1 Introduction

I introduce a logic of property modifiers modelled as a mapping from properties to properties,
such that the result of the application of a modifier to a property is another
property.^{1} I agree with Montague
that the result of modification does not depend on the state of the world, nor on
time.^{2} For instance, if one
applies the modifier *Skillful* to the property
*Surgeon*, they obtain the property of being a skillful surgeon.
On the other hand, a property is (modelled as) a mapping from a logical space of
possible worlds to a mapping from times to sets of individuals. A set of individuals
is in turn a characteristic function from individuals to truth-values. So, for an
individual to instantiate a property, whether modified or not, is to be an element
of its extension at a given *world* and *time*. The
novel contribution of this paper is a new definition of subsective and privative
modifiers in terms of *intensional essentialism*.

Kamp’s seminal [^{9}] seeks to draw a line between those adjectives whose meaning is a property and those adjectives whose meaning is a function that maps properties to properties. He (ibid., pp. 147ff) suggests that most adjectives have a property as their meaning. Yet he admits that it would seem that some adjectives must occur in attributive position and are incapable of occurring in predicative position. Their meaning is a property-to-property function:

The same can be said to be true […] of adjectives such as

fake,skillful, orgood. Where precisely we should draw the boundaries of the class of adjectives to which the second theory [property-to-property function] applies I do not know. For example, doesskillfulbelong to this class? Surely we must always ask ‘skillful what?’ before we can answer the question whether a certain thing or person is indeed skillful […].

(9, pp. 153-4)

I agree with Kamp’s linguistic observations. Kamp is concerned with a demarcation among
adjectives. We can hypothesize that his demarcation is in effect a demarcation
between those adjectives that represent *properties* and those that
represent *property modifiers*. Furthermore, some adjectives can
represent both, that is, they can occur both in attributive position and in
predicative position. For instance, if a is a round peg then a is round; not ‘round
what’ but ‘round simpliciter’.

As a starting point, here is a standard taxonomy of the three kinds of modifiers, with
rigorous definition coming afterwards. Let {…} be an operation forming sets from
properties,^{3}
*M* standing for a modifier, *M** for a property
corresponding to a modifier.^{4}

*Intersective*. “If a is a round peg, then a is round and a is a peg”:

Necessarily, i.e., in all worlds and times, the set of round pegs equals to the intersection of the sets of round objects and pegs.

Intersectivity is the least interesting form of modification, since antecedent and consequent,
or premise and conclusion, are equivalent. Still, even in the case of the apparently
logically trivial intersectives we cannot transfer *M _{i}*
from the premise to the conclusion. The reason is that a modifier cannot also occur
as a predicate; these are objects of different types. Hence

*M** instead of just

*M*.

Subsective. “If a is a *skillful* surgeon, then a is a surgeon.” :

Necessarily, i.e., in all worlds and times, the set of skillful surgeons is a subset of the set of surgeons.^{5}

The major difference between subsective and intersective modification is that subsectivity
bans this sort of argument: [*M _{s}P*](

*a*),

*Q*(

*a*) ∴ [

*M*](

_{s}Q*a*). Tilman may be a skillful surgeon, and he may be a painter too, but this does not make him a skillful painter. Or, Jumbo may be a small elephant, as well as a mammal, but this does not make Jumbo a small mammal. Jumbo is small as an elephant rather than as a mammal. Scalar adjectives like ‘small’, ‘big’ or ‘skillful’ represent subsective modifiers. On the other hand, to each intersective modifier Mi there is a unique ‘absolute’ property

*M** such that if a is an [

*M*

_{i}*P*] then a is

*M** not only as a

*P*but absolutely.

^{6}

*Privative*. “If a is a forged banknote, then a is not a banknote”:

Necessarily, i.e., in all worlds and times, the intersection of the set of forged banknotes and banknotes is empty.

Partee, in [^{11}], attempts to reduce privative modifiers
to subsective modifiers so that “the [linguistic] data become much more orderly”
(*ibid*.). In her case guns would divide into fake guns and real
guns, and fur into fake fur and real fur. Her argument is that only this reduction
can do justice to the meaningfulness of asking the following sort of question: “Is
this gun real or fake?” At first blush, however, it would seem the question
pre-empts the answer: if some individual is correctly identified as a gun, then
surely it is a real gun, something being a gun if, and only if, it is a real gun.
However, if we go along with the example, we think the argument is easily rebutted
by putting scare quotes around ‘gun’ so that the question becomes, “Is this ‘gun’
fake or real?” The scare quotes indicate that ‘gun’ is something ‘gun-like’,
including toy guns, which are not guns. If the answer is that the gun-like object is
a fake gun (hence not a gun), the scare quotes stay on. If the answer is that it is
a real gun (i.e., a gun), the scare quotes are lifted. Similarly with “Is this ‘fur’
fake or real?” A more direct way of phrasing the question would be, “Is this fur?”,
which does not preempt the answer and which does not presuppose that there be two
kinds of fur, fake and real. For an intuitive test, ask yourself what the sum is of
a fake €10 bill and a €10 bill.

If *Forged* is privative, then a forged banknote is not a banknote that is
forged, such that there would be two kinds of banknotes: those that are genuine and
those that are forged. The sum of four genuine banknotes and one forged banknote is
four banknotes and not five (though five pieces of paper).^{7} This is also to say that *Genuine*
is an idle modifier: anything is a genuine *F* iff it is an
*F*. This is not to say that the same material object may not be
genuine in one respect and fail to be genuine in another. For instance, an artefact
being passed off as a paper banknote may fail to be a banknote (being a forged
banknote), while being indeed made of paper (rather than polymer, say), thereby
being a paper artefact. (“The ‘banknote’ is fake, the paper is real”).

Modifiers are intersective, subsective and privative with respect to a property
*P*. One and the same modifier can be intersective *with
respect to a property P* and privative with respect to another property
*Q*. For instance, a stone bridge is stony and is a bridge, but a
stone lion is not a lion. We leave aside the question whether there are modifiers
privative with respect to *any* property. Most probably, yes,
modifiers like *faked*, forged, *false* appear to be
privative with respect to any property. Yet this issue is irrelevant to the main
goal of this paper, which is to define the *rule of
pseudo-detachment* (PD) and prove its validity for *any*
kind of modifiers. The applicability of (PD) presupposes the validity of existential
generalisation over properties and of substituting identical properties, something I
am not going to doubt.

My background theory is Transparent Intensional Logic (TIL) with its
*procedural* semantics that assigns abstract procedures to terms
of natural language as their context-invariant meanings. These procedures are
rigorously defined as TIL constructions that produce lower-order objects as their
products or in well-defined cases fail to produce an object by being improper.

The rest of the paper is organised as follows. Section 2 introduces the fundamentals of TIL necessary to deal with property modifiers, which is the issue I deal with in Section 3. Here in Section 3.1 the difference between non-subsective and subsective modifiers is defined in a novel way, and the main result presented, which is formulation of the rule of pseudo-detachment defined in Section 3.2. Concluding remarks can be found in Section 4.

2 Basic Notion of TIL

Definition 1 (construction)

(i) Variables

*x*,*y*, … are constructions that construct objects (elements of their respective ranges) dependently on a valuation*v*; they*v-construct*.(ii) Where

*X*is an object whatsoever (even a*construction*), 0X is the*construction Trivialization*that*constructs**X*without any change of*X*.(iii) Let

*X*,*Y*_{1},…,*Y*be arbitrary_{n}*constructions*.*Then Composition*[*X**Y*_{1}…*Y*] is the following construction. For any_{n}*v*, the*Composition*[*X**Y*_{1}…*Y*] is_{n}*v-improper*if at least one of the constructions*X**Y*_{1}…*Y*is_{n}*v*-improper, or if*X*does not*v-construct*a function that is defined at the*n*-tuple of objects*v-construct*ed by*Y*_{1}…*Y*. If_{n}*X*does*v-construct*such a function, then [*X**Y*_{1}…*Y*]_{n}*v-construct*s the value of this function at the*n*-tuple.(iv) (λ-)

*Closure*[λ*x*_{1}…*x*_{m}*Y*] is the following construction. Let*x*_{1},*x*_{2}, …,*x*be pair-wise distinct variables and_{m}*Y*a construction. Then [λ*x*_{1}…*x*_{m}*Y*]*v-construct*s the function f that takes any members*B*_{1}, …,*B*of the respective ranges of the variables_{m}*x*_{1}, …, xm into the object (if any) that is*v*(*B*_{1}/*x*_{1},…,*B*/_{m}*x*)-constructed by Y, where_{m}*v*(*B*_{1}/*x*_{1},…,*B*/_{m}*x*) is like v except for assigning_{m}*B*_{1}to*x*_{1}, …,*B*to_{m}*x*._{m}(v) Where

*X*is an object whatsoever,^{1}*X*is the*construction**Single Execution*that*v-construct*s what X*v-construct*s. Thus, if*X*is a*v-improper**construction*or not a construction as all,^{1}*X*is*v*-improper.(vi) Where

*X*is an object whatsoever,^{2}*X*is the construction Double Execution. If*X*is not itself a construction, or if*X*does not*v-construct*a construction, or if*X**v-construct*s a v-improper construction, then^{2}*X*is v-improper. Otherwise^{2}*X**v-construct*s what is*v-construct*ed by the construction*v-construct*ed by*X*.(vii) Nothing is a

*construction*, unless it so follows from (i) through (vi).

With constructions of constructions, constructions of functions, functions, and functional values in our stratified ontology, we need to keep track of the traffic between multiple logical strata. The ramified type hierarchy does just that. The type of first-order objects includes all non-procedural objects. Therefore, it includes not only the standard objects of individuals, truth-values, sets, mappings, etc., but also functions defined on possible worlds (i.e., the intensions typical of possible-world semantics). The type of second-order objects includes constructions of first-order objects and functions with such constructions in their domain or range. The type of third-order objects includes constructions of first- and/or second-order objects and functions with such constructions in their domain or range. And so on, ad infinitum.

**Definition 2 (ramified hierarchy of types).** Let *B* be a
*base*, where a base is a collection of pair-wise disjoint,
non-empty sets. Then:

**T _{1}** (

*types of order 1*):

(i) Every member of

*B*is an elementary*type of order 1 over B*.(ii) Let α, β

_{1}, ..., β_{m}(m > 0) be types of order 1 over*B*. Then the collection (α β_{1}, ..., β_{m}) of all m-ary partial mappings from β_{1}× ... × β_{m}into α 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) and (ii).

**C _{n}** (

*constructions of order n*):

(iv) Let

*x*be a variable ranging over a type of order*n*. Then x is a construction of order*n*over*B*.(v) Let

*X*be a member of a type of order*n*. Then^{0}*X*,^{1}*X*,^{2}*X*are constructions of order n over*B*.(vi) Let

*X*,*X*_{1}, ...,*X*(_{m}*m*> 0) be constructions of order*n*over*B*. Then [*X**X*_{1}…*X*] is a construction of order n over_{m}*B*.(vii) Let

*x*_{1}, ...,*x*,_{m}*X*(*m*> 0) be constructions of order*n*over*B*. Then [λ*x*_{1}…*x*_{m}*X*] is a construction of order n over*B*.(viii) Nothing is a construction of order n over

*B*unless it so follows from**C**(i)-(iv)._{n}

**T _{n+1}** (t

*ypes of order n*+1) Let *

*n*be the collection of all constructions of order

*n*over

*B*. Then

(i) *

_{n}and every type of order*n*are types of order*n*+1.(ii) If

*m*> 0 and α, β_{1}, ..., β_{m}are types of order*n*+1 over*B*, then (α β_{1}... β_{m}) (see T_{1}ii)) is a*type of order n*+1 over*B*.(iii) Nothing is a type of order

*n*+1 over*B*unless it so follows from (i) and (ii).

For the purposes of natural-language analysis, we are assuming the following base of ground types:

ο: the set of truth-values {

**T**,**F**};ι: the set of individuals (the universe of discourse);

τ: the set of real numbers (doubling as discrete times);

ω: the set of logically possible worlds (the logical space).

We model sets and relations by their characteristic functions. Thus, for instance, (οι) is the
type of a set of individuals, while (οιι) is the type of a relation-in-extension
between individuals. Empirical expressions denote *empirical
conditions* that may or may not be satisfied at the particular
world/time pair of evaluation. We model these empirical conditions as
possible-world-semantic (*PWS*-) *intensions*.
PWS-intensions are entities of type (βω): mappings from possible worlds to an
arbitrary type β. The type β is frequently the type of the chronology of α-objects,
i.e., a mapping of type (ατ). Thus α-intensions are frequently functions of type
((ατ)ω), abbreviated as ‘α_{τω}’. Extensional entities are entities of a
type α where α ≠ (βω) for any type β. Where w ranges over ω and t over τ, the
following logical form essentially characterizes the logical syntax of empirical
language: λ*w*λ*t*
[…*w*….*t*…].

Examples of frequently used PWS intensions are: *propositions* of type
ο_{τω}, *properties of individuals* of type
(οι)_{τω}, binary *relations-in-intension *between
individuals of type (οιι)_{τω}, *individual offices* (or
*roles*) of type ι_{τω}.

As mentioned above, we define property modifiers as mappings from α-properties to
α-properties, for any type α. In this paper we deal mostly with modifiers of
individual properties of type ((οι)_{τω}(οι)_{τω}), or sometimes
(ππ) for short.

Logical objects like *truth-functions* and *quantifiers* are
extensional: ∧ (conjunction), ∨ (disjunction) and ⊃ (implication) are of type (οοο),
and ¬ (negation) of type (οο). Quantifiers ∀^{α}, ∃^{α} are
type-theoretically polymorphic total functions of type (ο(οα)), for an arbitrary
type α, defined as follows. The universal quantifier ∀^{α} is a function
that associates a class *A* of α-elements with **T** if
*A* contains all elements of the type α, otherwise with
**F**. The existential quantifier ∃^{α} is a function that
associates a class A of α-elements with **T** if A is a non-empty class,
otherwise with **F**.

Below all type indications will be provided outside the formulae in order not to clutter the
notation. Moreover, the outermost brackets of the Closure will be omitted whenever
no confusion arises. Furthermore, ‘*X*/α’ means that an object
*X* is (a member) of type α. ‘*X* →_{v} α’
means that *X* is typed to *v-construct* an object of
type α, if any. We write ‘*X* → α’ if a valuation v does not matter.
Throughout, it holds that the variables *w* →_{v} ω and t
→_{v} τ. If C → α_{τω} then the frequently used Composition
[[*C*
*w*] *t*], which is the intensional descent (a.k.a.
extensionalization) of the α-intension *v-construct*ed by
*C*, will be encoded as ‘*C _{wt}’*. For
instance, if

*Student*/(οι)

_{τω}is the property of being a student, the procedure of extensionalizing this property to obtain its population in a given world w and time

*t*is the Composition:

for short.

Whenever no confusion arises, we use traditional infix notation without Trivialisation for truth-functions and the identity relation, to make the terms denoting constructions easier to read. Thus, for instance, instead of:

we usually write:

3 Property Modifiers and Essences of Properties

3.1 Privative vs. Subsective Modifiers

The fundamental distinction among modifiers is typically considered to be one between the
subsectives and the *non-subsectives*. The former group consists
of the *pure subsectives* (that are governed by the rule of
*right subsectivity*, which amounts to eliminating the
modifier and predicating the surviving property) and the
*intersectives* (that are governed by the rule of
*right subsectivity* and a rule of *left
subsectivity*).^{8} The
latter group consists of the modals and the privatives. Since I am not dealing
with modal modifiers here, I now want to define the distinction between
subsectives and privatives. At the outset this distinction between modifiers
subsective (*M _{s}*) and privative
(

*M*) with respect to a property

_{p}*P*has been characterized by the rules of right subsectivity as follows:

Now we have the technical machinery at our disposal to define these modifiers in a rigorous
way. To this end, I apply the logic of intensions based on the notions of
requisite and essence of a property, which amounts to *intensional
essentialism*.^{9} The
idea is this. Every property we countenance has a host of other properties
necessarily associated with it. For instance, the property of being a bachelor
is associated with the properties of being a man, being unmarried, and many
others. Necessarily, if *a* happens to be a bachelor then
*a* is a man and *a* is unmarried. We call
these adjacent properties *requisites* of a given property.

The requisite relations *Req* are a family of relations-in-extension between
two intensions, so they are of the polymorphous type
(οα_{τω}β_{τω}), where possibly α = β. Infinitely many
combinations of Req are possible, but for our purpose we will need the following
one: *Req* /(ο(οι)_{τω}(οι)_{τω}); a property of
individuals is a requisite of another such property.

TIL embraces *partial functions*.^{10} Partiality gives rise to the following complication.
The requisite relation obtains analytically necessarily, i.e., for all worlds
*w* and times *t*, and so the values of
intensions at particular 〈*w*, *t*〉-pairs are
irrelevant. But the values of properties are isomorphic to characteristic
functions, and these functions are amenable to truth-value gaps.

For instance, the property of having stopped smoking comes with a bulk of requisites like, e.g., the property of being a former smoker. If a never smoked, then the proposition that a stopped smoking comes with a truth-value gap, because it can be neither true nor false.

Thus, the predication of such a property *P* of a may also fail, causing
[^{0}*P _{wt}*

^{0}

*a*] to be v-improper. There is a straightforward remedy, however, namely the propositional property of being true at a given 〈

*w*,

*t*〉:

*True*/(οο

_{τω})

_{τω}. Given a proposition

*v-construct*ed by X, [

^{0}

*True*

_{wt}*X*]

*v*-constructs the truth-value

**T**if the proposition presented by

*X*is true at 〈

*w*,

*t*〉; otherwise (i.e., if the proposition

*v*-constructed by

*X*is false or else undefined at 〈

*w*,

*t*〉) the truth-value

**F**. Thus, we define:

**Definition 3** (*requisite relation between ι-properties*). Let
*P*, *Q* be constructions of individual
properties; *P*, *Q*/*_{n} →
(οι)_{τω}; *x* → ι. Then:

Gloss definiendum as, “*Q* is a requisite of *P*”, and
definiens as, “Necessarily, i.e. at every 〈*w*,
*t*〉, if it is true that whatever x instantiates
*P* at 〈*w*, *t*〉 then it is
also true that this x instantiates *Q* at 〈*w*,
*t*〉.”^{11}

Example. Let the property of being a person be a requisite of the property of being a
student. Then the hyperproposition that all students are persons is an analytic
truth. It constructs the proposition TRUE, which is the necessary proposition,
which takes value **T** at all world-time pairs. Wherever and whenever
somebody happens to be a student they are also a person. Formally:

Next, I am going to define the essence of a property. Our essentialism is based on the idea
that since no purely contingent property can be essential of any individual,
essences are borne by intensions rather than by individuals exemplifying
intensions.^{12} That a
property *P* has an essence means that a relation-in-extension
obtains a priori between the property *P* and a set Ess of the
requisites of *P*, that is, other properties such that,
necessarily, whenever an individual instantiates *P* at some
〈*w*, *t*〉 then the same individual also
instantiates any of the properties belonging to Ess at the same
〈*w*, *t*〉. Hence, our essentialism is based
on the requisite relation, couching essentialism in terms of a priori interplay
between properties, regardless of who or what exemplifies a given property.
Intensional essentialism is technically an algebra of individually necessary and
jointly sufficient conditions for having a certain property (or other sort of
intension). The 〈*w*, *t*〉-relative extensions of
a given property are irrelevant, as I said.

**Definition 4** (*essence of a property*). *Let* p,
*q* → (οι)_{τω} be constructions of individual
properties, and let *Ess*/((ο(οι)_{τω})(οι)_{τω})
be a function assigning to a given property *p* the set of its
requisites defined as follows.

Then the *essence of a property p* is the set of its requisites:

Each property has (possibly infinitely) many requisites. The question is, how do we know
which are the requisites of a given property? The answer requires an
*analytic definition* of the given property, which amounts to
the specification of its essence.^{13} For instance, consider the property of being a
bachelor. If we define this property as the property of being an unmarried man,
then the property of being an unmarried man is a requisite of the property of
being a bachelor. From this definition, it follows that, for instance, the
sentence “bachelors are unmarried men” comes out analytically true:

And since the modifier *Unmarried* is intersective, it also follows that
necessarily, each bachelor is unmarried and is a man:

Note, however, that *Unmarried’*/(οι)_{τω} and
*Unmarried*/((οι)_{τω}(οι)_{τω}) are entities
of different types. The former is a property of individuals uniquely assigned to
the latter, which is an intersective modifier.

With these definitions in place, we can go on to compare two kinds of subsectives against
privatives.^{14} Since these
modifiers change the essence of the root property, we need to compare the
essences, that is sets of properties, of the root and modified property. To this
end, we apply the set-theoretical relations of being a *subset*
and a *proper subset* between sets of properties, and the
*intersection* operation on sets of properties, defined as
follows.

Let π = (οι)_{τω}, for short, ⊆, ⊂/(ο(οπ)(οπ)), and let *a*,
*b* →_{v} (οπ); *p* →_{v} π.
Then:

Furthermore, the *intersection* function ∩/((οπ)(οπ)(οπ)) is defined on sets
of properties in the usual way:

In what follows I will use classical (infix) set-theoretical notation for any sets
*A*, *B*; hence instead of ‘[^{0}⊆
*A B*]’ I will write ‘[*A* ⊆
*B*]’, and instead of ‘[^{0}∩ *A B*]’
I will write ‘[*A* ∩ *B*]’.

**Definition 5** (*subsective vs. privative modifiers*). Let the
types be: *P* → (οι)_{τω}, *M* →
((οι)_{τω}(οι)_{τω}), *p* →
(οι)_{τω}, *x* → ι. Then

Example. The modifier *Stony*/((οι)_{τω}(οι)_{τω}) is
subsective with respect to the property of being a bridge,
*Bridge*/(οι)τω, but privative with respect to the property
of being a lion, *Lion*/(οι)_{τω}. Of course, a stony
bridge is a bridge, but the essence of the property [^{0}Stony
^{0}Bridge] is enriched by the property of being stony. This
property is a requisite of the property of being a stony bridge, but it is not a
requisite of the property of being a bridge, because bridges can be instead made
of wood, iron, etc.:

But a stony lion is not a lion. The modifier *Stony*, the same modifier that
just modified *Bridge*, deprives the essence of the property of
being a lion, *Lion*/(οι)_{τω}, of many requisites, for
instance, of the property of being an animal, having a bloodstream, a heartbeat,
etc. Thus, among the requisites of the property
[^{0}*Stony*
^{0}*Lion*] there are properties like not being a
*living thing, not having a bloodstream*, etc., which are
contradictory (not just contrary) to some of the requisites of the property
*Lion*. On the other hand, the property
[^{0}*Stony*
^{0}*Lion*] shares many requisites with the property of
being a lion, like the outline of the body, having four legs, etc., and has an
additional requisite of being made of stone. We have:

A modifier *M* is *non-trivially subsective *with respect to a
property *P* iff the modified property [*M P*] has
all the requisites of the property *P* and at least one another
requisite that is not a requisite of *P*. In other words, the
essence of the property *P* is a proper subset of the essence of
the property [*M P*]. For instance, a skillful surgeon is a
surgeon because the property of being a skillful surgeon must have all the
requisites of the property of being a surgeon, and the additional property of
being skillful as a surgeon, i.e., with respect to the property of being a
surgeon.

A modifier *M* is *trivially subsective* with respect to
*P* iff the modified property [*M P*] has
exactly the same requisites as the property *P*. In other words,
the essence of the property [*M P*] is identical to the essence
of the property *P*. These modifiers are trivial in that the
modification has no effect on the modified property and so might just as well
not have taken place.

For instance, there is no semantic or logical (but perhaps rhetorical) difference between the
property of being a leather and the property of being a *genuine*
leather. Trivial modifiers such as genuine, real, actual are pure subsectives.
As mentioned above, genuine leather things are not located in the intersection
of leather things and objects that are genuine, for there is no such property as
being genuine, pure and simple.^{15}

A modifier *M* is privative with respect to a property *P* iff
the modified property [*M P*] lacks at least one, but not all, of
the requisites of the property *P*. However, in this case we
cannot say that the essence of the property [*M P*] is a proper
subset of the essence of the property *P*, because the modified
property [*M P*] has at least one other requisite that does not
belong to the essence of *P*, because it contradicts to some of
the requisites of *P*. Hence, *M* is privative
with respect to a property *P* iff the essence of the property
[*M P*] has a non-empty intersection with the essence of the
property *P*, and this intersection is a proper subset of both
the essences of *P* and of [*M P*]. For instance,
a well-forged banknote has almost the same requisites as does a banknote, but it
has also another requisite, namely the property of being forged with respect to
the property of being a banknote.

As a result, if *M _{p}* is privative with respect to the property

*P*, then the modified property [

*M*] and the property

_{p}P*P*are contrary rather than contradictory properties:

It is not possible for x to co-instantiate [*M _{p}P*] and

*P*, and possibly x instantiates neither [

*M*], nor

_{p}P*P*.

The left-hand conjunct:

is the clause that [*M _{p}P*] and P are mutually

*exclusive*. This is because among the requisites of the properties

*P*and [

*M*] there is at least one pair of mutually contradictory properties. The second conjunct:

_{p}P

is the *contrariety* clause that the negation of one of the conjuncts
[[*M _{p}P*]

_{wt}

*x*], [P

_{wt}

*x*] does not entail the truth of the other one. This is because only some but not all the requisites of [

*M*] contradict some of the requisites of

_{p}P*P*, and the intersection of the essences of P and [

*M*] is non-empty. Since we are talking about non-trivial properties, it is possible that an individual x has none of the properties [

_{p}P*M*] and

_{p}P*P*.

3.2 The Rule of Pseudo-Detachment

The issue I am going to deal with now is left subsectivity.^{16} We have seen that the principle of left
subsectivity is trivially (by definition) valid for intersective modifiers. If
Jumbo is a yellow elephant, then Jumbo is yellow. Yet how about the other
modifiers? If Jumbo is a small elephant, is Jumbo small? If you factor out
*small* from *small elephant*, the conclusion
says that Jumbo is small, period. Yet this would seem a strange thing to say,
for something appears to be missing: Jumbo is a small *what*?
Nothing or nobody can be said to be small — or forged, skilful, temporary,
larger than, the best, good, notorious, or whatnot, without any sort of
qualification. A complement providing some sort of qualification to provide an
answer to the question, ‘a … *what*?’ is required. I am going to
introduce now the rule of pseudo-detachment that is valid for all kinds of
modifiers including subsective and privative ones. The idea is simple. From a is
an *MP* we infer that a is an *M-with respect to
something*.

For instance, if the customs officers seize a forged banknote and a forged passport, they may
want to lump together all the forged things they have seized that day,
abstracting from the particular nature of the forged objects. This lumping
together is feasible only if it is logically possible to, as it were, abstract
forged from a being a *forged A and b being a forged B* to form
the new predications that a *is forged* (something) and that
*b is forged* (something – possibly else), which are
subsequently telescoped into a conjunction.

Gamut claims that if Jumbo is a small elephant, then it does not follow that Jumbo is small [^{3, §6.3.11}]. I am going to show that the conclusion does follow. The rule of pseudo-detachment (PD) validates a certain inference schema, which on first approximation is formalized as follows:

where ‘*a*’ names an appropriate subject of predication while
‘*M*’ is an adjective and ‘*P*’ a noun phrase
compatible with *a*.

The reason why we need the rule of pseudo-detachment is that *M* as it occurs
in *MP* is a modifier and, therefore, cannot be transferred to
the conclusion to figure as a property. So, no actual detachment of
*M* from *MP* is possible, and Gamut is
insofar right. But (PD) makes it possible to replace the modifier
*M* by the property *M** compatible with a to
obtain the conclusion that a is an *M**. (PD) introduces a new
property *M** ‘from the outside’ rather than by obtaining
*M* ‘from the inside’, by extracting a component from a
compound already introduced. The temporary rule above is incomplete as it
stands; here is the full pseudo-detachment rule, SI being substitution of
identical properties (Leibniz’s Law).

To put the rule on more solid grounds of TIL, let π = (οι)_{τω} for short, M → (ππ)
be a modifier, *P* → π an individual property,
[*MP*] → π the property resulting from applying
*M* to *P*, and let
[*MP*]_{wt}
→_{v} (οι) be the result of extensionalizing
the property [*MP*] with respect to a world *w*
and time *t* to obtain a population of the property at the world
and time of evaluation, i.e. a set in the form of a characteristic function,
applicable to an individual *a* → ι. Further, let =/(οππ) be the
identity relation between properties, and let *p*
→_{v} π range over properties,
*x* →_{v} ι over individuals. Then
the proof of the rule is this:

Any valuation of the free occurrences of the variables *w*, *t*
that makes the first premise true will also make the second, third and fourth
steps true. The fifth premise is introduced as valid by definition. Hence, any
valuation of w, t that makes the first premise true will, together with step
five, make the conclusion true.

Additional type: ∃/(ο(οπ)).

Here is an instance of the rule.

(1’) a is a forged banknote,

(2’) forged* is the property of being a forged something,

(3’) a is forged*.

The schema extends to all (appropriately typed) objects. For instance, let the inference be,
“Geocaching is an exciting hobby; therefore, geocaching is exciting”. Then a is
of type π, *P* → (οπ)_{τω}, *M* →
((οπ)_{τω}(οπ)_{τω}), and *M** →
(οπ)_{τω}.

Now it is easy to show why this argument must be valid:

Types: Number_of/(τ(οι)); B*anknote, Passport, Forged**/(οι)_{τω};
*Have*/(οιι)_{τω};
*Forged*/((οι)_{τω} (οι)_{τω}).

There are three conceivable objections to the validity of (PD) that I am going to deal with now.

*First objection*. If Jumbo is a small elephant and if Jumbo is a big mammal,
then Jumbo is not a small mammal; hence Jumbo is small and Jumbo is not small.
Contradiction!

The contradiction is only apparent, however. To show that there is no contradiction, we apply (PD):

Types: *Small, Big*/(ππ); *Mammal, Elephant*/π;
*Jumbo*/ι; p, q → π.

To obtain a contradiction, we would need an additional premise; namely, that, necessarily, any individual that is big (i.e., a big something) is not small (the same something). Symbolically:

Applying this fact to Jumbo, we have:

This construction is equivalent to:

But the only conclusion we can draw from the above premises is that Jumbo is a small
something and a big *something else*:

Hence, no contradiction.

The conclusion ought to strike us as being trivial. If we grant, as we should, that nobody
and nothing is absolutely small or absolutely large, then everybody is made
small by something and made large by something else. And if we grant, as we
should, that nobody is absolutely good or absolutely bad, then everybody has
something they do well and something they do poorly. That is, everybody is both
good and bad, which here just means being good at something and being bad at
something else, without generating paradox (*Good,
Bad*/(ππ)):

But nobody can be good at something and bad *at the same thing*
simultaneously:

*Second objection*. The use of pseudo-detachment, together with an
innocuous-sounding premise, makes the following argument valid:

Yet it is not so. Well, it is necessarily true that if *x* is a small
something and y is a big object *of the same kind*, then
*y* is a bigger object of that kind than
*x*:

Additional type: *Bigger*/(οπιι)_{τω}: the relation of being bigger
with respect to a property which obtains between individuals and the
property.^{17} This cannot be
used to generate a contradiction from the above premises, because
*p* ≠ *q*:

Geach, in [^{4}], launches an argument similar to the one we just dismantled to argue against a rule of inference that is in effect identical to (PD). He claims that that rule would license an invalid argument. And indeed, the following argument is invalid:

ais a big flea, soais a flea and a is big;bis a small elephant, sobis an elephant andbis small; soais a big animal andbis a small animal. (Ibid., p. 33.)

But pseudo-detachment licenses no such argument. Geach’s illegitimate move is to steal the
property *being an anima*l into the conclusion, thereby making
*a* and *b* commensurate. Yes, both fleas and
elephants are animals, but *a*’s being big and
*b*’s being small follow from *a*’s being a flea
and *b*’s being an elephant, so pseudo-detachment only licenses
the following two inferences, *p* ≠ *q*:

And a big *p* may well be smaller than a small *q*, depending
on the values assigned to *p*, *q*.

*Third objection*. If we do not hesitate to use ‘small’ not only as a modifier
but also as a predicate, then it would seem we could not possibly block the
following fallacy:

But we can and must block it, for this argument is obviously not valid. The premises do not
guarantee that the property *p* with respect to which Jumbo is
small is identical to the property *Elephant*. As was already
pointed out, one cannot start out with a *premise* that says that
Jumbo is small (is a small something) and conclude that Jumbo is a small
*B*.

4 Conclusion

In this paper, I applied TIL as a logic of intensions to deal with property modifiers and properties in terms of intensional essentialism. Employing the essences of properties, I defined the distinction between non-subsective (that is privative) and subsective modifiers. While the former ones deprive the root property of some but not all of its requisites, the latter enrich the essence of the root property. The main result is the rule of pseudo-detachment together with the proof of its validity for any kind of modifiers. The next step is to examine iterated modifiers. While obviously, the iteration of subsective modifiers goes smoothly due to right subsectivity, iterated privatives are much more complicated. For instance, while a nice, skilful surgeon is a skilful surgeon and is a surgeon as well, a demolished damaged house is not a house, while a repaired damaged house is a house. On the other hand, a demolished damaged house is not a demolished damaged bridge. The definitions provided in this paper make it possible to examine iteration of modifiers within these ideas. The issue of iterated privation has been dealt with by Jespersen & Carrara whom later the author of this paper joined. For the results see [^{7}].