The most famous doctrine in William of Ockham's metaphysics is probably his nominalism.

One theme that defines Ockham's nominalism,² the one we will discuss in this paper, is his ontological elimination of eight Aristotelian categories - categories other than those of substance and quality. The main mechanism that brings forth the ontological elimination is his theory of connotation.

The theory of connotation is a theory about significations of terms in mental language. Ockham, following Boethius, distinguished three levels of language: written, spoken, and mental. Among the three, mental language is natural, while the other two conventional. That is to say, while spoken or written signs vary from one community to another, mental signs -concepts and their combinations - are at least ideally the same for all people, and while the meanings of mental signs are originally acquired through natural processes, the meanings of written and spoken signs are derived from those of the mental signs with which they are associated by convention. In Ockham's mental language, terms are divided into two groups: (1) categorematic terms, mental counterparts of nouns, adjectives, and verbs, that signify (make us think of) things in the world; (2) syncategorematic terms, mental counterparts of adverbs, conjunctions, and prepositions, that do not by themselves signify any distinct reality but modify the significations of the categorematic terms to which they are conjoined.³ Thus, the concepts "man," "white," "to kick" are categorematic; while "every," "but," "on" syncategorematic.

In connection with the connotation theory, a further demarcation among categorematic terms is introduced: there are absolute terms (e.g., "man"), terms of categories of substance and quality, corresponding to natural kind terms in modem terminology; and there are connotative terms (e.g., "white"), terms of categories of quantity, relation, action, passion, when, where, position, and habit. Despite its rich content, the central thesis of the connotation theory is simple: while absolute terms have no semantically equivalent definitions, every connotative term has a fully expanded nominal definition which consists of only absolute terms and syncategoremata, and which is semantically equivalent to the connotative term (e.g., the connotative term "white" has a semantically equivalent nominal definition "something having a whiteness"). What this thesis suggests is a reductional program in the semantics of mental terms: there are irreducible rock-bottoms - absolute terms, and all other categorematic terms - connotative terms - can be reduced to absolute terms and syncategoremata (as far as their significations are concerned). With the help of the related theory of "exposition" and "exponibles,"⁴ we can easily extend the reductional program in the connotation theory to a recursive semantics in modern jargon, i.e., the significations of all mental signs can be systematically accounted for by appealing to the significations of absolute terms and the semantical roles syncategoremata play.

We have said that there are correlations between absolute terms and substances and qualities, and between connotative terms and putative entities falling under other Aristotelian categories. We have also said that the connotation theory, together with the theory of "exposition" and "exponibles," enables us to have a reductional semantics. But these two still do not suffice to give us ontological reduction. To have ontological elimination, we need some additional principles that hook up the connotation theory with ontology, principles that provide us sufficient reasons to posit ontological entities.⁶ In Summa Logicae Ockham gave two such principles: (1) the epistemological one, that we have a sufficient reason to posit certain entities if they have to be appealed to in accounting for our acquisitions of mental signs;⁷ (2) the semantical one, that we have a sufficient reason to posit certain entities if they have to be appealed to in accounting for the significations of mental signs. Since Ockham's focus was mainly on (2), we shall follow Ockham and focus on the semantical one in this paper.

Notice that 'ontological elimination' can mean two different things: in the weak sense it means our refraining from asserting the existence of putative entities; while in the strong sense it means our denial of the existence of putative entities. Clearly the above principles only enable us

to have "ontological elimination" in the weak sense, for we cannot rule out the possibility that there might be a sufficient reason to posit the existence of some putative entities under another philosophical context. To have "ontological elimination" in the strong sense, we must strengthen the principles.⁸ It is not my aim here to determine which interpretation of "ontological elimination" is more appropriate - all I want to point out is that both are realizable by choosing the appropriate principle.

In the connotation theory, the key doctrine that takes Ockham to ontological elimination is the thesis that a connotative term has a semantically equivalent, fully expanded nominal definition, the "synonymy thesis" proposed by Paul Spade in his "Ockham's Distinctions between Absolute and Connotative Terms."⁹ Commentators have paid considerable attention to this thesis. While Spade and Marilyn Adams disagree about the degree of defensibility of the synonymy thesis,¹⁰ they agree that Ockham did hold this thesis and it provides at least a programmatic scheme for ontological elimination.¹¹ Following Spade and Adams, let's say that a good interpretation of the connotation theory must accommodate the synonymy thesis. Call it 'requirement (a)'.

There seem to be other requirements too. First, Ockham's mental language is a minimal one - it contains nothing that neither has significations nor contributes to the significations of others, and it has no synonym. This can be viewed as a result of the principle of parsimony known as "Ockham's Razor," that "Plurality should not be posited without necessity,"'² that "One ought not postulate many items when he can get by with fewer,"'³ that "What can happen through fewer [principles] happens in vain through more,"¹⁴ etc. So, a good interpretation of the connotation theory must also accommodate the thesis that there is no synonym in mental language. Call it 'requirement (b)'. Second, as Claude Panaccio pointed out in his "Connotative Terms in Ockham's Mental Language"¹⁵ (which is now widely accepted), Ockham did allow simple connotative terms in mental language, which, too, must be accommodated in our interpretation of the connotation theory. Call this 'requirement (c)'.

Unfortunately, among current interpretations of the connotation theory, none is able to satisfy all the above requirements (requirements [a], [b], and [c]). Spade and Adams' approach clearly satisfies requirement (a), but does it satisfy requirements (b) and (c)? In their approach they treat mental language as having a compositional syntax, i.e., absolute terms and syncategoremata are mutually independent simples, and connotative terms are just aggregates of simples (as appeared in their nominal definitions) (e.g., the connotative term "fireman" is literally an aggregate of simples "fire," "man," etc.). What this suggests is that a connotative term and its nominal definition are the same mental entity, not two distinct (distinct in type) entities that are semantically equivalent. Hence, their approach satisfies requirement (b). But since a connotative term is an aggregate of simples, it is complex. Hence, their approach does not satisfy requirement (c). Claude Panaccio takes requirements (b) and (c) seriously, but largely motivated by a worry about the semantical reduction of relational terms,¹⁶ he rejects the synonymy thesis. Martin Tweedale, in his "Ockham's Supposed Elimination of Connotative Terms and His Ontological Parsimony,"¹⁷ convincingly shows that Panaccio' 5 worry about relational terms is ungrounded, and reaffirms the synonymy thesis. But unfortunately he stipulates that there be mental synonyms among complex signs or between simple and complex signs in Ockham's mental language,¹⁸ and hence his account does not satisfy requirement (b).¹⁹

In what follows I shall offer an interpretation of the connotation theory which satisfies all three requirements.

The current prevailing interpretation of Ockham's view on the structure of mental signs is that mental propositions and some mental terms are complex - they have constitutive parts. In this section I argue that this interpretation is unfavorable. My strategy is to use both textual evidence and three arguments to show that mental propositions are simple for Ockham. Since my arguments for the simplicity of mental propositions can be transferred to mental terms, I shall conclude that all signs in Ockham's mental language are simple.

During the late mediaeval period, the issue concerning the structure of mental propositions was a controversial one. One common view, held by John Buridan and commonly attributed to Ockham, is that a mental proposition has constitutive parts in a way similar to a written or spoken proposition. Take, for example, the mental counterpart of the written proposition "Every man is (an)²⁰ animal." According to Buridan, the mental proposition consists of the universal quantifier "every," the concept "man," the mental copula "is," and the concept "animal." The alternative view, held by Gregory of Rimini and Peter of Ailly, is that all mental propositions are structureless mental acts which do not contain constitutive parts.²¹

Contrary to the prevailing interpretation, in at least two places of Commentary on Aristotle 's De interpretatione I, Prologue, § 6, Ockham seemed to espouse the view that mental propositions do not have constitutive parts, where he said:

and later in the same section:

Are these just accidental aberrations? Let's consider the problem carefully.

It is no doubt true that in several places Ockham said that a mental proposition is composed of mental terms,²⁴ and one possible interpretation (the prevailing one) is to take it as saying that a mental proposition is a complex having various constitutive parts. But there are other possible interpretations too. Notice that for the mature Ockham a mental sign is an act of thinking or understanding rather than a 'fictum" or thought-object (a view he held in the early stages of his career), analogous to the phenomenological "intentional object," having no real but a son of "intentional being. "²⁵ This view of mental signs allows us to have at least two other interpretations: First, we may say that a mental proposition is or can be formed by putting together several mental terms. And this leaves open the question whether a mental proposition itself as the product of combining several mental terms, is complex or not, because it is perfectly possible that by putting together several mental terms (acts of understanding) we get a new and simple entity (an analog can be found in chemical experiments: by mixing together different sons of stuff we get a new sort with a distinct molecular structure). Second, we may say that what we apprehend through an act of understanding which is a mental proposition is a complex, i.e., the signification of a mental proposition is equivalent to the sum of the significations of several mental terms, which also leaves open the question whether mental propositions themselves are complex. While it is not my business here to determine which of these two interpretations is more appropriate, I do want to show that the prevailing interpretation (that a mental proposition is complex) is unfavorable, by appealing to the following three arguments.

First, for Ockham mental language is the one in which God thinks. And since mental signs are acts of understanding, they have to be simple. For if they were not, God's thoughts would have to be complex - an unwelcome result.

Second, the view that mental propositions have constitutive parts is at odds with Ockham's principle of parsimony, that "one ought not postulate many items when he can get by with fewer. "26 It is quite clear that with respect to the structure of mental propositions, we posit fewer entities in the case when propositions are simple than in the case when they are complex, because in the former case we just posit the proposition itself - a single act of understanding, while in the latter case we need to posit the constitutive parts of the proposition - several acts of understanding.²⁷ So, Ockham's principle of parsimony is more compatible with the view that mental propositions are simple.

Third, there are the notorious word-order and word-binding problems²⁸ associated with the view that mental propositions have constitutive parts. The word-order problem is how we account for the mental counterpart of the word order in spoken and written language. And the word-binding problem (or the problem of the unity of a proposition) is how the various constituents of a proposition are bound together in the proposition. In the following let me show there are no easy solutions to both problems

Concerning the word-order problem, we know that mental word-order, if any, can be neither spatial nor temporal, since the acts of understanding are not spatial and the mind can produce a mental proposition instantaneously. But what can mental word-order be if it is neither spatial nor temporal? An easy solution is to adopt a "jigsaw" theory to account for mental word-order, i.e., let the characteristics of the constituents of a mental proposition themselves take care of mental word-order.²⁹ In other words, we register the syntactic information of word-order in spoken or written language, i.e., the syntactic position of a term in a proposition, in every mental term. Thus, to every categorematic term (e.g., "cat") in the old mental language, there correspond a number of distinct concepts (e.g., "subject-cat," "object-cat," etc.) in the new mental language, each of which bears a piece of syntactic information on the face of it. Accordingly, to find out mental word-order of a proposition in the new mental language, we only need to look at the constitutive terms in the proposition, extracting the syntactic information they bear. Now, it is

true that this theory does nicely account for mental word-order; however, it has some undesirable implications. First, this theory implies that every categorematic term in written or spoken language is necessarily equivocal, since a written or spoken categorematic term (e.g., 'cat') is subordinate to one concept (e.g., "subject-cat") when it is in the subject position, and another concept (e.g., "object-cat") when it is in the predicate position. But that does not seem to be held by Ockham. Second, according to this theory, in developing a theory of synonymy in spoken or written language Ockham's criterion that two spoken terms are synonymous if and only if they are subordinate to the same concept does not work here without qualification. Hence, the "jigsaw" theory, although interesting by itself, can hardly be attributed to Ockham.

Concerning the word-binding problem, an easy solution is to say that the mental copula "is" binds together the mental terms in the proposition. However, this solution, too, is not without problem. Consider a mere collection of mental terms including the mental copula "is" and a mental proposition which has exactly the same constituents. Certainly there is a difference between them, since a mere collection of terms is not a proposition. But if they have the same constituents, where does the difference come from? The only way to answer this question, it seems to me, is to appeal to the difference between mental copulas in the two cases.³⁰ We may say either that in the two cases the mental copulas are simply different concepts or that they are the same but behave differently. If we hold the former, how do we understand the binding mental copula in the mental proposition? Presumably the binding copula cannot occur outside a proposition, for if it did occur alone or in a mere collection of mental terms, it would not appear as a binding copula and hence would be a different concept. But Ockham does not seem to admit that there are mental terms (e.g., the binding mental copula) which cannot exist independently from mental propositions. If we hold the latter, how do we understand and explain the difference between behaviors of the same mental copula in the two cases? It cannot be psychological, because for Ockham the difference between a mere collection of terms and a proposition is not psychological, i.e., it does not vary from person to person, from language to language; and neither can it (the difference between behaviors of the same mental copula) be logical, since the same mental copula should have the same logical-syntactic employment.³¹

There might be other ways, compatible with Ockham' 5 theory, that satisfactorily solve the word-order and word-binding problems, but without working them out in detail, these two problems remain as severe challenges to the view that mental propositions have constitutive parts.

On the basis of the above three arguments, I suggest that Ockham's mental propositions are simple rather than complex, which is then compatible with Ockham's remarks in the Commentary. And if my arguments are sound, we can use similar ones to show that Ockham's mental terms, including connotative ones, are simple too. Taking these together, we then reach the conclusion that all signs in Ockham' 5 mental language, propositions and terms, are simple.

If every sign in Ockham's mental language is simple, i.e., Ockham's mental language is really non-compositional, a common worry is that we cannot have a recursive semantics for the non-compositional mental language,³² since ordinarily we only think of recursive analysis of semantics in a compositional language (e.g., a spoken or written language where there are complex signs built out of simple ones), where we proceed from syntax to approach semantics -e.g., we calculate the meaning of a sign from the meanings of its syntactic constituents. For example, to diagnose the meaning of 'a cat is on a mat' in written language, we first diagnose its syntactic structure - it has constituents 'a', 'cat', 'is', 'on', 'a', 'mat', and then calculate its meaning from the meanings of the constituents.

But a compositional syntax and a recursive semantics do not have to go hand in hand. For a language to have a recursive semantics, all that is required is that the meanings of signs in that language can be calculated from the meanings of those signs that have been previously calculated and/or those "primitive" signs whose meanings are given individually, which does not seem to impose a particular requirement on the syntax of the language. Hence, Ockham's mental language being non-compositional does not seem to be an obstacle to its having a recursive semantics.

However, the above worry does contain some legitimate element, i.e., it is motivated by a correct observation: in our ordinary (written or spoken) language a compositional syntax and a recursive semantics always appear together, and we are used to following syntactic structure of a sign to grasp the recursive analysis of its meaning. But in Ockham's non-compositional mental language, such a strategy is clearly blocked. And this poses a problem: how, in Ockham's mental language, are we going to explicate the semantics of the mental language to ordinary people who, as laymen in theology and logic, are used to thinking of the recursive analysis of semantics in a compositional language?

It is in the context of solving this problem, I suggest, that Ockham and Peter of Ailly introduced the notion of "semantical equivalence." We know that Ockham took semantical equivalence as a relation between a simple proposition and a complex proposition (several acts of understanding),³³ and that Peter took it as a relation between a simple connotative term and a complex one.³⁴ But what do these moves signify? The idea behind these seems to be the introduction of an imaginary compositional mental language which is semantically equipotent to the real, non-compositional mental language. Particularly, the construction of the imaginary mental language can take the following steps: (1) we throw all absolute terms and syncategoremata there; (2) for each mental proposition X in the real mental language, we build a counterpart of it by simulating the syntax of the corresponding written or spoken sign of X; (3) for each connotative term Y in the real mental language, we first find a complex written or spoken sign Y* that is subordinate to Y and that contains only written or spoken counterparts of absolute terms and syncategoremata, and then build a counterpart of Y by simulating the syntax of Y*; (4) we let every sign in the imaginary mental language have the same signification as its counterpart in the real mental language.³⁵ Thus constructed, the imaginary mental language must be compositional, since its syntax is modeled after the syntax of "our" (actually it is Ockham's) written or spoken language; and it is semantically equipotent to the real mental language because of step (4). Now, the imaginary mental language, together with the semantical equivalence relation - what takes us from a sign in the real mental language to its counterpart in the imaginary mental language, enable us to explicate the semantics of the real mental language in a way familiar to ordinary people, because through the semantical equivalence relation we can explain the semantics of the real, non-compositional mental language in terms of the semantics of the imaginary, compositional mental language which is a common place to understand recursive analysis. This then solves the problem in the previous paragraph.

Under this interpretation, the fully expanded nominal definition of a connotative term, a complex sign consisting of only absolute terms and syncategoremata, is not a sign in the real mental language, but a sign in the imaginary mental language, a piece of theoretical apparatus we create to explicate the signification of the connotative term Thus, a connotative term and its nominal definition are not mental synonyms, simply because the nominal definition does not really exist in Ockham's non-compositional mental language. Hence, our interpretation satisfies requirement (b).

An additional advantage that our interpretation has is that it explains why Peter Geach found that Ockham' 5 account of mental language made it look suspiciously like Latin,³⁶ since, according to our interpretation, in the part of Ockham' 5 account Geach was thinking of, Ockham was talking about the imaginary compositional mental language which is modeled after the spoken and written language he knew - Latin.³⁷

To recapitulate, in the foregoing I have offered an interpretation of Ockham's connotation theory that accommodate three things: the synonymy thesis that a connotative term has a semantically equivalent, fully expanded nominal definition; the minimal-language requirement that there is no mental synonym; and Panaccio's observation that there are simple connotative terms. Through elaborating on the distinction between real, non-compositional mental language and imaginary, compositional mental language, we accommodate the synonymy thesis and the no-mental-synonym thesis; and by arguing that all signs in Ockham 5 mental language are simple, we accommodate Panaccio's observation. To keep the synonymy thesis is to save the possibility of a recursive semantics for the mental language, and accordingly the intelligibility of Ockham's ontological elimination; to take seriously the no-mental-synonym thesis is to pay homage to "Ockham's Razor;" to appreciate Panaccio's observation is to do justice to textual evidence. Whether or not my suggestions are ultimately satisfactory, a good interpretation should accommodate all.