Logical Constructions

First published Wed Nov 20, 1996; substantive revision Sat Jun 24, 2023

The term “logical construction” was used by BertrandRussell to describe a series of similar philosophical theoriesbeginning with the 1901 “Frege-Russell” definition ofnumbers as classes and continuing through his“construction” of the notions of space, time and matterafter 1914. Philosophers since the 1920s have argued about thesignificance of “logical construction” as a method inanalytic philosophy and proposed various ways of interpretingRussell’s notion. Some were inspired to develop their ownprojects by examples of constructions. Russell’s notion oflogical construction influenced both Carnap’s project ofconstructing the physical world from experience and Quine’snotion ofexplication, and was a model for the use of settheoretic reconstructions in formal philosophy later in the twentiethcentury.

It was only when looking back on his work, in the programmatic 1924essay “Logical Atomism”, that Russell first describedvarious logical definitions and philosophical analyses as“logical constructions”. He listed as examples theFrege-Russell definition of numbers as classes, the theory of definitedescriptions, the construction of matter from sense data and thenseries, ordinal numbers and real numbers. Because of the particularnature of Russell’s use of “contextual” definitionsof expressions for classes, and the distinctive character of thetheory of definite descriptions, he regularly called the expressionsfor such entities “incomplete symbols” and the entitiesthemselves “logical fictions”.

Logical constructions differ in whether they involve explicitdefinitions or contextual definitions, and in the extent to whichtheir result should be described as showing that the constructedobject is a mere “fiction”. Russell’s 1901definition of numbers as classes of equinumerous classes isstraightforwardly a case of constructing one sort of entity as a classof others with an explicit definition. This was followed by the theoryof definite descriptions in 1905 and the “no-classes”theory for defining classes inPrincipia Mathematica in 1910,both of which involved the distinctive technique of contextualdefinition. In a contextual definition apparent singular terms (eitherdefinite descriptions or class terms) are eliminated through rules fordefining the entire sentences in which they occur. Constructions whichare like those using contextual definitions are generally called“incomplete symbols”, while those like the theory ofclasses are called “fictions.” Russell included theconstruction of matter, space and time as classes of sense data at theend of his 1924 list. The main problem for interpreting the notion oflogical construction is to understand what these various examples havein common, and how the construction of matter is comparable to eitherof the early constructions of numbers as classes or the theory ofdefinite descriptions and “no-classes” theory of classes.None of the expressions “fiction”, “incompletesymbol” or even “constructed from” seems appropriatefor an analysis of the fundamental features of the familiar physicalworld and the material objects that occupy it.

1. Honest Toil

The earliest construction on Russell’s 1924 list is the famous“Frege/Russell definition” of numbers as classes ofequinumerous classes from 1901 (Russell 1993, 320). The definitionfollows the example of the definitions of the notions of limit andcontinuity that were proposed for the calculus in the precedingcentury. Russell did not rest content with adopting the Peano-Dedekindaxioms as the basis for the theory of the natural numbers and thenshowing how the properties of the numbers could be logically deducedfrom those axioms. Instead, he defined the basic notions of“number” , “successor” and “0” andproposed to show, with carefully chosen definitions of their basicnotions in terms of logical notions, that those axioms could bederived from principles of logic alone.

Russell defined natural numbers as classes of equinumerous classes.Anypair, a class with two members, can be put into a one toone correspondence with any other, hence all pairs areequinumerous. The numbertwo is then identified with theclass of all pairs. The relation between equinumerous classes whenthere is such a one to one mapping relating them is called“similarity”. Similarity is defined solely in terms oflogical notions of quantifiers and identity. With the natural numbersso defined, Peano axioms can be derived by logical means alone. Afternatural numbers, Russell adds “series, ordinal numbers and realnumbers” (1924, 166) to his list of constructions, and thenconcludes with the construction of matter.

Russell credits A. N. Whitehead with the solution to the problem ofthe relation of sense data to physics that he adopted in 1914:

I have been made aware of the importance of this problem by my friendand collaborator Dr Whitehead, to whom are due almost all thedifferences between the views advocated here and those suggested inThe Problems of Philosophy. I owe to him the definition ofpoints, and the suggestion for the treatment of instants and“things,” and the whole conception of the world of physicsas aconstruction rather than aninference. (Russell1914b,vi)

It is only later, in an essay in which Russell reflected on hisphilosophy that he also described his earlier logical proposals as“logical constructions.” The first specific formulation ofthis method of replacing inference with construction as a generalmethod in philosophy is in the essay “LogicalAtomism”:

One very important heuristic maxim which Dr. Whitehead and I found, byexperience, to be applicable in mathematical logic, and have sinceapplied to various other fields, is a form of Occam’s Razor.When some set of supposed entities has neat logical properties, itturns out, in a great many instances, that the supposed entities canbe replaced by purely logical structures composed of entities whichhave not such neat properties. In that case, in interpreting a body ofpropositions hitherto believed to be about the supposed entities, wecan substitute the logical structures without altering any of thedetail of the body of propositions in question. This is an economy,because entities with neat logical properties are always inferred, andif the propositions in which they occur can be interpreted withoutmaking this inference, the ground for the inference fails, and ourbody 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 tounknown entities’. (Russell 1924, 160)

Russell was referring to logical constructions in this frequentlyquoted passage from hisIntroduction to MathematicalPhilosophy. He objects to introducing entities with implicitdefinitions, that is, as being those things that obey certain axiomsor “postulates”:

The method of ‘postulating’ what we want has manyadvantages; they are the same as the advantages of theft over honesttoil. Let us leave them to others and proceed with our honest toil.(Russell 1919, 71)

He charges that we need a demonstration that there are any objectswhich satisfy those axioms. The “toil” here is the work offormulating definitions of the numbers so that they can be shown tosatisfy the axioms using logical inference alone.

The description of logical constructions as “incompletesymbols” derives from the use of contextual definitions thatprovide an analysis or substitute for each sentence in which a definedsymbol may occur. The definition does not give an explicit definition,such as an equation with the defined expression on one side that isidentified with adefiniendum on the other, or a universalstatement giving necessary and sufficient conditions for theapplication of the term in isolation. The connection between being afiction and expressed by an “incomplete symbol” can beseen in Russell’s constructions of finite cardinal and ordinalnumbers by means of the theory of classes. That“no-classes” theory, via the contextual definitions forclass terms, makes all the numbers “incomplete symbols”,and so numbers can be seen as “logical fictions”.

The notions of construction and logical fiction appear together inthis account from Russell’s “Philosophy of LogicalAtomism” lectures:

You find that a certain thing which has been set up as a metaphysicalentity can either be assumed dogmatically to be real, and then youwill have no possible argument either for its reality or against itsreality; or, instead of doing that, you can construct a logicalfiction having the same formal properties, or rather having formallyanalogous formal properties to those of the supposed metaphysicalentity and itself composed of empirically given things, and thelogical fiction can be substituted for your supposed metaphysicalentity and will fulfill all the scientific purposes that anyone candesire. (Russell 1918, 144)

Incomplete symbols, descriptions, classes and logical fictions areidentified with each other and then with the “familiar objectsof daily life” in the following passage from earlier in thelectures:

There are a great many other sorts of incomplete symbols besidesdescriptions. There are classes… and relations taken inextension, and so on. Such aggregations of symbols are really the sameas what I call “logical fictions”, and they embracepractically all the familiar objects of daily life: tables, chairs,Piccadilly, Socrates, and so on. Most of them are either classes, orseries, or series of classes. In any case they are all incompletesymbols, i.e. they are aggregations that only have a meaning in useand do not have any meaning in themselves. (Russell 1918, 122)

In what follows these various features of logical constructions willbe disentangled. The result appears to be a connected series ofanalyses sharing at least a family resemblance with each other. Thecommon feature is that in each case some formal or “neat”properties of objects that had to be postulated in axioms before couldnow be derived as logical consequences of definitions. The replacedentities are variously “fictions”, “incompletesymbols” or simply “constructions” depending on theform that the definitions take.

2. Logical Analysis and Logical Construction

It would be a mistake to see Russell’s logical constructions asthe product of the converse operation of a method that begins withlogical analysis. Analysis was indeed the distinctive method ofRussell’s realist and atomistic philosophy with the method ofconstruction appearing only later. Russell’s new philosophy wasself-consciously in opposition to the Hegelianism prevailing inphilosophy at Cambridge at the end of the nineteenth century (Russell1956, 11–13). Russell first needed to defend the process ofanalysis, and to argue against the view of the idealists that complexentities are in fact “organic unities” and that anyanalysis of these unities loses something, as the slogan was“analysis is falsification”. (1903, §439) The subjectof our analysis is reality, rather than merely our own ideas:

All complexity is conceptual in the sense that it is due to a wholecapable of logical analysis, but is real in the sense that it has nodependence on the mind, but only upon the nature of the object. Wherethe mind can distinguish elements, there must be different elements todistinguish; though, alas! there are often different elements whichthe mind does not distinguish. (1903, §439)

As ultimate constituents of reality are what is discovered by logicalanalysis, logical construction cannot be the converse operation, forundoing the analysis by putting things back together only returns usto the complex entities with which we began. What then is the point ofconstructing what has already been analyzed?

The distinction made here between analysis and constructiondeliberately side-steps an important discussion among scholars ofFrege and Russell about the nature of analysis. Frege held, in hisFoundations of Arithmetic (1884, §64), that aproposition about identity of numbers could be also analyzed as oneabout the similarity of classes. He describes this as“recarving” one and the same content in different ways.Later Frege asserted that the same thought could be viewed as theresult of the application of a function to an argument in differentways. As the logical form of a thought is the result of theapplication of concepts to arguments, this means that distinct logicalforms are assigned to the same thought. To resolve the apparentconflict with Frege’s famous thesis ofcompositionality, that a thought is built up from itsconstituents in a fashion that by and large follows its syntacticform, Michael Dummett (1981, chapter 15) distinguishes two notions ofanalysis in Frege, one as “analysis” proper, the other as“decomposition”. Peter Hylton (2005, 43) argues that thereis a problematic notion of analysis in Russell, with it being verydifficult to say that sentences containing definite descriptions havethe complicated quantificational structures assigned to them in“On Denoting” (1905) as their “realstructure”. Michael Beaney, in his introduction to (2007, 8)gives the names “decompositional” and“transformative” to two kinds of analysis in hisintroduction to papers that discuss the significance of thisdistinction for Russell. James Levine claims that in fact the firstform of analysis, by which the project is to find the ultimateconstituents of propositions, belongs to an early project of“Moorean Analysis” that Russell abandoned early. Indeed,by the time of the account of numbers as classes of equinumerousclasses, Russell had already adopted what Levine calls“Russell’s Post-Peano Analysis ”.

This debate is certainly relevant to the study of Frege’sphilosophy, and its connections with Russell’s role as a founderof Analytic Philosophy as a movement, but it is perhaps out of keepingwith Russell’s own use of the terminology of“analysis”. While Peter Strawson, in his “OnReferring” (1950) makes numerous allusions to Russell’s“analysis” of definite descriptions, in fact the term doesnot appear in “On Denoting”. Russell refers to his“theory” of descriptions, and acknowledges that it is nota proposal that will be recognized immediately as what we have alwaysmeant by such sentences, but instead says of his somewhat complicateduse of quantifiers and identity symbols that:

This may seem a somewhat incredible interpretation: but I am not apresent giving reasons, I am merely stating the theory. (Russell 1905,482)

He then goes on to defend his theory by “dealing” with thethree puzzles including the famous example of whether “Thepresent King of France is bald” is true or false. At no pointdoes he appeal to what a speaker may have in mind upon uttering one ofthese sentences. As a result of these facts, it seems thatRussell’s methodology is best understood by analogy with thelogical approach to scientific theories. On this model the result of“logical analysis” will be the definitions and primitivepropositions or axioms from which the laws of a formalized scientifictheory can be derived by logical inference. The reduction of onetheory to another consists of rewriting the axioms of the targettheory using the language of the reducing theory, and then provingthem as theorems of that reducing theory. Construction, then, is bestseen as the process of choosing definitions so that previouslyprimitive statements can be derived as theorems. (See Hager 1994 andRussell 1924.)

This picture fits best with this linguistically oriented notion of“theory construction” rather than the project ofphilosophical analysis. It also follows the use of the notion ofconstruction in the tradition of mathematics. Euclid prefaces eachdemonstration with a “construction” of a figure thatfeatures in the following proof. Gottlob Frege begins every proof inhisBasic Laws of Arithmetic (1893) with an“Analysis”, which informally explains the notions used inthe theorems and the strategy of the derivation, followed by theactual, gapless proof, which is called the “Construction”.Historically, then, there is no notion of a construction as asynthetic stage following an analytic stage as two processes of acomparable nature, but leading in opposite directions.

Even when described in terms of stages of theory construction,analysis and logical construction are not simply converse operations.Russell stresses that the objects discovered and distinguished inanalysis are “real” as are their differences from eachother. Thus there is a constraint on the “choice” ofdefinitions and primitive propositions with which to begin. Therelationships between a deductive system and a realistic ontologydiffer among the various cases that Russell lists as examples oflogical constructions. Propositions and “complexes” suchas facts are analyzed in order to find the real objects and relationsof which they are composed. A logical construction, on the other hand,results in a theory from which truths follow by logical inferences.The truths that are part of a deductive system resulting from logicalconstruction are only “reconstructions” of some of the“pre-theoretic” truths that are to be analyzed. It is onlytheir deductive relations, in particular their deducibility from theaxioms of the theory, that are relevant to the success of aconstruction. Logical constructions do not capture all of the featuresof the pre-theoretic entities with which one begins.

Much of the attention to logical construction has focused on whetherit is in fact a unified methodology for philosophy that will introducea “scientific method in philosophy” as Russell says in thesubtitle of (Russell 1914b). Commentators from Fritz (1952) throughSainsbury (1979) have denied that Russell’s variousconstructions fit into a unified methodology, as well as questioningthe applicability of the language of “fiction” and“incomplete symbol” to all examples. Below it will beshown how, nevertheless, constructions do fall into several naturalfamilies that are described by various of these terms with aconsiderable degree of accuracy.

3. Natural Numbers

Russell’s definition of natural numbers as classes ofsimilar, or equinumerous, classes, first published in(Russell 1901), was his first logical construction, and was the modelfor those that followed. Similar classes are those that can be mappedone to one onto each other by some relation. The notion of a“one-to-one relation” is defined with logical notions:\(\rR\) is one-one when for every \(x\) there is a unique \(y\) suchthat \(x \rR y\), and for every such \(y\) in the range of \(\rR\)there is a unique such \(x\). These notions of existence anduniqueness come from logic, and so the notion of number is thusdefined solely in terms of classes and of logical notions. Russellannounced the goal of his logicist program inThe Principles ofMathematics: “the proof that all pure mathematics dealsexclusively with concepts definable in terms of a very small number offundamental logical concepts, and that all its propositions arededucible from a very small number of fundamental logicalprinciples…” (Russell 1903,xv). If class is also shown to be a logical notion, then this definition wouldcomplete the logicist program for the mathematics of natural numbers.

Giuseppe Peano (Peano 1889, 94) had stated axioms for elementaryarithmetic, which were later formulated by Russell (1919, 8) as:

0 is a number.
The successor of any number is a number.
No two numbers have the same successor.
0 is not the successor of any number.
If a property belongs to 0 and belongs to the successor of \(x\)whenever it belongs to \(x\), then it belongs to every number.

For Peano these were the axioms of number, which, along with axioms ofclasses and propositions, describe the properties of these entitiesand lead to the derivation of theorems that express the otherimportant properties of those entities.

Richard Dedekind (Dedekind 1887) had also listed the properties ofnumbers with similar looking axioms, using the notion ofchain, an infinite sequence of sets, each a subset of thenext, that is well ordered and has the structure of the naturalnumbers. Dedekind then proves that the principle of induction (Axiom 5above) holds for chains. (See entry onDedekind). Although Russell finds it “most remarkable thatDedekind’s previous assumptions suffice to demonstrate thistheorem” (Russell 1903, §236), he compares the twoapproaches, of Peano and Dedekind, with respect to simplicity andtheir differing ways of treating mathematical induction, and concludesthat:

But from a purely logical point of view, the two methods seem equallysound; and it is to be remembered that, with the logical theory ofcardinals, both Peano’s and Dedekind’s axioms becomedemonstrable. (Russell 1903, §241)

It was Peano and Dedekind that Russell had in mind when he laterspeaks of “the method of ‘postulating’” whenhe compares the “advantages” of their method overconstruction as those of theft over honest toil.

To complete his project Russell needed to find definitions and some“very small number of fundamental logical principles”(Russell 1903,xv) and then produce the required derivations.Finding an adequate definition of classes with the “no-classestheory” and the principles of logic needed to derive theproperties of numbers and classes was only completed withPrincipia Mathematica (Whitehead and Russell 1910–13).This construction of numbers was a clear example of defining entitiesas classes of others so as to be able to prove certain properties astheorems of logic rather than having to rest with the theft ofhypotheses. With the device of contextual definition from the theoryof descriptions Russell then eliminated classes too, taking asfundamental the logical notion of a propositional function and soshowing that the principles of classes where a part of logic.

4. Definite Descriptions

Definite descriptions are the logical constructions that Russell hasin mind when when he describes them as “incompletesymbols”. The notion of a “logical fiction”, on theother hand, applies most straightforwardly to classes. Otherconstructions, such as the notions of the domain and range of arelation, and of one to one mappings that are crucial to thedevelopment of arithmetic, are only “incomplete” in anindirect sense due to their being defined as classes of a certainsort, which are in turn constructions.

Russell’stheory of descriptions was introduced in hispaper “On Denoting” (Russell 1905) published in thejournalMind. Russell’s theory provides the logicalform of sentences of the form ‘The \(F\) is \(G\)’ where‘The \(F\)’ is called adefinite description incontrast with ‘An \(F\)’ which is anindefinitedescription. The analysis proposes that ‘The \(F\) is\(G\)’ is equivalent to ‘There is one and only one \(F\)and it is \(G\)’. Given this account, the logical properties ofdescriptions can be deduced using just the logic of quantifiers andidentity. Among the theorems in ∗14 ofPrincipiaMathematica are those showing that, (1) if there is just one\(F\) then ‘The \(F\) is \(F\)’ is true, and if there isnot, then ‘The \(F\) is \(G\)’ is always false and then,(2) if the \(F = \text{the } G\), and the \(F\) is \(H\), then the\(G\) is \(H\). These theorems show that proper (uniquely referring)descriptions behave like proper names, the “singularterms” of logic. Some of these results have been controversial— Strawson (1950) claimed that an utterance of ‘Thepresent King of France is bald’ should be truth valueless sincethere is no present king of France, rather than “plainly”false, as Russell’s theory predicts. Russell’s reply toStrawson in (Russell 1959, 239–45) is helpful for understandingRussell’s philosophical methodology of which logicalconstruction is just a part. It is, however, by assessing the logicalconsequences of a construction that it is to be judged, and soStrawson challenged Russell in an appropriate way.

The theory of descriptions introduces Russell’s notion ofincomplete symbol. This arises because no definitionalequivalent of ‘The F’ appears in the formal analysis ofsentences in which the description occurs. The sentence ‘The\(F\) is \(H\)’ becomes:

\[\exists x [ \forall y (Fy \leftrightarrow y=x) \ \&\ Hx ]\]

of which no subformula, or even a contiguous segment, can beidentified as the analysis of ‘The F’. Similarly, talkabout “the average family” as in “The average familyhas 2.2 children” becomes “The number of children infamilies divided by the number of families = 2.2”. There is nosegment of that formula that corresponds to “the averagefamily”. Instead we are given a procedure for eliminating suchexpressions from contexts in which they occur, hence this is anotherexample of an “incomplete symbol” and the definition of anaverage is an example of a “contextual definition.”

It is arguable that Russell’s definition of definitedescriptions was the most prominent early example of the philosophicaldistinction between surface grammatical form and logical form, andthus marks the beginnings of linguistic analysis as a method inphilosophy. Linguistic analysis begins by looking past superficiallinguistic form to see an underlying philosophical analysis. FrankRamsey described the theory of descriptions as a “paradigm ofphilosophy” (Ramsey 1929, 1). While in itself surely not a modelfor all philosophy, it was at least a paradigm for the other examplesof logical constructions that Russell listed when looking back on thedevelopment of his philosophy in 1924. The theory of descriptions hasbeen criticized by some linguists and philosophers who seedescriptions and other noun phrases as full-fledged linguisticconstituents of sentences, and who see the sharp distinction betweengrammatical and logical form as a mistake. (See the entry ondescriptions.)

Following Gilbert Ryle’s (1931) influential criticisms ofMeinong’s theory of non-existent objects, the theory ofdescriptions has been taken as a model for avoiding ontologicalcommitment to objects, and so logical constructions in general areoften seen as being chiefly used to eliminate purported entities. Infact, that goal is at most peripheral to many constructions. Theprincipal goal of these constructions is to allow the proof ofpropositions that would otherwise have to be assumed as axioms orhypotheses. Nor need the introduction of constructions always resultin the elimination of problematic entities. Yet other constructionsshould be seen more as reductions of one class of entity to another,or replacements of one notion by a more precise, mathematical,substitute.

5. Classes

Russell’s “No-Class” theory of classes from∗20 ofPrincipia Mathematica provides a contextualdefinition like that of the theory of definite descriptions. One ofRussell’s early diagnoses of the paradox of the class of allclasses that are not members of themselves was that it showed thatclasses could not be individuals. Indeed Russell seems to have comeacross his paradox by applying Cantor’s famous diagonal argumentto show that there are more classes of individuals than individuals.Hence, he concluded, classes could not be individuals, and expressionsfor classes such as ‘\(\{x: Fx \}\)’ cannot be thesingular terms they appear to be. Inspired by the theory ofdescriptions, Russell proposed that to say something \(G\) of theclass of \(F\)s, \(G\) \(\{x: Fx \}\), is to say that there is some(predicative) property \(H\) coextensive with (true of the same thingsas) \(F\) such that \(H\) is \(G\). The restriction to predicativeproperties, or those which are not defined in terms of quantificationover other properties, was a consequence of theramificationof the theory of types to avoid intensional or “epistemic”paradoxes which motivated the theory of types in addition to the settheoretical “Russell’s Paradox” (see Whitehead andRussell 1910–13, Introduction, Chapter II). These predicativeproperties are intensional, however, in the sense that two distinctproperties might hold of the same objects. (See the entry on thenotation in Principia Mathematica.) That classes so defined have the feature of extensionality is thusderivable, rather than postulated. If \(F\) and \(H\) are coextensivethen anything true of \(\{x:Fx \}\) will be true of \(\{x:Hx \}\).Features of classes then follow from the features of the logic ofproperties.

Because classes would at first 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 JeremyBentham’s notion of “legal fictions.” (Hart 1994,84) (See entry onlaw and language). That a corporation is a “person” at law was for Benthammerely a fiction that could be cashed out in terms of the notion oflegal standing and of limits to the financial liability of realpersons. Thus any language about such “legal fictions”could be translated in other terms to be about real individuals andtheir legal relationships. Because statements attributing a propertyto particular classes are replaced by existential sentences sayingthat there is some propositional function having that property, thisconstruction also can be characterized as showing that classexpressions, such as ‘\(\{x:Fx \}\)’, are incompletesymbols. They are not replaced by some longer formula expressing aterm. On the other hand, the definition should not be seen as avoidingontological commitment entirely, as showing that something isliterally a “fiction”. Rather it shows how to reduceclasses to propositional functions. The properties of classes arereally properties of propositional functions and for every class saidto have a property there really is some propositional function havingthat property.

6. Series, Ordinal Numbers and Real Numbers

Whitehead and Russell define aseries in volume II ofPrincipia Mathematica at ∗204.01 as the classSer of all relations which is transitive, connectedand irreflexive. A relation \(R\) istransitive when, if\(xRy\) and \(yRz\) then \(xRz\). It isconnected when forany \(x\) and \(y\) for which it is defined, either \(xRy\) or\(yRx\). Finally, anirreflexive relation is one such thatfor all \(x\), it is not the case that \(xRx\). Any relation that hasthose properties forms aseries of the things that itrelates. Such relations are now called “linear orderings”or simply, “orderings”. Here the “logicalconstruction” simply consists of an implicit definition of acertain property of relations. There is certainly no thought thatseries are merely invented “fictions”, and the symbol‘Ser’ for them is“incomplete” only in that it can be explicitly defined asthe intersection of other classes (a class of classes) and classes arethemselves “incomplete”.

Russell’s definitions of ordinal numbers and real numbersresemble the definitions of natural numbers. Ordinal numbers are aspecial case of relation numbers. Just as a cardinal numbercan be defined as a class of similar classes where the similarity issimply equinumerosity, the existence of a one to one mapping betweenthe two classes, a relation number is a class of similar classes whichare ordered by some relation. Ordinal numbers are the relation numbersof well-ordered classes. “Relation-Arithmetic” is thesubject of Part IV of Volume II ofPrincipia Mathematica,chapters ∗150 to ∗186. All of the properties of thearithmetic of ordinal numbers are derived from the more generalarithmetic of relation numbers. Thus, for example, the addition ofordinal numbers is not commutative. The first infinite ordinal\(\omega\) is the relation number of the well-ordered classes similarto \(1, 2, 3, \ldots\) etc. The sum \(1 + \omega\) will be therelation number of ordered classes which result from adding oneelement at the beginning of the ordering, say \(0, 1, 2, 3, \ldots\)etc., which has the same ordinal number \(\omega\). Thus \(1 + \omega= \omega\). On the other hand, adding an element at the“end” of such a well ordered class will give an orderingthat is not similar: \(1, 2, 3, \ldots \text{etc.}, 0\). Consequently,\(1 + \omega \ne \omega + 1\). On the other hand addition of ordinals,and indeed relation numbers in general, is associative, that is,\((\alpha + \beta) + \gamma = \alpha + (\beta + \gamma)\), which isproved with certain restrictions in ∗174. Ordinal numbers arethus defined exactly as natural numbers, as classes of similarclasses, in such a way that all the desired theorems can be proved.Describing ordinal numbers as “fictions”,“incomplete symbols” and “constructions”applies in the same way as in the case of natural numbers.

The class of real numbers, Θ, is defined in Volume III ofPrincipia Mathematica at ∗310.01 as consisting of“Dedekindian series” of rational numbers, which are inturn relation numbers of “ratios” of natural numbers.Whitehead and Russell follow the account of real numbers asDedekind cuts of the rational numbers, and only differ frommore standard developments of the numbers in contemporary set theoryby treating rational numbers as relation numbers of a certain sort,rather than ordered pairs of integers (the “numerator” and“denominator”). Like the construction of relation numbersas classes of similar classes, the “logical construction”of real numbers differs from the theory of definite descriptions andclasses in general in not defining “incomplete symbols” orby showing that these numbers are really “fictions”. Theyare best characterized as definitions that allow for the proof oftheorems about these numbers that would otherwise have to bepostulated as axioms. They are the product of the “honesttoil” that Russell prefers.

7. Mathematical Functions

Mathematical functions are not mentioned by Russell in the 1924 listof “logical constructions” although the analysis ofmathematical functions is the principal application of the theory ofdefinite descriptions in PM. The basic “functions” of PMare propositional functions. The Greek letters \(\phi, \psi, \theta,\ldots\) are variables for propositional functions, and, withindividual variables \(x, y, z, \ldots\) go together to form opensentences \(\phi(x), \psi(x,y)\), etc. This is the familiar syntax ofmodern predicate logic. Mathematical functions, such as thesine function and addition, are represented as termforming operators such as \(\sin x\), or \(x + y\). Incontemporary logic they are symbolized by function letters that arefollowed by the appropriate number of arguments, \(f(x),g(x,y)\), etc.In chapter ∗30 Whitehead and Russell propose a directinterpretation of such expressions for mathematical functions in termsof definite descriptions, which they call “descriptivefunctions”. Consider the relation between a number and its sine,the relation which obtains between \(x\) and \(y\) when \(y = \sinx\). Call this relation “\(\text{Sine}(x,y)\)” or moresimply, “\(\bS(x,y)\)”, as a two-place relation. Themathematical function can then be expressed with a definitedescription, interpreting our expression “the sine of\(x\)” not as “\(\sin(x)\)”, but literally as“the Sine of \(x\)”, with a definite description,or “the \(y\) such that \(\text{Sine}(x,y)\)”. Using thenotation of the theory of definite descriptions, this is‘\((\iotax)\bS(x,y)\)’. The effect ofthis analysis is that Whitehead and Russell can replace allexpressions for mathematical functions with definite descriptionsbased on relations. This definition involves relationsinextension, which are represented with upper case Roman lettersand with the relation symbol between the variables. The definition inPM is: ∗30.01. \(R`y = (\iota x)xRy\), with the notation\(R`y\) to be read as “the \(R\) of \(y\).” As with thetheory of descriptions, the result of this definition is to facilitatethe proofs of theorems which capture the logical properties ofmathematical functions that will be needed in the further work ofPM.

The logical analysis of function expressions in PM presents them as aspecial case of definite descriptions, “the \(R\) of\(x\)”. In the Summary of ∗30 we find:

Descriptive functions, like descriptions in general, have no meaningin isolation, but only as constituents of propositions. (Whitehead andRussell 1910–13, 232)

Mathematical or descriptive functions are thus explicitly includedamong the incomplete symbols ofPrincipia Mathematica.

8. Propositions and Propositional Functions

InPrincipia Mathematica Russell’smultiplerelation theory of judgment is introduced by presenting anontological vision:

The universe consists of objects having various qualities and standingin various relations. (Whitehead and Russell 1910–13, 43)

Russell goes on to explain the multiple relation theory of judgment,which finds the place of propositions in this world of objects andqualities standing in relations. (See the entry onpropositions.)

Russell’s multiple relation theory, that he held from 1910 toaround 1919, argued that the constituents of propositions, say‘Desdemona loves Cassio’, are unified in a way that doesnot make it the case that they constitute a fact by themselves. Thoseconstituents occur only in the context of beliefs, say, ‘Othellojudges that Desdemona loves Cassio’. The real fact consists of arelation of Belief holding between the constituents Othello,Desdemona, and Cassio; \(B(o,d,L,c)\). Because one might also havebelieved propositions of other structures, such as \(B(o,F,a)\) thereneed to be many such relations \(B\), of different“arities”, or number of arguments, hence the name“multiple relation” theory. Like the construction ofnumbers, this construction abstracts from what a number of occurrencesof a belief have in common, namely, a relation between a believer andvarious objects in a certain order. The account also makes theproposition an incomplete symbol because there is no constituent inthe analysis of ‘\(x\) believes that \(p\)’ thatcorresponds to ‘\(p\)’. As a result Russell concludesthat:

It will be seen that, according to the above account, a judgment doesnot have a single object, namely a proposition, but has severalinterrelated objects. That is to say, the relation which constitutesjudgment is not a relation of two terms, namely the judging mind andthe proposition, but is a relation of several terms, namely the mindand what we call the constituents of the proposition…
Owing to the plurality of the objects of a single judgment, it followsthat what we call a “proposition” (in which it is to bedistinguished from the phrase expressing it) is not a single entity atall. That is to say, the phrase which expresses a proposition is whatwe call an “incomplete” symbol; it does not have meaningin itself, but requires some supplementation in order to acquire acomplete meaning. (Whitehead and Russell 1910–13,43–44)

Although bound variables ranging over propositions hardly occur inPrincipia Mathematica (with a prominent exception in∗14.3), it would seem that the whole theory of types is atheory of propositional functions. Yet following on the claim thatpropositions are “not single entities at all”, Russellsays the same for propositional functions. In theIntroduction toMathematical Philosophy, Russell says that propositionalfunctions are really “nothing”, but “nonethelessimportant for that” (Russell 1919, 96). This comment makes bestsense if we think of propositional functions as somehow constructed byabstracting them from their values, which are propositions. Thepropositional function “\(x\) is human” is abstracted fromits values “Socrates is human”, “Plato ishuman”, etc. Viewing propositional functions as constructionsfrom propositions, that are in turn constructions by the multiplerelation theory, helps to make sense of certain features of the theoryof types of propositional functions inPrincipia Mathematica.We can understand how propositional functions seem to depend on theirvalues, namely propositions, and how propositions in turn canthemselves be logical constructions. The relation of this dependenceto the theory of types is explained in the Introduction toPrincipia Mathematica in terms of the notion of“presupposing”:

It would seem, however, that the essential characteristic of afunction isambiguity… We may express this by sayingthat “\(\phi x\)” ambiguously denotes \(\phi a, \phi b,\phi c,\) etc., where \(\phi a, \phi b, \phi c,\) etc. are the variousvalues of “\(\phi x\).” … It will be seen that,according to the above account, the values of a function arepresupposed by that function, not vice versa. It is sufficientlyobvious, in any particular case, that a value of a function does notpresuppose the function. Thus for example the proposition“Socrates is human” can be perfectly apprehended withoutregarding it as a value of the function “\(x\) is human.”It is true that, conversely, a function can be apprehended without itsbeing necessary to apprehend its values severally and individually. Ifthis were not the case, no function could be apprehended at all, sincethe number of values (true and false) of a function is necessarilyindefinite and there are necessarily possible arguments with which weare not acquainted. (Russell 1910–13, 39–40)

The notion of “incomplete symbol” seems less appropriatethan “construction” in the case of propositional functionsand propositions. To classify propositions and even propositionalfunctions as instances of the same logical phenomenon as definitedescriptions requires a considerable broadening of the notion.

The ontological status of propositions and propositional functionswithin Russell’s logic, and in particular, inPrincipiaMathematica, is currently the subject of considerable debate. Oneinterpretation, which we might call “realist,” issummarized in this footnote by Alonzo Church in his 1976 study of theramified theory of types:

Thus we take propositions as values of the propositional variables, onthe ground that this is what is clearly demanded by the background andpurpose of Russell’s logic, and in spite of what seems to be anexplicit denial by Whitehead and Russell in PM, pp. 43–44.
In fact, Whitehead and Russell make the claim: “that what wecall a ‘proposition’ (in the sense in which this isdistinguished from the phrase expressing it) is not a single entity atall. That is to say, the phrase which expresses a proposition is whatwe call an ‘incomplete symbol’ …” They seemto be aware that this fragmenting of propositions requires a similarfragmenting of propositional functions. But the contextual definitionor definitions that are implicitly promised by the “incompletesymbol” characterization are never fully supplied, and it is inparticular how they would explain away the use of bound propositionaland functional variables. If some things that are said by Russell inIV and V of his Introduction to the second edition may be taken as anindication of what is intended, it is probable that the contextualdefinitions would not stand scrutiny.
Many passages in [(Russell 1908)] and [(Whitehead and Russell1910–13)] may be understood as saying or as having theconsequence that the values of propositional functions are sentences.But a coherent semantics of Russell’s formalized language canhardly be provided on this basis (notice in particular, that, sincesentences are also substituted for propositional variables, it wouldbe necessary to take sentences as names of sentences.) And since thepassages in question seem to involve confusions of use and mention orkindred confusions that may be merely careless, it is not certain thatthey are to be regarded as precise statements of a semantics. (Church1976, n.4)

Gregory Landini (1998) has proposed that there is indeed a coherentsemantics for propositions and propositional functions in PM, whichtreats functions and propositions as linguistic entities. Landiniproposes that this “nominalist semantics” is the intendedinterpretation of PM and is what remains of Russell’s earlier“substitutional theory.” He argues that Russell was led tothis nominalism after first rejecting the reality of classes, then ofpropositional functions, and finally the reality of propositions. Thisrejection, according to Landini, leaves us with only a nominalistmetaphysics of individuals and expressions as the interpretation ofRussell’s logic. See also Cocchiarella (1980), who describes a“nominalist semantics” for ramified type theory, butrejects it as Russell’s intended interpretation. Sainsbury(1979) describes a “substitutional” interpretation of thequantifiers over propositional functions, but combines this with atruth-conditional semantics that does not require the ramification ofthe theory of types that is central to Russell’s interpretationin PM.

Propositions and propositional functions are unlike definitedescriptions and classes in that there are no explicit definitions ofthem in PM. It is unclear what it means to say: a symbol for aproposition — such as a variable \(p\) or \(q\) — has“no meaning in isolation”, but the meaning can be given“in context”. No such contextual definition seems possiblein a logic in which propositions and propositional functions appear asprimitive notions.

9. The Construction of Matter, Space, and Time

Whether or not they are provided with contextual definitions byWhitehead and Russell, logical constructions do not appear as thereferents of logically proper names, and so by that accountconstructions are not a part of the fundamental“furniture” of the world. Early critical discussions ofconstructions, such as Wisdom (1931), stressed the contrast betweenlogically proper names, which do refer, and constructions, which werethus seen as ontologically innocent.

Beginning withThe Problems of Philosophy in 1912, Russellturned repeatedly to the problem of matter. As has been described byOmar Nasim (2008), Russell was stepping into an ongoing discussion ofthe relation of sense data to matter that was being carried on by T.P.Nunn (1910), Samuel Alexander (1910), G.F. Stout (1914), and G.E.Moore (1914), among others. The participants of this “Edwardiancontroversy”, as Nasim terms it, shared a belief that directobjects of perception, with their sensory qualities, were nonethelessextra-mental. The concept of matter, then, was the result of a looselydescribed social or psychological “construction”, goingbeyond what was directly perceived. A project shared by theparticipants in the controversy was the search for a refutation ofGeorge Berkeley’s idealism, which would show how the existenceand real nature of matter can be discovered. In TheProblems ofPhilosophy (Russell 1912) Russell argues that the belief in theexistence of matter is a well supported hypothesis that explains ourexperiences. Matter is known only indirectly, “bydescription”, as the cause, whatever it may be, of our sensedata, which we directly know by “by acquaintance”. This isan example of the sort of hypothesis that Russell contrasts withconstruction in the famous passage about “theft” and“honest toil”. Russell saw an analogy between the case ofsimply hypothesizing the existence of numbers with certain properties,those described by axioms, and hypothesizing the existence ofmatter.

The need for some sort of account of the logical features of matter,what he called “the problem of matter”, had alreadyoccupied Russell much earlier. While we distinguish the certainknowledge we may have of mathematical entities from the contingentknowledge of material objects, Russell says that there are certain“neat” features of matter that are just too tidy to haveturned out by accident. Examples include the most generalspatiotemporal properties of objects, that no two can occupy the sameplace at the same time, which he calls “impenetrability”,and so on. InThe Principles of Mathematics (Russell 1903,§453) there is a list of these features of matter including“indestructibility”, “ingenerability” and“impenetrability”, which were all characteristic of theatomic theory of the day. Russell followed the progression through theexact sciences from logic through arithmetic, and then real numbersand then to infinite cardinals. There followed a discussion of spaceand time, with the book ending with a last part (VII) on Matter andMotion, chapters §53 to §59. In them Russell discusses whathe calls “rational Dynamics as a branch of puremathematics” (Russell 1903, §437). ThisrationalDynamics, would involve justifying many of the fundamentalprinciples of physics with pure mathematics alone, from definitionsthat yield the geometry of space and time and the formal properties ofits occupants, quantities of matter and energy. In this respect theconstruction of matter most resembles the construction of numbers asclasses as an effort to replace the “theft” of postulatingaxioms with the “honest toil” of devising definitions thatwill validate those postulates.

In the later project of constructing matter, from 1914 on, beginningwithOur Knowledge of the External World (Russell 1914b),material objects come to be seen as collections of sense data, then of“sensibilia”. Sensibilia are potential objects ofsensation, which, when perceived become “sense data” forthe perceiver. Influenced by William James, Russell came to defend aneutral monism by which matter and minds were both to beconstructed from sensibilia, but in different ways. Intuitively, thesense data occurring as they do “in” a mind, are materialto construct that mind, the sense data derived from an object fromdifferent points of view to construct that object. Russell saw somesupport for this in the theory of relativity, and the fundamentalimportance of frames of reference in the new physics.

In the passage inOur Knowledge of the External World quotedabove, Russell acknowledges that Whitehead initiated the project of“constructing” points and instants of time as classes ofoverlappingevents, that is, regions of space and intervalsof time. Whitehead had developed the logical foundations of theoriesof mathematical physics in his “On Mathematical Concepts of theMaterial World” (Whitehead 1906). There he consideredalternative definitions based on different conceptions of projectivegeometry, in which lines can be seen as classes of points, or elsepoints as where lines intersect. Whitehead then went on to propose themethod of extensive abstraction (Whitehead 1920, Chapter 4)which follows the construction of numbers as classes of equivalenceclasses. Whitehead’s project was to construct points of thespace-time of relativity theory from nested classes of classes ofevents. Discussions of difficulties for this project began in the1920s, including a variant that Whitehead himself proposed inProcess and Reality (1929). See de Laguna (1922), Bostock (2010)and Varzi (2021) for the history of these proposals.

Russell first constructed moments of time as classes:

The assumptions made regarding time-relations in the above are asfollows:
I. In order to secure that instants form a series, we assume: (a) Noevent wholly precedes itself. (An “event” is defined aswhatever is simultaneous with something or other.) (b) If one eventwholly precedes another, and the other wholly precedes a third, thenthe first wholly precedes the third. (c) If one event wholly precedesanother, it is not simultaneous with it. (d) Of two events which arenot simultaneous, one must wholly precede the other.
II. In order to secure that the initial contemporaries of a givenevent should form an instant, we assume: (e) An event wholly aftersome contemporary of a given event is wholly after someinitial contemporary of the given event.
III. In order to secure that the series of instants shall be compact,we assume: (f) If one event wholly precedes another, there is an eventwholly after the one and simultaneous with something wholly before theother.
This assumption entails the consequence that if one event covers thewhole stretch of time immediately preceding another event, then itmust have at least one instant in common with the other event;i.e. it is impossible for one event to cease just before anotherbegins. I do not know whether this should be regarded as inadmissable.For a mathematico-logical treatment of the above topics, cfN. Wiener, “A Contribution to the Theory of relativeposition,” Proc. Camb. Phil. Soc., xvii 5, pp.441–449. (Russell 1914b, 120n)

See Anderson (1989) for a discussion of Norbert Wiener’scontribution to this account in (Wiener 1914b). Wiener (1921)approached the issue of the sensation of intensive qualitiesusing the idea of just noticeable differences originallyproposed by Fechner (1860). (See the entry onmeasurement in science.) The difference of loudness of sounds, intensity of heat sensations,or brightness of lights was previously contrasted with extensivequalities such as length and weight. Fechner’s idea wasthat intensive qualities could be seen as composed of parts, detectedas just noticeable differences, and their number counted to measurethe quality. Wiener made use of Russell’s ontology ofsensibilia, or potential sense data, counting the shortest paththrough a space of just noticeable differences.

10. From Logical Constructions to Measurement Theory

At just this time, Norman R. Campbell (1920) introduced the ideas ofwhat is known asMeasurement Theory which can be seen as analternative to the logical construction of physical quantities.Campbell proposed the direct measurement of length, forexample, by correlating objects with numbers in a scale, say meters,through performing a number of operations of laying down meter sticksuntil the combination is longer than the given object.Campbell’s direct measurement of weight can beperformed by placing the measured object on one pan of a balancescale, and adding unit weights, say of one gram each, in the other panuntil the pan no longer tilts to the side of the weighed object. Ineach case there is an operation (concatenating measuring rods oradding weights to a pan) and a relation (of overlapping with length ortipping the scale with weight).

The logical construction of an extended object in Russell’sapproach specifies a class of sensibilia that occupy a given region ofspace. Measurement would thus attribute a quality to that class.Campbell’s account of direct measurement, however, is completelynon-committal about the nature of the object of measurement, andsimply describes the operations and relations that allow one to assignnumbers to the values.

In his summary of Wiener (1921), Henry Kyburg notes thatWiener’s suggested treatment of measurement did not change thedirection of the theory, that is, away from the notion of measurementas the assignment of numbers to physical entities in the manner ofCampbell (1920). Kyburg’s assessment is at Wiener (1976, 86).Campbell’s lengthy 1920 book in which he presents his theory ofmeasurement does not include any references to previous writers, butdoes dedicate the book to Russell and with a qualification: “Butfor the general train of thought which inspires the whole I can makeacknowledgement to my masters, Henri Poincaré and Mr BertrandRussell; but I fear that the latter (at any rate) will think his pupilanything but a credit to him.” (Campbell 1920,vii)

11. Successors to Logical Construction

In the 1930s Susan Stebbing and John Wisdom, founding what has come tobe called the “Cambridge School of Analysis,” paidconsiderable attention to the notion of logical construction (seeBeaney 2003). Stebbing (1933) was concerned with the unclarity overwhether it was expressions or entities that are logical constructions,and with how to understand a claim such as “this table is alogical construction” and indeed what it could even mean tocontrast logical constructions with inferred entities. (See the entry onSusan Stebbing.)Russell had been motivated by the logicist project of findingdefinitions and elementary premises from which mathematical statementscould be proved. Stebbing and Wisdom were concerned, rather, withrelating the notion of construction to philosophical analysis ofordinary language. Wisdom’s (1931) series of papersinMind interpreted logical constructions in terms of ideasfrom Wittgenstein’sTractatus (1921).

Demopoulos and Friedman (1985) find an anticipation of the recent“structural realist” view of scientific theories in(Russell 1927),The Analysis of Matter. They argue that thelogical constructions of sense data in Russell’s earlierthinking on the “problem of matter” were replaced byinferences to the structural properties of space and matter frompatterns of sense data. We may sense patches of color next to eachother in our visual field, but what that tells us about the causes ofthose sense data, about matter, is only revealed by the structure ofthose relationships. Thus the color of a patch in our visual fieldtells us nothing about the intrinsic properties of the table thatcauses that experience. Instead it is the structural properties of ourexperiences, such as their relative order in time, and which arebetween which others in the visual field, that gives us a clue as tothe structural relationships of time and space within the materialworld that causes the experience. The contemporary version of thisaccount, called “structural realism”, holds that it isonly the structural properties and relations that a scientific theoryattributes to the world about which we should be scientific realists.(See the entry onstructural realism.)

According to this account, Russell’s initial project ofreplacing inference with logical construction was to find for eachpattern of sense data some logical construction that bears a patternof isomorphic structural relations. That project was transformed,Demopoulos and Friedman argue, by replacing inference from the givenin experience to the cause of that experience with an inference to therather impoverished, structural, reality of the causes of thoseexperiences. Russell’s matter project was interpreted in thisway by others, and led, in 1928, to G.H. Newman’s apparentlydevastating objection. Newman (1928) pointed out that there is alwaysa structure of arbitrarily “constructed” relations withany given structure if only the number of basic entities, in this casesense data, is large enough. According to Demopoulos and Friedman,Newman shows that there must be more to scientific theories thantrivial statements to the effect that matter has some structuralproperties isomorphic to those of our sense data. The project ofThe Analysis of Matter does indeed face a serious difficultywith “Newman’s problem”, whether or not thosedifficulties arise for the earlier project of logical construction(see Linsky 2013).

The notion of logical construction had a great impact on the futurecourse of analytic philosophy. One line of influence was via thenotion of a contextual definition, or paraphrase, intended to minimizeontological commitment and to be a model of philosophical analysis.The distinction between the surface appearance of definitedescriptions, as singular terms, and the fully interpreted sentencesfrom which they seem to disappear was seen as a model for makingproblematic notions disappear upon analysis. Wisdom (1931) proposedthis application of logical construction in the spirit ofWittgenstein. In this way the theory of descriptions has been viewedas a paradigm of philosophical analysis of this“therapeutic” sort that seeks to dissolve logicalproblems.

A more technical strand in analytic philosophy was influenced by theconstruction of matter. Rudolf Carnap quotes (Russell 1914a, 11) asthe motto for his “Aufbau”, theLogicalStructure of the World (1967):

The supreme maxim in scientific philosophizing is this: Wheneverpossible, logical constructions are to be substituted for inferredentities. (Carnap 1967, 6)

In theAufbau the construction of matter from“elementary experiences”, and later Nelson Goodman (1951)continued the project. Michael Friedman (1999) and Alan Richardson(1998) have argued that Carnap’s project of construction owedmuch more to his background in neo-Kantian issues about the“constitution” of empirical objects than withRussell’s project. See, however, Pincock (2002) for a responsethat argues for the importance of Russell’s project ofreconstructing scientific knowledge in (Carnap 1967). More generally,the use of set theoretic constructions became widespread amongphilosophers, and continues in the construction of set theoreticmodels, both in the sense of logic where they model formal theoriesand to provide descriptions of truth conditions for sentences aboutentities.

The most faithful successor to Russell’s notion of logicalconstruction is to be found in Willard van Orman Quine’s“explication”. Quine presents his methodology inWordand Object (1960) beginning with an allusion to Ramsey’sremark in the title of section 53: “The Ordered Pair asPhilosophical Paradigm”. The difficulties with apparentlyreferring expressions that motivated Russell’s theory ofdescriptions are presented as a more general problem:

A pattern repeatedly illustrated in recent sections is that of thedefective noun that proves undeserving of objects and is dismissed asan irreferential fragment of a few containing phrases. But sometimesthe defective noun fares oppositely: its utility is found to turn onthe admission of denoted objects as values of the variables ofquantification. In such a case our job is to devise interpretationsfor it in the term positions where, in its defectiveness, it had notused to occur. (Quine 1960, 257)

The notion of a “defective noun” that is to be“dismissed as an irreferential fragment” clearly echoesthe description of constructions as logical fictions and theirexpressions as mere incomplete symbols that so aptly describe thecontextual definitions for definite descriptions and classes. The taskof “devising interpretations” is more like the positiveaspect suggested by the term “construction” andillustrated in the cases of the construction of numbers and matter.After concluding that the expression “ordered pair” wassuch a “defective noun”, Quine says that the notion of anordered pair \(\langle x,y \rangle\) of two entities \(x\) and \(y\)does have “utility” and is limited only in having tofulfill one “postulate”:

(1): If \(\langle x,y \rangle = \langle z,w \rangle\) then \(x = z\)and \(y = w\).

In other words, that ordered pairs are distinguished by having uniquefirst and second elements. Quine then continues:

The problem of suitably eking out the use of these defective nouns canbe solved once for all by systematically fixing upon some suitablealready-recognized object, for each \(x\) and \(y\), with which toidentify \(\langle x,y \rangle\). The problem is a neat one, for wehave in (1) a single explicit standard by which to judge whether aversion is suitable. (Quine 1960, 258)

Again Quine echoes Russell’s language with his mention of a“neat” property that calls out for a“construction” from known entities. Quine distinguisheshis project, which he calls “explication”, by the factthat there are alternative possible ways to fix the notion. AlthoughWhitehead and Russell give an account in PM ∗55, where they arecalled “ordinal couples”, the first proposal to treatordered pairs as classes of their members is from Norbert Wiener(1914a) who identifies \(\langle x,y \rangle\) with \(\{\{ x \}, \{ y,\Lambda \}\}\), where \(\Lambda\) is the empty class. From thisdefinition it is easy to recover the first and second elements of thepair, and so Quine’s (1) is an elementary theorem. Later,Kazimierz Kuratowski proposed the definition \(\{\{ x \}, \{x,y\}\}\),from which (1) also follows. For Quine it is a matter of choice whichdefinition to use, as the points on which they differ are“don’t-cares” (1960, 182), issues which give aprecise answer to questions about which our pre-theoretic account ismute. An explication thus differs considerably from an“analysis” of ordinary, or pre-theoretic language, both ingiving a precise meaning to the expression where it might have beenobscure, or perhaps simply silent and in possibly differing frompre-theoretic use, as suggested by the name. This fits well with theasymmetries we have noted between analysis and construction, withanalysis aimed at the discovery of the constituents and structure ofpropositions which are given to us, and construction which is more amatter of choice, with the goal being the recovery of particular“neat” features of the construction in a formal theory.The ordered pair is thus a “philosophical paradigm” forQuine just as Russell’s theory of descriptions was a paradigm ofphilosophy for Ramsey, and each is a “logicalconstruction”.

Bibliography

Primary Literature: Works by Russell

1901, “The Logic of Relations”, (in French)Rivista di Matematica, Vol. VII, 115–48. Englishtranslation in Russell 1956, 3–38 and Russell 1993,310–49.
1903,The Principles of Mathematics, Cambridge: CambridgeUniversity Press; 2^nd edition, 1937, London: Allen &Unwin.
1905, “On Denoting”,Mind 14 (Oct.),479–93. In Russell 1956, 39–56 and Russell 1994,414–27.
1908, “Mathematical Logic as Based on the Theory ofTypes”,American Journal of Mathematics 30,222–62. In van Heijenoort 1967, 150–82 and Russell 2014,585–625.
1912,The Problems of Philosophy, London: Williams andNorgate. Reprinted 1967 Oxford: Oxford University Press.
1914a, “The Relation of Sense Data to Physics”,Scientia, 16, 1–27. InMysticism and Logic,Longmans, Green and Co. 1925, 145–179 and Russell 1986,3–26.
1914b,Our Knowledge of the External World: As a Field forScientific Method in Philosophy, Chicago and London: Open Court.
1918, “The Philosophy of Logical Atomism” in TheMonist, 28 (Oct. 1918): 495–527, 29 (Jan., April, July1919): 32–63, 190–222, 345–80. Page references toThe Philosophy of Logical Atomism, D.F. Pears (ed.), LaSalle: Open Court, 1985, 35–155. Also in Russell 1986,157–244 and Russell 1956, 175–281.
1919,Introduction to Mathematical Philosophy, London:Routledge.
1924, “Logical Atomism”, inThe Philosophy ofLogical Atomism, D. F. Pears (ed.), La Salle: Open Court, 1985,157–181. Russell 2001, 160–179.
1927,The Analysis of Matter, London: Kegan Paul, Trench,Trubner & Co.
1956, Logic and Knowledge: Essays 1901–1950, R. C. Marsh(ed.), London: Allen & Unwin.
1959,My Philosophical Development, London: George Allen& Unwin.
1973,Essays in Analysis, D. Lackey (ed.), London: Allen& Unwin.
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1994,The Collected Papers of Bertrand Russell, vol. 4,Foundations of Logic: 1903–1905, A. Urquhart (ed.),London and New York: Routledge.
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Primary Literature: Works by Whitehead

1906, “On Mathematical Concepts of the MaterialWorld”, Philosophical Transactions of the RoyalSociety, Series A., Vol. 205, 465–525.
1910–13, A.N. Whitehead and B.A. Russell,PrincipiaMathematica, Cambridge: Cambridge University Press;2^nd edition, 1925–27.
1920, The Concept of Nature, Cambridge: CambridgeUniversity Press.
1929, Process and Reality: An Essay in Cosmology,Cambridge: Cambridge University Press.

Primary Literature: Works by N. Wiener

1914a, “A Simplification of the Logic of Relations”,Proceedings of the Cambridge Philosophical Society, 17:387–390; reprinted in van Heijenoort 1967, 224–227.
1914b, “A Contribution to the Theory of RelativePosition”,Proceedings of the Cambridge PhilosophicalSociety, 17: 441–449.
1921, “A New Theory of Measurement: A Study in the Logic ofMathematics”,Proceedings of the London Mathematical SocietySociety, 19: 181–205.
1976,Norbert Wiener: Collected Works, Volume I, P.Masani (ed.) Cambridge, Mass: MIT Press.

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Acknowledgments

I am grateful to Allen Hazen for explaining the significance ofQuine’s chapter on ordered pairs.

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