Is the relation of numerical identity contingent or necessary? It seems that in many cases identity statements are contingent. Gottlob Frege explained how it is possible that true identity statements can be informative. We may understand the meanings of names “a” and “b” without knowing that they refer to one and the same object, hence the statement “a = b” tells us more than the trivial truth that an object is identical with itself. It is tempting to interpret Frege’s result as implying that the identity “a = b” could be false, i.e. that there is a possible world in which “a” and “b” refer to different objects while retaining their original meanings. Saul Kripke famously questioned that suggestion. Kripke claims that all identity statements are in fact necessarily true, and he presents a formal argument in support of his claim. The argument is based on two premises. (1) Every object is necessarily identical with itself (For all x, it is necessary that x = x). (2) If x has property P and y is identical with y, then y has P. Premise (2) is a variant of Leibniz’s law. We can now use the formula “It is necessary that x = x” and, given that x = y, we can substitute y for x, obtaining “It is necessary that x = y”. In conclusion, (1) and (2) lead to the statement: (3) For all x, y, if x = y, then it is necessary that x = y. From (3) it follows that if we take any proper names “a” and “b”, then if only it is true that a = b, it is necessarily so.
But how can we reconcile this formal result with the intuition expressed at the beginning of the previous paragraph? In the actual world the names “Hesperus” and “Phosphorus” refer to the same object: the planet Venus. But couldn’t it be the case that in some other possible world Hesperus and Phosphorus were different objects? Kripke explains away this intuition by pointing out that the possible situation in which we would be tempted to say that Hesperus is not identical with Phosphorus can be reinterpreted in such a way that the identity will be preserved. According to Kripke, terms “Hesperus” and “Phosphorus” are so-called rigid designators, i.e. terms that refer to the same object in all possible worlds in which they refer to anything at all. “Hesperus” is not synonymous with the description “The brightest star on the morning sky”, nor is “Phosphorus” synonymous with “The brightest star on the evening sky”. These descriptions are used contingently in the actual world to fix the reference of both names. In another world the descriptions may not pick the object which is the referent of both terms (i.e. the planet Venus), but if Venus exists in this world, both terms “Hesperus” and “Phosphorus” will continue referring to it.
It may be pointed out that when we restrict the thesis of the necessity of identity statements to rigid designators, its truth becomes quite trivial. However, Kripke claims that his thesis has non-trivial consequences regarding for instance the mind-body controversy. Without going into too much detail, let us consider the Identity Theory, according to which mental events are numerically identical with some physical events. The identity theorists maintain that their claim is true but only contingently, i.e. in some possible worlds there are beings that possess particular neurological states but lack mental states (so-called zombies), and in other possible worlds there may be disembodied minds. But according to Kripke’s analysis, if the statement “This pain is identical with this stimulation of the nervous system” is true, it is necessary, and hence neither zombies nor disembodied minds are possible. But couldn’t we explain away these possible scenarios in a similar way we have redescribed the Hesperus-Phosphorus case? Namely, couldn’t we just say that the rigid designators “this pain” and “this stimulation of the nervous system” are contingently associated with some descriptions, which fail to pick the same object in alternative possible worlds? Unfortunately, as Kripke points out, the terms referring to mental states have no associated descriptions, because their reference is fixed directly by the person that is in a given mental state. So the case of the mind-body identification is different from the case of the Hesperus-Phosphorus identity.
What is the ontological status of possible worlds? One radical answer to this question is known under the name of modal realism (possibilism), which has been proposed by David Lewis. Modal realism consists of several claims. One of them is that possible worlds are made up of concrete, spatiotemporal objects. Possible worlds are not fictions or abstract constructions, but real places with flesh-and-blood inhabitants. Things contained in other possible worlds exist in the same fundamental sense as things in the actual world. Thus it is legitimate to assume that the variables of the existential quantifier range over all possible worlds. For the sake of convenience we may want to relativise the notion of existence to a particular world (speaking about “existing-in-a-world”), but this relativisation does not imply any significant ontological difference. The second element of the doctrine of modal realism is that there is nothing fundamental and absolute about the notion of actual world. The term “actual” is indexical (analogously to terms such as “here”, “now”, “I”, whose meaning depends on the context of utterance), which means that each possible world is actual from the perspective of its inhabitants. Finally, modal realism assumes that possible worlds are spatiotemporally and causally separated from each other. One important consequence of these assumptions is that one object cannot exist in more than one world. Transworld identity is an empty notion in modal realism. But how can we interpret the modal statement “This two-metre high tree could be five metres high” without assuming that this tree can exist in other possible worlds? Lewis solves this problem by introducing the notion of a counterpart. This tree has many counterparts in other possible worlds – trees that are sufficiently similar to it, but not numerically identical. An object could have a given property, if one of its counterparts possesses this property in some possible world.
The main motivation for modal realism comes from its radically reductive character. Lewis subscribes to nominalism and he uses the concept of concrete possible worlds to give a reductive analysis of various abstract entities, such as properties, propositions or meanings. For instance a proposition is simply defined as a class of possible worlds (the proposition “Snow is white” is the class of possible worlds in which snow is white). According to this definition, proposition p is true in a world w, if w is an element of the class of worlds p. Properties, in turn, are defined as functions which assign a set of objects to each possible world. These sets are intuitively understood as consisting of objects that possess a given property in a particular world. This definition avoids the well-known difficulty resulting from the fact that two numerically distinct properties can nevertheless have the same extension in the actual world. The property of being an elephant and the property of being the largest land animal living on Earth now may have the same extensions in the actual world, but there are possible worlds in which elephants are not the largest living land animals. However, reductions offered by modal realists are often criticised as not entirely adequate. For instance, it is pointed out that all necessary true propositions become identical. But we believe that there is a difference between the statements “2+2 = 4” and “It is snowing or it is not snowing”. Similarly, it can be maintained that the property of being triangular and being trilateral are different, and yet in all possible worlds their extensions are identical.
Modal realism is often rejected on grounds of its extravagant ontology. An alternative position is offered in the form of actualism (moderate realism). Alvin Plantinga suggests that possible worlds are mere theoretical constructions which enable us to formulate non-reductive explanations of modal notions. The only genuine world is the actual world, and the quantification in our language should be restricted to objects in the actual world. Possible worlds different from the actual world can be defined as complete and consistent sets of propositions, or better as complete and consistent states of affairs (situations). Each proposition corresponds to a given state of affairs. All states of affairs are abstract objects which exist in the actual world, but only some of them obtain (those that correspond to true propositions). False propositions describe existing states of affairs which nevertheless don’t obtain. According to actualism, the expression “actual” has an absolute meaning: it refers to one and only world that truly exists.
One important difference between actualism and possibilism regards the notion of transworld identity. Actualism admits that one object can exist in many possible worlds. But how are we to understand this statement, if possible worlds do not exist literally, but are mere constructions out of abstract objects? Plantinga proposes the following solution. That an object a exists in a possible world w means that if w were actual, a would exist in it. Note that the explicans is a counterfactual conditional, but it cannot be interpreted in terms of possible worlds, since this would require an introduction of second-order possible worlds (a possible world in which another possible world would be actual). This fact shows that Plantinga’s conception does not offer a fully reductive analysis of modal notions, and that some modalities have to be taken as primitive.
Readings:
M.J. Loux, "The necesary and the possible", pp. 153-186, Metaphysics: A Contemporary Introduction.
E.J. Lowe, "Necessity and identity", pp. 84-95, "Possible worlds", pp. 120-133, A Survey of Metaphysics.
Wednesday, January 20, 2010
Wednesday, January 13, 2010
Modality
Modal notions, such as possibility and necessity, play an important role in metaphysical considerations. Intuitively, we can distinguish two ways of speaking about possibilities. We can say that it is now possible that an object might change in the future. For instance this chair may be painted in a colour different from the one it has right now. This type of possibility can be called temporal. But in a different sense it is possible that the chair might have a different colour right now – if it had been painted this colour before. This kind of possibility, which applies to the present time as well as to the past (I can say that I might have been born in a different town), will be referred to as counterfactual possibility. It is interesting to notice that counterfactual and temporal notions of possibility are logically independent, i.e. one does not imply the other. From the fact that some state of affairs is counterfactually possible it does not follow that this state of affairs is possible temporarily. A given sculpture could have a different shape now, but once it receives its actual shape it cannot be turned into a different statue in the future (Rodin’s sculpture “The kiss” could have been “The thinker” in the counterfactual sense, but not in the temporal sense). Conversely, although a seed can grow into a tree in the future, it could not be a tree right now. It should be added that counterfactual possibility, in spite of what the term suggests, does not exclude actuality. Actual states of affair are considered possible in the counterfactual sense.
Counterfactual possibility is often presented in the language that uses the concept of possible worlds. A proposition is possible if it is true in some possible worlds. Possible worlds themselves are usually interpreted as complexes (sums) of situations (states of affairs). An example of a possible situation may be that Poland has a king now. Possible worlds have to satisfy two conditions: the condition of consistency and the condition of completeness. A situation s is consistent if there is no proposition p such that p and not-p are true in s. A situation s is complete if for all propositions p, either p is true in s or not-p is true in s. From these two conditions it follows that two numerically distinct possible worlds are mutually exclusive (incompatible), i.e. there is a proposition p such that p is true in one world, and p is false in the other one. Situations that are not complete don’t have to be exclusive. An example: that this ball is red and that this ball is round. The world that we live in is called the actual world, and it is interpreted in the same way as other possible worlds. It is natural to assume that the actual world is complete, i.e. every proposition is either true or its negation is true in the actual world.
Let us see how we can use the notion of possible worlds in order to explicate some modal terms, such as possibility, necessity and contingency. These notions can be applied to propositions as well as to objects. Proposition p is possible iff p is true in some possible world. Proposition p is necessary iff p is true in all possible worlds. And p is contingent iff p is true in some possible worlds and it’s false in some possible worlds. Similarly we can define possible, necessary and contingent objects. A possible object is an object that exists in some possible worlds. A necessary object exists in all possible worlds, and a contingent object exists in some worlds, but in some it does not. The usual examples of necessary truths are the laws of logic and of mathematics. Necessary beings, in turn, typically include mathematical objects and other abstract objects. Some also cite God as an example of a necessary being. It is open to a debate whether there are any spatiotemporal necessary objects (perhaps the universe as a whole can satisfy this requirement).
Let us make an important distinction between modality de re and de dicto. Modality de dicto applies to the entire sentence, whereas modality de re is attributed to a given object. The sentence “It is possible that some man is the present king of Poland” belongs to the de dicto type, whereas “Some man is possibly the king of Poland” is de re. The first sentence can be presented in a semi-formal way as “It is possible that for some x, x is a man and x is the king of Poland”, and this sentence in turn is true if and only if there is a possible world in which Poland has a king. The second sentence translates into “For some x, x is a man and it is possible that x is the king of Poland”, and in order for this proposition to be true, there has to exist someone in the actual world who, in another possible world is the king of Poland. (These explications presuppose of course that one and the same object can exist in different possible worlds.) The second proposition logically implies the first, but the implication in the opposite direction is a matter of some controversy (the validity of the so-called Barcan law). Another example illustrating the de re/de dicto distinction is as follows: “The number of planets in the solar system is necessarily divisible by 2” (de re) and “It is necessary that the number of planets in the solar system is divisible by 2” (de dicto). The first translates into “There is an x such that x is the number of planets in the solar system and it is necessary that x is divisible by 2”, and the second reads “It is necessary that there is an x such that x is the number of planets in the solar system and x is divisible by 2”. The truth of the first sentence follows from the simple arithmetical fact that 8 is (necessarily) divisible by 2, but for the second sentence to be true, the number of planets in all possible worlds would have to be even.
We can now define an important notion of an essential property. P is an essential property of object a iff for every possible world w, if a exists in w, a has P in w. Loosely speaking, if an object loses its essential property, it ceases to be itself. Napoleon’s essential property is being a human, but being the victor from Austerlitz belongs to his accidental properties (in some possible worlds Napoleon lost the battle of Austerlitz). It is interesting to ask whether things have individual essences, i.e. essential properties such that only one object can possess them all. More specifically, the individual essence of object a is a set S of essential properties of a such that in any possible world w, if x possesses all properties from S, x is identical with a. Some philosophers claim that the individual essence of an object a is the property of being identical with a. However, this interpretation prevents us from using the notion of essence in order to explicate transworld identity. According to a different view, an object’s individual essence is its origin, i.e. the cause of its existence. In the case of human beings, their essence would be determined by the zygote (the fertilized egg) that developed into a particular person. Yet another version of essentialism insists that an object’s essence is its constitution, i.e. all parts the object consists of.
Further reading:
E.J. Lowe, Chapter 5, “Necessity and identity”, pp. 79-84; Chapter 6 “Essentialism”, pp. 98-114, in: A Survey of Metaphysics.
Counterfactual possibility is often presented in the language that uses the concept of possible worlds. A proposition is possible if it is true in some possible worlds. Possible worlds themselves are usually interpreted as complexes (sums) of situations (states of affairs). An example of a possible situation may be that Poland has a king now. Possible worlds have to satisfy two conditions: the condition of consistency and the condition of completeness. A situation s is consistent if there is no proposition p such that p and not-p are true in s. A situation s is complete if for all propositions p, either p is true in s or not-p is true in s. From these two conditions it follows that two numerically distinct possible worlds are mutually exclusive (incompatible), i.e. there is a proposition p such that p is true in one world, and p is false in the other one. Situations that are not complete don’t have to be exclusive. An example: that this ball is red and that this ball is round. The world that we live in is called the actual world, and it is interpreted in the same way as other possible worlds. It is natural to assume that the actual world is complete, i.e. every proposition is either true or its negation is true in the actual world.
Let us see how we can use the notion of possible worlds in order to explicate some modal terms, such as possibility, necessity and contingency. These notions can be applied to propositions as well as to objects. Proposition p is possible iff p is true in some possible world. Proposition p is necessary iff p is true in all possible worlds. And p is contingent iff p is true in some possible worlds and it’s false in some possible worlds. Similarly we can define possible, necessary and contingent objects. A possible object is an object that exists in some possible worlds. A necessary object exists in all possible worlds, and a contingent object exists in some worlds, but in some it does not. The usual examples of necessary truths are the laws of logic and of mathematics. Necessary beings, in turn, typically include mathematical objects and other abstract objects. Some also cite God as an example of a necessary being. It is open to a debate whether there are any spatiotemporal necessary objects (perhaps the universe as a whole can satisfy this requirement).
Let us make an important distinction between modality de re and de dicto. Modality de dicto applies to the entire sentence, whereas modality de re is attributed to a given object. The sentence “It is possible that some man is the present king of Poland” belongs to the de dicto type, whereas “Some man is possibly the king of Poland” is de re. The first sentence can be presented in a semi-formal way as “It is possible that for some x, x is a man and x is the king of Poland”, and this sentence in turn is true if and only if there is a possible world in which Poland has a king. The second sentence translates into “For some x, x is a man and it is possible that x is the king of Poland”, and in order for this proposition to be true, there has to exist someone in the actual world who, in another possible world is the king of Poland. (These explications presuppose of course that one and the same object can exist in different possible worlds.) The second proposition logically implies the first, but the implication in the opposite direction is a matter of some controversy (the validity of the so-called Barcan law). Another example illustrating the de re/de dicto distinction is as follows: “The number of planets in the solar system is necessarily divisible by 2” (de re) and “It is necessary that the number of planets in the solar system is divisible by 2” (de dicto). The first translates into “There is an x such that x is the number of planets in the solar system and it is necessary that x is divisible by 2”, and the second reads “It is necessary that there is an x such that x is the number of planets in the solar system and x is divisible by 2”. The truth of the first sentence follows from the simple arithmetical fact that 8 is (necessarily) divisible by 2, but for the second sentence to be true, the number of planets in all possible worlds would have to be even.
We can now define an important notion of an essential property. P is an essential property of object a iff for every possible world w, if a exists in w, a has P in w. Loosely speaking, if an object loses its essential property, it ceases to be itself. Napoleon’s essential property is being a human, but being the victor from Austerlitz belongs to his accidental properties (in some possible worlds Napoleon lost the battle of Austerlitz). It is interesting to ask whether things have individual essences, i.e. essential properties such that only one object can possess them all. More specifically, the individual essence of object a is a set S of essential properties of a such that in any possible world w, if x possesses all properties from S, x is identical with a. Some philosophers claim that the individual essence of an object a is the property of being identical with a. However, this interpretation prevents us from using the notion of essence in order to explicate transworld identity. According to a different view, an object’s individual essence is its origin, i.e. the cause of its existence. In the case of human beings, their essence would be determined by the zygote (the fertilized egg) that developed into a particular person. Yet another version of essentialism insists that an object’s essence is its constitution, i.e. all parts the object consists of.
Further reading:
E.J. Lowe, Chapter 5, “Necessity and identity”, pp. 79-84; Chapter 6 “Essentialism”, pp. 98-114, in: A Survey of Metaphysics.
Wednesday, January 6, 2010
Reductionist theories of particulars
Realists admit a two-type ontology including universals and particulars, while nominalists insist that there is only one category of objects, namely particulars. Realists have sufficient resources to attempt to reduce the category of particulars to that of universals. One way of reducing particulars to universals is known as the bundle theory, according to which particulars are constituted by all of their properties. This approach is reminiscent of Berkeley’s conception of things as clusters of ideas (‘sensations’), except that properties are assumed to be independent of the perceiving subject. Another possibility is to reduce particulars to properties plus an extra object, called the bare substratum, whose role is to be the literal bearer of the properties without actually possessing them. This conception is not surprisingly called the substratum theory.
Let us consider the bundle theory in some detail. The first problem we have to face is that not every set of properties constitutes a particular object. For instance, the set consisting of the property of being a horse and the property of being winged does not identify any particular, because no object is a winged horse. In order to deal with this problem, the relation of co-instantiation (compresence, collocation) is introduced. The above-mentioned properties are not co-instantiated, hence they cannot constitute an individual. A particular is constituted by properties which are mutually co-instantiated. However, there is a small technical problem here. If we treat co-instantiation as a two-place relation, then the condition that every property in a set is co-instantiated with every other property from this set is not sufficient to ensure that all the properties are co-instantiated together. It is possible to find an example of three properties P, Q, and R such that P is co-instantiated with Q, Q is co-instantiated with R, and P is co-instantiated with R, and yet P, Q and R fail to occur together (for example we can choose R as the property of not being P or not being Q, and assume that P co-occurs with Q, P co-occurs with not-Q, and Q co-occurs with not-P). One solution can be to assume the hierarchy of higher-level relations of compresence binding the relations of compresence between properties, but this leads to the proliferation of numerically distinct relations of compresence. Another way out is to accept that the relation of co-instantiation can admit a varying number of arguments, but it is doubtful if this solution is formally correct. The substratum theory avoids this difficulty, because the substratum acts as the “glue” clumping together all the properties.
Another issue is completeness. Not every set of co-instantiated properties can be identified with one individual object. The set {redness, smoothness} is exemplified by many objects (for example some red apples). We can reduce particulars to complete sets of properties only. A complete set of properties can be broadly characterised as a set such that if we added a new property to it, we would get an inconsistent set. But it is unclear what type of inconsistency is involved in this definition: logical, nomological, or perhaps metaphysical. Also, it may be questioned whether sets of all properties possessed by particular objects are in this sense complete. Why does it create an inconsistency to add to all the properties possessed by this horse the property of having wings? The existence of inconsistency can be perhaps defended if we assumed that the set of all properties of a given object contains also negative properties (in the case of the horse the property in question would be the property of not being winged).
Several objections can be raised to the bundle theory, according to which complete sets of co-instantiated properties constitute particular objects. One such objection is that subject-predicate sentences about particulars become necessarily true. In the true sentence “This table is wooden” the noun phrase “this table” refers to a particular set of properties T, and the sentence can be translated as stating that the property of being wooden belongs to this set T. But each set possesses its elements necessarily, so the sentence cannot be false. If this table, understood as a set, didn’t have the property of being wooden, it would be a numerically different object. Another way of expressing this objection is that all properties of particulars become essential. We could try to circumvent the problem by modifying the interpretation of the above sentence so that its subject is identified with the set T minus the property of being wooden, and the thought expressed in the sentence is that the property of being wooden is co-instantiated with the rest of the properties in T. But a downside of this solution is that an attribution of a different property to the same table, for instance “This table is white”, will have to be interpreted as a sentence about a different object, i.e. the set T minus the property of being white. It is debatable whether the substratum theory is affected by a similar problem. If the subject of the sentence “This table is wooden” is identified as the set of all properties plus the bare substratum of the table, then the same argument in favour of the necessity of the property ascription goes through. However, the sentence can be interpreted as having the bare substratum as its subject, and in this case it will be contingent (the bare substratum exemplifies its properties contingently).
Another difficulty for the bundle theory is related to the problem of change. It is commonly accepted that objects can change in time by acquiring new properties or losing old ones, without losing their numerical identity. But sets of different properties are numerically distinct. The same problem seems to affect the substratum theory, unless we decide to explicate the relation of identity between various temporal stages of an object in terms of the preservation of its bare substratum (but in that case it seems that the identification of a particular object with the set of all its properties plus the substratum turns out to be vacuous – instead, we simply take the substratum as identical with the object). One possible solution may be to treat an object as the bundle of all its past, present and future properties indexed by moments of time. This suggestion corresponds well to the conception of the existence in time known as perdurantism, of which we will talk more later in the course.
Finally, the bundle theory is criticised for implying the necessary truth of the Principle of the Identity of Indiscernibles (PII). This follows immediately from the extensionality of sets: two sets containing the same elements are numerically identical. Hence there can be no numerically different bundles of the same properties. This consequence is unwelcome because of the strong arguments in favour of the possibility of the PII being false (and even in favour of its actual falsity) which we discussed earlier in the course. Again, the substratum theory has the upper hand because it is possible to have two individuals with exactly the same properties, if only their substrata are numerically distinct (the possibility of making the PII false seems to be the main advantage of the substratum theory). The bundle theory may be rescued if we agreed to replace properties with tropes. It is possible to have two individuals made up of perfectly similar tropes, hence the spirit of the PII is preserved.
The substratum theory assumes that when we “subtract” all properties from a given object what is left is a property-less pure object, called a bare substratum. The substratum cannot posses any properties, because metaphysically it has to be ready to accept any properties as their bearer. If the substratum possessed any properties, this would lead to a regress, since it would need its own bare substratum presumably with its properties and so on. The only attribution that can be made directly about the substratum regards its numerical distinctness from other substrata. Hence it can be maintained that bare substrata ground the numerical identity and distinctness of objects, known as their “thisness” or haecceity.
The main criticism of the substratum theory comes from the empiricists, who point out that because bare substrata lack any properties they cannot be perceived or in any way identified or known. It is also claimed that the notion of a bare substratum is inconsistent. On the one hand it is assumed that bare substrata don’t posses any properties, but on the other hand we characterize them in some ways: we say that they are ‘bare’, that they act as literal possessors of properties, that they ground numerical identity and difference. But aren’t these things properties of bare substrata?
A solution which tries to combine the strengths of the bundle theory and the substratum theory without their weaknesses is called the nuclear theory (proposed by Peter Simons). According to it, properties of every object can be divided into two parts: the inner nucleus and the outer fringe. The inner nucleus contains properties that are essential, and therefore such that an object cannot lose them without losing its identity (we will characterize precisely the notion of an essential property in the next lecture). Thus the nucleus plays the role of a bare substratum. The outer fringe, on the other hand, contains properties that can be lost and gained without a change in the numerical identity. But this solution has to be worked out in details in order to make sure that it copes with the problems we talked about earlier. In particular, if we elect to identify particular objects with their inner nuclei (this seems to be necessary in order to deal with the problem of necessary attributions and the problem of change), we have to make sure that no two distinct objects possess all the properties from the nucleus.
Reading:
M.J. Loux, Chapter 3 "Concrete particulars I", pp. 84-107, Metaphysics: A Contemporary Introduction
Also recommended:
J. Van Cleve, "Three Versions of the Bundle Theory".
P. Simons, "Particulars in Particular Clothing: Three Trope Theories of Substance".
Let us consider the bundle theory in some detail. The first problem we have to face is that not every set of properties constitutes a particular object. For instance, the set consisting of the property of being a horse and the property of being winged does not identify any particular, because no object is a winged horse. In order to deal with this problem, the relation of co-instantiation (compresence, collocation) is introduced. The above-mentioned properties are not co-instantiated, hence they cannot constitute an individual. A particular is constituted by properties which are mutually co-instantiated. However, there is a small technical problem here. If we treat co-instantiation as a two-place relation, then the condition that every property in a set is co-instantiated with every other property from this set is not sufficient to ensure that all the properties are co-instantiated together. It is possible to find an example of three properties P, Q, and R such that P is co-instantiated with Q, Q is co-instantiated with R, and P is co-instantiated with R, and yet P, Q and R fail to occur together (for example we can choose R as the property of not being P or not being Q, and assume that P co-occurs with Q, P co-occurs with not-Q, and Q co-occurs with not-P). One solution can be to assume the hierarchy of higher-level relations of compresence binding the relations of compresence between properties, but this leads to the proliferation of numerically distinct relations of compresence. Another way out is to accept that the relation of co-instantiation can admit a varying number of arguments, but it is doubtful if this solution is formally correct. The substratum theory avoids this difficulty, because the substratum acts as the “glue” clumping together all the properties.
Another issue is completeness. Not every set of co-instantiated properties can be identified with one individual object. The set {redness, smoothness} is exemplified by many objects (for example some red apples). We can reduce particulars to complete sets of properties only. A complete set of properties can be broadly characterised as a set such that if we added a new property to it, we would get an inconsistent set. But it is unclear what type of inconsistency is involved in this definition: logical, nomological, or perhaps metaphysical. Also, it may be questioned whether sets of all properties possessed by particular objects are in this sense complete. Why does it create an inconsistency to add to all the properties possessed by this horse the property of having wings? The existence of inconsistency can be perhaps defended if we assumed that the set of all properties of a given object contains also negative properties (in the case of the horse the property in question would be the property of not being winged).
Several objections can be raised to the bundle theory, according to which complete sets of co-instantiated properties constitute particular objects. One such objection is that subject-predicate sentences about particulars become necessarily true. In the true sentence “This table is wooden” the noun phrase “this table” refers to a particular set of properties T, and the sentence can be translated as stating that the property of being wooden belongs to this set T. But each set possesses its elements necessarily, so the sentence cannot be false. If this table, understood as a set, didn’t have the property of being wooden, it would be a numerically different object. Another way of expressing this objection is that all properties of particulars become essential. We could try to circumvent the problem by modifying the interpretation of the above sentence so that its subject is identified with the set T minus the property of being wooden, and the thought expressed in the sentence is that the property of being wooden is co-instantiated with the rest of the properties in T. But a downside of this solution is that an attribution of a different property to the same table, for instance “This table is white”, will have to be interpreted as a sentence about a different object, i.e. the set T minus the property of being white. It is debatable whether the substratum theory is affected by a similar problem. If the subject of the sentence “This table is wooden” is identified as the set of all properties plus the bare substratum of the table, then the same argument in favour of the necessity of the property ascription goes through. However, the sentence can be interpreted as having the bare substratum as its subject, and in this case it will be contingent (the bare substratum exemplifies its properties contingently).
Another difficulty for the bundle theory is related to the problem of change. It is commonly accepted that objects can change in time by acquiring new properties or losing old ones, without losing their numerical identity. But sets of different properties are numerically distinct. The same problem seems to affect the substratum theory, unless we decide to explicate the relation of identity between various temporal stages of an object in terms of the preservation of its bare substratum (but in that case it seems that the identification of a particular object with the set of all its properties plus the substratum turns out to be vacuous – instead, we simply take the substratum as identical with the object). One possible solution may be to treat an object as the bundle of all its past, present and future properties indexed by moments of time. This suggestion corresponds well to the conception of the existence in time known as perdurantism, of which we will talk more later in the course.
Finally, the bundle theory is criticised for implying the necessary truth of the Principle of the Identity of Indiscernibles (PII). This follows immediately from the extensionality of sets: two sets containing the same elements are numerically identical. Hence there can be no numerically different bundles of the same properties. This consequence is unwelcome because of the strong arguments in favour of the possibility of the PII being false (and even in favour of its actual falsity) which we discussed earlier in the course. Again, the substratum theory has the upper hand because it is possible to have two individuals with exactly the same properties, if only their substrata are numerically distinct (the possibility of making the PII false seems to be the main advantage of the substratum theory). The bundle theory may be rescued if we agreed to replace properties with tropes. It is possible to have two individuals made up of perfectly similar tropes, hence the spirit of the PII is preserved.
The substratum theory assumes that when we “subtract” all properties from a given object what is left is a property-less pure object, called a bare substratum. The substratum cannot posses any properties, because metaphysically it has to be ready to accept any properties as their bearer. If the substratum possessed any properties, this would lead to a regress, since it would need its own bare substratum presumably with its properties and so on. The only attribution that can be made directly about the substratum regards its numerical distinctness from other substrata. Hence it can be maintained that bare substrata ground the numerical identity and distinctness of objects, known as their “thisness” or haecceity.
The main criticism of the substratum theory comes from the empiricists, who point out that because bare substrata lack any properties they cannot be perceived or in any way identified or known. It is also claimed that the notion of a bare substratum is inconsistent. On the one hand it is assumed that bare substrata don’t posses any properties, but on the other hand we characterize them in some ways: we say that they are ‘bare’, that they act as literal possessors of properties, that they ground numerical identity and difference. But aren’t these things properties of bare substrata?
A solution which tries to combine the strengths of the bundle theory and the substratum theory without their weaknesses is called the nuclear theory (proposed by Peter Simons). According to it, properties of every object can be divided into two parts: the inner nucleus and the outer fringe. The inner nucleus contains properties that are essential, and therefore such that an object cannot lose them without losing its identity (we will characterize precisely the notion of an essential property in the next lecture). Thus the nucleus plays the role of a bare substratum. The outer fringe, on the other hand, contains properties that can be lost and gained without a change in the numerical identity. But this solution has to be worked out in details in order to make sure that it copes with the problems we talked about earlier. In particular, if we elect to identify particular objects with their inner nuclei (this seems to be necessary in order to deal with the problem of necessary attributions and the problem of change), we have to make sure that no two distinct objects possess all the properties from the nucleus.
Reading:
M.J. Loux, Chapter 3 "Concrete particulars I", pp. 84-107, Metaphysics: A Contemporary Introduction
Also recommended:
J. Van Cleve, "Three Versions of the Bundle Theory".
P. Simons, "Particulars in Particular Clothing: Three Trope Theories of Substance".
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