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".
Wednesday, December 16, 2009
Two notions of sets
Sets are considered fundamental mathematical objects, because in principle all other mathematical objects can be defined in terms of sets (can be reduced to sets). However, the notion of a set can be given at least two unequivalent interpretations. In one interpretation, the set of physical objects X is a physical complex whose spatiotemporal parts are all objects X. Sets of that sort are called mereological (or collective). Mereological sets have the following characteristic properties. First, the mereological set consisting of one object is identical with this object. Second, there is no mereological empty set (for instance, the collection of all centaurs does not exist). Third, two mereological sets built out of numerically different objects can nevertheless be identical. For instance, the mereological set of two hydrogen atoms is identical with the mereological set of two protons and two electrons constituting those atoms. This example also illustrates the fact that the relation of belonging to a mereological set is transitive. Clearly, this follows from the fact that the relation of being a member of a mereological set is identical with the part-whole relation, and the latter is transitive (if x is a part of y and y is a part of z, then x is a part of z).
The second interpretation of sets is called distributive (or set-theoretical). Distributive sets are analogous to linguistic concepts. The distributive set of all people has people as its only elements. No proper part of a person (such as a hand or a leg) belongs to this set, since proper body parts are not humans. As concepts differ from the objects subsumed under them, the set consisting of only one object is numerically different from this object. There is an empty set (a set with no elements), since there are empty concepts (such as the concept of a unicorn). Actually, as we will soon see, it can be proven that there is exactly one empty set. If you have two sets which have different numbers of elements, you can be sure that these sets are different. Hence the set of two hydrogen atoms is distinct from the set of their two protons and two electrons. Distributive sets satisfy the principle of extensionality: two sets are identical if and only if they have exactly the same elements. (Actually, mereological sets satisfy this principle too: two mereological sets are identical iff they have the same parts. But it is still possible to describe one and the same mereological set as consisting of numerically different objects, as in the example with two hydrogen atoms). From the condition of extensionality it follows that there is exactly one empty set. The relation of membership is not transitive in the case of distributive sets: if x is a member of y and y is a member of z, x does not have to be a member of z (although it may). Example: 1 є {1} and {1} є {{1}, {2}}, but it’s not the case that 1 є {{1}, {2}}.
From the ontological point of view it is important to ask what kind of objects sets are and whether they can be accepted by a nominalist. Mereological sets don’t create much of a problem, since they are just spatiotemporal objects, provided that their elements are spatiotemporal. The only contentious issue is whether we should admit the existence of arbitrary collections of objects. Is there an individual object that consists of my left pinkie, the planet Venus, and the left hind leg of some particular dinosaur? But the status of distributive sets is more controversial. Arguably, distributive sets cannot be identified with spatiotemporal objects. The singleton consisting of one physical object x is not identical with x, and because it cannot be identical with any other physical object y (since its existence would be contingent upon the existence of y, and the only acceptable ontological dependence of {x} is on x), hence {x} cannot be a physical object. Thus the most common interpretation of distributive sets is that they are abstract objects, and as such are not acceptable to the nominalist. However, nominalists can make use of the notion of distributive set in certain contexts, for which it is possible to give a nominalistic paraphrase. For instance, the statement “Socrates belongs to the set of all philosophers” can be interpreted nominalistically as “Socrates is a philosopher”. The sentence “The set of all philosophers is a subset of the set of all people” is interpreted as “All philosophers are humans”, and an analogous interpretation of the sentence “The set of all people is disjoint from the set of all elephants” can be given as “No humans are elephants”. Thus it can be claimed that the nominalist can accept first-order sets of physical objects (so-called classes). But higher-order sets, and especially those founded on the empty set, are not so easy to eliminate from the discourse.
At the beginning of the lecture we mentioned the fact that mathematical objects can be reduced to (distributive) sets. But, as Paul Benacerraf has famously noticed, such reductions are not unique. For instance, natural numbers can be interpreted as sets in at least two ways. One interpretation is given by following identifications: 0 = Ø, 1 = {Ø}, 2 = {Ø, {Ø}}, 3 = {Ø, {Ø}, {Ø, {Ø}}}, etc. But an alternative interpretation can look like this: 0 = Ø, 1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc. These two interpretations, taken literally, cannot be true, for this would imply mathematical falsehoods, such as {{Ø}} = {Ø, {Ø}} (you can prove that this identity is false, given the principle of extensionality and the assumption that {x} is different from x). But mathematical practice does not tell us which identification should be preferred. It looks like some questions regarding numerical identity between mathematical objects are fundamentally undecidable, which calls into question the ontological status of mathematical objects as independent entities. One solution to this problem is encompassed in the so-called structuralist interpretation of mathematics. According to this interpretation, the fundamental objects that mathematical theories speak about are whole structures, not individual objects. There is no number 1 as an entity that exists separately from the entire structure of natural numbers, hence it does not make sense to ask what this number is identical with. Mathematical objects are just positions in a given structure. There is more than one way to interpret one structure (e.g. the structure of natural numbers) within another structure (the structure of sets). But an interpretation is just a homomorphism, i.e. a mapping which preserves the structure. There is no identity involved. The essence of natural numbers is exhausted in the structure of a linear, discrete order.
The second interpretation of sets is called distributive (or set-theoretical). Distributive sets are analogous to linguistic concepts. The distributive set of all people has people as its only elements. No proper part of a person (such as a hand or a leg) belongs to this set, since proper body parts are not humans. As concepts differ from the objects subsumed under them, the set consisting of only one object is numerically different from this object. There is an empty set (a set with no elements), since there are empty concepts (such as the concept of a unicorn). Actually, as we will soon see, it can be proven that there is exactly one empty set. If you have two sets which have different numbers of elements, you can be sure that these sets are different. Hence the set of two hydrogen atoms is distinct from the set of their two protons and two electrons. Distributive sets satisfy the principle of extensionality: two sets are identical if and only if they have exactly the same elements. (Actually, mereological sets satisfy this principle too: two mereological sets are identical iff they have the same parts. But it is still possible to describe one and the same mereological set as consisting of numerically different objects, as in the example with two hydrogen atoms). From the condition of extensionality it follows that there is exactly one empty set. The relation of membership is not transitive in the case of distributive sets: if x is a member of y and y is a member of z, x does not have to be a member of z (although it may). Example: 1 є {1} and {1} є {{1}, {2}}, but it’s not the case that 1 є {{1}, {2}}.
From the ontological point of view it is important to ask what kind of objects sets are and whether they can be accepted by a nominalist. Mereological sets don’t create much of a problem, since they are just spatiotemporal objects, provided that their elements are spatiotemporal. The only contentious issue is whether we should admit the existence of arbitrary collections of objects. Is there an individual object that consists of my left pinkie, the planet Venus, and the left hind leg of some particular dinosaur? But the status of distributive sets is more controversial. Arguably, distributive sets cannot be identified with spatiotemporal objects. The singleton consisting of one physical object x is not identical with x, and because it cannot be identical with any other physical object y (since its existence would be contingent upon the existence of y, and the only acceptable ontological dependence of {x} is on x), hence {x} cannot be a physical object. Thus the most common interpretation of distributive sets is that they are abstract objects, and as such are not acceptable to the nominalist. However, nominalists can make use of the notion of distributive set in certain contexts, for which it is possible to give a nominalistic paraphrase. For instance, the statement “Socrates belongs to the set of all philosophers” can be interpreted nominalistically as “Socrates is a philosopher”. The sentence “The set of all philosophers is a subset of the set of all people” is interpreted as “All philosophers are humans”, and an analogous interpretation of the sentence “The set of all people is disjoint from the set of all elephants” can be given as “No humans are elephants”. Thus it can be claimed that the nominalist can accept first-order sets of physical objects (so-called classes). But higher-order sets, and especially those founded on the empty set, are not so easy to eliminate from the discourse.
At the beginning of the lecture we mentioned the fact that mathematical objects can be reduced to (distributive) sets. But, as Paul Benacerraf has famously noticed, such reductions are not unique. For instance, natural numbers can be interpreted as sets in at least two ways. One interpretation is given by following identifications: 0 = Ø, 1 = {Ø}, 2 = {Ø, {Ø}}, 3 = {Ø, {Ø}, {Ø, {Ø}}}, etc. But an alternative interpretation can look like this: 0 = Ø, 1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc. These two interpretations, taken literally, cannot be true, for this would imply mathematical falsehoods, such as {{Ø}} = {Ø, {Ø}} (you can prove that this identity is false, given the principle of extensionality and the assumption that {x} is different from x). But mathematical practice does not tell us which identification should be preferred. It looks like some questions regarding numerical identity between mathematical objects are fundamentally undecidable, which calls into question the ontological status of mathematical objects as independent entities. One solution to this problem is encompassed in the so-called structuralist interpretation of mathematics. According to this interpretation, the fundamental objects that mathematical theories speak about are whole structures, not individual objects. There is no number 1 as an entity that exists separately from the entire structure of natural numbers, hence it does not make sense to ask what this number is identical with. Mathematical objects are just positions in a given structure. There is more than one way to interpret one structure (e.g. the structure of natural numbers) within another structure (the structure of sets). But an interpretation is just a homomorphism, i.e. a mapping which preserves the structure. There is no identity involved. The essence of natural numbers is exhausted in the structure of a linear, discrete order.
Saturday, December 12, 2009
Abstract objects
The distinction between universals and particulars is parallel to the more general distinction between abstract objects and concrete objects. There is no universal consensus regarding the definition of abstract objects; however we can list some general properties that are usually attributed to them. Abstract objects are typically assumed to exist outside space and time, where the notion of “being outside” should not be interpreted in the spatiotemporal sense. This is usually explained in the form of the requirement that abstract objects cannot be subjects of true tensed predications which are also essential (excluding such predications as "It is true of number 6 that yesterday I thought about it", which is not essential for 6). One consequence of this assumption is that abstract objects cannot undergo genuine changes. But it may be claimed that there are objects which exist in space and time, and yet are not concrete things. An example can be the centre of mass of the solar system. However, the centre of mass lacks another important characteristic of concrete objects: it is namely not causally efficacious. Abstracta are assumed to be causally inert; they do not participate in causal interactions. This criterion of abstractness has to presuppose some philosophical conception of causation. According to the most popular approach causation is a relation between events. But this may suggest that things are not concrete, for they cannot literally cause one another. One solution is to extend the notion of causal interactions: a thing x participates in a causal interaction iff some event e which is constituted by x stays in the causal relation with some other events.
The third attribute of abstract objects is considered to be their ontological dependence on other objects. For instance, it can be claimed that the abstract object “direction” is ontologically derivative from and dependent on the existence of parallel lines. But this can be questioned by Platonists, who claim that if there is any ontological dependence at all, it goes in the opposite direction: it is concrete objects that depend on abstract objects. As we will see later, this approach can be further supported by the so-called bundle theory of particulars. Typical examples of abstract objects include properties, relations, meanings, propositions, values, and mathematical objects (numbers, sets). The status of tropes is somewhat controversial. Some insist that they are concrete, since they exist in space-time. But it is unclear whether they can interact causally.
Nominalists criticise the notion of abstract objects by applying the following two arguments. They point out that it is unclear how we can acquire knowledge about abstract objects (the epistemological problem) and how we can refer to them (the semantic problem). Underlying these two problems are the assumptions of the causal theories of knowledge and of reference. According to the first one, in order to know something about an object x we have to interact causally (directly or indirectly) with x. According to the causal theory of reference, for an expression t to refer to some objects, a causal link has to be established between one sample object belonging to the extension of t and the later utterances of the expression t. Because they are causally inert, abstract objects are excluded from causal theories of knowledge and reference. There is no consensus regarding what alternative theories can be adopted in the case of abstracta.
The most important category of abstract objects is the category of mathematical objects. A simple argument based on mathematical practice can be given in favour of the existence of mathematical objects:
(1) Mathematical statements are true,
(2) Mathematical statements imply that mathematical objects exist.
Therefore
(3) Mathematical objects exist
The nominalist can meet this challenge by denying either (1) or (2). Let us start with the strategy that tries to question (2). This is essentially to claim that mathematical theorems can be reformulated in such a way as to eliminate their ontological commitments to abstract objects. One possible way is to try to interpret mathematical statements as being about concrete things. This may work in the case of simple arithmetical truths, such as 2 + 3 = 5. This equation can be restated as expressing the fact that if there are two objects of the kind A and three objects of the kind B, and no object is both A and B, then there are five objects of the kind A or B. Crucial to the success of this strategy is the fact that statements of the sort “There are exactly (at most, at least) n objects of the kind A” can be expressed in first-order language without any reference to number n. For instance, the sentence “There are exactly two objects with property P” can be interpreted as “There is an x and a y such that x is distinct from y, x has P and y has P, and for all z, if z has P, then z is identical with either x or y”. This reformulation is more awkward, but does not contain any reference to the number 2. But this strategy cannot be directly applied to more abstract theorems, such as the statement that there is no greatest prime number.
A more general nominalistic method of paraphrase is possible. Let S be any mathematical theorem. Then the implication “If there are mathematical objects, then S” does not carry any commitments to mathematical objects. However, the problem is that a material implication is true if its antecedent is false, hence the nominalistic interpretations of even false mathematical statements will always be trivially true. This leads to the following modification: instead of material implication we should use strict implication “It is necessary that if there are mathematical objects, then S”. This is known as modal interpretation of mathematics. There are two main problems with this interpretation. Firstly, it is unclear whether a satisfactory semantic analysis of the modal operator of necessity can be given in purely nominalistic terms (without any reference to abstract objects). Secondly, in order to maintain that some strict implications of the above form are false we have to assume that the antecedent “There are mathematical objects” is not necessarily false. But what sense can the nominalist make of the hypothesis that abstract objects might exist? Under what conditions would this be true?
Fictionalism is the approach which denies premise (1). Mathematical statements are literally false, but they are useful. The main challenge to fictionalism is given in the form of the indispensability argument whose premise is that mathematical theories and notions are applied in empirical sciences (physics, chemistry, biology, etc.). If we confirm empirically a given scientific theory, this confirmation should also reach to its mathematical part. Thus we should conclude that the best explanation for the empirical successes of a scientific theory is that the mathematical theorems used in it (such as the theorems of mathematical analysis or group theory, etc.) are true. Hartry Field in his 1980 book Science without numbers set out to defend nominalism against the indispensability argument. His strategy, in rough outline, is to find, for a given physical theory T, two theories Tp and Tm such that Tp contains only physical, nominalistically acceptable notions, while Tm is a mathematical theory used in T. T has to be logically equivalent to the conjunction of Tp and Tm. If finding such Tp and Tm were possible, then in the next step we could appeal to the logical fact that all mathematical theories are conservative with respect to non-mathematical vocabulary. This means that whatever logical consequence A of Tp + Tm can be expressed in the non-mathematical vocabulary, A should follow logically from Tp itself. Thus Tm does not have to be considered true, and its role is reduced to a mere simplification of logical deductions. The main problem with Field’s strategy is to find the nominalistic version Tp of a given physical theory T. Field showed how to do this in the case of classical mechanics, but it is unlikely that his method could be applied to more sophisticated theories, such as quantum mechanics, quantum field theory or general theory of relativity.
Further reading:
E.J. Lowe, "The abstract and the concrete", pp. 366-385, A Survey of Metaphysics.
The third attribute of abstract objects is considered to be their ontological dependence on other objects. For instance, it can be claimed that the abstract object “direction” is ontologically derivative from and dependent on the existence of parallel lines. But this can be questioned by Platonists, who claim that if there is any ontological dependence at all, it goes in the opposite direction: it is concrete objects that depend on abstract objects. As we will see later, this approach can be further supported by the so-called bundle theory of particulars. Typical examples of abstract objects include properties, relations, meanings, propositions, values, and mathematical objects (numbers, sets). The status of tropes is somewhat controversial. Some insist that they are concrete, since they exist in space-time. But it is unclear whether they can interact causally.
Nominalists criticise the notion of abstract objects by applying the following two arguments. They point out that it is unclear how we can acquire knowledge about abstract objects (the epistemological problem) and how we can refer to them (the semantic problem). Underlying these two problems are the assumptions of the causal theories of knowledge and of reference. According to the first one, in order to know something about an object x we have to interact causally (directly or indirectly) with x. According to the causal theory of reference, for an expression t to refer to some objects, a causal link has to be established between one sample object belonging to the extension of t and the later utterances of the expression t. Because they are causally inert, abstract objects are excluded from causal theories of knowledge and reference. There is no consensus regarding what alternative theories can be adopted in the case of abstracta.
The most important category of abstract objects is the category of mathematical objects. A simple argument based on mathematical practice can be given in favour of the existence of mathematical objects:
(1) Mathematical statements are true,
(2) Mathematical statements imply that mathematical objects exist.
Therefore
(3) Mathematical objects exist
The nominalist can meet this challenge by denying either (1) or (2). Let us start with the strategy that tries to question (2). This is essentially to claim that mathematical theorems can be reformulated in such a way as to eliminate their ontological commitments to abstract objects. One possible way is to try to interpret mathematical statements as being about concrete things. This may work in the case of simple arithmetical truths, such as 2 + 3 = 5. This equation can be restated as expressing the fact that if there are two objects of the kind A and three objects of the kind B, and no object is both A and B, then there are five objects of the kind A or B. Crucial to the success of this strategy is the fact that statements of the sort “There are exactly (at most, at least) n objects of the kind A” can be expressed in first-order language without any reference to number n. For instance, the sentence “There are exactly two objects with property P” can be interpreted as “There is an x and a y such that x is distinct from y, x has P and y has P, and for all z, if z has P, then z is identical with either x or y”. This reformulation is more awkward, but does not contain any reference to the number 2. But this strategy cannot be directly applied to more abstract theorems, such as the statement that there is no greatest prime number.
A more general nominalistic method of paraphrase is possible. Let S be any mathematical theorem. Then the implication “If there are mathematical objects, then S” does not carry any commitments to mathematical objects. However, the problem is that a material implication is true if its antecedent is false, hence the nominalistic interpretations of even false mathematical statements will always be trivially true. This leads to the following modification: instead of material implication we should use strict implication “It is necessary that if there are mathematical objects, then S”. This is known as modal interpretation of mathematics. There are two main problems with this interpretation. Firstly, it is unclear whether a satisfactory semantic analysis of the modal operator of necessity can be given in purely nominalistic terms (without any reference to abstract objects). Secondly, in order to maintain that some strict implications of the above form are false we have to assume that the antecedent “There are mathematical objects” is not necessarily false. But what sense can the nominalist make of the hypothesis that abstract objects might exist? Under what conditions would this be true?
Fictionalism is the approach which denies premise (1). Mathematical statements are literally false, but they are useful. The main challenge to fictionalism is given in the form of the indispensability argument whose premise is that mathematical theories and notions are applied in empirical sciences (physics, chemistry, biology, etc.). If we confirm empirically a given scientific theory, this confirmation should also reach to its mathematical part. Thus we should conclude that the best explanation for the empirical successes of a scientific theory is that the mathematical theorems used in it (such as the theorems of mathematical analysis or group theory, etc.) are true. Hartry Field in his 1980 book Science without numbers set out to defend nominalism against the indispensability argument. His strategy, in rough outline, is to find, for a given physical theory T, two theories Tp and Tm such that Tp contains only physical, nominalistically acceptable notions, while Tm is a mathematical theory used in T. T has to be logically equivalent to the conjunction of Tp and Tm. If finding such Tp and Tm were possible, then in the next step we could appeal to the logical fact that all mathematical theories are conservative with respect to non-mathematical vocabulary. This means that whatever logical consequence A of Tp + Tm can be expressed in the non-mathematical vocabulary, A should follow logically from Tp itself. Thus Tm does not have to be considered true, and its role is reduced to a mere simplification of logical deductions. The main problem with Field’s strategy is to find the nominalistic version Tp of a given physical theory T. Field showed how to do this in the case of classical mechanics, but it is unlikely that his method could be applied to more sophisticated theories, such as quantum mechanics, quantum field theory or general theory of relativity.
Further reading:
E.J. Lowe, "The abstract and the concrete", pp. 366-385, A Survey of Metaphysics.
Thursday, December 3, 2009
Versions of nominalism
Metalinguistic nominalism proposes a more uniform method of paraphrasing statements containing abstract terms. The general idea is to replace terms referring to putative universals (properties, relations, kinds) by terms describing linguistic expressions. Thus the statement “This ball is red” can be explicated as “This ball satisfies predicate ‘red’”. “Triangularity is a shape” becomes “’Triangular’ is a shape predicate”, and the troublesome sentence “Courage is a moral virtue” gets translated into “’Courageous’ is a virtue predicate” (note that the word “virtue” is clearly ambiguous: in the first sentence it serves as a noun, and hence carries an unwanted commitment to properties, whereas in the second sentence it becomes an adjective, modifying the noun “predicate”). Similarly we can treat the sentence “This tulip and that rose have the same colour”, rephrasing it as “This tulip and that rose satisfy the same colour predicate”. But it is unclear what the ontological status of linguistic expressions is, and whether a nominalist can accept them in his ontology. First we have to make a distinction between types and tokens. A token of an expression is an individual inscription or utterance. Hence each word has more than one token which belong to one and the same type. In the above examples of metalinguistic paraphrases the subject terms are singular, not general, hence it looks like they refer to types, not tokens. But types resemble universals in all relevant aspects: they are entities that are common to all individual tokens of a given expression, hence they can be interpreted as the common property of all inscriptions (utterances).
Another problem with metalinguistic nominalism is that it trades the objective, independent notion of property for a language-dependent notion of predicate. But what with properties that are not expressed in any language? There are examples of properties that we discovered and named only recently, such as spin or charm. It is quite natural to expect that there are more properties of that sort which have yet to be discovered. Consequently, a metalinguistic nominalist can’t offer a satisfactory translation for the sentence “Every object has a property that we will never know”.
The initial motivation for metaphysical realism was provided by the existence of objective similarities between particulars. Resemblance nominalism tries to develop and apply the notion of similarity without any recourse to universals. It may be said for instance that to be red is to be sufficiently similar to a paradigmatic red object. But this simple interpretation won’t do. Clearly there may be non-red objects that are similar to a selected red thing (with respect to every property other than colour). A more sophisticated attempt to explain away the attribution of properties may be as follows. The resemblance nominalist may try to define resemblance classes which will roughly correspond to the realist’s properties. For instance, a resemblance class can be defined as a maximal class such that any two objects in this class are more similar to one another than they are to any object outside of the class (more formally this condition can be spelled out as follows: for all x, y, and z, if x and y belong to class K but z does not belong to K, then x is more similar to y than to z). The condition of maximality is needed, because we don’t want to qualify the class of two red objects as a resemblance class. However, three fundamental objections can be made against such a solution.
(1) As we have already indicated, it can be argued that you can find a non-red object which is more similar to a particular red object than this object is to another red thing. Think for example of a green sphere, a red sphere of exactly the same dimensions, and a red cube twice as big as the sphere. It can be argued that the spheres resemble each other more that the red one resembles the red cube.
(2) Let’s consider two properties P and Q such that all objects that have P have Q but not vice versa. In such a case P will not define a resemblance class, for the condition of maximality fails.
(3) The universal class (the class of all particulars) satisfies the condition of being a resemblance class. But it is debatable whether there is a (non-trivial) property that is common to all particulars.
It should be clear that the problems (1) and (2) are a direct consequence of the fact that the nominalist cannot distinguish between various aspects of the similarity relation (for instance we would like to say that the class of red objects is defined by the relation of similarity with respect to colour). One solution that promises to evade this difficulty is known as trope theory. It postulates a new kind of objects – tropes – that may be acceptable to nominalists. Tropes are individual properties: the redness of that rose, the shape of that tree. Two numerically distinct individuals can never share any tropes. However, their tropes can be similar. The idea is that resemblance classes can be defined on tropes, and not on particulars, so that the resemblance class corresponding to redness will contain all tropes of redness. It is easy to notice that problems (1) and (2) disappear in this approach. No non-red trope can be more similar to a particular trope of red than a different trope of red, because tropes don’t have any ‘aspects’: they are themselves aspects. If two tropes are similar, they are always similar in precisely one respect. Problem (2) disappears, because the class of tropes that correspond to one property is always disjoint from the class of tropes corresponding to a numerically distinct property, even if all objects that posses one property possess the other one as well.
An interesting question arises whether the Principle of the Identity of Indiscernibles can be reinterpreted in trope theory. A simple replacement of properties with tropes results in a trivialisation of the PII. In virtue of the definition of tropes, if two individuals share at least one trope, they are numerically identical. A more promising strategy is to reformulate the PII in the form of the requirement that if each trope of object x is similar to a trope of object y, and vice versa, then x is numerically identical with y. Another point worth mentioning is that trope theory cannot accommodate unexemplified universals, hence it is more appropriate for a reinterpretation of the Aristotelian version of realism rather than the Platonist one.
Further readings:
M.J. Loux, The Problem of Universals II, pp. 62-79 (Metaphysics. A Contemporary Introduction)
E.J. Lowe, Realism Versus Nominalism, pp. 355-365 (A Survey of Metaphysics)
Another problem with metalinguistic nominalism is that it trades the objective, independent notion of property for a language-dependent notion of predicate. But what with properties that are not expressed in any language? There are examples of properties that we discovered and named only recently, such as spin or charm. It is quite natural to expect that there are more properties of that sort which have yet to be discovered. Consequently, a metalinguistic nominalist can’t offer a satisfactory translation for the sentence “Every object has a property that we will never know”.
The initial motivation for metaphysical realism was provided by the existence of objective similarities between particulars. Resemblance nominalism tries to develop and apply the notion of similarity without any recourse to universals. It may be said for instance that to be red is to be sufficiently similar to a paradigmatic red object. But this simple interpretation won’t do. Clearly there may be non-red objects that are similar to a selected red thing (with respect to every property other than colour). A more sophisticated attempt to explain away the attribution of properties may be as follows. The resemblance nominalist may try to define resemblance classes which will roughly correspond to the realist’s properties. For instance, a resemblance class can be defined as a maximal class such that any two objects in this class are more similar to one another than they are to any object outside of the class (more formally this condition can be spelled out as follows: for all x, y, and z, if x and y belong to class K but z does not belong to K, then x is more similar to y than to z). The condition of maximality is needed, because we don’t want to qualify the class of two red objects as a resemblance class. However, three fundamental objections can be made against such a solution.
(1) As we have already indicated, it can be argued that you can find a non-red object which is more similar to a particular red object than this object is to another red thing. Think for example of a green sphere, a red sphere of exactly the same dimensions, and a red cube twice as big as the sphere. It can be argued that the spheres resemble each other more that the red one resembles the red cube.
(2) Let’s consider two properties P and Q such that all objects that have P have Q but not vice versa. In such a case P will not define a resemblance class, for the condition of maximality fails.
(3) The universal class (the class of all particulars) satisfies the condition of being a resemblance class. But it is debatable whether there is a (non-trivial) property that is common to all particulars.
It should be clear that the problems (1) and (2) are a direct consequence of the fact that the nominalist cannot distinguish between various aspects of the similarity relation (for instance we would like to say that the class of red objects is defined by the relation of similarity with respect to colour). One solution that promises to evade this difficulty is known as trope theory. It postulates a new kind of objects – tropes – that may be acceptable to nominalists. Tropes are individual properties: the redness of that rose, the shape of that tree. Two numerically distinct individuals can never share any tropes. However, their tropes can be similar. The idea is that resemblance classes can be defined on tropes, and not on particulars, so that the resemblance class corresponding to redness will contain all tropes of redness. It is easy to notice that problems (1) and (2) disappear in this approach. No non-red trope can be more similar to a particular trope of red than a different trope of red, because tropes don’t have any ‘aspects’: they are themselves aspects. If two tropes are similar, they are always similar in precisely one respect. Problem (2) disappears, because the class of tropes that correspond to one property is always disjoint from the class of tropes corresponding to a numerically distinct property, even if all objects that posses one property possess the other one as well.
An interesting question arises whether the Principle of the Identity of Indiscernibles can be reinterpreted in trope theory. A simple replacement of properties with tropes results in a trivialisation of the PII. In virtue of the definition of tropes, if two individuals share at least one trope, they are numerically identical. A more promising strategy is to reformulate the PII in the form of the requirement that if each trope of object x is similar to a trope of object y, and vice versa, then x is numerically identical with y. Another point worth mentioning is that trope theory cannot accommodate unexemplified universals, hence it is more appropriate for a reinterpretation of the Aristotelian version of realism rather than the Platonist one.
Further readings:
M.J. Loux, The Problem of Universals II, pp. 62-79 (Metaphysics. A Contemporary Introduction)
E.J. Lowe, Realism Versus Nominalism, pp. 355-365 (A Survey of Metaphysics)
Sunday, November 29, 2009
Nominalism and realism
In the previous lecture we saw that unrestricted realism leads to difficulties, including contradictions. Yet another argument can be presented in support of this claim. As we know, the realist interpretation of subject-predicate sentences relies on the notion of exemplification. Thus, the meaning of the statement “Socrates is courageous” is explicated as “Socrates exemplifies courage”. But now we can observe that the last statement can receive a treatment similar to that we applied to the sentence “Socrates is the teacher of Plato”. Namely, we should interpret it as stating that the pair (Socrates, courage) exemplifies the two-argument relation of exemplification E. But this manoeuvre can be repeated: now the last statement is explained as follows: the triple (Socrates, courage, exemplification) exemplifies the three-argument relation of exemplification E’. Clearly, this procedure can be repeated indefinitely, and hence it leads to an infinite regress (this regress is related to the so-called third-man argument considered by Plato and subscribed to by Aristotle, as well as to Bradley’s argument against the existence of irreducible relations). It is open to a debate whether this regress is vicious, but it can be maintained that it actually is, for according to the semantic rules accepted by the realist, the initial statement “Socrates is courageous” does not have a definite meaning unless it is explicated in terms of the meaningful sentence “Socrates exemplifies courage”. However, this last sentence is meaningful only if it can be explicated in terms of yet another meaningful sentence and so on. Consequently, sentences of natural language will never receive their proper meaning.
Another restriction that can be placed on realism stems from the relation between the universals and the particulars that exemplify them. A radical version of realism, called Platonism, insists that universals exist independently of whether they are exemplified. A more moderate view, originally proposed by Aristotle, rejects universals that are not exemplified. Thanks to this restriction, Aristotelians can maintain that universals exist in things (in space-time), and that the way we can discover them is through abstraction from individual objects. On the other hand, Platonists must assume that universals exist beyond space-time, because unexemplified universals do not have specific locations. Aristotelians argue that postulating things that have no power in the spatiotemporal world is useless. Platonists reply to this that unexemplified universals, such as the property of being a unicorn, are necessary to account for the meaning of certain statements (for instance the false sentence “This animal is a unicorn”). They also point out that the fact that some universals are not exemplified is a contingent matter (there could be unicorns), and the existence of universals should not be contingent.
Nominalists reject the existence of universals for many reasons. They point out that postulating universals violates the principle of ontological parsimony (Ockham’s razor). Nominalists emphasise that universals are troublesome entities. One particular problem is their location. If we agree with the Aristotelians that universals are located where the objects that exemplify them are located, then we have to accept some unintuitive consequences, such as the multilocation of properties. For instance, one property can be said to be located at a certain distance from itself. The problem of locating relations is even more difficult (they cannot be said to be wholly located where separate relata are located, or even where the mereological sums of the relata are located). If, on the other hand, we follow the Platonists in their view that universals have no location at all, the question arises how we can have any knowledge about them (without being able to causally interact with universals). Another criticism of universals is that they don’t admit clear-cut criteria of numerical identity. This follows from the fact that universals are not extensional. Two properties can be instantiated by exactly the same particulars, and yet be numerically distinct. An example can be: the property of being the greatest planet in the solar system, and the property of being the fifth planet from the Sun. It is quite clear that these are distinct properties, and yet they are exemplified by exactly the same object: the planet Jupiter.
Several solutions to the problem of the criterion of identity/distinctness for universals can be proposed. One proposal involves possible worlds: two properties are considered identical if they have the same extension in all possible worlds. But this solution still has an unintuitive consequence: two properties that are necessarily empty (such as being a square circle and being a triangular circle) will be treated as one. Another possibility is to stipulate that properties defined with the help of distinct fundamental universals are numerically distinct. According to this criterion the two above-mentioned properties are distinct because the property of being a square and the property of being a triangle are distinct (they are exemplified by distinct individuals). But the following case remains problematic: the property defined as being divisible by 4 and being divisible by 3, and the property of being divisible by 2 and being divisible by 6, are both the same property of being divisible by 12.
The most radical version of nominalism, austere nominalism, insists that there are only particular, individual objects: individual people, tables, trees. The phenomena of attribution agreement, and of objective similarities between individuals, do not require any explanation. The subject-predicate sentences are treated as primitive, not explainable with the help of any further statements. The main problem of the austere nominalist is how to account for abstract expressions in natural language. The only available strategy is to paraphrase the sentences involving abstract reference in the form of statements that are about individual objects only. A simple example of that sort of paraphrase is as follows: instead of saying that triangularity is a shape, we can express the same idea in the sentence “All triangular objects are shaped objects”. But austere nominalism does not offer any systematic way of making nominalist paraphrases. Each sentence has to be approached individually, on a case-by-case basis, and there is no guarantee that a satisfactory solution will be found. Examples of troublesome cases are: “Courage is a moral virtue”, “This tulip and that rose have the same colour”, “Each object possesses a property that we will never know”. The first one cannot be simply interpreted as the statement that all courageous people are morally virtuous, for this last sentence is obviously false. It may be true that all courageous people are morally virtuous ceteris paribus, but this just means that if two people possess the same moral virtues except that one is courageous and the other is not, then the one that is courageous is more virtuous than the other one. It is unclear how a nominalist can express this thought without relying on properties.
Reading:
M.J. Loux, The problem of universals I, pp. 30-43; The problem of universals II, pp. 46-62 (Metaphysics, A Contemporary Introduction).
Another restriction that can be placed on realism stems from the relation between the universals and the particulars that exemplify them. A radical version of realism, called Platonism, insists that universals exist independently of whether they are exemplified. A more moderate view, originally proposed by Aristotle, rejects universals that are not exemplified. Thanks to this restriction, Aristotelians can maintain that universals exist in things (in space-time), and that the way we can discover them is through abstraction from individual objects. On the other hand, Platonists must assume that universals exist beyond space-time, because unexemplified universals do not have specific locations. Aristotelians argue that postulating things that have no power in the spatiotemporal world is useless. Platonists reply to this that unexemplified universals, such as the property of being a unicorn, are necessary to account for the meaning of certain statements (for instance the false sentence “This animal is a unicorn”). They also point out that the fact that some universals are not exemplified is a contingent matter (there could be unicorns), and the existence of universals should not be contingent.
Nominalists reject the existence of universals for many reasons. They point out that postulating universals violates the principle of ontological parsimony (Ockham’s razor). Nominalists emphasise that universals are troublesome entities. One particular problem is their location. If we agree with the Aristotelians that universals are located where the objects that exemplify them are located, then we have to accept some unintuitive consequences, such as the multilocation of properties. For instance, one property can be said to be located at a certain distance from itself. The problem of locating relations is even more difficult (they cannot be said to be wholly located where separate relata are located, or even where the mereological sums of the relata are located). If, on the other hand, we follow the Platonists in their view that universals have no location at all, the question arises how we can have any knowledge about them (without being able to causally interact with universals). Another criticism of universals is that they don’t admit clear-cut criteria of numerical identity. This follows from the fact that universals are not extensional. Two properties can be instantiated by exactly the same particulars, and yet be numerically distinct. An example can be: the property of being the greatest planet in the solar system, and the property of being the fifth planet from the Sun. It is quite clear that these are distinct properties, and yet they are exemplified by exactly the same object: the planet Jupiter.
Several solutions to the problem of the criterion of identity/distinctness for universals can be proposed. One proposal involves possible worlds: two properties are considered identical if they have the same extension in all possible worlds. But this solution still has an unintuitive consequence: two properties that are necessarily empty (such as being a square circle and being a triangular circle) will be treated as one. Another possibility is to stipulate that properties defined with the help of distinct fundamental universals are numerically distinct. According to this criterion the two above-mentioned properties are distinct because the property of being a square and the property of being a triangle are distinct (they are exemplified by distinct individuals). But the following case remains problematic: the property defined as being divisible by 4 and being divisible by 3, and the property of being divisible by 2 and being divisible by 6, are both the same property of being divisible by 12.
The most radical version of nominalism, austere nominalism, insists that there are only particular, individual objects: individual people, tables, trees. The phenomena of attribution agreement, and of objective similarities between individuals, do not require any explanation. The subject-predicate sentences are treated as primitive, not explainable with the help of any further statements. The main problem of the austere nominalist is how to account for abstract expressions in natural language. The only available strategy is to paraphrase the sentences involving abstract reference in the form of statements that are about individual objects only. A simple example of that sort of paraphrase is as follows: instead of saying that triangularity is a shape, we can express the same idea in the sentence “All triangular objects are shaped objects”. But austere nominalism does not offer any systematic way of making nominalist paraphrases. Each sentence has to be approached individually, on a case-by-case basis, and there is no guarantee that a satisfactory solution will be found. Examples of troublesome cases are: “Courage is a moral virtue”, “This tulip and that rose have the same colour”, “Each object possesses a property that we will never know”. The first one cannot be simply interpreted as the statement that all courageous people are morally virtuous, for this last sentence is obviously false. It may be true that all courageous people are morally virtuous ceteris paribus, but this just means that if two people possess the same moral virtues except that one is courageous and the other is not, then the one that is courageous is more virtuous than the other one. It is unclear how a nominalist can express this thought without relying on properties.
Reading:
M.J. Loux, The problem of universals I, pp. 30-43; The problem of universals II, pp. 46-62 (Metaphysics, A Contemporary Introduction).
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