# The Language of the TheoryExamples

The examples provided in this document are all presented in the following format. First we present a numbered sentence or phrase expressed in technical English that uses the primitive notions of the theory. This is followed by the formal representation of that sentence or phrase. After the formal representation, there is some explanation and discussion.

1) Every object that exemplifies the property of having a spatiotemporal location fails to encode properties.

This is a theorem of the theory. The predicate `E! ' denotes the property of having a spatiotemporal location. Objects like you, me, your computer, the planet Earth, the Solar System, the Milky Way, rocks, trees, etc., exemplify the property of having a spatiotemporal location. So the theory asserts that such objects do not encode properties. Properties will be encoded by abstract objects only.

2) Every object that possibly exemplifies the property of having a spatiotemporal location necessarily fails to encode properties.

This is an axiom of the theory. The idea is that not only are there objects in spacetime, but there might have been other such objects. That is, there are other possible worlds and at those worlds there are objects in spacetime there; the properties they exemplify there are analogous to the properties we exemplify here. They are determinate with respect to the properties they exemplify at these worlds---for every property F, they either exemplify F or they exemplify the negation of F. This axioms tells us that these objects do not encode properties either, and necessarily so. We shall call the objects that have a spatiotemporal location at some world or other `ordinary objects'. We shall use the predicate `O! ' to denote the property of being ordinary. The definition is simply this:
In other words, the property of being ordinary is just the property of being an object x such that it is possible that x has a spatiotemporal location.

3) There are objects that couldn't possibly have a location in spacetime.

This is a theorem of the theory. In fact, we will define abstract objects to be objects that exemplify the property of not possibly having a location in spacetime. That is, an abstract object is not the kind of thing that could have a location in spacetime. Our theory will assert that there is a wide variety of such objects.

4) The property of being abstract is, by definition, the property of being an object x such that x could not possibly have a location in spacetime.

This introduces a new symbol. The formal predicate `A! ' is introduced as an abbreviation for the complex predicate to the right of the definition sign. The complex predicate, which is delimited by the brackets and begins with a Greek lambda, may be read as follows: being an x such that it is not possible for x to exemplify the property of having a spatiotemporal location. This complex predicate, therefore, denotes a complex relation (see the next item). The definition of the symbol `A! ' is found in the monograph in the item Language/Ordinary and Abstract Objects. It follows the definition of the new predicate `O! ' which is to denote the property of being an ordinary object. An ordinary object is an object that possibly has a location in spacetime. So, whereas ordinary objects are those that are spatiotemporally located at some possible world, abstract objects are spatiotemporally located at no possible world.

5) An object x exemplifies the complex property of not-possibly-being-located-in-spacetime if and only if it is not possible that x is located in spacetime.

This is a simple truth of logic. It ensures that our complex predicate works in just the way we should expect. From this truth of logic and the previous definition, we now know that anything exemplifying the property denoted by the predicate `A! ' is an object that couldn't possibly be located in spacetime.

6) If it is possible that an object encodes a property, it does so necessarily.

This is the logical axiom that governs encoding. It captures not only the idea that the properties that an abstract object encodes are necessarily encoded, but also the stronger claim that any property that an abstract object encodes at any possible world is encoded at every possible world. The properties an abstract object encodes are essential to it, in precisely this sense. Whereas ordinary objects have a presence in spacetime to which their properties are anchored, abstract objects have to be constituted by properties. The properties that constitute an abstract object (i.e., the ones they encode) are part of its nature, and so even when that object is considered from the point of view of some other possible world, it encodes there its defining properties.

7) If x and y are both abstract objects, then x=y if and only if necessarily, x and y encode the same properties.

This is a theorem of the theory. The identity symbol is defined notation, not primitive! It is defined disjunctively in the item Language/Identity. Basically, the definition says that objects x and y are identical if and only either (1) they are both ordinary objects and they necessarily exemplify the same properties, or (2) they are both abstract objects and they necessarily encode the same properties.

8) The object Augustus Caesar exemplifies the worshipped relation to the object Jupiter.

This provides the technical analysis of the ordinary English sentence `Augustus Caesar worshipped Jupiter'. We represent the name `Augustus Caesar' with the letter `a ', represent the name `Jupiter' with the letter `j ', and represent `worshipped' with the letter `W '. This sentence is not a consequence of the theory (for it is a contingent fact), but it is consistent with the theory.

9) The object Jupiter does not exemplify the worshipped relation to the object Augustus Caesar.

This is the technical analysis of the English sentence `Jupiter didn't worship Augustus'. The sentence that analyzes the English is consistent with the theory. However, it is a consequence of a non-logical postulate that governs the worshipping relation, namely, if one object worships another, then the first object is a concrete object (the second object need not be). Since Jupiter will be construed as an abstract object, i.e., one that couldn't be concrete, it follows that Jupiter doesn't worship anything, and in particular, doesn't worship Augustus. The non-logical postulate governing worhipping is a intuitive consequence of under a more general auxiliary hypothesis to the effect that abstract objects do not exemplify ordinary properties (such as having a spatiotemporal location, having a shape, having a texture, having a length, being a student, being a human, being a building, having mass, etc.), stand in ordinary relations (such as meet, kick, stand between, taller than, etc.), or stand in the first position of intentional relations (such as loves, admires, worships, search for, etc.).

10) Holmes exemplifies the property of being a detective.
11) Holmes encodes the property of being a detective.

We use `h ' to represent the name `Holmes' and `D ' to denote the propery of being a detective. The English sentence `Holmes is a detective' is ambiguous and the above sentences are the two ways in which it can be disambiguated. If construed as an exemplification predication, it is false. The reason is that an auxiliary hypothesis of the theory tells us that abstract objects do not exemplify such ordinary properties as being a detective. If construed as an encoding predication, however, the sentence is true. The fact that Holmes encodes the property of being a detective is a consequence of the facts that: (1) in the (lengthy) story in which Holmes originates, the property of being a detective is attributed to Holmes, and (2) Holmes is theoretically identified as the abstract object that encodes exactly the properties attributed to him in the story in which he originates. Facts (1) and (2) figure prominently in the section of the monograph entitled The Theory of Fiction.

12) Every object that exemplifies being real and being a detective exemplifies the admiration relation to the object Sherlock Holmes.

This sentence analyzes the English sentence `Every real detective admires Sherlock Holmes'. We use the predicate `E! ' (which denotes the property of having a spatiotemporal location) to analyze the English notion of being real. This is a simple example of a quantified, molecular formula involving universal quantification and a conditional. It might have been true, but it is no doubt false. Nevertheless, it is consistent with the theory and has a simple logical analysis. We could have omitted the `E!x ' conjunct in the antecedent since it seems reasonable to assume the hypothesis that anything that exemplifies the property of being a detective is real. We consider this in the next example.

13) Every object that exemplifies the property of being a detective exemplifies the property of having a spatiotemporal location.

This sentence captures the belief every object that `is' a detective (in the exemplification sense) `is' real (in the exemplification sense). On the assumption that the property of being a detective is an ordinary property, it is a consequence of the auxiliary hypotheses that abstract objects do not exemplify this property.

14) Some objects that exemplify the property of having been Greek exemplify the worshipped relation to Dionysus.

This is a technical representation of the historical fact `Some Greeks worshipped Dionysus'. It is a simple example of a quantified, molecular formula, involving the defined `existential quantifier' and conjunction. As an historical fact, it is consistent with the theory and it has a simple logical analysis.

15) No object exemplifies every property that Holmes encodes.

This sentence is the way to represent our ordinary intuition that Holmes doesn't exist. This intuition is derived from the fact that when we look around in spacetime, we find nothing that exemplifies Holmes's properties.

16) It is possible that some object exemplifies every property Holmes encodes.

This sentence is the way to represent our ordinary intuition that Holmes might have existed. The diamond at the beginning of the formal representation expresses the notion `it is possible that'. The notion `it is possible that ...' is defined as `it is not necessary that not ...'. Intuitively, this means that in some possible world, ... is true. So our target sentence tells us that in some possible world: there is an object that exemplifies there every property that Holmes encodes there. Of course, the object at that other possible world will be a concrete, spatiotemporal object at that world, just like you and I are concrete, spatiotemporal objects at this world. So it will be determinate down to the last detail.