This page has three main sections:
Though the following metadefinition can be stated formally with base and recursive clauses for the notions of formula and term, we can summarize the various categories and subcategories as follows:
The examples provided below are all presented in the following format. First we present a numbered, formal sentence of the theory. This is followed by a bulleted sentence expressed in technical English that uses the primitive or defined notions of the theory. After the reading in technical English, there is some explanation and discussion.
(1) ∀x(E!x → ¬∃FxF)
This is a theorem of the theory. More colloquially, we can read it as: Concrete objects don't encode properties. The predicate ‘E!’ denotes the property of being concrete. Objects like you, me, your computer, the planet Earth, the Solar System, the Milky Way, rocks, trees, etc., exemplify the property of being concrete. So the theory asserts that objects that are concrete (in the exemplification sense) do not encode properties. Properties will be encoded only by abstract objects (i.e., objects that couldn't possibly be concrete – see below).
(2) ∀x(◇E!x → □¬∃FxF)
This, too, is a theorem of the theory. Colloquially, it says that any object that might have been concrete couldn't encode properties. In semantic terms, any object that is concrete at some possible world fails to encode properties at every possible world. Intuitively, possibly concrete objects are objects like us at other possible worlds; they just exemplify properties at those worlds—for every property F, they either exemplify F or they exemplify the negation of F. This theorem tells us that these objects do not encode properties either, and necessarily so. We call the objects that are possibly concrete ‘ordinary objects’. We 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: being an object x such that it is possible that x is concrete.
(3) ∃x¬◇E!x
This is a theorem of the theory. In fact, we will define abstract objects to be objects that exemplify the property of not possibly being concrete. That is, an abstract object is not the kind of thing that could exemplify concreteness. Our theory will assert that there is a wide variety of such objects.
(4) A! =df [λx ¬◇E!x]
This is a definition introduces a new property and a symbol for that property. The formal property constant ‘A!’ is introduced as an abbreviation for the complex property term to the right of the definition sign. The complex property term, which is delimited by the brackets and begins with a Greek λ, may be read as follows: being an x such that it is not possible for x to exemplify the property of being concrete. This complex property term, therefore, denotes a property (see the next item). The definition of the symbol ‘A!’ is found in the monograph in Chapter 7, Section 7.2 (“Definitions Extending the Object Language”). It follows the definition of the new property term ‘O!’ which denotes the property of being ordinary. An ordinary object is an object that is possibly concrete. So, whereas ordinary objects are those that are concrete at some possible world, abstract objects are concrete at no possible world.
(5) [λx ¬◇E!x]x ≡ ¬◇E!x
[Aside: Notice that the first two occurrences of the variable x in (5) are bound by the λ, whereas the second two occurrences of x are not. So by the Rule of Generalization, we can infer from (5):
And when we instantiate this universal claim, say, to a simple constant like ‘a’, the constant becomes substituted only for the two free occurrences, so that we may infer:
This asserts that a exemplifies not-possibly-exemplifying-concreteness if and only if it is not possible that a exemplify concreteness. End of Aside]
(5) is a simple theorem governing relations. An axiom guarantees that the λ-expression [λx ¬◇E!x] has a denotation and we symbolize this fact in object theory by writing [λx ¬◇E!x]↓. And another axiom of the theory (β-Conversion) requires that we can reduce complex λ-expressions if they denote; the unary case of β-Conversion asserts generally:
Clearly, as an instance of this we have:
So since it is axiomatic that [λx ¬◇E!x]↓, sentence (5) above is a consequence. It also follows, from the fact that [λx ¬◇E!x]↓, that the definition of the property term A! in (4) is well-defined: if the definiens has a denotation, then so does the definiendum.
(6) ◇xF → □xF
This is a theorem that governs the modal logic of encoding. In S5 modal logic, it follows from the axiom xF → □xF (“if an object encodes a property, it does so necessarily”). But (6) expresses the stronger claim that if an object possibly encodes a property, it does so necessarily; intuitively, if an abstract object encodes a property at any possible world, it encodes that property at every possible world. The properties an abstract object encodes are essential to it, in precisely this sense. Whereas ordinary objects have a concrete presence to which their properties are anchored, abstract objects have are constituted by the properties they encode. The properties that constitute an abstract object 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 its defining properties there.
(7) A!x & A!y → (x=y ≡ □∀F(Fx ≡ Fy))
This is a theorem of the theory. The identity symbol is defined notation, not primitive! It is defined disjunctively in Chapter 7, Section 7.2 (“Definitions Extending the Object Language”). Basically, the definition of x=y 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. So (7) is a consequence of this general definition of x=y.
(8) Wcj
This assumes we have added the constant ‘c’ to denote Augustus Caesar, added the constant ‘j’ to denote Jupiter, and added the binary relation constant ‘W’ to denote the relation of worships. (8) provides the technical analysis of the historical fact ‘Augustus Caesar worships Jupiter’. (8) is not a theorem of the theory (for it is a contingent fact), but it can be consistently added to the theory.
(9) ¬Wct
This is the technical analysis of the historical fact ‘Augustus doesn’t worship Thor’. Since ‘Augustus worships Jupiter’ is true and ‘Augustus worships Thor’ is false, we know that the expressions ‘Jupiter’ and ‘Thor’ must be contributing something to the meaning of the sentences that is partly responsible for the truth of the first and the falsity of the second.
Note also that one can adopt the principle that, necessarily, if x worships y, then x is a concrete object. (The object of worship need not be concrete, but the worshipping can only have a concrete object as a subject, no matter what possible world we are in – abstract objects can't worship anything.) So the principle to be adopted would be represented as
Now if we identify the Roman god Jupiter as an abstract object that encodes just the properties F that satisfy the open statement “In the Roman myth, Jupiter exemplifies F”, then it would follow that Jupiter is an abstract object, i.e., that A!j. So by definition (4) and β-Conversion, ¬◇E!j. And by classical modal logic, □¬E!j. Then by the T schema of modal logic, it follows that ¬E!j. Thus, from the principle about worshipping represented above and our identification of Jupiter as an abstract object, we can infer that ¬Wja, i.e., that it is not the case that Jupiter worshipped Augustus.
Our principle governing worhipping is a intuitive consequence of a more general auxiliary hypothesis to the effect that abstract objects (which are provably necessarily nonconcrete 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.), or stand in ordinary relations (such as meet, kick, stand between, taller than, etc.), or stand in the first position of intensional relations (such as loves, admires, worships, search for, etc.).
(10) Dh (false)
(11) hD (true)
Where ‘h’ denotes Sherlock Holmes, and ‘D’ denote the property being a detective, then (10) and (11) can be read, respectively, as:
The English sentence ‘Holmes is a detective’ is (10) and (110 are the two ways in which it can be disambiguated. If analyzed as an exemplification predication as in (10), the English sentence 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 the English sentence is analyzed as an encoding predication as in (11), however, the sentence is true. The fact that Holmes encodes the property of being a detective is a consequence of the facts that: (a) in the (lengthy) story in which Holmes originates, the property of being a detective is attributed to Holmes, and (b) Holmes is theoretically identified as the abstract object that encodes exactly the properties attributed to him in the story in which he originates. Facts (a) and (b) figure prominently in Chapter 12, Section 12.7 of the monograph, entitled “Stories and Fictional Individuals”.
(12) ∀x(E!x & Dx → Axh)
This sentence analyzes the English sentence ‘Every real detective admires Sherlock Holmes’. We use the predicate ‘E!’ (which denotes the property of being concrete) to analyze the ordinary sense of what it is to be real. This is a simple example of a quantified 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 conjunct ‘E!x’ 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) ∀x(Dx → E!x)
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) ∃x(Gx & Wxd)
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) ¬∃x∀F(hF → Fx)
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) ◇∃x∀F(hF → Fx)
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.