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Semantic Compositionality, Predicates and Properties
Antonio Rauti
0. Introduction
This paper concerns the difficulties faced by any view of language that is committed to all the
following ideas:
(A) language is semantically compositional and compositionality essentially involves a
correspondence between syntactic rules and semantic rules;
(B) predicates have semantic values; these values are (or are constituents of) truth-makers;
(C) what properties there are is not to be determined solely by semantic inquiry—there are no
linguistic short-cuts in ontology.
Of the three, (A) is most widely accepted, and the reasons for embracing it are numerous. The
second is likely to be embraced by authors who favor a realist approach to semantics that concerns
itself not only with the question of what, properly, is to be described as true or false—what the truthbearers are—but also with the issue of the ontological grounds for truth—what the truth-makers are.
The third is quite simply a reasonable position for such an approach to semantics.
As appealing as each of these propositions is by itself, they generate a challenge, which I am
going to set out. This project may remind one of attempts to show the inconsistency of a given set of
sentences. This is not quite fitting. (A)-(C) are vague, to different degrees. In adding precision, one
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will inevitably introduce additional ideas and constraints. But when (A)-(C) are interpreted
according to some of these constraints a difficulty arises: (A) commits us to the idea that all
sentences of subject-predicate form,  is  share the same kind of truth-conditions, to be given
one general formulation; (B) encourages us to state the truth-conditions in a way that involves a
reference to the semantic value of the predicate, thus introducing the need for a formulation that is
sensitive to the type of value—e.g., if the value is a property the formulation will read ‘true iff [[]]
instantiates/has [[]]’; (C) encourages us to think that not all predicates have the same type of value
and this needs to be reflected in the formulation of the truth-conditions. But—by (A)—for all
sentences of the form  is we can only have one formulation.
I will show in some detail how the difficulty arises and discuss some strategies for solving it. I
believe that none of the strategies is appealing but it would be futile to rule them all out as nonviable; not all theoretical needs and interests can be anticipated and unappealing moves might not
look so bad from certain angles. My focus in this paper is on setting out the difficulty and on the
costs to possible solutions.
The following part is devoted to sharpening (A)-(C) and to showing that they do not lack
plausibility . The second part is devoted to the problems.
PART 1
1. Compositionality
It is widely assumed that natural language is semantically compositional, that is, that the meaning of
a complex expression is determined by the meaning of its parts and the way in which they are put
together. This assumption is appealing because only semantic theories which incorporate it seem
capable of accounting for our ability to understand any novel sentence in our language, as long as
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we know the meaning of the components and the syntactic structure (thus our ability to understand
any of a virtually infinite number of sentences). The assumption, however, is vague and can evolve
into different specific claims.
A prominent approach that succeeds in turning the assumption into a precise claim is the kind of
truth-conditional semantic theory that originates with [Davidson 1965, ’67, ’70, ‘73] and uses the
apparatus employed by Tarski in his definition of truth for formal languages. It is with this type of
theory in mind that most authors formulate the principle of compositionality as follows: the truthconditions of a sentence are a function of, and only of, the meaning of is parts and the syntactic
structure.1
To appreciate the compositional character of the theory we need only look at the semantics for a
simple sentential logic language, where we find the essential elements in a simple setting: 1) an
interpretation (I), that is, an assignment of a truth value to every atomic sentence of the language,
and 2) a definition of truth(-in-I). The latter is what allows the spreading of truth values from the
atomic to the complex sentences, and accounts for the fact that the meaning (truth-value) of a
complex sentence is a function of that of its parts and the structure.
Form (the syntax) plays an important role in this kind of definition.2 There is a tight connection
between the semantic rules that compose the definition and the forms of the sentences. This is an
essential feature. The parallelism between syntactic and semantic rules is often expressed by
formulating each semantic rule as a conditional, where the antecedent says: ‘if the sentence is of the
following form, …’ thus making reference to a syntactic structure, while the consequent introduces
the semantic rule to use in evaluating the sentence. The antecedent could also be formulated this
way: ‘if the sentence has been formed by means of such and such syntactic rule, …’ and then
perhaps it would be more obvious that we have a correspondence between rules of one kind and
rules of another kind. This correspondence is a function and is an essential part of the apparatus:
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“adherence to [the principle of compositionality] leads us to construct our syntax and semantics so
that they work in tandem” [Dowty, Wall & Peters, 1981, p. 42].
This tandem work is traditionally described as follows: any given syntactic rule S which applies
to arguments e1, ... en to yield the well formed expression S(e1, ... en), is in correspondence with a
semantic rule that applies to the semantic values of e1, ... en (in an order specified by S) and yields
the semantic value of S(e1, ... en). (This rule-to-rule correspondence can be described mathematically
as a homomorphism from syntax to semantics).
The parallelism between syntax and semantics involves a clear commitment: sameness of form
implies sameness of semantic rule. All sentences of the same form, e.g., all conditionals, are to be
evaluated by the same rule. A theory of meaning for English under the guise of a truth-definition
will carry the same commitment. All sentences of a given form will have to share a semantic rule. It
is this commitment I will be focusing on.
2. Predicates have semantic values
A long tradition assigns semantic values to predicates and general terms. A variety of candidates
have been suggested for service as values, for example, abstract ideas, concepts, properties, sets,
intensions. This variety does not compel us to choose only one item. After rejecting a unified and
generic notion of meaning, as we should, it is quite reasonable to associate predicates with more
than one type of value, each one intended to perform different tasks. These values allow us to
preserve the idea that predicates contribute some one entity to the meaning of the sentences in which
they occur.3 Frege and Russell embraced the idea. It is standard in the extensional model-theoretic
semantics for formal languages—with sets as values—and it surfaces in contemporary discussions
in which the notion of a Russellian proposition is invoked (for example in some of the literature on
direct reference).4
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In the following, I address the position that the values of predicates are among the grounds (e.g.,
as constituents of states of affairs) for the truth or lack of truth of statements made by using
sentences in which the corresponding predicates occur (to simplify I’ll talk of sentences, instead of
statements).5
A concern with the grounds of truth within semantic theorizing is essential to those interested in
developing, in a realist vein, a conception of truth as correspondence. It is explicitly advocated, for
example, by K. Mulligan, P. Simon and B. Smith in reaction to the fact—as they judge it—that “In
the wake of Tarski, philosophers and logicians have largely turned their attentions away from the
complex and bewildering difficulties of the relations between language and the real world, turning
instead to the investigations of more tractable set-theoretic surrogates” [Mulligan et al., 1984, p.
288].
2.1. Truth-makers
A surrogate is a proxy for the real thing, and the rejection of surrogates that is advocated by
Mulligan et al. must be based on the confidence that we can theorize on the authentic grounds for
the truth of sentences. This confidence presupposes the acceptance of a powerful intuition, namely
that for every truth there is something in the world that makes it a truth. It is an old intuition that has
enjoyed much attention in recent years. It is often expressed in the form of a truth-maker principle,
for example:
(TM) Every true sentence of type T has a truth-maker [Oliver, 1996 p. 69].
(as Oliver observes, by specifying T we can restrict the principle to only some categories of
sentences, e.g., the contingently true ones). The relation between a true sentence and its truth-maker
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is a matter of debate. Most authors view it as a form of necessitation: in all counterfactual
circumstances in which the truth-maker exists, the relevant sentence is true.6 (TM) does not entail
that there exists only one truth-maker for every true statement (each cat makes true ‘there are cats’),
or that there is only one type of entity that truth-makers can be.
My concern is not with the details of truth-making theory but with the general idea that the
values of predicates will have a role to play in an account of truth-making. More precisely, this is
the idea that the values are to be among the entities whose existence (co-)guarantees the truth of the
sentence. Let us call these values ‘properties’. Property is primarily an ontological notion and the
identification of values with properties promises to provide us with the connection between
semantics and ontology that a commitment to truth-making theory makes desirable.
As properties can be conceived in many ways, we do not have a very precise view in focus yet. I
intend to restrict my attention to an interpretation of (B) according to which properties are
universals, i.e., are such that their instances have something strictly identical in common. There are
two main alternatives to this conception: one is to view properties as tropes and the other is to view
them as sets. The former does not allow us to identify the semantic value of a predicate with a
property, and forces us to define a more complex relation between a predicate and the tropes that in
some sense correspond to it. This conception, therefore, falls outside the scope of my discussion.
The latter alternative deserves some comments. Properties may very well be identified with
intensions (functions from possible worlds to sets, i.e., to the extensions of the predicates whose
intensions they are) or with sets of Lewisian possibilia. Yet, if the values of predicates are to have
the role of (co)truth-making, then we will have to rule out their identification with set-theoretical
entities.
This may not be obvious, for such entities would be well-enough suited to play the role of
properties for the exclusive purposes of semantic theory. At the very least they satisfy the following
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general conditions of adequacy for a conception of properties (conditions proposed in [Oliver 1996,
pp. 20-21]):
(i) the conception must offer some account of the instances of  instantiates , where ‘’ and
‘’ are place-holders for names of particulars and properties (this involves saying something helpful
about the nature of the referents and, to some extent, about the predicate ‘… instantiates …’);
(ii) it must preserve these two claims as truths: (a) different particulars can instantiate the same
property and (b) one particular can instantiate several properties;
(iii) it must avoid entailing that two properties instantiated by the very same actual items are in
reality the same property.
These conditions are easily satisfied; for example, if we are ready to follow D. Lewis, we can
assign to the monadic predicate ‘is a donkey’ the set of all donkeys, including “not only the actual
donkeys of this world we live in, but also all the unactualised, otherworldly donkeys” [Lewis 1983,
p. 344]. Sets collecting both the actual and the unactualised are adequate by Oliver’s standards. Sets
(if not possibilia) are well understood and we can interpret  instantiates  as  belongs to .
Several particulars can all be members of a given set and one particular can belong to several sets.
Finally, by allowing unactualised particulars into sets we avoid the identification of properties which
have the same actual instances (e.g., being a creature with a heart and being a creature with
kidneys): these sets will not be disjoint but they need not be the same set. Set-theoretical resources
may even enable us to distinguish between necessarily co-extensional properties, e.g. triangularity
and trilaterality.
However, it is arguable that in many cases these properties are not adequate as truth-makers.
Certainly, one can insist that the truth-maker of ‘Rufus is a donkey’ is the state of affairs of Rufus’
being a member of the set of actual and non-actual donkeys, and the set itself—the “property”—
would be a truth-maker indirectly.7 But this is bound to leave many dissatisfied. The dissatisfaction
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is sometimes voiced as a charge of idle circularity. We ask: What makes ‘Pa’ true? and we are told
that ‘Pa’ is true iff a belongs to the extension of ‘P’. We do not seem to receive a real answer
because, as Mulligan, et al. put it: “Whatever their formal advantages, approaches of this kind do
nothing to explain how sentences about the real world are made true or false. For the extension of
‘P’ is simply the set of objects such that, if we replace ‘x’ in ‘Px’ by a name of the object in
question, we get a true sentence. Set-theoretic elucidations of the basic truth-relation can, it would
seem, bring us no further forward” [Mulligan et al., 1984, p. 288].
I think that this charge does not go to the heart of the matter. It is true that if the extension of ‘P’
is identified as the set of objects such that replacing ‘x’ in ‘Px’ by a name of any one of them yields
a true sentence, then we do not accomplish much in elucidating the truth of ‘Pa’ as above. But one
could still retort that we have not shown that the ground of truth for an atomic sentence cannot be
membership in a set. Suppose we identify the extension of ‘P’ without mentioning truth, or suppose
we simply say that ‘P’ has an extension. What is wrong with the claim that a’s belonging to the set
is an ontological ground for the truth of ‘Pa’? Or with the more general claim that an item’s
belonging to a set is a ground for the truth of claims about it? We do have the feeling that the set and
the relation of membership are not adequate ontological material for this kind of grounding, or that
being in a set does not make anything be what it is (the other way ‘round, rather). But this is just a
vague intuition (and ‘grounding’ itself is vague).
Yet I do trust the intuition. It is beyond the scope (and the purpose) of my paper to mount a full
defense of it but I think it is best defended by pointing out that even if membership may be a
ground, it is not the deepest ground. Why do Rufus and every other donkey belong to their asinine
set? There does seem to be a “common ground” to all the members (at least the actual ones) and the
set seems more unified, more natural than an arbitrary collection of items. What accounts for the
unified character of the set (for the naturalness of the property) seems to be precisely the ground for
the truth of ‘Rufus is a donkey’.
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The naturalness of the set has to do with the resemblance between any two members. Do we
have the resources to account for naturalness within the set-theoretical approach? No. In D. Lewis’
words: “properties [Lewis means sets] do nothing to capture facts of resemblance” [Lewis, 1983,
346]. As he points out, any two things, whether they be very much alike or utterly dissimilar, are comembers of infinitely many sets and fail to be co-members of an equal number of sets. In other
words they both share and fail to share infinitely many properties. Therefore we cannot account for
their relative similarity or dissimilarity by appeal to the number of properties they both have (to
which they both belong) or fail to have.8 Other accounts are not apparent.
This does not yet disqualify set membership as a truth-maker of ‘Rufus is a donkey’, it only
suggests that in many cases (but not necessarily in all cases) there are grounds for membership itself
and these are deeper truth-makers for our sentence (so we may want to distinguish between deeper
and shallower truth makers). This leaves room for a fair question: if we are interested in truthmakers why should we have a preference for the deeper ones (simply put: for the non set-theoretical
ones)? The answer, I think, ought to be methodological. Although I am not going to argue for it,
there are other property-roles that set-theoretical constructs are not well suited to play. For example
as grounds for the causal powers of things, as ingredients in some accounts of scientific laws or,
more classically, as the answer to the problem of one-over-many.9 After choosing the deeper-level
truth-makers we might be able to identify them with the same entities that can fulfill these other
roles and thus achieve a nice level of theoretical integration.
Thus I take (B) to express a view that has a certain plausibility: some predicates have properties
(universals) as semantic values and these properties are not set-theoretical entities. This, of course,
does not amount to excluding the latter from the ontology.
3. What properties are there?
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(C) expresses the view that finding out what properties there are can never be as easy as consulting
the dictionary; it is the result of scientific and rational inquiry. At best, semantic considerations are
auxiliary to the enterprise of articulating actual ontology.
This attitude supports the further claim that there is no a priori reason to believe that predicates
and properties match up. This claim is not (or need not be) a form of revisionism. It does not entail
the prescription to change the way we talk once we have satisfied ourselves that, in some cases,
apparent discourse about properties—by way of the use of predicates—is not best interpreted as
about properties. Nor does it require accommodating our uses of certain predicates as systematic but
inconsequential errors. It simply relieves predicates of the responsibility of always introducing a
property.
The attitude I just described can be sharpened into slightly more precise views. For example, D.
M. Armstrong embraces it in the form of a posteriori realism: “it is to natural science … that we
should look for knowledge, or perhaps just more or less rational belief, of what universals there are
… The theory of universals may have to be developed in an a priori manner. But the theory of what
universals there are must be an a posteriori matter.” [Armstrong 1997, p. 25].
Armstrong is quite explicit in drawing from this the conclusion that predicates and properties
cannot be expected to stand in a one-to-one correspondence. To the contrary, there are many-many
relations between them. One predicate may have corresponding to it many properties (‘is a game’),
one property (‘has negative charge’) or no properties (‘is phlogisticated’). On the other hand one
property may have corresponding to it, many predicates (‘has gravitational rest mass M’ and ‘has
inertial rest mass M’), one predicate (though we can obviously add more) or no predicate (in the
case of unknown properties) (the examples are from [Armstrong 1997, p. 26]). 10
In general, those who are interested in truth-makers have a fundamental reason to embrace (C):
avoiding the trivialization of their inquiry. Whether trivialization is a serious threat actually depends
on the precise aim of the inquiry. An inquiry into truth-makers could be confined to the task of
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defining the most general features of the relation between a sentence and that which makes it true (Is
it a relation of necessitation? How should we understand it? Do only contingently true statements
require a truth-maker? and so on). But it could also extend into an investigation on the ontological
constitution of what makes true statements true. In that case, saying that, e.g., ‘Jerusalem is holy’ is
true, and it is so in virtue of Jerusalem’s being holy (or Jerusalem’s holiness), i.e., using the details
of the sentence to construct an automatic answer to the question ‘What makes it true?’, does not
bring us any closer to a real answer. We may agree that there is such a truth-maker as Jerusalem’s
being holy. But that there is a truth-maker is already a consequence of the assumption that the
statement is true (together with the general principle that a true statement is made true by a truthmaker). Describing the truth-maker as the truth-maker consisting of Jerusalem’s being holy does not
say much more about its constitution than the sentence itself does. In the same way, we can say that
‘Chess is a game’ is true in virtue of chess’ being a game and that does not settle what the truthmaker is: is it instantiating a complex property or being a member of a certain historically defined
class that is not unified by a property? In both cases—trivially—the truth-maker is chess’ being a
game. If an answer about constitution is obtained by linguistic transformation the inquiry is
trivialized.
Thus, I take (C) as well to express a view that does not lack initial plausibility: there is no
automatic answer to the question whether a property is the semantic value of a predicate.
PART 2
4. Semantic rules and propertiesIt is time to get to my argument. Recall that a Davidsonian theory
of meaning (in English) for a language L is organized as a set of axioms sufficient to derive all the
instances of the schema
S is true (in L) if and only if p
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where S is a placeholder for a structural description of a sentence of L and p is to be replaced by an
English sentence that has the same truth-value as the sentence described (if English contains L then
the sentence ought to be the same as the sentence described).11 The aspect of the theory on which I
want to focus is the division of labor between axioms that specify the meanings, or reference
conditions, of the atomic expression of L and the axioms that formulate the semantic rules for
evaluating sentences, i.e., axioms that formulate the truth-conditions of the sentences. To simplify
the discussion I will only consider subject-predicate sentences of the simplest form, such as ‘Tom is
fat’, i.e. sentences whose standard translation in first order logic languages is of the kind Fa.
Clearly, an interpretation that assigns Tom to ‘Tom’ and the property of being fat to ‘is fat’ is
not sufficient to yield the truth-conditions of ‘Tom is fat’. We need a semantic rule (a compositional
axiom) that says that if a sentence is of the form
 is 
then it is true iff something is the case. The rule will have to mention the semantic values of the
singular term and the predicate, and will have to specify circumstances involving them such that
when they obtain the sentence is true. A natural way to do it is by saying:
 is  is true iff [[ instantiates/has [[]]
(where ‘[[…]]’ abbreviates ‘the semantic value of …’). This formulation is not the most familiar. It
states the truth conditions of the target sentence in terms of a relation between semantic values.
Appeal to this relation introduces an ontological notion (instantiating/having/possessing) that is
usually absent in more familiar formulations, e.g., in this one:
A sentence coupling a name with a predicate is true iff the object denoted by the name satisfies
the predicate. [Evans 1981, p. 123]
An object’s satisfying a predicate—or a predicate’s being true of an object—is the key notion of this
formulation and can be used without being explicated by means of any more specific ontological
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notion. In fact it is not meant to be explicated in such a way. Appeal to satisfaction, or truth, is all
we need in the rule, in virtue of the fact that in theories containing this type of rule the axioms for
the predicates, let us say for ‘is fat’, are stated thus:
An object satisfies ‘is fat’ (‘is fat’ is true of an object) iff the object is fat.
This, however, is not the way in which we deal with ‘is fat’ after assuming that predicates have
semantic values (properties). The axiom for ‘is fat’ simply assigns the property of fatness to it. It
could be stated thus:
‘is fat’ denotes/expresses fatness12
a statement perfectly analogous to that by which we dealt with ‘Tom’:
‘Tom’ denotes Tom.
Thus we are led to the rule I formulated above. Notice that the rule is analogous in a crucial respect
to the rule we would employ if we decided that the semantic value of ‘is fat’ is the set of fat people.
In that case the rule would be
 is  is true iff [[  [[]].
This too is a rule that states the truth conditions of the target sentence in terms of a relation between
semantic values. Rules like these are peculiar because they are stated in a way that reflects the
nature of the value assigned to the predicate and thus give ontology a foothold in semantics. The
nature of the predicate’s value is regarded as a fundamental constraint on the rule. For example, it
would be wrong to say that a sentence of the form  is  is true iff [[ instantiates [[]] when
[[]] is a set, or that it is true iff [[ is a member of [[]] when the latter is a property.
In a sense, these rules satisfy a limited request for conceptual breakdown, but this provides more
trouble than insight. My argument is quite simple. According to (A) compositionality requires that a
certain form of sentence be paired with one and only one semantic rule. By (B) we are induced to
pair the subject-predicate form with a rule like
13
 is  is true iff [[instantiates/has [[]]
(let us call it the property rule) thus presupposing the semantic value of every predicate to be a
property and its instantiation the key to the truth of the sentence. But if we adopt the attitude
expressed by (C), i.e., the rejection of linguistic short-cuts in ontology, we cannot be confident that
this is going to make sense always, because it may very well be the case that not all predicates are
plausibly taken to have a property as their value. To some of them it may be best to assign a set.
That is, a set may be available as a value while no property is. 13 In other words, the rule is
ontologically committed in a way that we, forgoing short-cuts, find irresponsible and cannot
endorse. A different rule, perhaps one stated in terms of set membership, would raise the same kind
of difficulty. We could eschew the commitment by admitting a plurality of rules, but rule-to-rule
compositionality requires that we choose only one rule.
To simplify a bit, the core of the problem is the possible heterogeneity of semantic values for
predicates. There is no ontological category which we can apply comfortably to all such values. If
we have rules that probe the relation between the value of a singular term and that of a predicate,
then the heterogeneity on the predicate side ought to be accommodated within the theory by making
the rule take account of it. But then it is not the case that one rule fits all.
I consider this is a bit of a simplification for a reason that I will mention shortly (in section 5.1).
But before I do so, I would like to put aside one problem that some readers might have already
thought of as germane to the discussion: how to deal with predicates, if there be any, (i) to which no
property corresponds and (ii) for which no other value may seem to be available (empty predicates).
I do not think this is an irrelevant issue and I put it aside for no better reason than that there are too
many other issues I prefer to focus on.
5. DiscussionA host of things may be said at this point. Let us consider them one by one.
The following considerations have to do mostly with two subjects: the predicates and the rule.
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5.1. PredicatesTo commit oneself to (B) is to adopt a view of properties (universals) as suitable
values for predicates, while embracing (C) leads to the expectation that ordinary language is
peppered with predicates whose value is not a property. It would be nice to see examples of such
spoiler predicates. But, of course, it is hard to point to good examples if we do not select a complete
theory of universals; so I will limit myself to mentioning one example.
Armstrong’s sparse theory of properties (universals) leads him to the provisional claim that
properties like having mass, having length or having color, i.e., determinables, actually “are not
universals, or more cautiously, need not be universals, but are classes of universals united by all
sorts of complex partial identities. To attribute a determinable property to a first-order particular is
to assert that the particular instantiates one member of that class.” [Armstrong 1997, p. 57].14
If the claim is correct, the truth-conditions of  is colored are not atomic. We cannot say that
the sentence is true iff [[]] instantiates the semantic value of ‘is colored’ and we cannot say that it
is true iff [[]] is a member of the semantic value of ‘is colored’. The sentence has quantified truthconditions: it is true iff there is a certain member of a certain set, that of color properties, such that
[[]] instantiates it. This set is “introduced into” (or becomes relevant to) the truth-conditions as the
semantic value of ‘is colored’. This case exposes the simplification I warned about. The decision to
associate a set of color properties with the predicate ‘is colored’ obviously prevents a formulation of
the semantic rule in terms of simple instantiation; the complication is that we cannot formulate the
rule in terms of set membership either. We are pushed towards a formulation according to which the
relation between the value of the singular terms and the set of color properties is fairly complex. So,
we cannot reduce our dilemma to a choice between two types of values for predicates—properties
and sets—each one of which brings in its own semantic rule.
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5.2. The rule
What about the semantic rule for sentences of the form  is ? Perhaps we can find a way of
stating it that avoids the difficulties. To simplify, let us assume that predicates have semantic values
of just two different kinds: properties and sets.
Two strategies seem available to avoid the problems:
1) One is to reformulate the rule generally enough to avoid constrictive commitments but not so
generally as to make the statement of truth-conditions too generic. This is a possible reformulation:
 is  is true iff [[]] fulfills/meets the condition imposed by [[]]
We can call this the ‘fulfillment rule’.
2) The other strategy is to adopt a disjunctive rule, something like
 is  is true iff either [[]] instantiates [[]] or [[]] belongs to [[]] or …
Let us call this the ‘disjunctive rule’.
5.2.1. The fulfillment rule
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While the property rule presupposes a specific type of semantic value for a predicate, the fulfillment
rule does not do so. It simply exploits the idea that whatever semantic value a predicate has, it can
be seen as imposing or defining a condition. The notion of a condition to be fulfilled should be
thought of as something general that can be specified in different ways; sometimes the condition to
be met is possessing a property, sometimes it is belonging to a set, sometimes it is being at the
center of a type of circumstances, or what have you. Thus the rule would not commit us to saying
that all predicates have the same type of semantic value.
This might seem promising but it is not. Suppose we are given a sentence, Tom is , plus an
interpretation of the language, the fulfillment rule, and knowledge of all the facts about Tom. We
would still not be in a position to evaluate the sentence. The simple reason is that all the information
given does not entail a specification of what condition Tom must meet in order for the sentence to
be true. Suppose [[]] is a property. One may be excused for making the default assumption that the
condition is instantiating the property. But it must be noticed that:
i) the property itself does not impose the condition;
ii) there is a variety of conditions we could associate with a property; one possible condition is that
of having [[]] until 3001 and not having it afterwards. Another is that of being the first object to
have [[]], and yet another that of being the only object to have [[]]. It is interesting to notice that
two other conditions, that of not having [[]] and that of having had [[]] sometime are such that
we chose not to adopt them, in favor of separating negation and tense as functions represented
syntactically. Still, it is conceptually legitimate to include them among the conditions that we could
have chosen to see [[]] as imposing;
iii) the default assumption that the condition is simply instantiating the property is not embedded
anywhere in the theory.
17
Thus the fulfillment rule by itself is insufficient. This is a problem if one makes a fundamental
assumption, namely that if we know the semantic rule associated with a sentence we ought to be in
such a position that the only other information we need, in order to establish the truth value of the
sentence, is of two kinds: knowledge of the semantic value of the words used (e.g., of the value of
the predicate) and knowledge of the world. Thus, if we are ignorant of the truth value of the
sentence while we do know the semantic rule and the meaning of the component words, we must be
lacking in worldly knowledge, we must not know all the relevant facts. But if, by hypothesis, we
know everything worldly there is to know, then our failure to evaluate the sentence means that the
rule is inadequate.
Now, there are at least two ways to respond to these observations about the fulfillment rule.
5.2.2. More bang for your semantic values
First, it can be countered that we need to see more in the fact that  has a semantic value. For
example, if  expresses a property, the latter is only part of the semantic value of . There is
another component, something associated with  that tells us: for a sentence like Tom is  it is
the having of the property expressed by  on which the truth value depends.
The idea is that it falls to the interpretation to tell us not only what item corresponds to the
predicate, but also what exact condition it must be seen as imposing. Then we’ll be able to rely
simply on the fulfillment rule when it comes to evaluating a sentence of the kind  is , for any
. This suggestion of a “rich” semantic value for predicates is still a bit vague and can be
incorporated into the theory in different ways.
One way is to distinguish two elements in the semantic value of a predicate, one of which might
as well be ‘externalized’ as a specific rule, e.g., a rule saying that the condition imposed by the value
18
of the predicate is that it be instantiated. Proceeding this way has the advantage that we keep a sharp
distinction between the notions of semantic value and that of semantic rule. Semantic values are
“inert” so to speak, they do not “tell us” anything; semantic rules tell us something concerning these
values.
The whole proposal, then does not amount to more than this: we distinguish two levels of
semantic rules, a general one and a specific one. At the general level we find a general rule like the
fufillment rule. All sentences of the form  is  will be associated with it. Up to this point we
keep the tight connection between form and semantic rule. But at the specific level we find a
plurality of (specific) rules, e.g.:
(a)  is  is true iff [[]] instantiates [[]] (or: … iff [[]] satisfies the condition of
instantiating [[]])
(b)  is  is true iff [[]] belongs to [[]] (or: … iff [[]] satisfies the condition of belonging
to [[]])
and so on. To complete the evaluation of an instance of the form  is  we will have to proceed
from the general rule to the specific rule that is relevant. The latter is determined by the type of
semantic value of the predicate.
This, however, entails that at the more specific level we have a collapse of compositionality, for
in fact not all sentences of the same form have the same (specific) truth-conditions. In other words,
at the specific level we lose the one-one correspondence between syntactic rules and semantic rules,
in fact, a sentence ‘a is P’ might be associated with (a) while a sentence ‘a is F’ might be associated
with (b).
19
An alternative way of incorporating the suggestion into the theory allows us to keep a one-one
correspondence by modifying the original one into a correspondence that relates the combination of
the subject-predicate structure plus a certain type of semantic value of the predicate with a certain
semantic rule, and relates the combination of the subject-predicate structure plus another type of
value with a different semantic rule, and so on.
The upshot of this modification is that semantic notions are present both in the domain (the set
of combinations) and in the range of the correspondence (the set of semantic rules). The original
correspondence could be formulated with just syntactic notions on one side and semantic notions on
the other. This feature is lost; semantic notions have seeped into the first side and we have an
“impure” form of parallelism: if the sentence is subject-predicate and the semantic value of the
predicate is of type 1, then… and if the sentence is subject-predicate and the semantic value of the
predicate is of type 2, then … .
This impure parallelism ties the prospect of a complete semantics for English to that of a
satisfactory metaphysical investigation that establishes all the possible types of semantic values for
predicates and the possible conditions that they may impose. It would not be sufficient to declare
that the theory of meaning acknowledges different types of values without specifying them, for that
would only give us the architecture of a possible theory. Moreover, the commitment to (C) forbids
us to use the semantic horse to draw the metaphysical cart, therefore the semantic intuition that a
certain sentence of the form  is  ascribes a property to an individual needs to await
metaphysical endorsement before we can be sure of how the theory should state the truth conditions
of that particular sentence.
There is a further potential complication that I will sketch but will not elaborate upon. The
impure parallelism might be a source of problems if we consider ascribing a tacit knowledge of the
theory to speakers. This ascription seems to presuppose that speakers have a tacit knowledge of the
20
ontological types of the values of predicates. Faced with a particular sentence of the form  is 
the speaker will first employ her tacit knowledge of the syntax to achieve the knowledge that the
sentence has the structure it has and then she will further employ her tacit knowledge of the type of
value of  in order to select the relevant truth-conditions. But except for a few philosophers,
speakers are not metaphysically competent. They might assume a folk ontology—which might or
might not be true—and this folk ontology might underlie some of their linguistic competence, but it
is not obvious that their entire linguistic competence depends on their ontology. For example, the
notion of property might not be relevant to linguistic competence. And in fact speakers seem to use
English well enough even when they do not have any (tacit) clue about what properties are and what
properties there are; e.g., it does not seem necessary that we (tacitly) know whether being lucky is
actually a property in order for us to use the predicate ‘is lucky’ adequately. Since the notion of tacit
knowledge can be cashed out in different ways, there might be some space for maneuvering. I limit
myself to pointing out the task.
5.2.3. Predicates as part of the recursive apparatus
There is a second way of solving the problem raised by the fulfillment rule. Let me set up the
discussion by taking a few steps back. Consider a simple sentential logic language (SL) again. Its
truth-definition is such that different forms of sentence are paired with different semantic rules. Now
this is a bit of an over-simplification. The structure of sentences of SL can be described in more or
in less detail. Very generally we might say that there are three categories of sentences: atomic ones;
sentences of the form connective + sentence (e.g., negations); sentences of the form sentence +
connective + sentence (e.g., conditionals). The latter is a form of sentence, in a very general sense.
21
Given the way I presented SL above (as having only two connectives, ‘~’ and ‘’) this would be
fine, but if we had more binary connectives the description needed in setting up the truth-definition
would have to be a little less generic. We would need to distinguish between the forms, say,
sentence + & + sentence and sentence +  + sentence, so as to be able to pair all the sentences of
one form with a semantic rule different from that with which we pair the sentences of the other
form.
Thus connectives are treated in a special way. They are expressions that give rise to forms of
sentence and in terms of which the truth-conditions are recursively assigned. It is the presence of
one connective rather than another that dictates what semantic rule is relevant to the evaluation of
the sentence. Connectives are part of the recursive apparatus.
This immediately suggests a possibility. Recall first what the problem is: if we grant that
sentences of the form Tom is  can assert different things, for example that they can assert that
(may have specific truth-conditions according to which the sentence is true iff) either:
(a) the designation of ‘Tom’ has the property expressed by ,
or
(b) the designation of ‘Tom’ belongs to the set designated by 
then it appears that the compositional project cannot be carried out because the principle that form
induces truth-conditions is violated. The form of the two sentences is the same but different
conditions are induced.
If we really want to stick to the devices of the compositional apparatus, however, we can block
this conclusion by including the adjective itself among the special expressions that determine truthconditions (together with form). Adjectives would be special symbols, like the connectives in SL.
22
For example, ‘Tom is blond’ and ‘Tom is lucky’ would be associated with different semantic rules
for the same reason that ‘A&B’ and ‘AB’ are: because of the different special symbols occurring
in them. As is the case for ‘A&B’ and ‘AB’, the two sentences ‘Tom is blond’ and ‘Tom is lucky’
would be of different form.
This is quite a radical move. For one thing, we should extend it to all adjectives. Every time, in
order to explain why Tom is  has certain truth-conditions, instead of others, we would have to
invoke  as the syntactic (co-)determiner of the conditions (which, again, means that we would
have to treat it as we treat connectives in SL). This amounts to having as many syntactic rules (or
forms of sentences) as there are adjectives. The correspondence between syntactic and semantic
rules would link many syntactic rules with the same semantic rule, therefore, we could tidy up
things by grouping adjectives, and obtain something like this:
(a') If the sentence is of the form  is  and  is one of these adjectives [list follows], then the
sentence is true iff the designation of  possesses the property expressed by 
Another would read:
(b’) If the sentence is of the form  is  and  is one of these adjectives [list follows], then the
sentence is satisfied iff the designation of  belongs to the set designated by 
And so on for all the kinds of adjectives that can form predicates inducing distinct truth-conditions.
This may strike us as a major sin against the spirit of compositionality, even if we group the
clauses. What allows us to group the clauses and formulate the kind of super-rule exemplified by (a')
and (b’) is the fact that many adjectives determine a link of their syntactic rule to the same semantic
23
rule. But this possibility cannot not hide the fact that there really is a syntactic rule for every
adjective. We form lists of them according to which semantic rule they are linked to.15
Of course, the mapping from syntax to semantics does not disappear. Now, instead of a one-toone correspondence between syntactic rules and semantic rules we simply have a many-to-one
correspondence.
This reorganization of the apparatus distinguishes between forms (syntactic rules) in the most
fine-grained way. So fine-grained indeed that it turns out that there are no sentences of the same
form (formed by the same syntactic rule) as ‘Tom is lucky’, except for ‘Tom is lucky’ itself, which
trivially reinstates the truth that all sentences of the same form as ‘Tom is lucky’ have the same
specific truth-conditions.
The trivial restoration of this truth comes at a price, however: an extravagant syntax that seems
to have no justification besides its ability to rescue the apparatus. This ad hoc syntax may not result
in an overly complicated apparatus because by grouping rules we manage to keep it tidy, as it were,
but this is all that can be said for it.16
5.3. The disjunctive rule
So much for the fulfillment rule. We may want to resort to the idea of a disjunctive rule:
(if  is  is subject-predicate then)  is  is true iff X or Y or Z or …
This gives us a disjunctive parallelism between syntax and semantics because it entails that either
the subject-predicate structure is associated with one type of truth-condition (true iff X) or it is
associated with another (true iff Y) or it is associated with another (true iff Z). This is equivalent to
24
saying: either if is is subject-predicate then it is true iff X or if is  is subject-predicate
then it is true iff Y, etc. Of course, this is still not as bad as a complete loss of parallelism, which we
would have if the ‘or’ were replaced by ‘and’.
Now, in this case the obvious question is: which disjunct holds? This is a legitimate question. It
is less clear whether it is the duty of a theory of meaning to provide an answer, but it does seem
reasonable to require that the theory have a way to say, in principle, how one disjunct comes to be
the relevant one. If this requirement were met, the theory could be used by someone well apprised of
the facts to determine if a given sentence is true or false. The only plausible way to make the theory
fulfill the requirement is to make the choice of the disjunct depend on the semantic value of the
predicate.
This leads again to an idea I have mentioned previously, namely that we may need to see more
in the fact that  has a semantic value than we have. For example, if  expresses a property, the
latter is only part of the semantic value of . The other element is what tells us that for a sentence
like Tom is  it is the having of the property expressed by  on which the truth-value depends.
However, the upshot of this is the same as the one discussed earlier: the theory must distinguish
between types of semantic values, and a semantic rule will be associated with a given sentence on
the basis of both the form of the sentence and the type of value of its predicate. Thus we cannot set
up a parallelism which on one side mentions only syntactic notions and on the other side mentions
only semantic notions.
6. Logical form
If, e.g.,  is square and  is lucky are associated with different semantic rules we can restore a
perfect parallelism between forms and rules by assigning the two sentences different forms, different
25
logical forms. A discussion of this strategy is beyond the scope of this paper. Two considerations
about it ought to be mentioned, however: a) the move is clearly ad hoc and b) there is no
independent syntactic motivation for postulating a difference between the logical forms of the two
sentences.
The second point is important if by ‘logical form’ we mean the level of syntactic representation,
often called ‘LF’, that describes the starting point for interpretation in standard approaches to syntax
(e.g. in Government and Binding theory). LF is also described as the interface between syntax and
semantics. As such it is not divorced from the rest of the theory. This means that claims about LF
must be accommodated within the general theory. It must be possible to describe in a principled way
the transformations that lead from an original structure to LF. I do not want to argue that it is
impossible to find an adequate accommodation, within a general syntactic theory, for the claim that
 is square and  is lucky have distinct LFs, but I emphasize that the confidence in this
possibility has no solid basis. If by ‘logical form’ we do not mean anything like LF but rather some
purely philosophical notion, then I limit my complaint to that of ad hocness. In any case, there is a
task before us and not yet a strategy to answer the concerns I articulated.
7. Conclusions
The three claims I have discussed are plausible and have been embraced, separately, by several
authors. They have not been presented together often, mostly because they are likely to surface in
discussions where there is no systematic focus on all three subject-matters (even though in
discussions where (B) and (C) are embraced the assumption of compositionality is probably a
default one). However, systematic attention to the notions involved is desirable. It also reveals a
difficulty.
26
In my argument I focused on a somewhat restrictive version of the second claim, according to
which the semantic values of predicates are properties, understood as universals. I also appealed to
the auxiliary claim that a theory of meaning ought to enable someone to evaluate a given sentence
when both the relevant facts and the semantic values of the components are known. We could reject
or refine in other ways any of the claims in order to avoid the difficulty and I think this can (and
must) be done. My preference is for rejecting the claim that predicates have a semantic value, but I
register this without argument.
Department of Philosophy
University of Wisconsin-Madison
27
Bibliography
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Armstrong D. M. 1992 ‘Properties’ in Language, Truth and Ontology, ed. Kevin Mulligan
(Dordrecht: Kluwer A. C.) 14-27.
Armstrong D. M. 1997 A World of States of Affairs (Cambridge Univ. Press: Cambridge, Mass.).
Cox D. 1997 ‘The Trouble with Truth-Makers’, Pacific Philosophical Quarterly 78, 45-62.
Davidson D. 1965 ‘Theories of Meaning and Learnable Languages’ now in Inquiries into Truth and
Interpretation (Oxford Univ. Press: Oxford, 1984), 3-15.
Davidson D. 1967 ‘Truth and Meaning’, in Inquiries, 17-36.
Davidson D. 1970 ‘Semantics for Natural Languages’, in Inquiries, 55-64.
Davidson D. 1973 ‘In Defense of Convention T’, in Inquiries, 65-75.
Dowty D. R., Wall R. E. and Peters S. 1981 Introduction to Montague Semantics (D. Reidel:
Dordrecht).
Evans G. 1981 ‘Reply: Semantic Theory and Tacit Knowledge’ in Wittgenstein: To Follow a Rule,
ed. S. H. Holzman and C. M. Leich (Routledge & Kegan Paul: London).
Evans G. and McDowell J. eds. 1976 Truth and Meaning (Oxford Univ. Press: Oxford).
Fox J. F. 1987 ‘Truthmaker’, Australasian Journal of Philosophy 65, 2, 188-207.
Hochberg H. 1994 ‘Facts and Classes as Complexes and as Truth Makers’, The Monist 77, 2, 170191.
Lewis D. 1983 ‘New Work for a Theory of Universals’, Australasian Journal of Philosophy 61, 4,
343-377.
Larson R. and Segal G. 1995 Knowledge of Meaning (The MIT Press: Cambridge, Mass.).
Mulligan K., Simon P. and Smith B. 1984 ‘Truth-Makers’, Philosophy and Phenomenological
Research, XLIV, 3, 287-321.
28
Oliver A. 1996 ‘The Metaphysics of Properties’, Mind 105, 417, 1-80.
Parsons J. 1999 ‘There is No “Truthmaker” Argument against Nominalism’, Australasian Journal of
Philosophy 77, 3, 325-334.
Pelletier F. J. 1994 ‘The Principle of Semantic Compositionality’, Topoi 13 11-24.
Pendlebury M. 1986 ‘Facts as Truthmakers’, The Monist 69 177-88.
Quine W. V. O. 1960 Word and Object (The MIT Press: Cambridge, Mass.).
Quine W. V. O 1982 Methods of Logic (Harvard University Press: Cambridge, Mass.).
Smith B. 1999 ‘Truthmaker Realism’, Australasian Journal of Philosophy 77, 3, 274-291.
Swoyer C. Fall 2001 “Properties”, The Stanford Encyclopedia of Philosophy, ed. E. N. Zalta, URL =
http://plato.stanford.edu/archives/fall2001/entries/properties/.
Notes
1
This formulation is essentially the one given in [Pelletier 1994]. It resembles most formulations closely except for the
qualification ‘and only of’, which Pelletier argues to be essential; other authors omit it. Nothing in my argument will
depend on it.
2
The form of a sentence can be described in more or less detail, and the difference in form between two sentences is
relative to the degree of detail chosen. This will be of some relevance later but it is not overly important here.
3
The tradition has dissenters. For example, Quine rejects the characterization of the usefulness of general terms as due
to their contributing some one “unified” value; general terms do not have a semantic value but rather a divided
reference, the items of which the term is true: “A word [a general term] can prove useful in such positions as to favor the
29
assumption of objects for it to be true of, without thereby favoring the assumption of objects related to it in other ways,
e.g., as extension or intension” [Quine, 1960, § 49, p. 239; also §19-20]. See also [Quine, 1982, pp. 288-89].
4
For example, F. Recanati writes: “This [the one expressed by ‘ is G’, where ‘’ is directly referential] is a ‘singular’
proposition, consisting of the reference of ‘’ and the property expressed by the predicate ‘G’ ” [Recanati, 1993, p. 27].
5
I will also simplify the discussion by focusing only on monadic predicates and on properties (as opposed to relations).
6
See [Fox 1987, p. 189], [Armstrong 1997, p. 115], [Cox 1997, p. 46]. [Smith 1999] denies that truth-making is just
necessitating and [Parson 1999] distinguishes between what Parson calls ‘truthmaker esentialism’ (truth-making as
necessitating) and a non-essentialist truth-making principle, according to which “The truth-maker for a sentence is that
thing that is intrinsically such that the sentence is true. Being intrinsically such that the sentence is true amounts to the
fact that there could not be a duplicate of that thing without the sentence being true” [Parson, 1999, p. 327]. This
formulation does not entail that in every possible world where the thing exists it makes the sentence true. Suppose a red
rose is the truth-maker of ‘this rose is red’ (according to the formulation quoted). As Parson points out, the same rose
could exist in another possible world and be pink or have each petal of a different colors. Relative to that world it would
not make the sentence true.
7
Or the truth-maker could be the set itself rather than the state of affairs of Rufus’ belonging to it. See [Hochberg 1994]
for an elaboration of the claim that statements of membership in a class can be taken to have just the class as their truthmakers.
8
This holds, naturally, if the universe is infinite. If it is not infinite but just large, the same point can be made if we
substitute ‘exceedingly many’ for ‘infinitely many’. In fact, we could still not account for (or even notice) facts of
similarity if they were to be reduced to facts of set membership involving exceedingly many sets.
9
This is a common enough point. For example, Chris Swoyer makes it in his entry “Properties” in The Stanford
Encyclopedia
of
Philosophy
(Fall
2001
Edition),
Edward
N.
Zalta
(ed.),
URL
=
http://plato.stanford.edu/archives/fall2001/entries/properties/.
10
The claim that no property (universal) corresponds to ‘phlogisticated’ is based both on the absence of things to which
the predicate applies and on the assumption that there are no uninstantiated universals. The latter is controversial.
11
Much of the debate on Davidson’s programme has centered on the constrains that the theory ought to meet in order to
ensure that the derived instances of the schema (the theorems) are interpretive, i.e., that they pair two sentences which
intuitively have the same meaning. I mention this component of the debate only to set it aside as irrelevant to the present
argument.
30
12
Nothing hinges on the choice between ‘denotes’ (or ‘names’) and ‘expresses’. In both cases all that is meant is that
fatness is the semantic value of ‘is fat’. See [Larson and Segal, 1995], chapter 4, for a presentation of an approach of
this kind (PCprop).
13
I argued before that facts of set membership are not deep enough when the set is a natural class whose members are
members on some ontological ground. When a property is not available to provide the ground and we only have the set,
we clearly have a ‘gerrymandered’ set. If it is the value of , then membership in the set, or simply the set itself is the
truth-maker of  is . In other words, membership in the set or the existence of the set is the necessary and sufficient
condition under which  is  is true.
14
[Pendlebury, 1986] endorses the same position—less tentatively.
15
A possible way to avoid this conclusion has been suggested to me by Andrew Hsu. Consider ‘alone’ and ‘lucky’. They
can both be used to form a predicate (‘is alone’, ‘is lucky’) but they differ in some syntactic respects. For example,
‘alone’ can occur in the construction ‘He alone was rescued’, whereas ‘lucky’ cannot. There may be other differences of
this kind. Let us call the syntactic possibilities of an adjective its syntactic profile; ‘alone’ and ‘lucky’ do not have the
same syntactic profile. This suggests that we may be able to group adjectives according to their syntactic profiles and, if
we are lucky, these groups may coincide with the lists I have mentioned. If they do, we can replace the list with the
profile and recover the generality we had lost. According to this solution, the semantic rule that is relevant for a given
instance of the form  is  is determined not only by the structure  is  but also by the syntactic profile of the
predicate replacing  (or we could say that two sentences like ‘Tom is lucky’ and ‘Tom is alone’ differ in structure just
in virtue of the difference in the profile of the adjectives).
It is clearly an empirical question whether a difference in the type of truth-conditions exists only when there is a
difference in syntactic profile. If there were a correlation of that kind in English, we would have to address a further
question: Should we expect to find it in other languages? And even if we found it, would it be incoherent to assert the
possibility of introducing new adjectives that destroy the correlation? Or is there some a priori reason for thinking that
this kind of correlation is unavoidable? (At least if a language is to have the versatility that natural languages have). My
reason to be sceptical of this suggestion is that on the approach I discuss, the difference in the type of truth-conditions is
determined by the difference in the nature of the semantic value, and Hsu's suggestion is tantamount to the suggestion
that perhaps there is a correlation between the syntactic profiles of adjectives and the metaphysical natures of the
semantic values of those adjectives. This could occur if the profile of an adjective dictated the type of semantic value
with which it can be associated or, viceversa, if the type of semantic value dictated its profile (its syntactic possibilities).
31
This idea of dictating (or determining) should be spelled out a bit more precisely but, in whatever sense, there seems to
be no a priori reason why metaphysics should determine syntax or viceversa. I confine myself to mentioning the
suggestions and the questions. The following conclusions ignore them.
16
Michael Byrd pointed out to me that this also suggests that no interesting notion of form is involved in the approach.
32