Some Remarks on Guptas Principle of Identity

Boris Hennig, University of Leipzig, Germany 1.7.96

There are several philosophical and semantical problems concerning modal logics. To which entity does an intensional property belong? If the range of modal functions is a domain of individuals, considered in a set of possible situations: How do I know which properties belong to the individual, as distinguished from the possible situation in which it is supposed to be? In the following I intend to investigate these issues using the concept of 'common nouns' introduced by Anil Gupta (See his book 'The Logic of Common Nouns, An Investigation in Quantified Modal Logics', New Haven, London 1980). I will report some criteria for 'being essentialistic', investigate the significance of the little word 'qua' in the given context, and report the changes of the concept of separateness in Guptas
In the course of the discussion I will develop an idea of my own a little further, a concept that is rather similar to what W. Stelzner calls 'Analysesprache',
- remark: The origins of which are found in his 'Epistemische Logik', Berlin 1984. but can be used in a wider domain.

When I refer briefly to 'Gupta' I always mean his book 'The Logic of Common Nouns, An Investigation in Quantified Modal Logics', New Haven, London 1980.
By referring to MLⁿ I quote Aldo Bressans 'General Interpreted Modal Calculus', New Haven, London, 1972.

Gupta's Challenge

In the first three chapters of his book Gupta develops several quantified modal logics featuring the notion of common nouns. The chapter I am concerned with (i.e. chpt. 4) contains a defence of the stated logics against Quine's objection, ''that quantified modal logis are committed to an unacceptible essentialism''.
- remark: Gupta, p. 86. However, Guptas arguments appear prima vista to be no real defence, but rather to accord with Quine.
Secondly, Gupta contrasts his account on quantification with the Fregean one, which is committed to neglect contingent identity.
Finally, the chapter yields a modification of Guptas logics L₁ - L₄ and a new concept of identity. This concept has something to do with the speaker's point of view.

Quine's Objection

The problem of Quinean essentialism is a result of placing modal operators before open sentences, or (equivalently) of abstracting frommodalized sentences. It can be divided into two aspects.
[first] Consider the following abstraction from an analytic sentence:

from 'for all x: Fx or not-Fx'
to the property of necessarily being (Fx or not Fx)

Such an abstraction would be just as harmless as useless. It is, however, the first step to essentialism, for a property is said to belong necessarily to a subject, and not: a statement is necessarily true.
[second] It gets more problematic to abstract modalized properties like 'necessarily being F',where 'for all x: Fx' does not hold. That is: a property is said to belong necessarily to something independent of a mode of designation. The crucial point is not, that a property belongs necessarily to a subject term, but that the very same property may belong to another subject term contingently. This is the kernel of the definitions of essentialism undertaken by Marcus and Parsons:

A property F is essential to x iff necessarily Fx, there is an Fy such that not necessarily Fy

(Cf. Ruth Barcan Marcus, Essentialism in Modal Logic, Nous 1, 1967, and Terence Parsons, Grades of Essentialism in Quantified Modal Logic, in a subsequent volume of the same journal.)
According to this definition, 'being essentialistic' can be defined for a logic:

A logic L is essentialistic iff it allows an essential property F.

It is not evident whether or not Guptas logics allow such properties.(I will postpone this question until I have -- at least -- -- intuitively -- introduced Guptas account on quantification.)
Here the necesstiy cannot lie in the analyticity of F but must be due to the x's being such-and-such.

This has often been called Aristotelian essentialism, first by Quine himself:

There is yet a further consequence, and a particular striking one: Aristotelian essentialism. This is the doctrine that some of the attributes of a thing (...) may be essential to the thing, and others accidental. For example, a man, or talking animal or featherless biped (...) is essential rational and accidentally two-legged and talkative, not merely qua man but qua itself.

(I take this quote from Parsons, but it is to be found in Quines 'Grades of Modal Involvement', The Ways of Paradox, New York 1966, pp. 173-74.) Obviously, according to Quine it may be not as bad to say something has a property qua x, than to ascribe it plainly to the 'thing' (qua entity). Quine's use of qua serves to state attributes explicitly, in order to make sentences like 'every featherless biped is rational' analytic in the second sense: 'every featherless biped, qua man, is rational'. Togehter with a definition of 'man' as 'rational animal', this is analytically true. Cf. his Two Dogmas of Empiricism, in: From a Logical Point of View. That may be a consequence of the denial of necessitas consequentis as distinguished from an admittable necessitas consequentiae: If e.g. a thing is green and two feet tall, its being green may be an analytic truth and thereby an 'analytic property' of it.
This necessitas consequentiae is sometimes said to be the same as necessity de dicto: Cf. I.M. Bochenskis edition of [pseudo-]Sancti Thomae Aquinatis de modalibus opusculum, in: Angelicum XVII, 1940, pp. 180-218, foreword. The linguistic entities introduced by Gupta, allowing necessity that is not in this sense de dicto, since the necessary statements are not analytic, are important for an adequate notion of de re - modality. W. Stelzner claims, that de re modality has to do with a certain 'nonlinguistic aspect' (De Dicto, De Re und Quantifikation in der alethischen Modallogik; Untersuchungen zur Logik und Methodologie 2, p. 54.
This aspect is already taken into account in such formulae like

there is an x = a such that necessarily Fx,

but here the x is quantified over the range of entities. It is not further described, so that in intensional contexts we would have to identify x in all worlds without guidance by any discernible property of x. The nonlinguistic aspect is, therefore, that we talk about an entity regardless of the attributes we ascribe to it.
Guptas introduces quantification exclusively over common nouns. In his logic of common nouns, quantifiers have to be followed by a K, that provides a principle of identity over (related) possible worlds, that is determining the intension and, thereby, a counterpart relation. (Gupta does not restrict accessability from world to world, but I have inserted the above for sake of generality. He defines 'counterpart' on p. 36. 'Counterpart' is defined as a unique corresponding object, that is, slightly different from Lewis' notion of 'counterpart'.
Guptas Logics

... are not committed to [the] idea that an object, independently of a principle of identity, has some of its properties essentially and others accidentally (Gupta, p. 88).

Part of the significance of Guptas use of 'qua ' (as distinguished from Quine's) lies in his distinction between 'principles' and 'criteria' of identity. According to Gupta, a principle is the metaphysical counterpart of its respectivecriterion (Gupta, p. 22). 'Metaphysical' indicates the 'non-mentality', that is, a principle is the res subiecta underlying a criterion of identity. (In what domain this res is supposed to be, could be clarified by distinguishing a quasi-objective language, the 'Analysesprache', from a de dicto language.)
When Gupta supposes the intensions of common nouns to be individual concepts, he obviously does not speak about Fregean 'Vorstellungen'.
(That is 'ideas' or 'imaginations'; maybe even the 'hyperintension').
To clarify the difference between the two qua -clauses, let us consider the following (quasi)formalizations of 'a, qua G, is essentially F':

(1) Ga and for all x: necessarily, if Gx, then Fx
(2) (there is G, x) [x=a and necessarily Fx]

Intuitively, to check the truth of (1), one has to look wether in every world, when some a is a G, it is also an F. This would be in accordance with Quine, for 'necessarily Gx, then Fx', if it is true, might be an analytic statement. In Gupta's notation, that is (2), (there is G, x) necessarily Fx may be an empriric truth. For instance, it is not evident, whether there is an a, that suffers from the hereditary desease y qua man. To conclude the brief discussion of the alleged 'non-linguistic' aspect in de re modality: Of course, being an instance of a common noun is a linguistic feature --- but the necessity is an empirical one.
The job of a principle of identity is to trace objects from world to world. That is, a principle of identity for a common noun K has to determine when an object a in a world w1 (at a time t) is the same K as a' in another world w2 (at a time t'). In order to do this, the intension of a common noun is given by a set of individual concepts that determine classes of K-counterparts in each world.Does Gupta allow predications that are essentailistic as defined by Marcus? As far as my investigation goes, it can be said, how such a property is formalized and when it is fulfilled.An essential property, according to Marcus and Parsons, must fulfill the sentence:

(there is K, x)(there is K, y)[necessarily Fx and Fy and not necessarily Fx]

What is it for a K to be necessarily F? In order to prove the possible truth of the sentence above, I present a plausible interpretation. Gupta allows common nouns to pick out different individal concepts - that is: different intesions - in each world. The common noun gathers several individual concepts, every single one of them providing its own 'tracing rule' by giveing classes of counterparts.
Let us imagine that only three persons are born in A, and one of them, a has the hereditary desease B. The concept 'human being' will include every single one of them in each world (regardless nonexistence), but the concept 'man born in A' might include few, in pairs different men in the regarded worlds. It is plausible that the DNA is somewhat essential to a man qua human being. Further, the desease can be supposed to be transmitted to another man born in A under certain circumstances. Then the property of 'suffering from B' could be inserted for F. There could be a possible world in which a infected b, and another in which only a has B. Thirdly, if K picks out individuals by their DNA, a has B in every world. So far the plausibility.

Conditioned Necessity

Briefly stated, the restriction of the quantifier via common noun provides an additional restriction of the range of counterparts (Since a furtherly undescribed x has too many counterparts qua x). Indeed, the idea of such a restriction is also developed in an article of Lewis about the relation of things to their counterparts:

In certain modal predications, the appropriate counterpart relation is selected not by the subject term but by a special clause. To say that something, regarded as a such-and-such, is such that it might have done so-and-so is to say that in some world it has a such-and-such counterpart that does so-and-so (Lewis, Counterparts of Persons and their Bodies, The Journal of Philosophy, 7, 1971, pp. 203-211).

(Gupta quotes an immediately following passage containing the word 'common noun' at p. 81.) The puzzle Lewis wants to solve with this distinction is as follows: If it is possible that I have switched my body remaining the same person, there should be a counterpart of mine, say, in a world w1, which is not identical to the counterpart of my body in w1. But, for I am identical with my actual body, it follows that my body (being identical with me) has a counterpart in w1 that is not the counterpart in w1 of my body. (I am not quite sure whether this identity is necessary. The thesis of Lewis spells: ''Necessarily, a person occupies a body at a time if and only if that person is identical with that body at that time'' (p. 203). This is a necessitas consequentiae. The Fregean analysis fails in not allowing for contingent identity; and that may be the point here.)
A restricted counterpart relation, however, yields a restricted equivalence-class of things being counterparts ofmyself or my body, so that intensional or transworld identity has to be guided by a trace; the trace being determined by the 'special clause'. This leads to the solution of the puzzle: the (personal) counterpart of mine cannot be replaced by my bodily counterpart, although I am actually identical with my body. The 'traces', i.e. the respective counterpart relations, are different.
In extensional contexts the introduction of common nouns would simply yield a restriction of quantification and can be omitted. An intensional logic, whether epistemic, modal or temporal, must include principles of identity in order to avoid paradoxes. (I use 'paradox' to denote unintuitive thoughts, self-contradicting sentences should be called 'antinomies'.) Gupta shows this by assuming the existence of a common noun 'thing'. The Fregean analysis of quantified sentences would look like this:

(for all things, x) Kx implies Fx
(there is thing, x) Kx and Fx

The allowance for contingent identity, as in the case of Lewis' body, will have to be analysed according to this example:

(there is thing, x) [Kx and (for all thing, x) possibly not x = y]

This, however, conflicts with the law (of which the following is an instance - Gupta, axiom AS10, p. 48).

(for all thing, x) (there is thing, y) necessarily x = y

This very principle, however, establishes the possibilty to trace an individual via its counterpart relation through possible worlds. The outcome of the discussion of Fregean analysis is not that it is useless or wrong, but rather that it is incompatible with contingent identity.

The Aristotelian thesis instead, mentioned by Gupta on p. 91, claims the possibility of tracing an individual by virtue of its being 'some particular thing' (to hekaston). Fregean analysis of quantification does quite well extensionally; and equally does in intensional contexts, given the quidditas of a thing (qua thing) can be found. According to Aristotle, there need to be no counterpart relation determined by the mode of designation or the intension of a part of speech.
(I am simplifying my discussion here by diminuishing the amount of discussed text; I will only treat two short passages from Aristotles' Categoriae.)
This (absolute nonlinguistic) trace is provided by the 'first substances' (hai prwtai ousiai),which seems to be similar to the Fregean 'Bedeutung': The first substance is the ultimate object of speech (Categ. 2a11-13), whithout being in any way linguistic itself. I suppose, however,that Aristotle did not distinguish between 'language' and 'world' in the modern (Kantian) way. For him it might have been evident that speech is directly about the world, that is, individual thingsmight have been the content of speech. Then the intension is the content of an expression only in a derived sense: it is the secondary substance. Sorts, kinds and common nouns are secondary substances, because they do underlie our speech about the world, but are usually (while using them) not spoken of.

'Like the primary substances behave towards the others, the eidos (kind) is related to its genus ; --- for the eidos underlies its genus ' (Categ. 2b17-20 - my translation, being not very elaborate).

I think the significance of this passage is as follows: For every language is universal, single words never cover one individual entity. But similarly, the genus, which is nevertheless a genus of real objects, does never cover a single individual. The relation of intensions to extensions is similar to the relation between classes and their members --- but obviously not the same.However, in order to deal with intensional contexts, it has become usual to identify the intension, i.e. the secondary substance, with a class of extensions, and that can no longer be the same as a class of real, discernible objects. The alleged Aristotelian directness, expressed in identifying the extension with the concrete things, is lost thereby. ('Alleged' only because I am not myself sure about this supposition.)

World-relativized Identity

Weak Separation

In ML^nu, there are two important properties of individual concepts:separation (def. 18.7, p. 66)and constancy (def. 13.2, p. 49).Bressan defines modal constancy as follows:

A concept F is modally constant iff
for all x, if and only if x falls under F in a world w, then x falls under F in every world.

This means actually, if an individual concept (intension) falls under F in one world, its members must be denoted by F in every world. Stelzner gives a similar definition for 'F is a rigid designator' and claims, that there is no difference between de dicto and de re modality if subject terms are denoted in this way. De Dicto, De Re (Cf. Stelzner, p. 55).
The Bressanian concept of modal separation is defined thus:

A concept F is modally separated iff
for all individual concepts i and i', if there there is a world w_i in which i(w_i) = i'(w_i), then i = i'.)

that is, if an object x_i falling under the individual concept F is the same as another object y_i in the actually considered world , then x_i and y_i must have the same counterparts qua F. A modally separated concept, therefore, establishes an intensional equivalence relation. It is a part of set theory, that, where xRy is an equivalence-relation and R < a > the respective equivalence class, then if the intersection ofR < a > and R < b > is not empty, then R < a > = R < b > --- and vice versa.
Objects, regarded as a token of the specified modally seperated type, cannot be contingently identical to another.
(Since I am not dealing with nonexistence, I have quoted the original definitions and not the modified ones of MLⁿ, p. 93 and 94 - Def 24.1 and 24.2.)
Gupta introduces mainly two different notions of separateness (regardless the modifications concerning nonexistence). First, he defines 'separation' slightly different from Bressanian modal separation:

An intensional property F in a model structure < W, D, i > is separated iff
for all worlds w, w' and all individual concepts i, i' that belong to F at w,w' respectively, if i(w₁) = i(w₁) at a world w₁, then i=i' (p. 29).

To illustrate the difference, I will use the following 'standard'-example: Let D = a,b,c be the domain and G = g1, g2, g3 the set of possible worlds. Then a concept that is Bressanian (modally) separated, but not 'Gupta-esque', is this:

g1	g2	g3
abc	acd	ddd
cda

In order to deal with a problem concerning transworld identity stated by Chisholm, Gupta restates his definition thus:

An intensional property F in a model structure < W, D, i > is separated in w iff all individual concepts i, i' that belong to F at w are such that if i(w₁) = i'(w₁) at any world w₁, then i=i'.
An intensional property F in a model structure < W, D, i > is quasi-separated iff F is separated in every world in W (p. 104).

Gupta constructs Chisholms problem as follows: It is intelligible, that a thing like a bicycle remains the same, when a little part of it is changed. Secondly, no one would like to claim that two bicycles are the same, if they are build up from different pieces at different places and times. But one can imagine a sequence of counterparts that are each identical to another, which leads to such a conclusion.
The difference in the last two definitions is, that original separation was defined for every world in the whole model structure < W, D, i >, whereas the modified version is defined for one particular world w. The principle of identity provided by separated concepts is world-relativized thereby. This means further, that the equivalence class of objects falling under the concept F is determined relative to each world.
For four worlds w1 , w2 , w3 and w4 the respective principle of identity may pick out existing entities in three other worlds, and determine the remaining object in the 'most distant' world to be the non-existing entity (For further illumination see Gupta, p.98, 105).

World-restricted identity is somewhat paradox. Let us imagine a concept F, which in separated in every single world g1, g2, g3:

g1	g2	g3
abc	adc	ddd
cdd	bbd	cbc

There are apparently inconsistent identity-relations:

(i) a in g1 = b in g2
(ii) but not: b in g2 = a in g1
(iii) c in g1 = d in g3
(iv) but not d in g3 = c in g1
(v) c in g3 = c in g1

The important addition to each equation is, from which point of view it is made. (i), for instance, is made from the point of view of g2. Similarly, (iii) is perfectly right, if one reads it thus: 'Go to world g1 and see what is determined by F to be the counterpart of a in g2 '.

Underlying Common Nouns

Let [K] be the common noun that underlies K and fulfills the condition [of modal constancy]. Now we understand intensional properties thus:
Proposition 2. If i belongs to the intension of K at w, then i(w) is a K in w and i(w') in w' is the same [K] as i(w) in w (p. 28, The square brackets in [K] are Guptas, the others not).

How do we know when a certain concept is modally constant? Concerning the concept of mass in physics, this question is easily answered: To do physics, it is inevitable to suppose that masses of masspoints are constant. Concepts may be constant by definition.
This can be used to deal with some puzzles about de dicto and de re modality. The most prominent example in Peter Abailards treatise on modalities is this:

Videntur autem duobus modis exponi posse, veluti si dicam 'possibile est stantem sedere'. (...) [A]lius est sensus per divisionem alius per conpositionem: per conpositionem vero est si stare et sedere simul in eodem subiecto coniungat, ac si dicamus 'possibile est stantem sedere manentem stantem' id est 'sedere simul stare' (ac si dicamus 'possibile est ita contingere ut hec propositio dicit "stans sedet" ') (...) Si vero accipitur quod ita accipiatur quod is qui stat possit sedere quandoque, non coniungimus tunc opposita; et ad rem ipsam, non ad propositionem, 'possibile' referimus dicentes 'rem que stat posse quandoque sedere'.
(Published by Minio-Paluello in: Twelfth Century Logic, Texts and Studies II, Abaelardiana inedita, Roma 1985, p. 13f. [18]).

Abailard gives here a method for interpreting one and the same sentence 'possibile est stantem sedere' de dicto (per conpositionem) or de re (per divisionem). If one has to deal intensionally with the expression 'stans', one has to introduce a tracing principle. In a non-Fregean analysis, 'stans' may be interpreted as the common noun A, just like 'homo' in 'homo sedet' has to be translated by a common noun H. Then 'stans sedet' has to be read as (there is A, x) Ex. This would lead to an interpretation per conpositionem: The proposition, that some standing entity is sitting manentem stantem, is asserted.
(It may seem natural not to interpret 'stans' as a common noun, but to insert 'man'. In other contexts, however, this is not as easy. The problem is that sometimes we have to insert a deliberatly chosen tracing principle, in order not to commit 'Aristotelian essentialism'.)
The reading per divisionem has to divide the tracing principles.First, 'stans' must be understood as a concept that picks out an actually standing object, say a in g1. Then an underlying concept A_u must be introduced to trace a not qua standing object, but for instance qua man. The divisio, therefore, is done thus:

for world g1: (there is A, x)(there is A_u, y) x=y and (there is W, gi) Ey

(I could have said possibly Ex instead of (there is W, gi) Ex, which is to be read: there is a world gi such that y is E; 'x=y' means in this context: 'x is actually identical to y'.)
To a certain degree, this resembles Stelzners characterisation of de re modality:

A formula (there is an x that is r-necessarily P) is assigned 'true' by f in w1 iff
there is an assignment f' in w1 that differs from f in w1 at most concerning the interpretation of x, and an assignment f'^* in w2 that differs only from f' in w2 in assigning f' in w1 of x instead of f'^* in w2 of x to x, such that:
for all (related) worlds wi , f'^* wi assigns 'true' to P(x),
(De Dicto, De Re ..., p. 69)

In Stelzners formalism the sentence reads even more complicated:

In S gilt: f w1 (ex. x r-notwendig P(x)) = T gdw. es ein f' w1(x) aus U mit f' w1 =(x) f w1 gibt, sodaß für alle w2 (w1 R w2 impliziert f'^* w2 (P(x)) = T), wobei sich f'^* w2 (P(x)) von f' w1 (P(x)) nur dadurch unterscheidet, daß für f'^* w2 (x) = f' w1 (x) gesetzt wird, ...

where 'r-necessary A' means 'A is de re necessary'.
The assignment f'^* wi (x) in a way 'transposes' the object x from the actual world wa into every other world wi. If one denotes a certain object a by 'stans' in g1, and proceeds to say 'stans potest quandoque sedere' (that is: there are circumstances, under which 'stans' is sitting) the reference in the latter proposition goes to 'stans'-in-g1 and there is claimed to be a world gi in which this very 'stans' is sitting. The value of 'stans'-in-g1 is inserted respectively. This procedure, indeed, is tantamount to inserting a (modally constant) underlying concept. But if one happened to take the phrase ''insert the value of stans'' seriously, no propress will be made. Given no further information, an instance of 'standing object' will have to be inserted. Of course, a tracing principle has to be introduced, that differs from 'standing object'.
In epistemic Logics, such a procedure is called 'stating circumstances in the analysis-language', as opposed to the object-language. Iterating propositional attitudes, someone might say: ''Peter believed that his murder is a trustworthy person''. A de re interpretation can be indicated with ''Peter believed his murder to be a trustworthy person'', which means actually: there is some person a which we call 'Peters murder', and Peter believed a to be a trustworthy one (W. Stelzner, Epistemische Logik, Berlin 1984, p. 61 ff.). There emerges a problem concerning quasi separated concepts. (Gupta defines a quasi separated concept to be separated in every world. That there is a modally constant concept A underlying another concept B means: the object, that falls extensionally under B can be traced qua A (from each world into each other). Suppose that A is only quasi-separated and instanciated by a in g1. Then the underlying concept B might yield abc (open to the possibility that a=b=c) as an intensional denotation for a in G = g1, g2, g3. The claim that underlying every common noun, there is another, that fulfills the condition of modal constancy may be stated thus:

(for all K, x) (there is K_u, y) [x=y and (possibly y=y' implies necessarily y=y')]

(For x is not traced by the same K as y in this formula, x=y does not (have to) yield 'necessarily x=y'.)
Now the problem about transworld identity can be developed once again: If x qua K_u could have had slightly different properties, a sequence of worlds can be supposed, such that x is identical to a completely different object z. For the underlying concept A is constant, it cannot be only quasi-separated. Either it is not separated, or it is separated in G. If A is not separated, no unique underlying tracing principle will be given. It could happen that a is traced by two concepts, say a,b,c and a,d,e in g1 (where not b = d or not c = e). Such a tracing principle will propably not establish an equivalence class: the identity relation given by a nonseparated A has the same features as the 'objective identity in case of quasi-separated', which Gupta defines as

There is a world w such that --- is the same [K] as ... from the point of view of w.

(This is because, if a nonseparated, but modally constant concept defines a counterpart in one world, then it has to define it in every world.)On the other hand, if A is separated, it cannot contain two different tracing principles for a in any two worlds of G.To conclude, if an underlying concept is used in analysing statements (that is in de re readings), the logician will have to make decisions.