There are four axioms for the logic of encoding, namely, the closures of all the following axioms and axiom schemata:
Axiom 1 asserts: objects x1, … , xn encode the relation Fn just in case: (1) x1 encodes the property: being a y such that y, x2, … , xn exemplify Fn, (2) x2 encodes the property: being a y such that x1, y, x3, … , xn exemplify Fn, and … and (n) xn encodes the property: being a y such that objects x1, … , xn−1, y exemplify Fn.
Intuitively, in terms of possible worlds, the axiom O!x → ¬∃FxF tells us that any object that might be concrete is one that doesn't encode properties. So, concrete objects don't encode properties, and neither do objects that are concrete at other possible worlds (whether or not they are concrete here). It will follow from this axiom and Axiom 2 that:
O!x → □¬∃FxF
For Axiom 2, discussed below, will imply, as a consequence, that ¬xF → □¬xF, i.e., that if an object fails to encode a property it necessarily fails to encode that property. So if x is ordinary, then by Axiom 1, it doesn't encode any properties, i.e., x fails to encode every property. So if x fails to encode every property, then x necessarily fails to encode every property, since for any property it fails to encode it necessarily fails to encode that property, by the consequence of Axiom 2. So, if x necessarily fails to encode every property, it follows that necessarily, it is not the case that there is a property x encodes.
Intuitively, in terms of possible world talk, this axiom asserts: if an object x encodes property F, then x encodes F in every possible world. In other words, encoding a property is not relative to any circumstance.
Intuitively, let us assume some set theory and picture our object terms as naming or ranging over a fixed domain of objects and picture our relation terms as naming or ranging over a fixed domain of relations. Let us further imagine that there is a domain of possible worlds (our primitive notion ‘it is necessary that’ is, intuitively, a universal quantifier that ranges over the domain of possible worlds). Since objects can exemplify properties and stand in relations at this world but not exemplify those same properties and stand in those same relations at every possible world, we may suppose that each n-place relation R in the domain of relations has an ‘exemplification extension’ at each possible world. The exemplification extension of a relation R at a possible world w is a set of n-tuples, and each member of this set represents an ordered group of objects that exemplify or stand in the relation R at w. Now suppose that the variables x1,…,xn are assigned, as values, the objects o1,…,on, respectively, and that the predicate ‘F’ is assigned, as value, the n-place relation R. Then we may say that the atomic sentence ‘Fx1…xn’ is true at world w just in case the n-tuple ⟨o1,…,on⟩ is an element of the exemplification extension of the relation R at w. In the unary case, where ‘x’ is assigned, as value, the object o and ‘F’ is assigned, as value, the property P, we may say that the atomic sentence ‘Fx’ is true at world w if and only if o is a member of the exemplification extension of the property P at w.
Now we can describe the truth conditions of encoding formulas as follows. Suppose that each n-ary relation in the domain of relations receives, in addition to an exemplification extension that varies from world to world, an ‘encoding extension’ that does not vary from world to world. The encoding extension of a relation R is just a set of n-tuples, and each member of this set represents an ordered group of objects that encode the relation R. Now again suppose that the variables x1,…,xn are assigned, as values, the objects o1,…,on, respectively, and that the predicate ‘F’ is assigned, as value, the n-place relation R. Then we may say that the atomic sentence ‘x1…xnF’ is true at world w just in case the n-tuple ⟨o1,…,on⟩ is an element of the encoding extension of the relation R. In the unary case, where ‘x’ is assigned, as value, the object o and ‘F’ is assigned, as value, the property P, we may say that the atomic sentence ‘xF’ is true at world w if and only if o is a member of the encoding extension of the property P. Each object that is in the encoding extension of property P represents the fact that the object encodes P.
Notice that, given these truth conditions, the following is a fact: if an atomic encoding formula is true at any world w, it is true at every world w, since truth conditions for encoding formulas at a world are indepedentn of the possible worlds. This fact, when expressed from the point of view of our language in which modality is primitive, is just our logical axiom for encoding, for that axiom asserts that if an atomic encoding formula is possibly true, it is necessarily true.
Finally, we'll be able to prove from this axiom that:
Thus, encoding formulas are ‘modally collapsed’, since all three of the following formulas are materially equivalent: xF, ◇xF, and □xF.
Here are some examples of Axiom 3:
(1) asserts, for an arbitrary object y, that there exists an abstract object that encodes exactly the properties F that y exemplifies. (2) asserts, for an arbitrary property G, that there exists an abstract object that encodes exactly the properties F that are materially equivalent to G. (3) asserts that there exists an abstract object that encodes exactly the properties F that are necessarily implied by the property T. (4) asserts that there exists an abstract object that encodes exactly the properties F that either y encodes or that z encodes. (5) asserts that there exists an abstract object that encodes exactly the properties F that have the form being such that p, for some true proposition p.
Since (1) above has the variable y free, it is subject to the Rule of Generalization (GEN) and so it follows that:
In other words, for every object y whatsoever, there is an abstract object that encodes exactly the properties that y exemplifies.
(5) above tells us that, in an extended sense of ‘encodes’, there are abstract objects that encode just propositions by encoding only propositional properties. We say that a property F is a propositional property just in case F is the property being such that p, for some proposition p, i.e.,
In the property-denoting expression [λz p], ‘p’ is a variable ranging over propositions, and so the variable z bound by the λ doesn't bind any other variables in the expression. But this property [λz p] logically well-behaved, for any proposition p, for it is subject to the unary instance of λ-Conversion (β-Conversion):
In other words, an object x exemplifies being such that p if and only if p is true. So we can extend our notion of encoding by saying that an object x encodes proposition p whenever x encodes [λz p].
As another example of comprehension, here is an instance that asserts the existence of an abstract object that encodes just the two properties being round (R) and being square (S):
This instance of comprehension asserts that there is an abstract object that encodes both R and S and no others. Why is it that a disjunction (‘or’) is used in the right side of the biconditional rather than a conjunction (‘and’)? Well, consider the following description of a set: {x | x=1 ∨ x=2}. How many elements does this set have? The answer is: two. It describes the set {1, 2}, for only the numbers 1 and 2 satisfy the defining condition ‘x=1 ∨ x=2’. Similarly, the set {F | F=R ∨ F=S} is a description of the set containing both the properties of being round and being square and no others; i.e., {R, S}. So if we take the condition ‘F=R ∨ F=S’ and use it as the right condition of the biconditional in an instance of the comprehension axiom, we assert the existence of an abstract object encoding two properties. NOTE: This abstract object is not inconsistent with the law of geometry that says: necessarily everything round fails to be square. This law is a law that governs objects that exemplify roundness, not those that encode it. It is rendered into formal notation as: □∀x(Rx → ¬Sx). (6) asserts only that there is an abstract object that encodes roundness and squareness; it is logically compatible with the above law of geometry for it does not follow from the fact that an object encodes a property that it also exemplifies that property.
One final instance of comprehension is worth noting. Suppose we have a formal analysis of the story operator ‘In the story s, p’ Then consider the Conan Doyle story A Study in Scarlet, and call that s, and let the name ‘Sherlock Holmes’ be represented by the constant h. Then an instance of comprehension asserts that there is an abstract object that encodes all and only the properties that Holmes exemplifies in the story s:
Let us assume that there are ordinary English sentences of the form ‘In to the story A Study in Scarlet, Holmes is F’ are true. Then the above instance of the comprehension axiom for abstract objects asserts that there is an abstract object that encodes just those properties F that satisfy these sentences. We don’t have to agree on what those sentences are – we might read the stories differently. But we should agree that Sherlock Holmes is an object that is defined by the properties ascribed to him in the story. The above instance of comprehension asserts that there is an object that encodes just the properties ascribed to Holmes in the story and since that object is unique, we can identify Holmes as that abstract object. So, Holmes is, in some sense, an incomplete object: there are properties F such that neither F nor the negation of F are attributed to Holmes in the story, and so there are properties that Holmes fails to encode. However, like every object whatsoever, no matter whether ordinary or abstract, Holmes is complete with respect to the properties he exemplifies. For every property F, either Holmes exemplifies F or exemplifies the negation of F. He exemplifies such properties as not being a detective, not living in London, not being Watson’s friend, etc., because he is an abstract object and abstract objects do not exemplify such ordinary properties. However, Holmes will exemplify such properties as being admired by some real detectives, being conceived by Conan Doyle, being more famous than any real detective, etc.