## Further Explanation of the Distinction Underlying the Theory

Exemplification and Classical First-Order Logic

Mally's distinction between exemplifying and encoding a property is formally represented in the theory as the distinction between the atomic formulas `Fx' (`x exemplifies F') and `xF' (`x encodes F'). The formula `Fx ' is well known from classical first-order logic; when we use Fx to represent such sentences as `John is happy', `Clinton is president', and `Socks is a cat', we are assuming that in each case, the predicate `F' (`is happy', `is president', `is a cat') denotes a property, and that the object term `x' (`John', `Clinton', `Socks') denotes an object. The formal notation `Fx' expresses the fact that the property F is predicated of the object; the mode of predication is exemplification. Exemplification can be generalized. Objects x and y can exemplify the 2-place relation R, and when that happens, we write `Rxy'. Examples are `John loves Mary' (`Ljm '), `Clinton met Yeltsin' (`Mcy'), and so on. Similarly, objects x, y, and z can exemplify a 3-place relation R, and when that happens, we write `Rxyz'. Examples are `x wrote y to z' (`Wxyz'), `x gives y to z' (`Gxyz'), etc. The general notation, Rx_1...x_n, which expresses the fact that objects x_1 through x_n exemplify the n-place relation R, forms the basis of the language underlying the logical system known as the predicate calculus.

Encoding Extends First-Order Logic: An Example

In our theory, however, we are extending classical first-order logic by adding the new mode of predication x encodes F, or, in symbolic terms: xF. This is the mode of predication that should be used to predicate the properties by which fictional and other abstract objects are identified and individuated. For example, we use the property of being a detective to identify Sherlock Holmes and distinguish him from other fictional characters. However, Holmes doesn't really exemplify the property of being a detective. Mally apparently supposed (as a principle) that if a medium-scale physical object really exemplifies a property such as being a detective, it must be a concrete object that has a spatiotemporal location, has a body with a shape, has a surface with a texture, has a mass, etc. But none of this is true when we consider Holmes. If Sherlock Holmes had a determinate spatiotemporal location, a body with a certain shape, etc., we would have been able to meet him, hire him to solve our cases, pay him money, find his grave, etc. So Mally would analyze the sentence "Holmes is a detective" as: Holmes encodes the property of being a detective. Formally, we would represent this sentence as `hD' instead of `Dh'. The same analysis applies with respect to Holmes and such properties as living in London, solving crimes, being brilliant, having Watson as a friend, etc., for these are the properties which define Holmes as an object and sitinguish him from other abstract objects. Note, however, that Holmes does exemplify properties: he exemplifies the property of being thought about by Conan Doyle; he exemplifies the property of being fictional; he exemplifies the property of not being a detective (he is an abstract object, after all), etc.

The logic of encoding will extend first-order logic because it is consistent with, and indeed assumes, all of the laws of classical logic. For example, it assumes that, for every object x and every property F, either x exemplifies F or x exemplifies the negation of F. However, this principle does not hold for encoding. For example, it is not determinate whether Sherlock Holmes has a mole on his left foot. So the theory allows us to say both that Holmes does not encode the property of having a mole on his left foot and that Holmes does not encode the property of not having a mole on his left foot. In this way, the logic of encoding extends and preserves the laws of classical first-order logic.

Two Further Examples

Another way to get a handle on the notion of encoding is to think about the content of certain mental images that we may have. For example, take the content of my mental image of Mark Twain. In my image, Twain is wearing a white western suit, with bowtie, and sports a walrus moustache. Now, we can ask, what is the relationship between the content of my mental image and the property of having a walrus moustache? The property of having a walrus moustache is essential to the content of that particular image---without that property, the content would be the content of some other mental image. Further, the property of having a walrus moustache characterizes the content image in some important way. However, the content of the image doesn't exemplify having a walrus moustache; rather Mark Twain himself exemplifies this property. We may say, however, that the content of the image encodes this property.

When we dream about a monster, there is no object that exemplifies the property of being a monster (for such monsters don't exist). Nevertheless, we can accurately report our experience by saying that we dreamed about an object of a certain kind, and that the object was, in some sense, a monster, otherwise why did we wake up screaming in the middle of the night? We can therefore explain our experience of fear if there is some mode of predication, some way of having a property, and some sense of `is' by which the dream object `is' a monster. Encoding is this mode of predication. English sentences of the form `x is F ' are therefore ambiguous. They can be formally represented as either `Fx' (x exemplifies F) or as `xF' (x encodes F).

Extending Mally's Ideas: Further Examples

Mally's ideas can be extended to any abstract object whatsoever. Whereas the identity of a concrete object is grounded in its location in spacetime, the identity of an abstract object must be grounded in some other way, for abstract objects are not the kind of thing that could have a location in spacetime. Encoding provides the means of grounding the properties by which an abstract object is conceived. The theory postulates that for any group of properties whatsoever, there is an abstract object that encodes just the properties in that group. Here are some examples:

• We can think of the natural numbers as abstract objects that encode just the properties attributed to them in Peano Number Theory. This is the group of properties by which we conceive and identify the natural numbers, and so the theory treats these properties as the properties these numbers encode.
• We can think of an object x of any mathematical theory T as the abstract object that encodes just the properties attributed to x in the theory T. Mathematical objects, therefore, encode all and only their structural, mathematical properties.
• We can think of possible worlds as abstract objects. Take the actual world, for example. It is not identified just by the objects in spacetime, but by what goes on---what properties those objects have annd how they are related to other objects. The world, to quote early Wittgenstein, is all that is the case. We can identify the actual world as an abstract object as follows: take each true proposition p; consider the property of being such that p; now take the actual world to be that abstract object that encodes all and only the properties of this sort constructed out of true propositions. The other possible worlds can be identified in a similar manner.

These are just some of the ways in which the theory can be applied in the analysis of abstract objects. Return to The Theory of Abstract Objects for a fuller description of the theory and its applications.