Properties As Functions

Properties as Functions

I have always thought that there is something right in Frege’s idea that concepts are functions from objects to truth-values. Bear in mind that for Frege concepts are not psychological or subjective entities, nor are they senses, but rather belong at the level of reference, like objects. They are individuated extensionally and exist independently of the mind (or even of Fregean thoughts). His idea is that these objective entities act as—indeed are–functions with arguments and values, just like mathematical functions. The arguments are objects (but sometimes other concepts) and the function takes us to other objects as values, truth-values being objects for Frege. In other words, the semantic value of a predicate is a function from objects to truth-values—Frege’s main interest being semantics. But we can interpret the view so as to cut out language, holding simply that facts are composed of objects and functions, since concepts in Frege’s sense are the objective correlates of linguistic expressions. Frege never speaks of properties in the intuitive sense in his semantic theory, but there is nothing to stop us formulating his insight in terms of properties: properties are functions too, forming facts. What are they functions from and to? From objects evidently (except when second-order), but are their values truth-values? That seems stretched given that truth-values attach to thoughts or propositions or sentences and properties exist outside the realm of sense.[1] For example, the property white is a function that can take snow as an argument, but it can’t yield the True as a value unless there exists a proposition that says that snow is white—but that is not the same thing as snow simply being white. As a metaphysical claim, the function theory is a theory about the structure of non-linguistic facts, so we don’t want to define it by reference to language and truth-values. So how do we transpose Frege’s insight from the semantic level to the ontological level? What does a function theory of objective mind-independent non-linguistic properties look like?

We have two options, one of which is implicit in a theory Frege considers and rejects. This is the theory that sentences denote states of affairs or facts and not truth-values, which would enable us to say that properties are functions from objects to states of affairs or facts—as Frege could have said that concepts are functions from objects to states of affairs considered as the reference of sentences. The function theory is quite independent of Frege’s decision to make the values of concept functions truth-values, taken to be the reference of sentences. So let us say that properties are functions from objects to facts, to put it briefly: the property white is a function that gives the fact that snow is white as value for the object snow as argument. According to this theory, the values of property functions are specific to the objects and properties we are considering, not general entities like the True and the False. The second option, however, seeks to mimic Frege’s decision by introducing special entities to serve as values of property functions—I will call these the Real and the Unreal. Thus the value of white for snow is the Real, while the value of white for coal is the Unreal. Intuitively, snow is really white—this is part of Reality—while coal is not really white—not part of Reality. The Real is the ontological counterpart to the True, which applies properly to representational entities not to facts. There are real states of affairs and unreal ones, and the Real and the Unreal are the entities that encompass (somehow!) these. We thus preserve parity with Frege’s economical apparatus: properties are functions from objects to reality-values, either the Real or the Unreal. This is neat theoretically, though a bit jarring to normal sensibility, just like Frege’s parallel stipulation; I like it as a concise means of formulation, though it is good that we have the fact-based theory to fall back on. We can keep both options in mind, while preferring the second option on grounds of elegance. The important point is that both values are distinct from truth-values and that they exist at the level of reality not our representation of reality—they are metaphysically objective. Then we can say that a fact consists of a function applied to an object (or sequence of objects in the case of relations) that gives the Real as its value (or the fact in question if we prefer that formulation). The thrust of the theory is less the nature of the value of the function as the fact that properties act as—and are—functions defined over objects. That is, the concept of a mathematical function can be generalized to all properties, thus delivering a metaphysical theory of properties.

Mathematics came to be understood as a theory of functions with arithmetical operations taken as paradigms. Addition, say, is a function from pairs of numbers to another number: its nature is to map numbers to numbers. We speak of this as giving, yielding, delivering—as a type of abstract action. Functions do something. The structure of a mathematical fact is a functional structure—argument, function, and value. Frege proposed to extend this structure beyond mathematics, purporting to discover that non-mathematical entities were also function-like, such as the concept denoted by the predicate “is white”. Later theorists extended the notion of function in developing formal semantics, postulating that intensions are functions from possible worlds to extensions (including truth-values): that is, they suggested that meanings are functions taking worlds as arguments and extensions as values. This is already a significant enlargement, since it applies the concept of function beyond its original home in mathematics; the concept is allowed to be not strictly mathematical but to have wider application. Semantics accordingly becomes the study of a certain type of function, somewhat like mathematics. What I am proposing is that the concept be extended yet further to furnish us with a general metaphysics of properties, and hence of reality generally. Facts are function-argument structures, whether physical, mental, legal, ethical, or what have you. The world is the totality of function-argument structures.

This substantially affects the way we think of both properties and objects. Let’s switch terminology and speak of universals and particulars: then we can say that universals are essentially entities (what other word is there?) that contain a slot or gap for objects—the argument-place of the universal—and that particulars are essentially entities that occur as arguments in such functions. Addition is essentially an operation that takes us from numbers to numbers, and numbers are essentially things that are subject to such operations; similarly whiteness is essentially an operation that takes us from objects to objects (specific facts or binary reality-values), and objects are essentially things that are subject to such operations. Universals are intrinsically argument-directed and particulars are ripe for occurring as such arguments—each feeds into the other. No function without arguments, no argument without functions. Just as addition is for numbers and numbers are for addition (among other functions), so whiteness is for objects like snow and objects are for universals like whiteness. The two are made for each other; they link harmoniously together, forming a logical unity. A fact is not a list but an organic unity (such phrases are unavoidable): the fact of snow being white is the result of snow and whiteness standing in the argument-function relation. And let us take this very literally: it is not just that we can represent facts this way—this is what they are. Reality consists of functions applied to arguments: not just an object and a property but an object acting as an argument to a property acting as a function. Frege liked to say that applying a function to an argument “saturates” the function, conceived as an incomplete entity: it made the function complete. As it were, a function with only a variable in its argument place suffers lonely incompleteness, which a willing object could satisfactorily saturate; well, that applies to the whole of reality, given that properties are functions and objects are arguments. Objects saturate incomplete properties thereby acquiring them. To put it poetically: objects yearn for properties to slot into and properties yearn for objects to saturate them. Less poetically: objects act as arguments to properties and properties act as functions defined over objects. The function-argument nexus is the root of reality. The world is the totality of saturated functions. Plato tended to oppose universals and particulars, locating them in different places, elevating one over the other, but he should have realized that they are partners in crime, natural bedfellows, man and wife. A universal is defined as a function over certain particulars; a particular is an argument destined for certain functions.[2] And not just any argument or function will do: functions will not accept any old argument and become saturated, and arguments will not offer themselves to any function in order to perform the act of saturation. For example, we cannot add John to a carrot and get a certain number as value, and 2 and 5 will not serve to saturate the function of being to the left of. Properties and objects are selective in their partners, only accepting certain potential mates in the saturation relation. In other words, the function-argument structure is both intimate and exclusive. The vision of reality it promotes is that of interlocking elements designed to fit harmoniously together. No doubt this was Frege’s intuition in the case of predicates and names; I am extending the point to objects and properties generally. Thus we derive a metaphysical conception of reality as a whole, indeed of the very structure of any reality. For it is not to be doubted that any conceivable reality containing objects and properties will exhibit the function-argument form: this is a necessary truth about reality as such. To rephrase, properties are essentially entities that take something as input and give something as output, just as addition takes pairs of numbers as input and gives a number as output. There is something action-like about functions, almost as if they have a purpose. A theist might see in this metaphysical picture an argument for the existence of a Creator: the architect of the world made it so that objects and properties would be capable of joining together—a non-trivial problem in ab initio universe design—and the solution she hit upon was to impose the function-argument structure on reality (she already had extensive experience in mathematical design). Surely, our theist might insist, it cannot be an accident that the universe is so cunningly designed—properties must have been made that way. We can sympathize with the theist’s starting point, though we may demur from his conclusion: it is certainly a very fortunate fact that properties are functions or else the world would have had trouble fitting coherently together. As it is, objects and properties nestle tightly together in the nurturing womb of the function-argument structure.

Frege’s motivations for his theory that concepts are functions were partly theoretical and partly intuitive. Theoretically, this idea might help with the problem of the unity of the thought or sentence; and it promises an enormous simplification in the semantic apparatus, as well as extending the reach of mathematical concepts (agreeable to a mathematician like Frege). Intuitively, a predicate joins with a singular term to form a sentence that is true or false, so its job is to ascend from one kind of expression to another concerned with truth. Likewise, the extension of this theory to properties is motivated both by theoretical and intuitive considerations. Theoretically, it might help with the problem of the unity of the fact; and it promises a striking uniformity in our metaphysical apparatus, as well as extending mathematical concepts still further. Intuitively, it seems natural to suppose that properties create facts from objects: facts are combinations of objects and properties, and the function theory enables to grasp the nature of this mode of combination. Frege’s overall theory strikes one at first as quite bold and counterintuitive in some ways, but it grows on one and one starts to see language through its theoretical lens (everyone should go through a dedicated Frege phase). Likewise, the theory I am proposing may seem startling and counterintuitive at first, but after a while one starts to see the world through its eyes—one looks at objects and their properties and sees them as function-argument structures. Certainly the theory affords an enormous simplification as well as other theoretical benefits, so any intuitive resistance it encounters should be regarded with suspicion. It is a nice theory, combining a sense of real discovery with an intuitively comprehensible foundation (viz. properties generate facts from objects).

Once we have absorbed the theory we can apply it further. Frege took logical connectives to denote truth functions construed as mappings from truth-values to truth-values (taken as objects). We can follow his path and understand conjunctive facts (say) as involving reality functions: conjoining two simple facts as arguments gives as value a further complex fact, or the reality-value the Real. Negation takes us from a fact to its opposite, or from the Real to the Unreal (or vice versa). So we can apply the theory to complex (“molecular”) facts not just to simple (“atomic”) facts.[3] For second-order functions we simply mimic the standard story: existence, say, is a function from properties to properties: the fact that dogs exist is analyzed as consisting in the first-order property-function dog as argument yielding the fact that that function is instantiated, where instantiation is a second-order function from functions to existential facts or reality-values. The instantiation function as applied to the dog function gives as value the fact that dogs exist, or simply the contrived theoretical entity the Real. Reality is composed of functions all the way up the hierarchy of properties, which is to say it is also composed of arguments to functions. All facts consist of arguments and functions locking together. If mathematical facts consist of function-argument combinations, the same is true of empirical reality. Thus in a certain sense all of reality is mathematical, since it has a structure characteristic of mathematics. The form of an empirical fact is the same as the form of a mathematical fact.

The value of a function is uniquely determined by its arguments, so that it depends on those arguments: the value of the addition function is fixed by the arguments inserted into it—if the arguments are 2 and 3, then the value must be 5. But the same is true of facts and properties: the fact is uniquely determined by the objects that occur as arguments and the identity of the function. Given that we are considering John and Mary and the marriage function, say, it is determined that the fact is that John and Mary are married (assuming they are): it could not be that the value of this function for those arguments is the fact that Paris is in France—the fact is a function of the function and its arguments. Intuitively, once you have the objects and properties the fact (state of affairs) is fixed; you can’t get a fact involving other objects and properties. So again facts, objects and properties relate just as mathematical functions and their arguments and values relate: the value of the property function for those arguments is uniquely such and such a fact, just as the value of the addition function for those arguments is uniquely a certain number. The dependency we observe in the mathematical case is mirrored in the non-mathematical case–one more reason to trust the analogy.

Finally, the humble variable achieves metaphysical significance: for it marks the place at which arguments are inserted into functions, and this operation is the key to constructing worlds.[4]

C

[1] This difficulty applies to Frege’s own theory, since concepts and truth-values are supposed to lie on the side of reference not sense, and yet truth-values are the reference of thoughts. We can suppose that Fregean concepts could exist without senses (they are close to sets of objects), but could truth-values? Aren’t truth-values precisely what thoughts stand for? I suppose Frege could maintain that truth-values, being objects, could exist in the absence of thoughts, and hence be the values of concept functions applied to objects; but that is certainly a very odd view and not one that we find Frege enunciating. Here we have the True and the False but with no thoughts to denote them! Note that if Frege had chosen states of affairs as the reference of sentences, instead of truth-values, he would not have faced this problem.

[2] We might see in this conception reconciliation between Plato’s view of universals and Aristotle’s view: by all means bring particulars into our account of universals (Aristotle), but don’t assimilate the two (Plato). Functions and arguments form a natural pair, but we need not deny that functions have being of their own.

[3] We thus obtain Russell-style logical atomism along with an injection of Frege-style function-argument theory.

[4] Allow me to add that the perspective developed here came to me as something of a revelation: the empirical world has more in common with mathematics than I had supposed (the same thing could be said about meaning understood in terms of functions). The abstract structure of the empirical world mirrors the abstract structure of arithmetic—how Pythagorean!