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One very important heuristic maxim which Dr.Whitehead and I found, by experience, to be applicable in mathematical logic,and have since applied to various other fields, is a form of Occam's Razor. When some set of supposed entities has neat logical properties, it turns out, in a great many instances, that the supposed entities can be replaced by purely logical structures composed of entities which have not such neat properties. In that case, in interpreting a body of propositions hitherto believed to be about the supposed entities, we can substitute the logical structures without altering any of the detail of the body of propositions in question.This is an economy, because entities with neat logical properties are always inferred, and if the propositions in which they occur can be interpreted without making this inference, the ground for the inference fails, and our body of propositions is secured against the need of a doubtful step. The principle may be stated in the form: ‘Whenever possible, substitute constructions out of known entities for inferences to unknown entities’ (1924, p.160)Russell was speaking of logical constructions in this memorable passage from his Introduction to Mathematical Philosophy: "The method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil." (1919, p. 71)
The notion of logical construction appears frequently with the idea that what is defined is a "logical fiction", and an "incomplete symbol". The latter term derives from the use of contextual definitions, providing an analysis of each sentence in which a defined symbol may occur without, however, giving an explicit definition, an equation or universal statement giving necessary and sufficient conditions for the application of the term in isolation. The terms "fiction" and "incomplete symbol" apply with differing aptness to various constructions.
Russell's first use of construction, and the model for later constructions, is the Frege/Russell definition of numbers as classes. This follows the kind of definitions used in the arithmetization of analysis of the preceding century, in particular, Dedekind's earlier construction of real numbers as bounded classes in the rational numbers. Russell's logicist program could not rest content with postulates for the fundamental objects of mathematics such as the Peano Axioms for the natural numbers. Instead numbers were to be defined as classes of equinumerous classes. Russell also refers to this method as "abstraction", now known as the abstraction of an equivalence class. The definition of equinumerosity, or of the existence of a one to one mapping between two classes, also called "similarity", is solely in terms of logical notions of quantifiers and identity. With the numbers defined, for example, two as the class of all two membered sets, or pairs, the properties of numbers could be derived by logical means alone.
The theory of descriptions introduces Russell's notion of incomplete symbol. Definite descriptions ‘The F’ do not show up in the formal analysis of sentences in which they occur, thus ‘The F is H’ becomes:
(∃x) [∀y(Fy ↔ y=x) & Hx]of which no subformula, or continuous segment, can be identified as the analysis of ‘The F’. Much as talk about "the average family" as in "The average family has 2.2 children" becomes "The number of children in families divided by the number of families = 2.2", there is no portion of that analysis that corresponds with "the average family". Instead we have a formula for eliminating such expressions from contexts in which they occur, hence the notion of "incomplete symbol" and the related "contextual definition". It is standard to see in this the origins of the distinction between between surface grammatical form and logical form, and thus the origin of linguistic analysis as a method in philosophy which operates by seeing past superficial linguistic form to underlying philosophical analysis. The theory of descriptions has been criticized by some linguists who see descriptions and other noun phrases as full fledged constituents of sentences, and who see the sharp distinction between grammatical and logical form as a mistake.
The theory of descriptions is often described as a model for avoiding ontological commitment to objects such as Meinongian subsistent entities, and so logical constructions in general are often seen as being chiefly aimed at ontological goals. In fact, that goal is at most peripheral to most constructions. Rather the goal is to allow the proof of propositions that would otherwise have to be assumed as axioms or hypotheses. Nor need the ontological goal be always elimination of problematic entities. Other constructions should be seen more as reductions of one class of entity to another, or replacements of one notion by a more precise, mathematical, substitute.
Russell's "NoClass" theory of classes from *20 of Principia Mathematica provides a contextual definition like the theory of descriptions. One of Russell's early diagnoses of the paradoxes was that they showed that classes could not be objects. Indeed he seems to have come across his paradox of the class of all classes that are not members of themselves by applying Cantor's argument to show that there are more classes of objects than objects. Hence, he concluded, classes could not be objects. Inspired by the theory of descriptions, Russell proposed that to say something G of the class of Fs, G{x: Fx}, is to say that there is some property H coextensive with (true of the same things as) F such that H is G. Extensionality of sets is thus derivable, rather than postulated. If F and H are coextensive then anything true of {x: Fx} will be true of {x: Hx}. Features of sets then follow from the features of the logic of properties, the "ramified theory of types". Because classes would seem to be individuals of some sort, but on analysis are found not to be, Russell speaks of them as "logical fictions", an expression which echoes Jeremy Bentham's notion of a "legal fiction". Because statements attributing a property to particular classes are analyzed by existential sentences saying that there is some propositional function having that property, this construction should not be seen as avoiding ontological commitment entirely, but rather of reducing classes to propositional functions. The properties of classes are really properties of propositional functions and for every class said to have a property there really is some propositional function having that property.
Beginning with The Problems of Philosophy in 1912, Russell turned repeatedly to the problem of matter. Part of the problem is to find a refution of Berkeleyan idealism, of showing how the existence and real nature of matter can be proved. In Problems Russell argues that matter is a well supported hypothesis that explains our experiences. Matter is known only indirectly, "by description", as the cause, whatever it may be, of our sense data, which we know "by acquaintance". This is the notion of hypothesis which Russell contrasts with construction in the passage above. Russell saw an analogy between the case of simply hypothesizing the existence of numbers with certain properties, those described by axioms, and hypothesizing the existence of matter. While we distinguish the certain knowledge we may have of mathematical entities from the contingent knowledge of material objects, Russell says that there are certain "neat" features of matter which are just too tidy to have turned out by accident. Examples include the most general spatiotemporal properties of objects, that no two can occupy the same place at the same time, and so on. Material objects are now to be seen as collections of sense data. Influenced by William James, Russell defended a "neutral monism" by which matter and minds were both to be constructed from sense data, but in different ways. Intuitively, the sense data occuring as they do "in" a mind, are material to construct that mind, the sense data derived from an object from different points of view to constructthat object. Russell saw some support for this in the theory of relativity, and the fundumental importance of frames of reference in the new physics.
These prominent examples are not the only use of the notion of construction in Russell's thought. In Principia Mathematica the multiple relation theory of propositions is introduced by saying that propositions are "incomplete symbols". Russell's multiple relation theory, that he held from 1910 to 1919 or so, argued that the constitituents of propositions, say ‘Desdemona loves Cassio’, which is false, are unified in a way that does not make it the case that they constitute a fact by themselves. Those constituents occur only in the context of beliefs, say, ‘Othello judges that Desdemona loves Cassio’. The real fact consists of a relation of Belief holding between the constituents Othello, Desdemona and Cassio, thus B(o, d, L, c). Because one might also have believed propositions of other structures, such as B(o, F, a) there need to be many such relations B, thus the "multiple" relation theory. Like the construction of numbers, this construction abstracts out what a number of occurrences of a belief have in common, a believer and various objects in a certain order. The analysis also makes the proposition an incomplete symbol because there is no constituent in the analysis of ‘x believes that p’ that corresponds to ‘p’.
Russell also suggests that propositional functions are logical constructions when he says that they are "nothing", but "nonetheless important for that". (1918, p. 96) Propositional functions are abstracted from their values, propositions. The propositional function ‘x is human’ is abstracted from its values ‘Socrates is human’, ‘Plato is human’ , etc. Viewing propositional functions as constructions from propositions which are in turn constructions by the multiple relation theory helps to make sense of the theory of types of propositional functions in Principia Mathematica. The notion of "incomplete symbol" does not make as much sense as "construction" when applied to propositional functions and propositions. This usage requires a broadening of the notion.
The notion of logical construction had a great impact on the future course of analytic philosophy. One line of influence was via the notion of a contextual definition, or paraphrase, intended to minimize ontological commitment and to be a model of philosophical analysis. The distinction between the surface appearance of definite descriptions, as singular terms, and the fully analyzed sentences from which they seem to disappear was seen as a model for making problematic notions disappear upon analysis. The theory of descriptions has been viewed as a paradigm of philosophical analysis.
A more technical strand in analytic philosophy was influenced by the construction of matter. Rudolf Carnap was attempted to carry out the construction of matter from sense data, and later Nelson Goodman continued the project. More generally, however, the use of set theoretic constructions became widespread among philosophers, and continues in the construction of set theoretic models, both in the sense of logic where they model formal theories, and as objects of interest in their own right.
Bernard Linsky bernard.linsky@ualberta.ca 