## Further Explanation of the Objectives of the Theory

1. To describe the logic underlying thought and reasoning by extending classical propositional, predicate, and modal logic. No matter whether we are natural scientists or literary critics, mathematicians or theologians, our thoughts have a common logical structure and, despite the differences in subject matter, proper reasoning takes the same form across disciplines. For example, the thoughts expressed by the following sentences all seem to have a similar structure:

Pensacola is smaller than Boston.
John loves Mary.
3 > 2.
Achilles fought Hector.
God is more powerful than Gabriel.
Each sentence has the following logical structure: (1) the first and last words are names that denote objects, (2) the words in between constitute a predicate that denotes a relation, and (3) the sentence as a whole asserts that the objects denoted by the names exemplify or stand in the relation denoted by the predicate. For example, the third sentence asserts that the object denoted by the name `3' stands in the greater than relation (which is denoted by the predicate `is greater than') to the object denoted by the name `2'. In general, the thought expressed by each sentence seems to have the following form: objects x and y (in that order) exemplify the relation R. The canonical notation for this is: Rxy. The relations in question are all 2-place relations because, logically speaking, the relations smaller than, loves, greater than, fought and is more powerful than require that there be two objects related. Of course, there are 3-place relations such as the following:
x gives y to z.
x is between y and z.
x taught the subject y to z
The canonical notation for 3-place relations is Rxyz. And there are 1-place relations, which are more naturally called `properties', such as the following:
x is a city.
x loves Mary.
3 > x.
x is prime.
x has mass.
In each case, the predicate denotes a property that some objects exemplify and other objects fail to exemplify. Some objects exemplify the property of being a city, while other things fail to exemplify this property. Some numbers exemplify the property of being prime, while other numbers fail to exemplify this property. Finally, notice that many of our thoughts have the form of an identity statement, in which certain things are said to be identical:
2 + 3 = 5.
Superman just is Clark Kent.
Mark Twain is Samuel Clemens.
Bill is Debbie's father.
Lightning is electrical discharge.
Heat is molecular motion.
F = ma.
In the first four sentences, a certain object is identified with another object---they are asserted to be the same object. In the next two sentences, certain phenomena (heat, lightning) are identified in terms of scientific descriptions. In the last sentence, a law of nature is expressed: the force on an object is equal to its mass multiplied by its acceleration. Sentences such as the above play an important role in the advance of knowledge.

The theory of abstract objects extends classical propositional and predicate logic to account for the logic of these sentences. It seems that no matter what subject matter we are thinking about, our thoughts can all be analyzed either in terms of objects standing in relations to other objects, or in terms of objects having certain properties, or in terms of things being identical, or some combination or variant (modality) of these. This is just a feature of the way we think. So the notions object, relation, exemplify, identical, etc., seem to play a fundamental role in explaining the content of our thoughts. The first objective of the theory of abstract objects is to systematize these notions by axiomatizing them. We have already begun the process of axiomatization by introducing the variables `x ', `y ', `R ', and the formal sentences `Rxy ' and `Rxyz '.

2. To describe the laws governing universal entities such as properties, relations, and propositions.When we develop scientific or other theories, or make claims about the world, we often classify objects into groups on the basis of shared similarities. We say, for example, that objects x and y, though differents pieces of rock, are both quartz; or that objects w and z, though different individuals, are both humans; or that objects t and u, though different particles, are both electrons; or that the numbers 2 and 7, though different, are both prime. Many philosophers and logicians (begining with Plato) have explained such classifications and thoughts by theorizing that, in each case, there is something that each of these pairs of objects have in common, namely, a property. In the first case, x and y are classified together because they both exemplify the property of being quartz; in the second case, w and z both exemplify the property of being a homo sapiens; etc. We came across properties in the previous section, and now we are thinking about them seriously as entities in their own right. In some sense, they are the common structural feature or quality that different individuals may share. Plato called them `Forms'. These properties are what the predicates `is quartz', `is a human', `is an electron', and `is prime' signify. But whereas the predicates are pieces of language, the properties they signify are something more abstract. The second objective of the theory of abstract objects, then, is to develop a theory of properties. Under what conditions do properties exist? Does every predicate denote a property? Do the predicates of discredited scientific theories, such as `is phlogiston' and `are simultaneous' denote properties? When are properties F and G identical?

Another important reason to investigate properties is that they appear to play a role in the content of scientific laws. Scientific laws are very general statements about regularities in the natural world. These laws do not specifically mention each object constrained by the law by name, but rather specify in a general way which properties and relations among objects are always found together (by some sort of natural necessity). But these properties and relations are not themselves in spacetime. To see why, consider an example. The laws governing the ligation of oligonucleotides into DNA molecules exist independently of the instances of oligonucleotides in spacetime. How can the laws `about nucleotides' exist independently of any nucleotides? One answer is that the laws involve the property of being an oligonucleotide and the ligation relation, not the nucleotides themselves, for the laws specify that if there are any free-standing instances of this property meeting certain conditions, they will become related through ligation in a certain way. Natural laws, therefore, seem to have the form: it is a necessity of nature that if something has a certain structure and is in a certain condition or situation, it will act or be acted upon in certain ways by (possibly other) things having a certain other structure and in a certain condition or situation. Thus, talk about `properties' is simply a more convenient way of talking about the forms or shapes that the elements of nature can take. But when we enquire about the nature of properties as entities in their own right, we are no longer doing natural science---you won't find `property detectors' in physics labs. Instead, such an enquiry concerns the conceptual framework presupposed by the natural sciences. The notion of a property makes sense of what scientists do: they classify the natural world according to the properties that things exemplify and they attempt to express laws that relate some properties of the natural world to others. In other words, the logic that is is embedded in the laws governing concrete objects such as galaxies, planets, stones, squirrels, cells, molecules, atoms, quarks, etc., is a logic of properties. This logic underlies classification and scientific laws themselves. If the logic of properties is to be precise, we must develop a theory of properties.

3. To identify theoretical mathematical objects and relations as well as the natural mathematical objects such as natural numbers and natural sets. Mathematical objects are another kind of object presupposed by scientists but which are not part of the concrete world---you won't find machinery in a physics lab that can detect natural numbers, real numbers, sets, etc. But such mathematical objects play an important role in science; in fact, we can't seem to do science without using them. Advances in science are often tied to advances in mathematics. But when we think of mathematical objects as entities in their own right, questions arise. What are they? Which ones are there? Does every consistent mathematical theory describe a realm of abstract mathematical objects? How do we acquire knowledge about such objects if they are not part of the concrete universe?

The third objective of the theory of abstract objects, then, is to say something precise about mathematical objects and mathematical relations, and to develop a systematic connection between mathematical theories and the realm of abstract mathematical objects which they are about.

4. To analyze the distinction between fact and fiction and to analyze the various relationships between stories, characters, and other fictional objects. We have numerous true beliefs about what appear to be fictional objects. But if these objects don't exist, how can we have beliefs about `them'? Here are some examples of beliefs that we have about fictional objects. Most of our beliefs about fictions would appear somewhere in the following catalogue:

• Historical facts:
• The ancient Greeks worshipped Zeus.
• Sherlock Holmes still inspires modern criminologists.
• Ponce de Leon searched for the fountain of youth.
• Ponce de Leon searched for something.
• Teams of scientists have searched for the Loch Ness monster, but since it doesn't exist, no one has ever found it.
• Literary criticism:
• Some fictional characters are interesting because they find themselves in situations in which they appear to be able to choose their identity, though it inevitably turns out that factors beyond their control, antecedent to the moment of choice, had already determined what they would do.
• The ancient Greeks and Romans worshipped the same gods, though they called them by different names.
• Bill thinks about things that will never happen.
• The monster Shirley dreamed about last night really frightened her.
• True Sentences Beginning with the Prefix: In the story (s), ...
• In Faulkner's Light in August, Joe Christmas was victimized by racial prejudice even as a child.
• In Günther Grass's The Tin Drum, Oskar Mazerath decides to stop growing at the age of 3.
We also reason in a perfectly valid way about nonexistent objects. Consider this argument.
Augustus worshipped Jupiter. Jupiter is a mythical character. Mythical characters don't exist.
Therefore, Augustus worshipped something that doesn't exist.
The fourth objective of the theory of abstract objects is to analyze these thoughts and inferences. This requires that we define the notion of a story and the notion of a fictional character, and say under what conditions a proposition is true in a story. The simplest way to proceed, from a logical point of view, is to treat names like `Zeus', `Joe Christmas', etc., as names of abstract objects. One of the tasks of the theory is to say precisely what the abstract objects denoted by such names are like.

5. To systematize our modal thoughts about possible (actual, necessary) objects, states of affairs, situations and worlds. We often think of what might be the case but which isn't in fact the case; that is, we often think of what's possible but not actual. Possibilities are not fictions because fictions are highly incomplete whereas possibilities are determinate down to the last detail (there are innumerable questions about Sherlock Holmes' life, heritage, etc., that are left unanswered by the Conan Doyle novels and so as a fictional object, Sherlock Holmes is just not determinate with respect to those details; as such, he is not possible in the strict sense because a genuinely possible object has to be determinate with respect to every detail). Our principal understanding of what is possible and what is necessary derives from the true beliefs that we have which take the following forms:

• It is possible that...
• It is not possible that...
• It is necessary that...
• It is not necessary that...
For example, it is possible that humans travel to Mars; it is not possible that both `Every person is mortal' and `Socrates is a person' are true and `Socrates is mortal' false; it is necessary that: either it is raining here now or it is not; etc. Logicians are interested in the notions of possibility and necessity in part because they allow for a precise characterization of deductively valid arguments: a deductively valid argument is one in which it is impossible for the premises to be true and the conclusion false. Deductively valid arguments have the strongest logical connection between the premises and conclusion.

The sentential prefixes `it is possible that' and `it is necessary that' are called modal operators, because they specify a way or mode in which the rest of the sentence can be said to be true. The logic of these modal operators was first discussed in a systematic way by Aristotle in De Interpretatione, but the study of `modal logic' began to flourish only in the present century. Aristotle noticed that the notions of necessity and possibility were interdefinable. A state of affairs p is possible if and only if its negation not-p is not necessary. Similarly, a state of affairs is necessary if and only if its negation is not possible.

Another contribution that Aristotle made to modal logic was to point out that from the separate facts that p is possible and that q is possible, it does not follow that the conjunction `p and q ' is possible. Similarly, it does not follow from the fact that a disjunction is necessary that that the disjuncts are necessary, i.e., it does not follow from `Necessarily, p or q ' that `Necessarily p or necessarily q '. For example, it is necessary that either it is raining or it is not raining. But it doesn't follow from this either that it is necessary that it is raining, or that it is necessary that it is not raining. This simple point of modal logic has been verified by recent techniques in modal logic, in which the claim `Necessarily, p ' has been analyzed as: p is true in all possible worlds. Using this analysis, it is easy to see that from the fact that p or not-p is true in all possible worlds, it does not follow either that p is true in all worlds or that not-p is true in all worlds. And more generally, it does not follow from the fact that p or q is true in all possible worlds either that p is true in all worlds or that q is true in all worlds.

The fifth objective of the theory of abstract objects, then, is to systematize our notions of possibility and necessity. The theory is based on a particular axiomatization of these notions (System S5), and a definition of `possible world' is then constructed in terms of these and other notions of the theory. We derive as a consequence of the axioms of the theory both that a proposition is necessarily true if and only if it is true in all possible worlds, and that a proposition is possibly true if and only if true in some possible world.

6. To account for the deviant logic propositional attitude reports, explain the informativeness of identity statements, and give a general account of the objective and cognitive content of natural language. Gottlob Frege (b. 1848, d. 1925) made a seminal contribution to the philosophy of language when he noticed that one cannot account for the meaningfulness of some sentences simply on the basis of the denotations of the terms. He noticed a problem with the simple analysis of language that we described in the explanation of the first objective. Recall that a sentence of the form
`a = b ' is true if and only if the object denoted by `a ' is the same as the object denoted by `b '. For example, `Clark Kent = Superman' is true iff the object denoted by `Clark Kent' is the same as the object denoted by `Superman'. Since there is only one extraterrestrial being who performs valiant feats of strength under the name `Superman' but who disguises himself as the reporter named `Clark Kent', the identity statement is indeed true. The analysis only requires that, for the sentence to be true, the names flanking the identity sign have a denotation and that they both denote the same object. But Frege noticed that this is no different from the truth conditions of the sentence `Clark Kent = Clark Kent'. Both of the names flanking the identity sign have a denotation, and they both denote the same object. Frege concluded that the simple analysis of language, therefore, assigns the same meaning to `Clark Kent = Superman' and `Clark Kent = Clark Kent'. This seems to be a mistake. The two sentences can't have the same meaning, for we can know that `Clark Kent = Clark Kent' is true simply by inspecting it. If it had the same meaning as `Clark Kent = Superman', then we could know that the latter was true by inspecting it. But, of course, we can't know the truth of `Clark Kent = Superman' simply by inspection of the sentence. For Lois Lane could then know that Clark Kent is Superman simply by inspecting the sentence `Clark Kent = Superman'. This, it seems clear, is something no one could do.

This problem is closely related to the problem of propositional attitude reports. A propositional attitude is a psychological relation between a person and a proposition. Belief, desire, intention, discovery, knowledge, etc., are all psychological relationships between persons, on the one hand, and propositions, on the other. When we report the propositional attitudes of others, these reports all have a similar logical form:

x believes that p
x desires that p
x intends that p
x discovered that p
x knows that p
To see the problem posed by the analysis of propositional attitude reports like those displayed immediately above, let us consider what appears to be a simple principle of reasoning. If a name, say n appears in a true sentence S, and the identity sentence n=m is true, then we should be able to substitute the name m for the name n in S without affecting the truth of S. For example, let S be the sentence `Mark Twain was an author', let n be the name `Mark Twain', and let m be the name `Samuel Clemens'. Then since the identity sentence `Mark Twain = Samuel Clemens' is true, we can substitute `Samuel Clemens' for `Mark Twain' without affecting the truth of the sentence. And indeed, the resulting sentence `Samuel Clemens is an author' is true. Similarly, if S is the sentence `4 > 3', n is the name `4', and m is the name `8/2', then since 4 = 8/2, we may substitute `8/2' for `4' in S and the resulting sentence `8/2 > 3' is true. This principle seems to capture the idea that if we say something true about an object, then even if we change the name by which we refer to that object, we should still be saying something true of that same object.

Now this principle tells us that if S is `Superman can leap tall buildings in a single bound', n is `Superman' and m is `Clark Kent', then given that Superman = Clark Kent, we can conclude `Clark Kent can leap tall buildings in a single bound'. And this is true, because Superman doesn't lose his powers when disguised as Clark Kent. But notice the following counterexample to this principle. Let S be the following sentence:

(a) Lois Lane believes that Superman can leap tall buildings in a single bound.
Now given that Superman = Clark Kent, we should be able to substitute the name `Clark Kent' for `Superman' in the sentence (a) and the resulting sentence should remain true. But in fact it does not work this way! If we perform the substitution on (a), we get sentence (b):
(b) Lois Lane believes that Clark Kent can leap tall buildings in a single bound.
Sentence (b) is clearly false. Lois Lane has no such belief. So our simple principle of substitution, which seemed like a good rule of inference, turns out to get us into trouble.

To explain this puzzle, Frege supposed that names and other expressions of language have a second kind of meaning, in addition to their denotation, which he called their `sense'. We may call the sense of a name its `cognitive content', whereas the denotation of a name is its `objective content'. Other expressions of language also have both sense (cognitive content) and denotation (objective content). The sixth objective of the theory of abstract objects, then, is to systematize propositional attitude reports, explain the informativeness of identity statements, and in general, give an account of the objective and cognitive content of natural language. The theory offers a precise treatment of both the objective content and the cognitive content of linguistic expressions. It allows the cognitive content of these expressions to play a role in the analysis and logic of propositional attitude reports.

7. To axiomatize the philosophical objects postulated by other philosophers. Over the centuries, philosophers have postulated various abstract objects. Plato discussed Forms; Leibniz discussed concepts, monads, and possible worlds; Frege discussed senses, concepts, and extensions of concepts; Meinong discussed nonexistent objects; Husserl discussed noemata; early Wittgenstein thought of the world as a large state of affairs (not as just a concrete object); many philosophers have discussed moments of time; etc. To give a systematic account of these entities, one must develop a precise definition for each one, prove that something falls under the definition, and prove that the objects that fall under the definition obey the principles alleged to govern that kind of object. To take an example, one must define possible world, prove that there are such things, and prove that worlds obey the principles of modality, such as that a proposition is necessary if and only if it is true in all possible worlds. The construction of such definitions and proofs is the seventh objective of the theory of abstract objects.