
rule twice over in order to derive [[Non [Non F]]wt a]. The rule [[Mp F]wt a]
[[Non F]wt a] is applicable only in case a is an element of the extension of the
property [Mp F], and not when a is an element of the extension of the property
[Non [Mp F]]. To see why, consider the extensions of the properties in question.
The following necessary relations hold:
wt [[Mp F]wt [Non F]wt \Fwt];
wt [[Mp [Mp F]]wt [Non [Mp F]]wt \[Mp F]wt];
wt [[Non [Non F]]wt \[Non F]wt \[Mp F]wt]
Obviously, these relations do not guarantee that if a [Mp [Mp F]]wt then also
a [Non [Non F]]wt. This is due to Def. 8 of Non. Since necessarily the extension
i i of [Non F] is the set-theoretical union of the extensions of [Mp F], where Mp are
modifiers privative with respect to F, we have: wt [Mp F]wt [Non F]wt, hence
wt \[Non F]wt \[Mp F]wt, but not vice versa. Hence this conjunction is a
logical possibility: a [Non [Mp F]]wt a [Non [Non F]]wt.
For higher-order double privation, when the premise is [[[Mp* Mp] F]wt a], boolean
negation cannot replace Mp* or Mp, of course. The admissible operand for is a truth-value rather than a property modifier, whereas the admissible operand for
Mp* is a property modifier rather than . Hence [ Mp] and [Mp* ] both come
49 out ill-formed, because ill-typed. Since Mp* is privative with respect to Mp in
[[[Mp* Mp] F]wt a], the only conclusion seems to be that a is not an [Mp F]:
[[[Mp* Mp] F]wt a] [[Mp F]wt a]
This is too weak to be of much logical interest. Instead it might be interesting to
ask which sort of modifier emerges from applying Mp* to Mp. The answer, however, is that there is no definite answer, in that no particular sort of modifier, not even
a modal one, can be predicted to emerge from applying Mp* to Mp. The only
thing we can know is that the emerging modifier is not going to be Mp. There can be no logical rule dictating which sort of modifier will always emerge. This negative result will be substantiated at the end of this Section and reappear in
Section 5 ad rule (8).
The definition of second-order modifier privative with respect to a first-order property modifier is this.
DEFINITION 9 (second-order modifier privative with respect to a first-order modifier).
Let Mp* be a second-order modifier/(()()); let m, n (); f ; x ;
ɿ/(()(())): the function that associates a set S of property modifiers with its only element if S is a singleton, and is otherwise undefined; and let w w, t t.
Then M*p is privative with respect to a property modifier m iff
50
M* = m ɿn [[n = [M* m]]
xf [wt [[[m f]wt x] [[n f]wt x]] wt[[[m f]w t x] [[n f]w t x]]].
In order to generalize, we define the function C*p/(((()()))()) that associates a first-order modifier m with the class of second-order modifiers privative with respect to m as follows.
DEFINITION 10 (function from a first-order modifier m to the set of second-order modifiers privative with respect to m). Let non* (()()) be a variable ranging over second-order modifiers; m (); x ; f ; w w, t t. Then
C*p = m non [fx [wt [[[m f]wt x] [[[non* m] f]wt x]]
wt [[[m f]wt x] [[[non* m] f]wt x]]]].
Hence, for all second-order modifiers privative with respect to some first-order modifier M that modifies some property F and for all individuals x, it holds that
it is impossible that x be simultaneously an [M F] and a [[non* M] F], whereas it
is possible that x be neither an [M F], nor a [[non* M] F].
The definition of the general second-order modifier privative with respect to the first-order modifier m that modifies property f is as follows:
51
DEFINITION 11 (general privative second-order modifier). Let non* (()()) be a variable ranging over second-order modifiers; m (); f ; =/(()()): the identity relation defined over (()()). Then
Non* = m f [wt x non* [[[[non* m] f]wt x] [C*p non*]]]
is the general modifier/(()()) privative with respect to m.
Non* is the unique general second-order privative modifier, which takes a first- order modifier m to the first-order modifier Non* m, such that Non* m f is the general contrary property of m f. Figuratively speaking, Non* is the union of all second-order privative modifiers. For instance, if fake is a modifier of the property banknote, then [[Non* fake] banknote] is the general property contrary to the property of being a fake banknote.
The corresponding elimination rule for Mp* is this, M an arbitrary first-order modifier:
[Mp* M Fwt x]
[Non* M Fwt x]
52
The conclusion of this rule states that modification by M eludes F: M does not
get to modify F. The reason is because Mp* cancels out modification by M. The logical effect that Non* has on M, whether M be subsective or privative, is that M is checkmated and so rendered inoperative: no positive conclusion can be drawn about either M or Non* M. The elimination rule behaves such that if, for instance, x is a ((deceptively* simple) solution) then x is a ((Non*-simple) solution).
See Definitions 9 through 11, and rules (8) and (10) in Section 5.
4. MODAL MODIFICATION
Modal modifiers such as alleged (modifying a property) and maybe (modifying a proposition) are the odd ones out among property and propositional modifiers.
One fundamental logical fact about modal modifiers is that if a is, say, an alleged assassin then it can be neither ruled in nor ruled out that a is an assassin.
This feature renders them barely informative in terms of deriving deductively valid conclusions. It also explains why there can be no rule of absolute elimination for modal modifiers. Logic cannot prejudge what must be established empirically, namely whether, on a particular occasion, a given modal
modifier Mm behaves subsectively or else privatively with respect to a given
property F. If Mm is alleged then it is a strictly empirical question whether it is true that a given alleged assassin is an assassin.
Nor can the conclusion of the rule be simply the disjunction wt Fwt a
Fwt a. This classical tautology applies no less to the other modifiers,
53 whenever ‘a’ is a defined term. For instance, if a is a skillful surgeon then it follows that a is a surgeon or that a is not a surgeon. It is just that the first disjunct always holds and the second one never does. Or if b is a fake banknote then the first disjunct never holds and the second one always does.
What is unique about modal modifiers is that there are cases where the first disjunct holds and other cases where the second disjunct holds. If a is an alleged assassin at w, t then it is alethically possible that w, t be identical to some
w´,t´ at which a is an assassin, and it is alethically possible that w, t be identical to some alternative w´´, t´´ at which a fails to be an assassin. The open question is whether w, t is w´, t´ or else w´´, t´´.
No other modifier has the context-sensitive feature that its status depends on the
given w, t of evaluation. If the proposition wt Mm Fwt a is true then a may be an F at w, t and a may be a Non-F at w, t. By contrast, if the proposition
wt Ms Fwt a is true then a must be an F at w, t; if the proposition wt Mp
Fwt a is true then a must be a Non-F at w, t. Modifiers that are modal with respect to some particular property F share, moreover, the particular sort of non- empirical context-sensitivity that also applies to the subsective and privative ones, that the argument property, on a particular occasion, dictates whether the modifier, on that occasion, is subsective or privative or modal. For instance, an alleged proposition is unequivocally a proposition, provided an alleged proposition is a proposition that has been alleged to be true.
54
DEFINITION 12 (modifier modal with respect to a property). Let M/() be a first- order modifier; let x ; f, g ; ɿ/(()): the partial function from singletons whose elements are properties to their sole element; and let w’ w’’, t’ t’’. Then
M is modal with respect to a property f iff
M = f ɿg [[g = [M f]]
wt x [[gwt x] [w´t´ [fw’t’ x] w´´t´´[[Non f]w’’t’’ x]]]].
From Def. 12 we obtain the following elimination rule for Mm:
[[Mm F]wt a]
w´t´ [Fw’t’ a] w´´t´´ [[Non F]w’’t’’ a]
This is as far as elimination will go. We are not going to define higher-order modal modifiers, because we have no need for them in this paper.
5. TEN INFERENCE SCHEMAS WITH EXAMPLES
The ten rules have been generated on the basis of their respective premises according to combinations of first-order and second-order subsective and
privative modifiers: Ms, Ms*, Mp, Mp*.
55
The first two rules are the familiar rule of single subsection and the revised rule of single privation, respectively.
(1)
[[Ms F]wt a]
[Fwt a]
Example. a is a larges elephant, hence a is an elephant.
(2)
[[Mp F]wt a]
[[Non F]wt a]
Example. a is a mockp turtle, hence a is a Non-turtle.
(3)
[[Ms [Ms´ F]]wt a]
[[Ms F]wt a]
(1)
[F wt a]
56
Example. a is a greens (Italians´ banknote), hence a is a green banknote, hence a is a banknote.30 It also follows that a is an Italian banknote, hence that a is a banknote, but this pattern is a matter of applying rule (1) twice over. (3) is a case of a subset within a subset within a subset, in the style of a Russian Easter egg.
Equivalently, it is a case of intersection between two subsets of a set of banknotes: a set of green banknotes and a set of Italian banknotes.
(4)
[[Ms [Mp F]]wt a]
[[Mp F]wt a]
(2)
[Non Fwt a]
Example. a is a greens (fakep banknote), hence a is a fake banknote, hence a is a Non- banknote. What is compromised is the property F (i.e. banknote) and not the
modifier Ms (i.e. green). It also follows from [[Ms [Mp F]]wt a] that a is a green Non- banknote, but this pattern requires a different rule, which is a variant of (2):
[Ms [Mp F]]wt a Ms Non Fwt a.
30 Abdullah and Frost (2005, §3.3.1) distinguish between what they call wide- scope and narrow-scope readings of the modified noun phrase ‘deep blue ocean’ as it occurs in “Oceania is a deep blue ocean”: deep modifies blue ocean or deep modifies ocean.
57
(5)
[[Mp [Ms F]]wt a]
[[Non [Ms F]]wt a]
Example. A fakep (cashs dispenser) is a Non-(cash dispenser). Such a thing is either
any other kind of dispenser or no dispenser at all. The premise says that Mp
modifies the property [Ms F], not the property F, which is another than [Ms F],
unless Ms is a trivial subsective modifier. That is why [[Non F]wt a] does not
follow. Nor does [[Mp F]wt a] follow, for [Mp F] is a Non-F, while [Mp [Ms F]] may still be an F.
(6)
[[Mp [M´p F]]wt a]
[[Non [M´p F]]wt a]
Examples. A malfunctioningp (fakep banknote) is a Non-(fake banknote). A
malfunctioningp (fakep banknote) amounts to something like a foiled scam.
Someone who is currently a (formerp (apparentp heir)) is either someone who finally made it to monarch or else is someone who is no longer even a prospective monarch. This goes to show that there are instances of (6) where one
starts out with [[Mp [M´p F]]wt a] and ends up with [Fwt a]. Whatever property
58
[M´p F] may be, anything and anyone with property [Mp [M´p F]] fails to be an
[M´p F], due to Mp.
(7)
[[[Ms* Ms] F]wt a]
[[Ms F]wt a]
(1)
[Fwt a]
Example. a is a ((verys* talls) building), hence a is a (tall building), hence a is a building.
(8)
[[[Mp* Mp] F]wt a]
[[[Non* Mp] F]wt a]
Example. The double privation [Non* Mp] in the conclusion can go either way, on
the obvious assumptions that Mp* is privative with respect to Mp and Mp is privative with respect to F. The individual examples may be more or less contentious, but we would suggest the following. On the understanding that
‘completed’ be understood privatively as ‘consumed’, ‘exhausted’, etc, from
something being an ((almost*p completedp) journey) we infer that it is a journey (an
59
31 event still in the process of unfolding). From something being ((almostp* halfp) pound), i.e. almost half a pound, we infer that it is a Non-pound. From someone
being a ((formerp* fallenp) angel) we infer they have rejoined the ranks of the angels. But, importantly, all of these three specific inferences rely for their validity on more than the amount of logic provided by rule (8). We feel comfortable drawing these inferences, because we bring semantic background knowledge to bear. Let us examine the example of ((almost* half) pound):
[[[almost* half] pound]wt a]
[[[Non* half] pound]wt a]
All the conclusion [[[Non* half] pound]wt a] tells us is that half does not get to modify pound. Still this is non-trivial, for the conclusion cannot be that a is half a pound. Since the conclusion is a negative proposition, is it logically possible that something instantiating the property [[Non* half] pound] may be more than half a pound? Yes; for this property applies equally to objects being more than half a pound and objects being less than half a pound. Excluded are only those objects that are exactly half a pound.
31 For a recent statement of TIL’s ontology of events and processes, see X* et al.
(2011).
60
(9)
[[[Ms* Mp] F]wt a]
[[Mp F]wt a]
[[Non F]wt a]
Example. a is an ((elegantlys* forgedp) banknote), hence a is a forged banknote, hence a is a Non-banknote. This example contrasts with this passage in XY (2010, §3.2):
The modifier Well-made needs to qualify Forged Banknote, otherwise one
ends up with the infelicitous ((Well-made Forged) Banknote). Whether Well-
made modifies Forged Banknote or Forged, Well-made is a subsective
modifier, and we do not want to extract well-made forged banknotes from
a set of banknotes.
Nobody wants to do that, of course. However, [[elegantly* forged] banknote] is not doing what it should not be doing, namely presupposing that elegantly forged banknotes be banknotes. In [[elegantly* forged] banknote] the modifier [elegantly* forged] applies to the property of banknote instead of a set of banknotes. Since elegantly* is subsective we can eliminate it from [elegantly* forged] and apply the surviving privative modifier forged to banknote, as in [forged banknote], which is a property only one kind of Non-banknotes can possibly have.
61
(10)
[[[Mp* Ms] F]wt a]
[[[Non* Ms] F]wt a]
32 Example. a is a (fakep* greens) banknote, hence a is a (Non*-green) banknote. Note
that the conclusion [Fwt a] (in this case, that a is a banknote) is not forthcoming,
although what is compromised is Ms (green) and not F (banknote). To see why,
consider this counterexample to [Fwt a] following from [[[Mp* Ms] F]wt a]. An
((almostp* successfuls) escape) is not an escape (a particular sort of event) that is almost successful, but an event that is a Non-escape coming close to being an
escape. Nor can [Non Fwt a] be inferred, however. For from [[[Non* Ms] F]wt a] it
follows only that the qualification Ms with respect to F eludes a (i.e. a is not the
sort of F-object that is an [Ms F], without presupposing that a be an F-object to
32 In colloquial speech we may say “This painting is fake” or “This painting is a fake” (‘fake’ occurring either as adjective or noun), where the conclusion is not that the referent of ‘this’ is not a painting. We suggest that both sentences are elliptical for “This painting is a fake Vermeer”, when it is understood by the parties to the discourse that the issue is whether a painting before them is a
Vermeer or a fake Vermeer (‘Vermeer’ itself being elliptical for ‘painting by
Vermeer’). A fake Vermeer painting is still a painting, for fake qualifies Vermeer and not painting. The scope distribution is fake* Vermeer painting in the vein of rule (10).
62 begin with) and not that a fails to be an F-object altogether. There will be cases
where an instance of [[[Mp* Ms] F]wt a] comes down on the side of [Fwt a] and
other cases that come down on the side of [Non Fwt a]. What is inferable from
[[[Mp* Ms] F]wt a] as a single premise is merely that a is a [[Non* Ms] F]. As with rule (8), any inference that goes beyond the cancellation of the respective
argument modifier Mp, Ms must bring semantic input to bear.
CONCLUSION
We set out to provide the tools to calculate whether an individual a having a doubly modified property F is an F. The tools we presented were a set of rules of inference formed from subsective and privative modifiers. We observed that there are cases where the extensional logic set out here does not suffice to decide whether an individual instantiating a doubly modified property F is an F.
Our inquiry has yielded two results. One is that the received rule of privation holds only for single privation, while being too strong for first-order double privation and inapplicable to higher-order double privation. So we generalized the rule by replacing boolean negation by property negation: from a being an
Mp F infer that a is a Non F. This enabled us to operate on the contraries of properties. We proposed and justified the hypothesis that the logic of double privation is a logic of contraries.
63
The other, related, result is that whether we have a pair of first-order privative modifiers or a pair consisting of a higher-order privative modifier and an arbitrary modifier (possibly privative) of one order lower, only the outermost modifier (the first element of the pair) can be eliminated, leaving the innermost modifier (the second element of the pair) intact. This goes to show that the logic of double privation is fairly feeble. This is only as it should be, though, because a slightly stronger logic would yield undesirable results. Any logic of double privation, it seems, must leave room for bifurcation: the individual instantiating the modified property may, or may not, instantiate the root property. The cookie may crumble in either of two ways. This is not to say that either of these two kinds of pairs of modifiers amounts to one modal modifier, for modal modifiers exhibit a different logical behaviour.
Nonetheless, we noted that it can often be inferred which way the cookie crumbles, as soon as semantic background knowledge is brought to bear. For instance, any competent speaker of English knows that a former fallen angel is an angel whereas almost half a pound is not a pound. One noteworthy exception was the property of being a former heir apparent, but that was because the predicate ´is a former heir apparent´ has a built-in bifurcation of its own.
The obvious way to carry this research forward will consist in proceeding from the present set-theoretic approach, aimed at classification, toward a semantic approach, aimed at characterization and description. This will be the only way to provide sufficient theoretical underpinning for such inferences as include
64 explicit semantic information about the logical effect a particular pair of first- order privatives or a pair consisting of a higher-order privative and a lower-order modifier has on a particular property.
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