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.
Wednesday, December 16, 2009
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).
Thursday, November 19, 2009
Universals
The concept of universals arises very naturally from the observation that things are similar to one another and that thanks to these similarities they can be grouped in various categories. Classification of objects is the foundation of all knowledge, and without it we wouldn’t be able to discover and describe any regularities. Most probably even the language that we speak would not be possible without things being similar in different respects. A natural metaphysical explanation of the fact that two things of a given kind (trees, chairs, electrons) have “something in common” is that there exists a certain object that stands in a particular relation to those things. For instance, two red flowers can be said to resemble one another because there is a third object, redness, that is somehow present in both flowers.
Thus the notion of a general object – a universal – is born. The fundamental relation that is supposed to hold between ordinary things (which we will now refer to as “particulars”) and universals can be called “instantiation” or “exemplification”. The colour red is instantiated, or exemplified, by various red objects. Hence we can broadly characterize universals as entities that are exemplified (more formally: u is a universal if and only if there is an x such that x exemplifies u), while particulars can exemplify but are never exemplified themselves. We can distinguish two categories of universals: monadic and polyadic. Monadic universals are instantiated by individual objects, while polyadic universals are exemplified by pairs, or more generally n-tuples of objects. Monadic universals are primarily properties, although some insist to identify a separate category of monadic universals, namely kinds. Polyadic universals, on the other hand, comprise relations of many arguments (such as being the father of, or being a child of two parents). We can also distinguish universals of the first order, which are instantiated by particulars only, and universals of higher orders, instantiated by other universals.
Postulating the existence of universals helps us analyzing some semantic features of basic subject-predicate statements of the natural language. If we consider the statement “Socrates is courageous”, it can be said to be true when the object described as “Socrates” exemplifies the property represented by the predicate “is courageous” (i.e. courage). However, the predicate “is courageous” cannot be said to refer to (to be the name of) the property of being courageous. Two different semantic functions have to be distinguished: that of denoting (referring, naming) and that of connoting. The predicate “is courageous” denotes all courageous people, but connotes courage. On the other hand, the name “Socrates” denotes an individual, but connotes nothing (there is no property with respect to which we call an individual “Socrates”). Thus, the semantic analysis of statements of the form “a is P” is as follows: this statement is true if the object denoted by “a” exemplifies the universal connoted by “P”.
The assumption of the existence of universals can also account for the phenomenon of abstract reference of natural language. Abstract reference is visible in sentences of the following sort: “Redness is a colour”, “Courage is a moral virtue”, “This rose and this tulip have the same colour”, “Each object has a property that no one will ever know”, “The relation of being an ancestor is the smallest relation satisfying the following conditions: if x is a parent of y, x is an ancestor of y, and if x is an ancestor of y and y is an ancestor of z, then x is an ancestor of z”. The terms “redness”, “courage”, “colour”, “moral virtue”, “property”, “the relation of being an ancestor”, can be naturally interpreted as denoting (not connoting!) universals. (Exercise: read all the past lectures and find sentences containing abstract reference there.)
The ontological position accepting the existence of universals is called metaphysical realism (realism in short), and its negation is referred to as nominalism. The unrestricted version of realism maintains that for every meaningful predicate of natural language there is a universal connoted by it. But it is highly unlikely that this version of realism could be true. Let us consider the following example: the predicate “has inertial mass m or electric charge q”. While it is natural to expect that the separate predicates “has mass m” and “has charge q” indeed connote some properties, the disjunctive predicate is unlikely to pick a new property. After all, it is unintuitive to assume that an object that has mass m and no electric charge, and another object that has a different mass m’ but electric charge q”, have something in common by virtue of this very fact. Moreover, disjunctive properties typically do not figure in causal explanations or laws of nature.
There is a more formal argument that can be made against the unrestrained version of realism. Let us consider the predicate “is not exemplified by itself”. If this predicate connotes a property, then a contradiction ensues, for it will be true that this property exemplifies itself if and only if it does not exemplify itself. This argument is a version of Russell’s famous paradox of the class of all classes that are not their own elements.
This lecture is based on Chapter 1 "The Problem of Universals I" of Michael Loux's book Metaphysics: A Contemporary Introduction. You are advised to read the entire chapter.
Thus the notion of a general object – a universal – is born. The fundamental relation that is supposed to hold between ordinary things (which we will now refer to as “particulars”) and universals can be called “instantiation” or “exemplification”. The colour red is instantiated, or exemplified, by various red objects. Hence we can broadly characterize universals as entities that are exemplified (more formally: u is a universal if and only if there is an x such that x exemplifies u), while particulars can exemplify but are never exemplified themselves. We can distinguish two categories of universals: monadic and polyadic. Monadic universals are instantiated by individual objects, while polyadic universals are exemplified by pairs, or more generally n-tuples of objects. Monadic universals are primarily properties, although some insist to identify a separate category of monadic universals, namely kinds. Polyadic universals, on the other hand, comprise relations of many arguments (such as being the father of, or being a child of two parents). We can also distinguish universals of the first order, which are instantiated by particulars only, and universals of higher orders, instantiated by other universals.
Postulating the existence of universals helps us analyzing some semantic features of basic subject-predicate statements of the natural language. If we consider the statement “Socrates is courageous”, it can be said to be true when the object described as “Socrates” exemplifies the property represented by the predicate “is courageous” (i.e. courage). However, the predicate “is courageous” cannot be said to refer to (to be the name of) the property of being courageous. Two different semantic functions have to be distinguished: that of denoting (referring, naming) and that of connoting. The predicate “is courageous” denotes all courageous people, but connotes courage. On the other hand, the name “Socrates” denotes an individual, but connotes nothing (there is no property with respect to which we call an individual “Socrates”). Thus, the semantic analysis of statements of the form “a is P” is as follows: this statement is true if the object denoted by “a” exemplifies the universal connoted by “P”.
The assumption of the existence of universals can also account for the phenomenon of abstract reference of natural language. Abstract reference is visible in sentences of the following sort: “Redness is a colour”, “Courage is a moral virtue”, “This rose and this tulip have the same colour”, “Each object has a property that no one will ever know”, “The relation of being an ancestor is the smallest relation satisfying the following conditions: if x is a parent of y, x is an ancestor of y, and if x is an ancestor of y and y is an ancestor of z, then x is an ancestor of z”. The terms “redness”, “courage”, “colour”, “moral virtue”, “property”, “the relation of being an ancestor”, can be naturally interpreted as denoting (not connoting!) universals. (Exercise: read all the past lectures and find sentences containing abstract reference there.)
The ontological position accepting the existence of universals is called metaphysical realism (realism in short), and its negation is referred to as nominalism. The unrestricted version of realism maintains that for every meaningful predicate of natural language there is a universal connoted by it. But it is highly unlikely that this version of realism could be true. Let us consider the following example: the predicate “has inertial mass m or electric charge q”. While it is natural to expect that the separate predicates “has mass m” and “has charge q” indeed connote some properties, the disjunctive predicate is unlikely to pick a new property. After all, it is unintuitive to assume that an object that has mass m and no electric charge, and another object that has a different mass m’ but electric charge q”, have something in common by virtue of this very fact. Moreover, disjunctive properties typically do not figure in causal explanations or laws of nature.
There is a more formal argument that can be made against the unrestrained version of realism. Let us consider the predicate “is not exemplified by itself”. If this predicate connotes a property, then a contradiction ensues, for it will be true that this property exemplifies itself if and only if it does not exemplify itself. This argument is a version of Russell’s famous paradox of the class of all classes that are not their own elements.
This lecture is based on Chapter 1 "The Problem of Universals I" of Michael Loux's book Metaphysics: A Contemporary Introduction. You are advised to read the entire chapter.
Thursday, November 12, 2009
Identity and Indiscernibility
What logical relations connect qualitative identity (indiscernibility) and numerical identity? First of all, there is Leibniz's law, or the principle of the indiscernibility of identicals, which states that numerical identity implies indiscernibility. More formally, this principle can be written as follows: For all x, y, if x = y, then for all P, if x has P, then y has P. (If you are worried that this formalization can only ensure that all x's properties will be y's properties, note that you can swap x with y and get the conclusion that all y's properties are x's properties as well). Alternatively, we may put this idea like that: if x has a property that y doesn't have, x must be distinct from y. Leibniz's law seems to be rather uncontroversial; however some possible counterexamples to it are conceivable. Let us consider the property of number 8 that it is necessarily greater than 6 (as a matter of mathematical necessity, 8 has to be greater than 6). But 8 is numerically identical to the number of planets in the solar system (excluding Pluto). But is it really necessary that the number of planets is greater than 6? (We will analyse this problem in more detail later in the course.) This example teaches us that perhaps not every proposition true of a given object picks out its property. The expression 'it is necessary that' is an example of an intensional context, which means that the truth of a statement built with its help depends not only on the reference of the terms included in the statement, but also on their meaning (the meaning of 'number 8' is obviously different from the meaning of 'the number of planets in the solar system').
A more controversial principle connecting numerical identity with qualitative identity is the principle of the identity of indiscernibles (PII for short). It states that all indiscernible objects are identical, or more formally that for all x and y, if x and y have the same properties, x is numerically identical with y. Alternatively, PII can be presented as the principle of the discernibility of distinct, i.e. if x is not numerically identical with y, x and y are discernible (there is at least one property that they don't have in common). It is important to notice that the meaning and status of PII depends on how broad the category of properties which can discern individual objects is. For instance, suppose that we agree that for any object a being identical with a is a property of a. In such a case PII becomes trivially satisfied, for if a and b agree with respect to all their properties, b will have to possess the property of being identical with a which settles the issue of their numerical identity. In order to avoid this trivialisation, we should restrict the range of acceptable properties to so-called qualitative properties (properties that do not involve numerical identity, or any form of 'labelling'). On the other hand, it is acceptable to discern objects with the help of relational properties (for instance, being at a given distance from a third object c).
At first sight it looks like PII should be satisfied at least by all material, spatiotemporal objects, for numerically distinct things are discernible by their different space-time locations. This conclusion can be accepted if we assume that material objects are impenetrable, i.e. that they cannot occupy the same space at the same time. In spite of this argument, it turns out that plausible counterexamples to PII can be presented. One of them is the famous Max Black example involving two spheres. Suppose that the universe consists only of two objects which happen to be two perfect spheres made of pure iron of exactly the same dimensions. In such a case the spheres seem to be indiscernible even by their location, because there is no third object in this entirely symmetrical universe which would stand in different spatiotemporal relations to the spheres. Black's counterexample does not show that PII is actually false (obviously our universe is not even similar to this impoverished world), but at least proves that PII can be only contingently true (that the falsity of PII is conceivable).
But a stronger claim can be made that PII is actually false in our world. According to modern physics, subatomic particles do not possess definite trajectories, thus their discernibility in terms of individual locations is not guaranteed. Moreover, quantum physics introduces the so-called symmetrization postulate which puts severe constraints on possible states of many-particles systems of the same type. The postulate requires that the state of many particles be either a symmetric, or an antisymmetric function* (the first case is when the particles in questions are bosons, e.g. photons; the second when they are fermions). From the postulate it follows that all measurable properties of particles of the same type assume the same values (either deterministic or probabilistic). This fact is interpreted by many as a clear sign that PII is violated in the case of elementary particles of the same type.
*) A symmetric function is a function that gets transformed onto itself when we swap its arguments. An antisymmetric function changes its sign under this operation.
Appendix
Recently an argument in favour of the discernibility of quantum particles has been proposed which involves the notion of weak discernibility. Two objects x and y are said to be weakly discernible if there is an irreflexive relation R (a relation that does not hold between an object and itself) which connects these objects. It can be argued that fermions are connected by an irreflexive relation "having opposite spin directions". But it is doubtful whether weak discernibility presents a case of genuine discernibility. It can be easily verified that the weak discernibility of two objects x and y can only guarantee that they can be discerned in the following way: the relation R connects x with y, but does not connect x with x, hence x discerns relationally x from y. But this is the same sort of 'discernibility' as in the case of the property of being identical with x, and therefore should not be accepted.
A more controversial principle connecting numerical identity with qualitative identity is the principle of the identity of indiscernibles (PII for short). It states that all indiscernible objects are identical, or more formally that for all x and y, if x and y have the same properties, x is numerically identical with y. Alternatively, PII can be presented as the principle of the discernibility of distinct, i.e. if x is not numerically identical with y, x and y are discernible (there is at least one property that they don't have in common). It is important to notice that the meaning and status of PII depends on how broad the category of properties which can discern individual objects is. For instance, suppose that we agree that for any object a being identical with a is a property of a. In such a case PII becomes trivially satisfied, for if a and b agree with respect to all their properties, b will have to possess the property of being identical with a which settles the issue of their numerical identity. In order to avoid this trivialisation, we should restrict the range of acceptable properties to so-called qualitative properties (properties that do not involve numerical identity, or any form of 'labelling'). On the other hand, it is acceptable to discern objects with the help of relational properties (for instance, being at a given distance from a third object c).
At first sight it looks like PII should be satisfied at least by all material, spatiotemporal objects, for numerically distinct things are discernible by their different space-time locations. This conclusion can be accepted if we assume that material objects are impenetrable, i.e. that they cannot occupy the same space at the same time. In spite of this argument, it turns out that plausible counterexamples to PII can be presented. One of them is the famous Max Black example involving two spheres. Suppose that the universe consists only of two objects which happen to be two perfect spheres made of pure iron of exactly the same dimensions. In such a case the spheres seem to be indiscernible even by their location, because there is no third object in this entirely symmetrical universe which would stand in different spatiotemporal relations to the spheres. Black's counterexample does not show that PII is actually false (obviously our universe is not even similar to this impoverished world), but at least proves that PII can be only contingently true (that the falsity of PII is conceivable).
But a stronger claim can be made that PII is actually false in our world. According to modern physics, subatomic particles do not possess definite trajectories, thus their discernibility in terms of individual locations is not guaranteed. Moreover, quantum physics introduces the so-called symmetrization postulate which puts severe constraints on possible states of many-particles systems of the same type. The postulate requires that the state of many particles be either a symmetric, or an antisymmetric function* (the first case is when the particles in questions are bosons, e.g. photons; the second when they are fermions). From the postulate it follows that all measurable properties of particles of the same type assume the same values (either deterministic or probabilistic). This fact is interpreted by many as a clear sign that PII is violated in the case of elementary particles of the same type.
*) A symmetric function is a function that gets transformed onto itself when we swap its arguments. An antisymmetric function changes its sign under this operation.
Appendix
Recently an argument in favour of the discernibility of quantum particles has been proposed which involves the notion of weak discernibility. Two objects x and y are said to be weakly discernible if there is an irreflexive relation R (a relation that does not hold between an object and itself) which connects these objects. It can be argued that fermions are connected by an irreflexive relation "having opposite spin directions". But it is doubtful whether weak discernibility presents a case of genuine discernibility. It can be easily verified that the weak discernibility of two objects x and y can only guarantee that they can be discerned in the following way: the relation R connects x with y, but does not connect x with x, hence x discerns relationally x from y. But this is the same sort of 'discernibility' as in the case of the property of being identical with x, and therefore should not be accepted.
Sunday, November 1, 2009
Identity
Two notions of identity have to be distinguished: qualitative identity and numerical identity. Qualitative identity can be defined in a relatively simple way: two objects are qualitatively identical if and only if they have all properties in common. Formally, we can write this as follows: x is qualitatively identical with y iff for all properties P, x has P if and only if y has P. It remains to be decided what is the range of the property variable P, but let's put this aside for the moment. However, the notion of numerical identity presents a challenge. It is not trivial to come up with a direct definition of this concept. One suggestion may be to characterize numerical identity as the relation that holds between each object and itself, and only itself. But this definition only appears to be correct. In fact it is not, since the condition that every object is identical only to itself is in fact empty, as it can be presented equivalently as follows: if y is distinct from x, then x is not identical with y. But this last expression is tautologous, hence does not restrict the notion of identity in any way. Consequently, we are left with the characteristic that identity holds between every object and itself, and this is definitely too broad.
Another possible definitional characteristic of numerical identity is similarly bound to fail. Suppose that we stipulate that for x to be numerically identical with y there has to be exactly one object that is both x and y. But the expression "exactly one object" contains hidden reference to the notion of identity. The standard way of interpreting the expression "There is exactly one x such that Px" is "There is an x such that Px and for all y, if Py, then y is identical to x". The proposed definition of identity turns out to be circular.
It is commonly accepted that the relation of numerical identity should satisfy the following conditions:
1. Reflexivity. For all x, x is identical to x
2. Symmetricity. For all x, y, if x is identical to y, y is identical to x
3. Transitivity. For all x, y, z, if x is identical to y, and y is identical to z, then x is identical to z
But obviously not all relations satisfying 1-3 are identities. The conditions 1-3 define what is called equivalence relations. However, equivalence relations are closely connected with identity. It turns out that each equivalence relation can be turned into numerical identity if we change appropriately the domain of the relation. To give an example, let us consider the relation of being parallel defined on the set of all straight lines on a plane. This relation is clearly not identity, but if we introduce the abstract notion of a direction (meant either as the property common to all mutually parallel lines, or as their set), then it can be claimed that this relation is reducible to identity when defined on the set of all directions (two "parallel" directions are actually one and the same direction).
In order to obtain the relation of identity, we have to add to 1-3 the following condition:
4. Identity is the smallest relation satisfying conditions 1-3
where the condition 4 is understood in such a way that the relation of identity must be included in any relation satisfying 1-3.
Another possible definitional characteristic of numerical identity is similarly bound to fail. Suppose that we stipulate that for x to be numerically identical with y there has to be exactly one object that is both x and y. But the expression "exactly one object" contains hidden reference to the notion of identity. The standard way of interpreting the expression "There is exactly one x such that Px" is "There is an x such that Px and for all y, if Py, then y is identical to x". The proposed definition of identity turns out to be circular.
It is commonly accepted that the relation of numerical identity should satisfy the following conditions:
1. Reflexivity. For all x, x is identical to x
2. Symmetricity. For all x, y, if x is identical to y, y is identical to x
3. Transitivity. For all x, y, z, if x is identical to y, and y is identical to z, then x is identical to z
But obviously not all relations satisfying 1-3 are identities. The conditions 1-3 define what is called equivalence relations. However, equivalence relations are closely connected with identity. It turns out that each equivalence relation can be turned into numerical identity if we change appropriately the domain of the relation. To give an example, let us consider the relation of being parallel defined on the set of all straight lines on a plane. This relation is clearly not identity, but if we introduce the abstract notion of a direction (meant either as the property common to all mutually parallel lines, or as their set), then it can be claimed that this relation is reducible to identity when defined on the set of all directions (two "parallel" directions are actually one and the same direction).
In order to obtain the relation of identity, we have to add to 1-3 the following condition:
4. Identity is the smallest relation satisfying conditions 1-3
where the condition 4 is understood in such a way that the relation of identity must be included in any relation satisfying 1-3.
Tuesday, October 27, 2009
The ontological argument for God's existence
The final blow to the property view of existence is dealt by the so-called ontological argument for the existence of God. The argument, whose original version is due to St Anselm, can be presented as follows. Let us define God as a being than which nothing greater can be conceived. As you should recall, underlying the property view is the assumption that to each description there corresponds an object which can be either existent or non-existent. Suppose that the object which corresponds to the above description is a non-existent entity (i.e. that God does not exist). In that case it can be argued that a greater being can be conceived, namely God that possesses the property of existence. Hence we have a contradiction, since the object which was to satisfy the description ("the greatest conceivable being") in fact does not satisfy it. The cause of the contradiction was the assumption that God does not exist, thus we conclude that God has to exist.
Analogously we could argue that there exists the greatest imaginable island, or any other object for that matter. It is even possible to eliminate the controversial assumption that existence is greater than non-existence. We could simply define the following concept: God who possesses the property of existence. By the same reasoning we can show that the assumption that the referent of this description is a non-existent object leads to a contradiction. But it cannot be so easy to prove the existence of God or any other object you could think of. The culprit was the assumption that each description has a referent - existent or non-existent. If we abandon it, the argument can't even get off the ground, since we cannot argue that it is possible to conceive a greater being than non-existent God. There is no non-existent God, hence we cannot compare it with anything else.
The quantifier view of existence deals with the problem easily. If God existed, it would be the greatest being conceivable, but if God does not exist, there is simply no object in the whole universe than which nothing greater can be conceived (it is not true that for some x, x is God). In a similar way, Kant stressed that existence is not a property whose absence can be detrimental to the greatness of an object. Kant's view should not be interpreted as implying that an existing coin does not differ from a non-existent one. Rather, a non-existent coin is not an object at all, so "it" cannot differ from anything. If we formulate a description of an object, adding to its definitional characteristic that it should exist does not make the description more informative.
Required reading:
B. Garrett, "The Ontological Argument", in What is this thing called metaphysics?, pp. 4-7
Analogously we could argue that there exists the greatest imaginable island, or any other object for that matter. It is even possible to eliminate the controversial assumption that existence is greater than non-existence. We could simply define the following concept: God who possesses the property of existence. By the same reasoning we can show that the assumption that the referent of this description is a non-existent object leads to a contradiction. But it cannot be so easy to prove the existence of God or any other object you could think of. The culprit was the assumption that each description has a referent - existent or non-existent. If we abandon it, the argument can't even get off the ground, since we cannot argue that it is possible to conceive a greater being than non-existent God. There is no non-existent God, hence we cannot compare it with anything else.
The quantifier view of existence deals with the problem easily. If God existed, it would be the greatest being conceivable, but if God does not exist, there is simply no object in the whole universe than which nothing greater can be conceived (it is not true that for some x, x is God). In a similar way, Kant stressed that existence is not a property whose absence can be detrimental to the greatness of an object. Kant's view should not be interpreted as implying that an existing coin does not differ from a non-existent one. Rather, a non-existent coin is not an object at all, so "it" cannot differ from anything. If we formulate a description of an object, adding to its definitional characteristic that it should exist does not make the description more informative.
Required reading:
B. Garrett, "The Ontological Argument", in What is this thing called metaphysics?, pp. 4-7
Thursday, October 22, 2009
Existence part II. Quantifiers
The conception of non-existent objects creates more problems than it solves. But there is an alternative approach. We can insist that all objects are existent and still solve the semantic problem of negative existential statements. One such solution is based on the observation that "existence" is a second-order notion, i.e. it applies not to things but to concepts. The sentence "Vulcan does not exist" is in fact about the concept of "Vulcan", and it states that the concept is empty (and not that it does not exist - obviously we have to assume that concepts exist).
Another solution is as follows. When we state that Vulcan does not exist, we in fact talk not about (non-existent) Vulcan, but about all objects in our universe - and we say that there is no Vulcan among them. Hence the sentence can be understood as being equivalent to "No object is Vulcan", which clearly does not carry the presupposition that Vulcan is an object. Using the definition of the term "Vulcan" we can express the same thought in an even more straightforward fashion as "No planet is closer to the Sun than Mercury", which leaves no trace of suggestion that it is some non-existent object that we talk about.
According to the quantifier conception of existence, all existential statements have to be reformulated in order to reveal their genuine logical structure, different from their surface grammar. The sentence "Elephants exist" does not attribute the property of existence to elephants, but rather speaks in a general fashion about all objects in the universe, claiming that some of them are elephants. Thus the literal form of the sentence should be "There is an x such that x is an elephant", where expression "there is" is known as the existential quantifier, "x" is a variable that can take any value from the domain of all objects, and "is an elephant" is a predicate. The formalization of the statement "Vulcan does not exist" leads to either "It is not the case that there is an x such that x is Vulcan" or, equivalently "For all x, x is not Vulcan" ("for all" is another quantifier, called universal).
It has to be stressed that the expression "x is Vulcan" is to be understood as consisting of the predicate "is Vulcan", and not as being equivalent to the identity statement x = Vulcan. In logic you can apply the identity symbol only to variables and terms called "proper names", of which it is assumed that they represent exactly one object. "Vulcan" cannot be interpreted as a proper name, since there is no object that is represented by that name. Generally, no empty name can be interpreted as a proper name, but only as a predicate.
One consequence of the quantifier view of existence is that certain expressions in natural language become "ontologically committing" (this notion was introduced by Willard V.O. Quine). For instance, the use of the word "some" indicates that we have to accept the existence of certain objects. If I say "Some thoughts of philosopher A are hard to understand", my utterance should be literally interpreted as "There is an x such that x is a thought of philosopher A and x is hard to understand", which implies that there are objects which are thoughts. In order to avoid unwanted ontological commitments, we can try to paraphrase our statements so that they no longer imply the existence of dubious entities.
Some critics point out that the quantifier view of existence commits us to the truth of the statement "Everything exist", which is clearly absurd. While it is true that this is a consequence of the quantifier view, it is by no means absurd. "Everything exists" means "For all x, x exists", and if we accept the quantifier interpretation of existence this statement has to be interpreted as "For all x there is a y such that x = y", which states a trivial truth that everything is identical with something. Instead of saying that not everything exists (which is self-contradictory on any interpretation that eliminates non-existent objects) we should better express our intuition in the statement "Some concepts are empty (non-referring)".
Further reading:
B. Garrett, "What is existence?" in What is this thing called metaphysics?
Another solution is as follows. When we state that Vulcan does not exist, we in fact talk not about (non-existent) Vulcan, but about all objects in our universe - and we say that there is no Vulcan among them. Hence the sentence can be understood as being equivalent to "No object is Vulcan", which clearly does not carry the presupposition that Vulcan is an object. Using the definition of the term "Vulcan" we can express the same thought in an even more straightforward fashion as "No planet is closer to the Sun than Mercury", which leaves no trace of suggestion that it is some non-existent object that we talk about.
According to the quantifier conception of existence, all existential statements have to be reformulated in order to reveal their genuine logical structure, different from their surface grammar. The sentence "Elephants exist" does not attribute the property of existence to elephants, but rather speaks in a general fashion about all objects in the universe, claiming that some of them are elephants. Thus the literal form of the sentence should be "There is an x such that x is an elephant", where expression "there is" is known as the existential quantifier, "x" is a variable that can take any value from the domain of all objects, and "is an elephant" is a predicate. The formalization of the statement "Vulcan does not exist" leads to either "It is not the case that there is an x such that x is Vulcan" or, equivalently "For all x, x is not Vulcan" ("for all" is another quantifier, called universal).
It has to be stressed that the expression "x is Vulcan" is to be understood as consisting of the predicate "is Vulcan", and not as being equivalent to the identity statement x = Vulcan. In logic you can apply the identity symbol only to variables and terms called "proper names", of which it is assumed that they represent exactly one object. "Vulcan" cannot be interpreted as a proper name, since there is no object that is represented by that name. Generally, no empty name can be interpreted as a proper name, but only as a predicate.
One consequence of the quantifier view of existence is that certain expressions in natural language become "ontologically committing" (this notion was introduced by Willard V.O. Quine). For instance, the use of the word "some" indicates that we have to accept the existence of certain objects. If I say "Some thoughts of philosopher A are hard to understand", my utterance should be literally interpreted as "There is an x such that x is a thought of philosopher A and x is hard to understand", which implies that there are objects which are thoughts. In order to avoid unwanted ontological commitments, we can try to paraphrase our statements so that they no longer imply the existence of dubious entities.
Some critics point out that the quantifier view of existence commits us to the truth of the statement "Everything exist", which is clearly absurd. While it is true that this is a consequence of the quantifier view, it is by no means absurd. "Everything exists" means "For all x, x exists", and if we accept the quantifier interpretation of existence this statement has to be interpreted as "For all x there is a y such that x = y", which states a trivial truth that everything is identical with something. Instead of saying that not everything exists (which is self-contradictory on any interpretation that eliminates non-existent objects) we should better express our intuition in the statement "Some concepts are empty (non-referring)".
Further reading:
B. Garrett, "What is existence?" in What is this thing called metaphysics?
Tuesday, October 13, 2009
Existence part I. Non-existent objects
Existence is one of the most fundamental notions of metaphysics. Hence it may be advisable to start our course with a precise definition of this term. Unfortunately this task is not so easy to complete, and some even claim that the term "exist" is primitive and cannot be defined. Others come up with fancy and long-winded definitions of existence that explain nothing. Here, instead of providing a direct definition, we will focus on the way the term "exist" functions in philosophical language and its logical connections with other, closely related terms.
Let us begin with a seemingly straightforward observation: apparently we should agree that not everything exists. We can give numerous examples of fictional objects (objects that don't exist): dwarfs, fairies, centaurs, Santa Claus. But the list is not limited to mythological creatures. In science we often discover that something does not exist, for instance aether or Vulcan (hypothetical planet located inside the orbit of Mercury). In math we can rigorously prove that the greatest prime number does not exist.
But there is a logical problem here. Let us consider the sentence "Vulcan does not exist". What is this sentence about? If, as its grammatical structure suggests, it is a sentence about Vulcan, then we have a contradiction here, because Vulcan has to exist in order for the sentence to be true. One way out of trouble is to divorce the notion of an object from the concept of existence. The initial sentence is about a particular object - Vulcan, that is - but this object does not have to exist. In fact Vulcan is a nonexistent object. We have arrived at a metaphysical conception according to which the set of all objects splits into two parts: existing and non-existing ones. The word "exist" is no longer synonymous with "there is" (there is Santa Claus, but he doesn't exist). Existence becomes a property of only some objects. The main proponent of this theory is the Austrian philosopher Alexius Meinong.
The conception of non-existent objects encounters serious difficulties. Some of them are listed below.
1. What is the extent of the domain of non-existent objects? It turns out that this domain has to be pretty large, vastly outnumbering the domain of existing things. To each description must correspond an object - in the majority of cases a non-existent one. Let's consider for instance the following description: "an x-foot high golden mountain". For each real number x there is a different non-existent object that satisfies this description, so we have easily created a continuum of non-existent objects. Morever, some of the non-existent objects are contradictory: compare a square circle. But this means that we have to accept a pair of contradictory statement: "the square circle is a circle" and "the square circle is not a circle".
2. Non-existent objects are incomplete. Vulcan can only be said to possess two properties: being a planet, and being closer to the Sun than any other planet (this is a relational property). But Vulcan is indeterminate with respect to any other properties that typically characterize planets: its mass, density, period of revolution, period of rotation, etc. Consequently, non-existent objects do not admit unambiguous criteria of identity and distinctness. It is impossible even in principle to decide whether Vulcan whose diameter equals 10000 km is identical with or distinct from the Vulcan whose period of revolution around its axis equals 20 hours.
3. Non-existent objects do not admit answers to the "How many?" questions. How many Vulcans are there? What if we define Vulcan as the only planet that orbits the Sun closer than Mercury? Then it looks like it is a definitional property of Vulcan that there is only one Vulcan, but on the other hand we can have two incompatible descriptions "The unique Vulcan whose diameter equals 1000 km" and "The unique Vulcan whose diameter equals 2000 km". Both non-existent referents of these descriptions obviously satisfy the definition of Vulcan simpliciter. If there are two unique non-existent Vulcans, we have a contradiction.
4. In what sense do non-existent object possess their properties? Vulcan is said to be an object that possesses the property of being a planet and the (relational) property of being closer to the Sun than any other planet, including Mercury. But if Vucan is literally located inside the orbit of Mercury, then it should be possible to see it there. Moreover, it should gravitationally affect Mercury and any other planet. But this is absurd. No non-existent object can gravitationally interact with existent ones. Some reply to this objection that non-existent Vulcan possesses its properties in a different sense than existent objects. But when we defined the concept of Vulcan, we wanted it to denote an object that literally possesses its definitional properties. So non-existent Vulcan understood in that way cannot be claimed to be a referent of our description.
5. If we admit non-existent objects, all laws of nature become literally false. The statement "All metals conduct electricity" is falsified by non-existent metals that do not conduct electricity. One possible solution: to limit the quantifiers in the laws of nature to existent objects only. But in that case we lose an ability to distinguish between laws and accidental generalizations.
Required reading:
Brian Garrett, "Non-existent objects", in What is this thing called metaphysics?, pp.27-31.
Let us begin with a seemingly straightforward observation: apparently we should agree that not everything exists. We can give numerous examples of fictional objects (objects that don't exist): dwarfs, fairies, centaurs, Santa Claus. But the list is not limited to mythological creatures. In science we often discover that something does not exist, for instance aether or Vulcan (hypothetical planet located inside the orbit of Mercury). In math we can rigorously prove that the greatest prime number does not exist.
But there is a logical problem here. Let us consider the sentence "Vulcan does not exist". What is this sentence about? If, as its grammatical structure suggests, it is a sentence about Vulcan, then we have a contradiction here, because Vulcan has to exist in order for the sentence to be true. One way out of trouble is to divorce the notion of an object from the concept of existence. The initial sentence is about a particular object - Vulcan, that is - but this object does not have to exist. In fact Vulcan is a nonexistent object. We have arrived at a metaphysical conception according to which the set of all objects splits into two parts: existing and non-existing ones. The word "exist" is no longer synonymous with "there is" (there is Santa Claus, but he doesn't exist). Existence becomes a property of only some objects. The main proponent of this theory is the Austrian philosopher Alexius Meinong.
The conception of non-existent objects encounters serious difficulties. Some of them are listed below.
1. What is the extent of the domain of non-existent objects? It turns out that this domain has to be pretty large, vastly outnumbering the domain of existing things. To each description must correspond an object - in the majority of cases a non-existent one. Let's consider for instance the following description: "an x-foot high golden mountain". For each real number x there is a different non-existent object that satisfies this description, so we have easily created a continuum of non-existent objects. Morever, some of the non-existent objects are contradictory: compare a square circle. But this means that we have to accept a pair of contradictory statement: "the square circle is a circle" and "the square circle is not a circle".
2. Non-existent objects are incomplete. Vulcan can only be said to possess two properties: being a planet, and being closer to the Sun than any other planet (this is a relational property). But Vulcan is indeterminate with respect to any other properties that typically characterize planets: its mass, density, period of revolution, period of rotation, etc. Consequently, non-existent objects do not admit unambiguous criteria of identity and distinctness. It is impossible even in principle to decide whether Vulcan whose diameter equals 10000 km is identical with or distinct from the Vulcan whose period of revolution around its axis equals 20 hours.
3. Non-existent objects do not admit answers to the "How many?" questions. How many Vulcans are there? What if we define Vulcan as the only planet that orbits the Sun closer than Mercury? Then it looks like it is a definitional property of Vulcan that there is only one Vulcan, but on the other hand we can have two incompatible descriptions "The unique Vulcan whose diameter equals 1000 km" and "The unique Vulcan whose diameter equals 2000 km". Both non-existent referents of these descriptions obviously satisfy the definition of Vulcan simpliciter. If there are two unique non-existent Vulcans, we have a contradiction.
4. In what sense do non-existent object possess their properties? Vulcan is said to be an object that possesses the property of being a planet and the (relational) property of being closer to the Sun than any other planet, including Mercury. But if Vucan is literally located inside the orbit of Mercury, then it should be possible to see it there. Moreover, it should gravitationally affect Mercury and any other planet. But this is absurd. No non-existent object can gravitationally interact with existent ones. Some reply to this objection that non-existent Vulcan possesses its properties in a different sense than existent objects. But when we defined the concept of Vulcan, we wanted it to denote an object that literally possesses its definitional properties. So non-existent Vulcan understood in that way cannot be claimed to be a referent of our description.
5. If we admit non-existent objects, all laws of nature become literally false. The statement "All metals conduct electricity" is falsified by non-existent metals that do not conduct electricity. One possible solution: to limit the quantifiers in the laws of nature to existent objects only. But in that case we lose an ability to distinguish between laws and accidental generalizations.
Required reading:
Brian Garrett, "Non-existent objects", in What is this thing called metaphysics?, pp.27-31.
Thursday, October 8, 2009
Introductory remarks
Philosophy is traditionally divided into metaphysics (ontology), epistemology (the study of knowledge) and axiology the study of values (ethics + aesthetics). Metaphysics is variously characterized as a study of reality itself, of what exists, of being. It is the study of reality as opposed to appearances.
The word “metaphysics” derives from the Greek “ta meta ta physica” (literally "what comes after physics"), which was the title of one of Aristotle’s treatises that followed the one devoted to physics. Aristotle never used this term. However, he used the term “first philosophy” and he explained it twofold.
First: as a knowledge of first causes. The central notion of Aristotle's first philosophy was God, or the Unmoved Mover.
Second: as the study of ‘being qua being” (being as such). This is a universal science that considers all the objects that there are. It also deals with very general notions that apply to all beings: identity, similarity, difference. Moreover, it delineates general classes of being, called categories.
Metaphysics is concerned with what is required for something to exist: the nature of things. One of the central questions of metaphysics is: what kinds of things are there? Are there any types of things other than ordinary physical objects? For instance properties: things that physical objects have in common. Do they exist, or is the talk about properties merely metaphorical?
Examples of categories of objects that are analyzed by ontology are: concretes, abstracts, universals, particulars, sets, numbers, events, minds.
An important part of metaphysics is an analysis of the fundamental structure of the material world. The fundamental categories with the help of which we describe the physical world are the categories of time, space, and causation. Philosophy of time forms a central part of modern metaphysics. It considers, among others, the following questions: Is the passing of time real or illusory? Do past and future events exist? Is time an independent substance, or does it depend ontically on the material world? In considering these and other metaphysical questions about the nature of time we should takie into account the advancements of modern science, in particular physics. Especially the development of the special and general theories of relativity had a significant impact on the modern metaphysics of time.
Tuesday, September 22, 2009
Syllabus
Ontology (also referred to as metaphysics) is one of the three fundamental branches of philosophy, the remaining two being epistemology (the theory of knowledge) and axiology (the theory of value). Traditionally, ontology is characterised as the most general study of what exists (of being qua being, in Aristotle’s terminology), while metaphysics is understood somewhat broader as reaching into the field of so-called philosophical theology (the inquiry into the nature of the supreme being). Ontology investigates the most general and ubiquitous features of reality such as existence, identity, objects and its properties, time and space, causation, determinism and free will. One of the fundamental ontological problems is the delineation and analysis of the basic categories of beings, such as things, properties, classes, facts, states of affairs, events. In its investigations, ontology draws upon a variety of methods, including conceptual analysis, empirical generalisations (based on scientific theories), and the formal methods of mathematics and logic.
This course is intended as an introduction into the problems and methods of modern ontology. The primary approach of the course will be non-historic, and the main emphasis will be put on the current developments of ontology in the analytic Anglo-American tradition of philosophy. Unavoidable references to past philosophical schools and thinkers will be made through the prism of modern investigations of the subject. Thus the students will have an opportunity to learn about state-of-the-art contributions to the subject of ontology and metaphysics by eminent contemporary philosophers. Each particular group of ontological problems will be presented in a way that is accessible and engaging to a beginner-to-intermediate audience. The method of presentation of successive topics will typically be three-tiered. On the first level a problem (or problems) will be stated in a uniform conceptual framework using carefully selected and precisely defined terms. On the second level possible solutions to the problems will be laid down. The final, and most advanced level will present arguments for and against selected solutions which will subsequently be evaluated.
The main topics covered in the lecture will include: the notions of existence, objecthood and identity; the problem of universals and particulars; events, facts, and states of affairs; possible objects and worlds; time and space-time; persistence in time; causation; determinism; free will.
Readings:
Brian Garrett, What Is This Thing Called Metaphysics?, Routledge (Taylor & Francis) 2006
Michael J. Loux, Metaphysics. A Contemporary Introduction, Third edition, Routledge (Taylor & Francis) 2006
E.J. Lowe, A Survey of Metaphysics, Oxford University Press 2002
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