In philosophy, mathematics, and logic, a property is a characteristic of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property however differs from individual objects in that it may be instantiated, and often in more than one thing. It differs from the logical/mathematical concept of class by not having any concept of extensionality, and from the philosophical concept of class in that a property is considered to be distinct from the objects which possess it. Understanding how different individual entities (or particulars) can in some sense have some of the same properties is the basis of the problem of universals. The terms attribute and quality have similar meanings.
YouTube Encyclopedic

1/5Views:101 5471 207 76145 5701 72530 798

Intro to the Philosophy of Mathematics (Ray Monk)

Is Math a Feature of the Universe or a Feature of Human Creation?  Idea Channel  PBS

Properties of Relations

The Nature of Properties & Particulars: Three Disputes (David Armstrong)

PHILOSOPHY  Metaphysics: Emergence
Transcription
I'm going to talk about, give a general introduction to, the philosophy of mathematics. And I'm not going to assume any knowledge at all of either philosophy or mathematics. So I'm sorry if it seems a bit basic to some of you. Okay, I mean, the first thing to say, I think, is that mathematics and philosophy, which I think most people tend to think are poles apart, are actually closer than you might think, and a lot of the great philosophers have also been great mathematicians. And I think that's not an accident. I think there's a close affinity actually between the two subjects in several respects, the most noticeable perhaps is that both require and demand thinking on a very abstract level. But as I hope to show today, there's a lot more to the relationship between the two than that, and in particular, that mathematics has furnished philosophy both with a model of a certain kind of knowledge and with a set of deep and interesting philosophical problems. But the two go together, Plato knew this. It's said that at the opening of Plato's, at the gate of Plato's Academy, it said: "let no one into here who knows no geometry." Interesting, I think, that it's geometry rather than arithmetic, and there's a story there which I'm going to tell which is to do with the fact that the ancient Greeks regarded geometry rather than arithmetic as the more foundational, the superior branch of mathematics. And one of the main reasons for that was the unhappy story of Pythagoras. Most of you I think will know the name Pythagoras from his famous theorem which we all learned at school which says that the square on the hypotenuse of a right angle triangle is equal to the sum of the squares on the other two sides. Well, that very theorem presented a problem to Pythagoras and his followers. I don't know whether many of you know this but Pythagoras was a sort of cult figure in ancient Greece, and he had a band of followers who were dedicated to the mysticism of numbers. It was one of their precepts that everything in the world could be expressed as number, and in particular, as the ratio between two whole numbers. Well now, if you go back to Pythagoras's theorem, it's a consequence of that theorem actually that that belief is unsustainable. And the reason for that is that reflecting upon that theorem leads you straight into what are called irrational numbers, numbers that cannot be expressed as the ratio between two whole numbers. To see this, imagine a right angle triangle with length 1 here and length 1 here. So you've got a square on here of 1, a square on here of 1. The square on the hypotenuse therefore is going to be the square root of 2. And Pythagoras discovered to his horror that the square root of 2 cannot be expressed as a ratio of two numbers, it's not a rational number. Pythagoras was so horrified by this he swore his followers to secrecy about it, they weren't allowed to mention the irrationality of the square root of 2. But it did undermine the faith in numbers as the foundational view of mathematics, and that's why geometry is regarded as superior because in geometry you can express the hypotenuse, you can draw it. But in numbers, you can't express it as a ratio. Okay. So, mathematics and philosophy have always gone handinhand. I said one reason for that is that mathematics provides a model of knowledge. It provides a model of knowledge of a particular kind. That's to say, philosophers since the ancient Greeks, for over 2,000 years, have regarded mathematical knowledge as somehow special and something to which all knowledge could possibly aspire. What makes mathematical knowledge special? Well, several things. One is, unlike other knowledge, it's certain. If something is true in mathematics and if you know it, you're not going to doubt it, you're not going to say, "well, I'm fairly sure that 2 + 3 = 5", you know that 2 + 3 = 5 with absolute certainty. That's the first thing. The second thing is that knowledge in mathematics seems to be incorrigible, that's to say, it's not going to be corrected by anything you might subsequently learn. 2 + 3 = 5 now, 2 + 3 = 5 two thousand years ago, and you might say, even if there weren't any people on earth, it would still be the case that if 2 apples fell to the ground and 3 more apples fell to the ground, then you would have 5 apples on the ground, altogether. So mathematical knowledge is certain, it's incorrigible. Another thing is, that's related to its incorrigibility, it's eternal, it's always true. If something is true in mathematics, it's not true for the time being, it's always true and always has been true. And a fourth thing about mathematical knowledge is that it seems that mathematical truths seem to be not just contingently true, but necessarily true. Southampton happens to be on the coast of England, but that's not a necessary truth, it doesn't have to be. I happen to be wearing black trousers, but again, that's not necessary. But 2 + 3 could not equal anything but 5, it's necessarily equal to 5. So, philosophers have looked at this example of mathematical knowledge and they've thought two things, one is: How is it that knowledge in mathematics has those characteristics? What is it about mathematics that gives its knowledge those characteristics? And the second thing that's occurred to several philosophers, including Plato, Hobbes, Russell was: Why can't other kinds of knowledge be like that? Maybe we could have a system of physics, for example, that made our knowledge of physics as incorrigible, eternal, certain and so on as our knowledge of mathematics. So, mathematics has provided a model of knowledge. On the other hand, it's inherently puzzling. Mathematics has provided philosophers with a number of deep puzzles that we have been scratching our heads over for over 2,000 years. The central one of which is the most basic of which, which is: What is mathematics about? So, if you know that 2 + 3 =5, you know with certainty, incorrigibility, and so on. But what is it that you know something about? Well, as I said if you add 3 apples to 2 apples, you'll get 5 apples. If you add 3 lemons to 2 lemons, you'll get 5 lemons. But 2 + 3 = 5 is not about apples, and it's not about lemons. It's applicable to those things but it's not about those things. So what is it about? Well, it's about numbers. But what are numbers? It's when you ask that questionit's one of those philosophical questions that the more you think about it, the less clear it gets. What are numbers? Numbers seem to be the content of objective truths. If I say that 2 + 3 = 5, I haven't just made it up, I'm not getting it from anybody; it is objectively true. And most of the things that we have objective truths of are objects. So are numbers objects? Well, unlike apples and lemons, you can't see a number, a number does not reflect light; you can't smell a number, you can't touch a number. So if a number is an object it's a peculiar kind of object. It doesn't exist in space and time, it doesn't corrode, it doesn't get old; it makes no sense to ask what its physical size is and so on. So if it is an object, it's a peculiar kind of object, it's an object that doesn't exist in space and time; it's not part of our spatial temporal world. Plato had a theory to account for this, which is the famous theory of Forms. According to Plato, numbers are Forms, and Forms are abstract, objectively existing objects. And the fact that they're not spatiotemporal, they're not part of our world of space and time didn't bother Plato a bit. On the contrary, it confirmed him in his opinion that reality is formal. The world of Forms is the reality of which our spatiotemporal world is but a shadow, according to Plato. And according to Plato, this accounts for the corrigibility of everything we get from our senses everything we see, everything we touch, everything we smellthe knowledge we get from our senses is always open to revision. We might look again & see something different. Whereas, formal knowledge is not. And what that shows, according to Plato, is the superiority of our reason over our senses. We can't see Forms, we can't touch Forms, but we can grasp them intellectually. We can get knowledge of Forms, knowledge of arithmetic by thinking. And according to Plato, that explains the characteristics of mathematical truth: it's necessarily true, it's not contingently true, it's not contingently true because it's not about the contingent world, the spatiotemporal world. Now, the downside of this theory  the upside of this theory is that it explains a lot about mathematical truth and mathematical knowledge  the downside is that it requires us to believe in something that a lot of us have trouble believing in, which is: an objectively existing world of Forms. And even if we could persuade ourselves that such a world existed, we'd have an apparently insolvable problem which is: How do we, so to speak, reach it? Given that we can't see it, we have no sensory awareness of it, how do we bridge the apparently unbridgeable gulf between that world and us? After all, we do exist in the spatio temporal world. So, for those reasons, a lot of philosophers, I would say most philosophers, have had trouble persuading themselves that Plato's world of Forms really exists, and therefore, that the socalled mathematical realm that was supposed to be part of the world of Forms, a lot of philosophers have trouble persuading themselves that that exists as well. And these thoughts had occurred to Aristotle who was a pupil of Plato's, who didn't believe in the world of Forms. He believed that mathematics is not about objects in the mathematical realm, there is no such realm. Mathematics is about our world. And okay, we can't see a number, but we might regard a number as a property of things that we can see. So okay, numbers are not objects, but we can understand them as features of objects. So we look at a field we see 4 cows, it's not that we see the cows and then we see 4, it's that 4 is a property of that collection of cows that we see. Well philosophers since Aristotle have put forward powerful objections to that way of looking at numbers, the most powerful of which were put by a German mathematician come philosopher, Gottlob Frege, who was writing in the 19th century. And he said look, when we know something about numbers, we know it objectively. But numbers, Frege said, cannot objectively be properties of other things. And the reason he said for that is that which number belongs to a collection of things will depend upon how we conceptualize it. So think of a deck of cards. A deck of cards has 52 cards in it. It has 4 kings, it has 4 suits; 4 suits, 52 cards. Depending on whether  so we have a deck of cards in front of us  depending on whether we're thinking in terms of cards or of suits of cards, different numbers will belong to that particular collection of things. So, does that collection of things have the property 52? Or does it have the property 4? It has the property 52 if we're thinking of cards. It has the property 4 if we're thinking of suits. At a simpler level imagine a pair of shoes. It's 1 pair of shoes, but 2 shoes. So, as an object, as a physical object, which number belongs to that? Is it the number one or the number two? So, Frege says this is in general true that objects, objectively, so to speak, do not have numbers as properties, they acquire numbers as properties when we think of them in different ways, and this is inconsistent with regarding mathematics as objective. So for those reasons, the idea that numbers are properties of objects is one of those ideas in philosophy that is, by a lot of philosophers, been regarded as being refuted, refuted by Frege. We seem to be on the horns of a dilemma where, when we solve certain problems about mathematics we encounter others. If we do justice to the objectivity of mathematics, like Plato did, we seem lumbered with a kind of metaphysics, a metaphysics of abstract objects. And when we asked ourselves too deeply questions about what these abstract objects are supposed to be and what the world of Forms is supposed to be, we find that we can't give satisfactory answers to those questions. Where we can give satisfactory answers to those questions we seem to be impaled on the other horn, which is, we seem to have adopted a view which does away with the objectivity of mathematics. Okay so, fastforward now from the Ancient world to the 18th century, Europe in the 18th century, and you come across a great towering figure in philosophy, the German Immanuel Kant, whose great work was the Critique of Pure Reason, published in the 1790s. Kant put forward a theory of mathematics which became the most influential up until the 20th century. And Kant did so in a way that grasped, as it were, one horn of this dilemma, and put forward a theory that did away with the objectivity of mathematics. Kant's thinking about mathematics starts with a question that I didn't raise about mathematical knowledge but which is implicit in some of the things I did say about it, which is this: If something is true mathematically, it's necessarily true, and yet, mathematics works. 3 apples plus 2 apples really is 5 apples. The world, as it were, seems to conform to the laws of arithmetic, but the laws of arithmetic are not just true, they're necessarily true. And Kant's question was: How can we know something about the world which is necessarily true? So he introduced two distinctions which have since become part of the technical vocabulary of philosophy. The first distinguishes two kinds of sentence: an analytically true sentence and a synthetically true sentence. The difference is this: an analytically true sentence is necessarily true. So, for example, "All bachelors are unmarried", that's an analytic statement. It's an analytic statement because it's true by definition. Compare it with the statement, "All bachelors are unhappy". It might be true that all bachelors are unhappy, but it's not necessarily true, it's not part of the definition of a bachelor that bachelor is unhappy. But it IS part of the definition that a bachelor is unmarried. So the statement "All bachelors are unmarried" is necessarily true because it's true by definition. Kant called this an analytic statement as opposed to a synthetic statement such as "All bachelors are unhappy". The reason he chose those particular terms "analytic" and "synthetic" is to do with the question of whether you're dealing with one concept or two. The idea here is that if you say "All bachelors are unhappy", you're making a synthesis of two quite different concepts: the concept of being a bachelor & the concept of being unhappy. If you say "All bachelors are unmarried", you're not synthesizing two unrelated concepts, you're analyzing, so to speak, a feature of one concept. It's a feature that  you know, what does the word 'bachelor' mean? It means unmarried man. So if you analyze the concept 'bachelor', you can analyze it into the concepts 'unmarried' and 'male'. So that's why he said that's an analytic statement as opposed to a synthetic statement. So there's another distinction he drew which is in regard to how we know things to be true, and he used Latin titles for this: 'a priori' and 'a posteriori'. Something is a priori known if our knowledge of it is prior to any experience, any observation, any testing. So again, we know that all bachelors are unmarried, we don't have to do a survey, we don't have to do a test; we know that a priori, we know that prior to any testing, any surveying, and whatever. Whereas, "All bachelors are unhappy", if it's true at all, it's going to be true a posteriori, it's going to be true on the basis of doing some empirical research. We know that smoking causes cancer, we didn't always know that, but we do know that now. Why do we know it? Because we've done experiments, we've done tests, we've looked at people who smoke, we've made observations, and so on. So "smoking causes cancer" is a posteriori. Now, think of those two distinctions. It ought to be the case that the analytic goes with the a priori, and the synthetic goes with the a posteriori. But what Kant said about mathematics  and this is to do with it being necessarily true AND true of the world  is as he says in mathematics we've got this curious hybrid. He said mathematics is not analytically true, it's not true by definition that 5 + 7 = 12. So that's a synthetic statement according to Kant. And yet it's a priori, we don't have to do any experiments to find it out. So mathematics  and this, according to Kant, is its great single feature  mathematics is synthetic a priori. And his question was: How on earth how can we know things "synthetic a priori"? And his answer was the whole system of metaphysics that he puts in the Critique of Pure Reason, which is called "Transcendental Idealism"; at the heart of which is the idea that we don't know anything and cannot know anything about things in themselves. We can only know  so he called those things 'noumena'  we can only know things about what Kant called 'phenomena', which are not things as they are in themselves but things as they appear to us. Things as they appear to us, according to Kant, have been put through a kind of filter, which is the way we see the world. And mathematics, according to Kant, is that filter. In other words, we don't get mathematics from the world, we bring it to the world. So if you think of mathematics as divided into two parts  geometry and arithmetic  geometry, according to Kant, is the spatial form through which we see the world. It's the spatial glasses, as it were, that we look at the world. The world appears to us to be threedimensional Euclidean space. It's the world, the space described by the system of geometry that we got all those thousands of years ago from the ancient Greek geometer Euclid. And the reason those things are necessarily true, they're true a priori according to Kant, is that we didn't get them from the world, we brought them to the world. We look at the world through those spectacles. Where does arithmetic come in? Where do numbers come in? According to Kant, just as geometry is the form of our spatial awareness, our spatial intuitions, arithmetic is the form of our temporal intuitions. So at the heart of arithmetic is a sequence of numbers 1, 2, 3, 4, 5... It's a onedimensional sequence, which corresponds, according to Kant, to our experience of time. We experience space as threedimensional, we experience time as a onedimensional sequence, which is the sequence of numbers. So we get  so numbers are the way we organize time, one moment after another moment. Geometry is the way we organize space. And put together, they give us the framework of the spatiotemporal world that we experience. Everything we know is known about the spatiotemporal world. In other words, everything we know, we know through those spectacles which we ourselves have put to the world. Okay, well I said that was enormously influential theory of mathematics. It was opposed in the late 19th and early 20th century by the man I mentioned earlier, Gottlob Frege & by a British philosopher and mathematician, Bertrand Russell. What Frege and Russell together sought to do was replace the Kantian view of mathematics with a view that did justice to the objectivity of mathematics. And they went right back to Plato, with this proviso: that they reverse the ancient Greek priority about geometry and arithmetic. The ancient Greeks regarded geometry as foundational, Russell and Frege regarded arithmetic as foundational. And this is for two reasons. One is in the middle of the 19th century, alternatives to Euclid's system of geometry were discovered: Riemann's system and Lobachevsky's system. And what these systems did is they dropped the assumption that parallel lines will never meet. In these systems of geometry, parallel lines do meet. And what that means is that in these systems, space is curved. If you think of the space of the outside of a globe, think of drawing two parallel lines, they're going to meet at the top and meet at the bottom. So in these systems, parallel lines can meet, which means that space is curved. And what really threw the cat among the pigeons it was bad enough for Kant's theory that there were alternatives to Euclid because now the question arises, well which pair of glasses should we wear? And according to Kant's theory, we shouldn't have a choice about that. But the cat was really put among the pigeons with Einstein in his theory of relativity, according to which physical space is Riemannian and not Euclidean. It's not just that you can  I mean, Riemann in the middle of the 19th century invented the system of geometry, as it were, just for the hell of it because he was a pure mathematician, he wanted to see what would happen if you drop the parallel postulate. But according to Einstein, it's not just of theoretical interest, the world is Riemannian, physical space is curved. Now in the light of those developments, together with a second development, which is that it was discovered in pure mathematics that you can build geometry upon arithmetic and algebra. And so Frege and Russell regarded arithmetic and not geometry as the foundational branch of mathematics. So for them the central question was about number: What is number? And they weren't very happy with Kant's answer to that, which is: number is something inside our heads that we bring to the world. They weren't very happy with that because it makes arithmetic about what's inside our heads. Whereas, for Frege and Russell, it was crucial that arithmetic is a body of objective knowledge. And so they went back to Plato, into Platonism, it's objective knowledge about forms. Now both of them, quite separatelyand this is a remarkable factquite separately, both of them had the same thought about that, which is: we can do justice to Plato's formal theory of arithmetic  the idea that arithmetic is really about things and these things are forms  we can do justice to that if we show that mathematics and arithmetic in particular is just logic. So their view is called "Logicism", and it's the view that arithmetic can be shown to be a branch of logic. Now this, of course, only leads to Platonism if you take a Platonic view of logical objects, which is exactly what Frege and Russell did. According to Frege and Russell, logic is about forms, and forms really exist. How do they make that plausible? They made that plausible by bringing together two things that previously had not been brought together, which were logic and arithmetic. It's been a hundred years now and over that hundred years we've become accustomed to the idea that logic and mathematics have got strong things in common with each other. But a hundred years ago that wasn't the case. A hundred years ago, logic belonged to the humanities. Logic was what you learned if you learned literature, poetry, rhetoric. It was part of the humanities. You learned Aristotle's system of logic. Learning logic went handinhand with studying classics at Oxford. Whereas, mathematics was what you learned if you were a scientist. And they were not considered to have very much to do with each other. Logic was to do with language. It was to do with using language to construct arguments, and the logic of Aristotle tells you which of those arguments are valid arguments and which are not valid arguments. Whereas mathematics gives you techniques that you can then use in science. Frege and Russell brought them together with a particular theory of number. And I think I've just got time to expound this theory. This theory makes use of the notion of a "class", a class of objects. All right, and the way you get to that notion is this: through language. So you start  I mean what they're doing is building a bridge between Aristotle's theory of logic and the study of languages, on the one hand, and arithmetic on the other. And the way that bridge works is this. You start with propositions, with sentences, with what is analyzed in logic. Okay so take some some sentences: "Plato is wise", "Aristotle is wise", "Socrates is wise". Those sentences all have the same form. Now you could capture that form by replacing the name with a variable, with 'x'. So now you have "x is wise". Now, a class is this. A class is all the things that would satisfy that sentence if you replace the x with a name. If you do that, you've got the class of wise people. So the class would have in it Aristotle, Plato, Socrates, all those people who could replace the x in "x is wise". So the jargon that they came up with for this was: "Plato is wise", "Socrates is wise", those are propositions, but "x is wise" is a propositional function. And the class is the extension of the propositional function. The class is all those wise things. Alright so now they used that notion to talk about numbers. Numbers are classes according to Frege and Russell. The number 4 is the class of all those things that have 4 members. So there are 4 points on a compass: north, south, east and west. There are 4 Beatles: John, Paul, George and Ringo. Collect together all those things that have 4 members, and that, according to Frege and Russell, is the number 4. Now notice that this is very different to the old property theory. It's not that the number 4 is a property of those collections. The number 4 is an object, but it's a particular kind of object, it's a class. And they built a whole system of logic on that notion of class. Then, and I'm not going to go into this because you've probably had enough of all this sort of thing, but I will go into it if you want, but we'll leave that for the question period. In 1901, Russell discovered a problem with that theory of classes which is called Russell's paradox. And he sent it to Frege. He said, look I've just read your work, your work is very similar to mine it makes use of class theory, have you thought about this problem? He showed the contradiction that arises in the theory of classes. Russell, at that time, was quite confident that the problem could be overcome, poor old Frege had a nervous breakdown. And after spending hospital, he came out and said he wasn't a Logicist anymore, he didn't believe Logicism was true. Russell persevered with it, but came up with a different view of logic  which he got from his pupil Wittgenstein  according to which logic is not the study of objectively existing forms. According to Wittgenstein, there aren't any forms, this is a myth. There aren't any forms. What you've got is language and ways of putting words together. And according to the rules for putting words together, sometimes what you end up with is what's called a tautology. So, if i say that it's raining outside, that's gonna be either true or false. If I say it's not raining outside, that's gonna be either true or false. But if I say either it's raining or it's not raining, that can only be true. That is a tautology; that is necessarily true. And so Russell took this notion from Wittgenstein and said that's what mathematical propositions are, mathematical propositions are just tautologies. The reason they're necessarily true is exactly the same as the reason that "All bachelors are unmarried" is necessarily true, they're true by definition. And so Russell towards the end of his life said that if there was a godnotoriously he didn't believe there was a god but if there was a god, he said, the truths of mathematics would have exactly the same profundity as the truth that a fourlegged animal is an animal. And on that note, I'll finish. Thank you.
Contents
Essential and accidental properties
In classical Aristotelian terminology, a property (Greek: idion, Latin: proprium) is one of the predicables. It is a nonessential quality of a species (like an accident), but a quality which is nevertheless characteristically present in members of that species. For example, "ability to laugh" may be considered a special characteristic of human beings. However, "laughter" is not an essential quality of the species human, whose Aristotelian definition of "rational animal" does not require laughter. Therefore, in the classical framework, properties are characteristic qualities that are not truly required for the continued existence of an entity but are, nevertheless, possessed by the entity.
Determinate and determinable properties
A property may be classified as either determinate or determinable. A determinable property is one that can get more specific. For example, color is a determinable property because it can be restricted to redness, blueness, etc.^{[1]} A determinate property is one that cannot become more specific. This distinction may be useful in dealing with issues of identity.^{[2]}
Lovely and suspect qualities
Daniel Dennett distinguishes between lovely properties (such as loveliness itself), which, although they require an observer to be recognised, exist latently in perceivable objects; and suspect properties which have no existence at all until attributed by an observer (such as being suspected of a crime)^{[3]}
Property dualism
Property dualism describes a category of positions in the philosophy of mind which hold that, although the world is constituted of just one kind of substance—the physical kind—there exist two distinct kinds of properties: physical properties and mental properties. In other words, it is the view that nonphysical, mental properties (such as beliefs, desires and emotions) inhere in some physical substances (namely brains).
Properties in mathematics
In mathematical terminology, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or equivalently, as the subset of X for which p holds; i.e. the set {x p(x) = true}; p is its indicator function. It may be objected (see above) that this defines merely the extension of a property, and says nothing about what causes the property to hold for exactly those values.
Properties and predicates
The ontological fact that something has a property is typically represented in language by applying a predicate to a subject. However, taking any grammatical predicate whatsoever to be a property, or to have a corresponding property, leads to certain difficulties, such as Russell's paradox and the Grelling–Nelson paradox. Moreover, a real property can imply a host of true predicates: for instance, if X has the property of weighing more than 2 kilos, then the predicates "..weighs more than 1.9 kilos", "..weighs more than 1.8 kilos", etc., are all true of it. Other predicates, such as "is an individual", or "has some properties" are uninformative or vacuous. There is some resistance to regarding such socalled "Cambridge properties" as legitimate.^{[4]}
Intrinsic and extrinsic properties
An intrinsic property is a property that an object or a thing has of itself, independently of other things, including its context. An extrinsic (or relational) property is a property that depends on a thing's relationship with other things. For example, mass is a physical intrinsic property of any physical object, whereas weight is an extrinsic property that varies depending on the strength of the gravitational field in which the respective object is placed.
Relations
A relation is often considered^{[by whom?]} to be a more general case of a property. Relations are true of several particulars, or shared amongst them. Thus the relation ".. is taller than .." holds "between" two individuals, who would occupy the two ellipses ('..'). Relations can be expressed by Nplace predicates, where N is greater than 1.
It is widely accepted^{[by whom?]} that there are at least some apparent relational properties which are merely derived from nonrelational (or 1place) properties. For instance "A is heavier than B" is a relational predicate, but it is derived from the two non relational properties: the mass of A and the mass of B. Such relations are called external relations, as opposed to the more genuine internal relations.^{[5]} Some philosophers believe that all relations are external, leading to a scepticism about relations in general, on the basis that external relations have no fundamental existence.^{[citation needed]}
See also
 Abstraction
 Doctrine of internal relations
 Identity of indiscernibles (or "Leibniz's law")
 Intension
 Unary relation
References
 ^ Stanford Encyclopaedia of Philosophy Determinate and Determinable Properties
 ^ Georges Dicker (1998). Hume's Epistemology & Metaphysics. Routledge. p. 31.
 ^ "Lovely and Suspect Qualities". Retrieved 3 August 2016.
 ^ Nelson, Michael (1 January 2012). Zalta, Edward N., ed. The Stanford Encyclopedia of Philosophy. Retrieved 3 August 2016 – via Stanford Encyclopedia of Philosophy.
 ^ George Moore, External and Internal Relations
External links
 Zalta, Edward N. (ed.). "Properties". Stanford Encyclopedia of Philosophy.
 MacBride, Fraser. "Relations". In Zalta, Edward N. Stanford Encyclopedia of Philosophy.
This article incorporates material from property on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.