User:Wvbailey/Logicism

This is just a scratch-pad re history and defining what "Logicism" really is.

Origin of the name "logicism"

Grattan-Guiness states that the French word 'Logistique' was "introduced by Couturat and others at the 1904 International of Congress of Philosophy', and was used by Russell and others from then on, in versions appropriate for various languages" (G-G 2000:4502).

Apparently the first (and only) usage by Russell appeared in his 1919: "Russell referred several time [sic] to Frege, introducing him as one 'who first succeeded in "logicising" mathematics' (p. 7). Apart from the mis-representation (which Russell partly rectified by explaining his own view of the role of arithmetic in mathematics), the passage is notable for the word which he put in quotation marks, but their presence suggests nervousness, and he never used the word again, so that 'logicism' did not emerge until the later 1920's" (G-G 2002:434).

About same time as Carnap (1929), but apparently independently, Fraenkel (1928) used the word: "Without comment he used the name 'logicism' to characterise the Whitehead/Russell position (in the title of the section on p. 244, explanation on p. 263)" (G-G 2002:269). Carnap used a slightly different word 'Logistik'; Behmann complained about its use in Carnap's manuscript so Carnap proposed the word "Logizisumus', but he finally stuck to his word-choice 'Logistik' (G-G 2002:501). Ultimately "the spread was mainly due to Carnap, from 1930 onwards." (G-G 2000:502)

Intent, or goal, of Logicism

Symbolic logic: The overt intent of Logicism is to reduce all of philosophy to symbolic logic (Russell), and/or to reduce all of mathematics to symbolic logic (Frege, Dedekind, Peano, Russell). As contrasted with Algebraic logic (Boolean logic) that employs arithmetic concepts, symbolic logic begins with a very reduced set of marks (non-arithmetic symbols), a (very-)few "logical" axioms that embody the three "laws of thought", and a couple construction rules that dictate how the marks are to be assembled and manipulated -- substitution and modus ponens (inference of the true from the true). Logicism also adopts from Frege's groundwork the reduction of natural language statements from "subject|predicate" into either propositional "atoms" or the "argument|function" of "generalization" -- the notions "all", "some", "class" (collection, aggregate) and "relation".

As perhaps its core tenet, logicism forbids any "intuition" of number to sneak in either as an axiom or by accident. The goal is to derive all of mathematics, starting with the counting numbers and then the irrational numbers, from the "laws of thought" alone, without any tacit (hidden) assumptions of "before" and "after" or "less" and "more" or to the point: "successor" and "predecessor". Gödel 1944 summarized Russell's logicistic "constructions", when compared to "constructions" in the foundational systems of Intuitionism and Formalism ("the Hilbert School") as follows: "Both of these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principal aims of Russell's constructivism" (Gödel 1944 in Collected Works 1990:119)

History: Gödel 1944 summarized the historical background from Leibniz's in Characteristica universalis, through Frege and Peano to Russell: "Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pure logic", whereas Peano "was more interested in its applications within mathematics". But "It was only [Russell's] Principia Mathematica that full use was made of the new method for actually deriving large prts of mathematics from a very few logical concepts and axioms. In addition, the young science was enriched by a new instrument, the abstract theory of relations" (p. 120-121).

Kleene 1952 states it this way: "Leibniz (1666) first conceived of logic as a science containing the ideas and priciples underlying all other sciences. Dedekind (1888) and Frege (1884, 1893, 1903) were engaged in defining mathematical notions in terms of logical ones, and Peano (1889, 1894-1908) in expressing mathematical theorems in a logical symbolism" (p. 43); in the previous paragraph he includes Russell and Whitehead as exemplars of the "logicistic school", the other two "foundational" schools being the intuitionistic and the "formalistic or axiomatic school" (p. 43).

Dedekind 1887 describes his intent in the 1887 Preface to the First Edition of his The Nature and Meaning of Numbers. He believed that in the "foundations of the simplest science; viz., that part of logic which deals with the theory of numbers" had not been properly argued -- "nothing capable of proof ought to be accepted without proof":

In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions of intuitions of space and time, that I consider it an immediate result from the laws of thought . . . numbers are free creations of the human mind . . . [and] only thourhg the purely logical process of building up the science of numbers . . . are we prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind" (Dedekind 1887 Dover republication 1963 :31).

Peano 1889 states his intent in his Preface to his 1889 Principles of Arithmetic:

Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source in the ambiguity of language. ¶ That is why it is of the utmost importance to examine attentively the very words we use. My goal has been to undertake this examination" (Peano 1889 in van Heijenoort 1967:85).

Frege 1879 describes his intent in the Preface to his 1879 Begriffshrift: He started with a consideration of arithmetic: did it derive from "logic" or from "facts of experience"?

"I first had to ascertain how far one could proceed in arithmetic by means of inferences alone, with the sole support of those laws of thought that transcend all particulars. My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed I had to bend every effor to keep the chain of inferences free of gaps . . . I found the inadequacy of language to be ano bstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the prsent ideography. Its first pupose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed" (Frege 1879 in van Heijenoort 1967:5).

Russell 1903 describes his intent in the Preface to his 1903 Principles of Mathematics:

"THE present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles.(Preface 1903:vi)

A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the philosophy of Dynamics. . . . [From two questions -- acceleration and absolute motion in a "relational theory of space"] I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and then, with a view to discovering the meaning of the word any, to Symbolic Logic" (vi-vii).

Epistemology

TBD: Epistemology of Dedekind and Frege clearly accepting as a priori the "laws of thought"; these laws would be sufficient in themselves for them to derive; classes created by generalizations and relations to allow Russell the realist: Russell's Realism served him as an antidote to British Idealism, with portions borrowed from European Rationalism and British Empiricism. To begin with, "Russell was a realist about two key issues: universals and material objects" (Russell 1912:xi). For Russell, tables are real things that exist independent of Russell the observer. Rationalism would contribute the notion of a priori knowledge, while Empircism would contribute the role of experiential knowledge (induction from experience). Russell would claim that "symbolic logic" and, therefore "mathematics" derive from a priori knowledge, innate knowledge "elicited and caused by experience . . . so direct[ing] our attention [so] that we see its truth without requiring any proof from experience" (Russell 1919:74). With respect to application to logical principles, Russell uses the word a priori rather than "innate" to indicate that babies' innate knowledge is not exactly a priori until it is deduced from experience. In particular is the a priori knowledge represented by the "Laws of thought" (he lists them as the law of identity, the law of contradiction and the law of excluded middle together with modus ponens -- "what follows from a true premiss is true" (cf 1912:72-73)).

"All pure mathematics is a priori, like logic." (Russell 1919:77)

"It seems strange that we should apparently be able to know some truths in advance about particular things of which we have yet no experience; but it cannot easily be doubted that logic and arithmetic will apply to such things" (1912:85)

But his epistemology about the innate is intricate. He would strongly, unambiguously express Platonic support for the "universals". And a priori knowledge (his example the law of contradiction) is about things and not merely about thoughts; these are somehow attached to our a priori knowledge.

Russell and the paradox: Russell had discovered of a "vicious circle" (the so-called Russell's paradox) in Frege's Begriffsschrift and he was determined not to repeat it in his 1903 Principles of Mathematics; in two Appendices that he tacked on at the last minute, he devotes 28 pages to a detailed analysis of , first Frege's theory against his own, and the second a fix for the paradox. Unfortunately he doesn't express optimism about the outcome:

"In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions requisite for the notion of class. And the contradiction discussed in Chapter x. proves that something is amiss, but what this is I have hitherto failed to discover. (Preface to Russell 1903:vi)"

When actual objects are counted, or when Geometry and Dynamics are applied to actual space or actual matter, or when, in any other way, mathematical reasoning is applied to what exists, the reasoning employed has a form not dependent upon the objects to which it is applied being just those objects that they are, but only upon their having certain general properties. In pure mathematics, actual objects in the world of existence will never be in question, but only hypothetical objects having those general properties upon which depends whatever deduction is being considered; and these general properties will always be expressible in terms of the fundamental concepts which I have called logical constants. Thus when space or motion is spoken of in pure mathematics, it is not actual space or actual motion, as we know them in experience, that are spoken of, but any entity possessing those abstract general properties of space or motion that are employed in the reasonings of geometry or dynamics. The question whether theSe propertie'l belong, as a matter of fact, to actual space or actual motion, is irrelevant to pure mathematics, and therefore to the present work, being, in my opinion, a purely empirical question, to be investigated in the laboratory or the observatory.

"On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. . . . The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. . . . Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour." (Preface 1903:viii)

"Fictionalism" and Russell's "no-class theory: Gödel in his 1944 would disagree with this last statement; Russell's mathematics failed (see Criticism section). While conclusively one can say this: Russell believed in a world of things external to our minds. There really are objects "out there" that Russell calls "tables", "chairs", "mirrors", "rabbits", Gödel also observes that Russell is a surprising "realist" with a surprising twist: he cites Russell's 1921:169 "Logic is concerned with the real world just as truly as zoology" (Gödel 1944:120). But he observes that "when he started on a concrete problem, the objects to be analyzed (e.g. the classes or propositions) soon for the most part turned into "logical fictions" . . . [meaning] only that we have no direct perception of them." (p. 121)

In an observation pertainent to logicism Perry remarks that Russell went through three phases of realism -- extreme, moderate and constructive (Perry 1997:xxv); as he was writing his 1912 and finishing up Principia Mathematica he would be in his moderate phase. In a few years (1914) he would "dispense with physical or material objects as basic bits of the furniture of the world. He would attempt to construct them out of sense-data" in his next book Our knowledge of the External World[1914]" (Perry xxvi).

These constructions in what Gödel would 1944 would call "nominalistic constructivism . . . which might be better called fictionalism" derived from Russell's "more radical idea, the "no-class theory":

"according to which classes or concepts never exist as real objects, and sentences containing these terms are meaningful only as they can be interpreted as . . . a manner of speaking about other things" (p. 125).

Commentary

John Perry's [Intro to Russell 1912: Says he and colleague G. E. Moore were "leading a revolt against the philosophical movement in which he was trained, British idealism ["that whatever exists, or at any rate whatever can be known to exist, must be in some sense mental], inaugurating the movement known as "analytical philosophy" (p. viii). He cites Russell's assertion that the table on which Russell puts his arm really exists "out there", is a material object independent of anyone's mind. And his rebellion against the spectrum of idealism from solipsism through "theistic idealism" (Berkeley) and "Absolutist idealism" F. H. Bradley) ??].

With respect to "realism" Perry states that "Russell was a realist on two key issues: universals and material objects". In the collection of philosophers "very enthusiastic about them" [universals] Perry places Plato and Russell (page xi). Perry discusses Russell's avowal of "relations", in particular, as universals: "Among the most important are spatial relations . . . temporal relations . . . and causation" (p. xii). In contrast to universals, Russell also was a realist about particulars (tables, chairs, etc) . "Two other important ingredients in Russell's ontology are facts and propositions", and he believes that these (particulars, universals (qualities and relations), with propositions debatable, are real things not to be identified with the mental or linguistic objects (p. xiv). Perry states that Russell "enunciates the fundamental princple of his epistemology" as follows:

Every proposition which we can understand must be composed wholly of consitutents of which we are acquainted" (p. 58)

where "acquainted" means by direct experience of universals ("brown") and particulars (this table). In particular when Russell sees "this table" as an oval brown patch, he experiences a sensation derived from the table's "sense data" (presumably derived from direct visual perception: oval, brown, patch) that comes from the object in his line of sight (inferred to be "a table"). (xxiii) In conclusion, Parson's view that Russell came to disavow propositions as "real things" squares with his "no class" theory, especially in the 2nd edition of PM and with Russell's "fictionalism", the description coined by Goedel 1944, quoted above. Parson's quotation of his epistemology also squares with Russell's Axiom of reducibility that claimed that given any predicate (proposition) of any type whatever (even infinite) it could be reduced, by extension, to the individual "atomic" propositions at the very bottom of the heap:

"Joe Dimaggio and the Yankees won the 1947 World Series"

would be decomposed into:

(i) "Joe Dimaggio won the 1947 World Series" #&# (ii)"The Yankees won the world series".

the sentence on the left is an "atom" because the particulars are Joe Dimaggio and the 1947 World Series. The sentence (ii) on the right can be reduced to its extensional []:

(ii)" 'Joe Dimaggio won the 1947 World Series' #&# 'Teamate Y2 won the 1947 world series' #&# 'Teamate Y3 won the 1947 World Series' #&# 'Teamate Yn won the 1947 World Series' ".

The first sentence can be subsumed into the second ('Joe Dimaggio won the 1947 World Series' in both (i), (ii)). And then the final result (ii) can be submitted to the real world for verification.

Richard Dedekind

"I regard the whole of arithmetic as a necessary, or at least natural, conseuqence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding . . ." (1858:4)

"In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. . . . numbers are free creations of the human mind. . . . It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number domain created in our mind. If we scrutinise closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensible " (1887:32)

"So from the time of birth, continually and in increasing measure we are led to relate things to things and thus to use that faculty of the mind on which the creation of numbers depends; by this practice continually occuring, though without definite purpose, in our earliest years and by the attending formation of judgements and chains of reasoning we acquire a stoe of real arithmetic truths to which our first teachers later refer as to something simple, self-evident, given in the inner consciousness; and so it happens that many very complicated notions (as for example that of the number of things) are erroneously regarded as simple."

Interestingly, he does not create the 0 first, but rather the unit (i.e. "1") (cf p. 35). Also, he does not want to follow Dirichlet and try to create the real numbers from the rationals, a "wearisome circumlocution" (p. 35) on the contrary, he will use his notion of "cut" [Schnitt] of the number line that he described in his 1853[8?].

In his 2nd Preface of 1893 he comments upon Frege's Grundlagen der Arithmetik: "it contains, particularly from § 79 on, points of very close contact with my paper, especially with my definition (44) . . . the positiveness with which the author speaks of the logical inference from n to n+1 . . . shows plainly that here he stands upon the same ground as me" (1893 preface to 1878:42-43).

"We are justified in calling numbers a free creation of the human mind. The relations or laws which are derived entirely from the conditions α, β, γ, δ in (71) and therefore always the same in all ordered simply infinite systems . . . form the first object of the science of numbers or arithmetic. [N' e N, N=1o, element 1 is not contained in N', transformation φ is similar], where N is a "infinite system" that is similar to a proper part of itself." (p. 67)

Gottlob Frege

"If it is one of the tasks of philosophy to break the domination of the word over the human spirit by laying bare the misconceptions that through the use of language . . . then my ideography . . . can become a useful toof for the philospher. To be sure, it too will fail to reproduce ideas in a pure form . . . inevitable when ideas are repesented by concrete means" (p. 7)

RE "truth": he divides it into two kinds, arrived at by "means of logic" and by facts of experience". He deduced that, at least as regards "arithmetic" my initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. But in doing so he found language to be an obstacle and he arrived at his "ideography" (p. 6). He hopes to be able to use express relations "that are independent of the particular characteristics of objects. I was also able to use the expession "formula language for pure thought" . . . modeled upon the formula language of arithmetic" (p. 6)

"Through the example [theory of sequences] we see how pure thought, irrespective of any content given by the senses or even by an intuition a priori, can, solely from the conent that results from its own constitution, bring for judgements that at first signt appear to be possible only on the basis of intuition . . . The propositions about sequences developed in what follows far surpass in generality all tose that can be derived from any intuition of sequences" (p. 55) "If therefore, one were to consider it moe appropriae to use an intuitive idea of sequence as a basis . . . the propositions thus obtained . . . would hold only in the domain of precisely that intuition upon which they were based." (p. 55)

In other words: because his derivation is from [a system? construction?] more primitive and therefore has wider applicability that one more narrowly focused.

Bertrand Russell

As a philosopher Russell's explanations are coherent and plainly written, without presumption or [familiarity with philosophic jargon]. . His 1912 The Problems of Philosophy, written at the same time as the first edition of Principia Mathematica.

"It is not by argument that we originally come by our belief in an independent external world . . . it is what may be called an instinctive belief . . . [doubts] leave undiminished our instinctive belief that there are objects corresponding to our sense-data (1912:24).

Knowledge can come by either acquaintance or by description; "The chief importance of knowledge by description is that it enables us to pass beyond the limits of our private experience" (cf chapter V Acquaintance and Description 1912:59). Repeated experiences and associations lead to a degrees of certainty called "induction" (cf Chapter VI On Induction) but there are certain "logical principles" (p. 71ff) that are "not trivial to the philosopher, for they show that we may have indubitable knowledge which is in no way derived from objects of sense" (p. 72), in particular the three "Laws of Thought": (1) the law of identity: 'Whatever is, is.' (2) The law of contradiction: 'Nothing cvan both be and not be.' (3) The law of excluded middle: "'Everything must be or not be.' "(p. 72) He also mentions as an alternate "law" (but not by the name) modus ponens: "what follows from a true premise is true". He observes that it is not the case that we think with these "laws" but rather "the fact that things behave in accordance with them" (p. 73). With this he descends into the thicket of a priori knowledge and concludes that it exists in the "Laws of Thought". He cites "the rationalists" Descartes and Leibniz as its proponents versus the empiricists Locke, Berkeley and Hume as its detractors (i.e. all knowldge is derived from experience) and concludes:

"It must be concluded that logical principles are known to us, and cannot be themselves proved by experience, since all proof presupposes them" (p. 74)

He observes that it would be absurd to believe that babies know the a priori exactly as grown men know it; rather, the a priori is "elicited" from experience. So he eschews the word "innate" and uses instead a priori "in the sense that the experience which makes us think of it does not suffice to prove it, but merely directs our attention that we see its truth wihtout requireing any proof from experience". And yet, in a cautionary observation, he grants the empiricists this: "Nothing can be known to exist except by the help of experience" (p. 74); all knowledge acquired a priori is "hypthetical" until confirmed by experience.

"All pure mathematics is a priori, like logic" (p. 77) He amplifies this in Chapter VIII How a piori Knowledge is Possible, opining: "It seems strange that we should apparently be able to know some truths in advance without particular things of which we have as yet no experience; but it cannot easily be doubted that logic and arithmetic will apply to such things" (p. 85). He discusses Kant's view that we have "a priori knowledge as to space and time and causality and comparision" (p. 86) but concludes that Kant's philosophy can be criticized by an objection that is "fatal": the certainy that the facts must confirm to "logic and arithmetic" (p. 87). "To say that logic and arithmetic are contributed by us does not account for this" (p. 87) Suppose that tomorrow that "our natures" change overnight so that two things and two more things make five things (i.e. 4 things plus one more things)" how can this be, when "Our nature is as much as a fact of the existing world as anything". In other words "the time-order of phenomena is determined by characteristics behind the phenomena" (p. 87). Another example: "the law of contradiction is about things and not merely about thoughts . . . a fact concrning the things in the world . . . thus our a priori knowledge . . . is applicable to whatever the world may contain, both what is mental and what is non-mental" but he places the entities of this a priori knowledge in neither the mental or in the physical world; they do not exist in either of these worlds. These entities are not "substantive" parts of speach" such as are "qualities" and "relations" (p. 90).

This nether-world is the Platonic world of ideals.

He gives the example of how, "in most questions of daily life such as whether our food is likely to be nourishing and not poisonous, we shall be driven back to the inductive pricple . . . beyond that, ther seems to be no further regress. . . . And the same holds for other logical priciples. Their truth is evident to us . . . [but] at lwast some of them, are incapable of demonstration." (p. 112)

"All our knowledge of truths depends upon our intuitive knowledge" (p. 109). Truth and falsity, in turn are derived from belief in correspondence with fact" (p. 123) "Thus a belief is true when it corresponds to a certain associated complex, and false withn it does not . . . the two terms and a relation, the terms being put in a certain order by the 'sense' of the believing, then if thet two terms in that order are united by the relation into a complex, the blief is true; if not, it is false." (p. 128) and this is his definition of truth and falsity.

"thus our inutiive knowledge, which is the source of all our other knowledge of truths, is of two sorts: pure empirical knowledge [acquaintanceship], and pure a priori knowledge, which gives us connexions between universals, and enables us to draw inferences from the particular facts given in empirical knowledege. [Thus] our derivatve knowledge always depends upon some pure a priori knowledge and also depends upon some pure empirical knowledge" (p. 149).

The Logistic construction of the natural numbers

Russell's attempt to construct the natural numbers is summarized succinctly by Bernays 1930-1931. But rather than use Bernays' precis, which is incomplete in the details, the construction is best given as a simple finite example together with the details to be found in Russell 1919.

In general the logicism of Dedekind-Frege is similar to that of Russell, but with significant (and critical) differences in the particulars (see Criticisms, below). Overall, though, the logicistic construction-process [Dedekind-Frege-Russell] is far different than that of contemporary set theory and the historically-important Formalist theories. In these theories the notion of "number" begins with intuition (or perhaps a priori (innate) knowledge) of time- or position-sequence and perhaps even of "counting", whereas in logicistic theories the only knowledge presummed is a priori (innate) "laws of thought" (see more in the section "Epistemology"). Whereas set theory assumes the axiom of pairing that leads to the definition of "ordered pair", and Formalism assumes the Peano axioms of number and induction no such axioms exist in the derivation of mathematics following the logicistic methodology.

Godel 1944 observed this stated it this way:

The logicistic derivation equates the cardinal numbers to the natural numbers, and these numbers end up all of the same "type" -- as equivalence classes of classes -- whereas in set theory each number is of a higher class than its predecessor (thus each successor contains its predecessor as a subset). The epistemology behind the two approaches is entirely different. Kleene observes that:

"The viewpoint here is very different from that [of Kronecker's supposition that 'God made the integers' plus Peano's axioms of number and mathematical induction,] where we supposed an intuitive conception of the number sequence and elicited from it the principle that, whenever a particular property P of natural numbers is given such that (1) and (2), then any given natural number must have the property P" (1952:44).

Whether the logistic or the intuitive construction (or both) is successful will be left to opinions voiced in the Criticism section, below.

The importance of the following derives from Russell's contention that "That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected" (1919:4). The derivation of the real numbers (rationals, irrationals) derives from the theory of Dedekind cuts on the continuous "number line". While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophic problems appear in the logistic attempt to derive the natural numbers, these problems will be sufficient to stop the program until these are fixed (see Criticisms, below).

Preliminaries

For Frege and Russell, collections (classes) are aggregates of “things” specified by proper names, that come about as the result of propositions (utterances about something that asserts a fact about that thing or things). Russell tears the general notion down in the following manner. He begin with "terms" in sentences that he decomposes as follows:

Terms: For Russell, “terms” are either “things” or “concepts”: “Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words, unit, individual, and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. is in some sense. A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term; and to deny that such and such a thing is a term must always be false” (Russell 1903:43)

'Things are indicated by proper names; concepts are indicated by adjectives or verbs: “Among terms, it is possible to distinguish two kinds, which I shall call respectively things and concepts; the former are the terms indicated by proper names, the latter those indicated by all other words . . .Among concepts, again, two kinds at least must be distinguished, namely those indicated by adjectives and those indicated by verbs.(1903:44)

Concept-adjectives are "predicates"; concept-verbs are "relations": "The former kind will often be called predicates or class-concepts; the latter are always or almost always relations.” (1903:44)

The notion of a "variable" subject appearing in a proposition: "I shall speak of the terms of a proposition as those terms, however numerous, which occur in a proposition and may be regarded as subjects about which the proposition is. It is a characteristic of the terms of a proposition that anyone of them may be replaced by any other entity without our ceasing to have a proposition. Thus we shall say that "Socrates is human" is a proposition having only one term; of the remaining component of the proposition, one is the verb, the other is a predicate.. . . Predicates, then, are concepts, other than verbs, which occur in propositions having only one term or subject.” (1903:45)

In other words, a “term” can be place-holder that indicates (denotes) one or more things that can be put into the placeholder. (1903:45).

Truth and falsehood: Suppose Russell were to point to an object and utter: “This object in front of me named “Emily” is a woman.” This is a proposition, an assertion of Russell's belief, true or false in the naïve sense, or in the Russellian sense, to be tested against the “facts” of the outer world: “Minds do not create truth or falsehood. They create beliefs . . . what makes a belief true is a fact, and this fact does not (except in exceptional cases) in any way involve the mind of the person who has the belief” (1912:130). If by investigation of the utterance and correspondence with “fact”, Russell discovers that Emily is a rabbit, then his utterance is considered “false”; if Emily is a female human (a female “featherless biped” as Russell likes to call humans), then his utterance is considered “true”.

If Russell were to utter a generalization about all Emilys then these object/s (entity/ies) must be examined, one after another in order to verify the truth of the generalization. Thus if Russell were to assert “All Emilys are women”, then the “All” is a tipoff that the utterance is about all entities “Emily” in correspondence with a concept labeled “woman” and a methodical examination of all creatures with human names would have to commence.

Classes (aggregates, complexes): "The class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate” (1909 p. 55). Classes can be specified by extension (listing their members) or by intension, i.e. by a "propositional function" such as "x is a u" or "x is v". But "if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential.” (1909 p. 66)

Propositional functions: "The characteristic of a class concept, as distinguished from terms in general, is that "x is a u" is a propositional function when, and only when, u is a class-concept." (1909:56)

Extensional versus intensional definition of a class: "71. Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. . . logically; the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal.(1909:69)

The definition of the natural numbers

The natural numbers derive from ALL propositions in this and all other possible worlds, that can be uttered about ANY collection of entities. Russell makes this clear in the second (italicized): In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration. In the second place, the collections having a given number of terms themselves presumably form an infinite collection: it is to be presumed, for example, that there are an infinite collectioa of trios in the world, for if this were not the case the total number of things in the world would be finite, which, though possible, seems unlikely. In the thid place, we wish to define "number" in such a way that infinite numbers may be possible; thus we must be able to speak of the number of terms in an infinite collection, and such a collection must be defined by intension, i.e. by a property common to all its members and peculiar to them." (1919:13)

To begin, devise a finite example. Suppose there are 11 propositions asserted “x is a child of Fn” applied to a collection households on a particular street of families F1, F2, . . .. This proposition regards whether or not a "childname" applies to a child in a particular household. The childrens’ names are the x, and the family name is F_n. To keep things simple all 26 letters of the alphabet are used up in this example, each letter representing the name of a particular child (there could be repeats). Notice that, in the Russellian view these collections are not sets, but rather "aggregates" or "collections" or "clases" --listings of names that satisfy the predicates F1, F2, . . ..

Step 1: Assemble ALL the classes: What exactly is a “class”? For Dedekind and Frege, a class is a distinct entity all its own, a “unity” that can be identified with all those entites x that satisfy the propostional function F( ). For example, a particular “unity” could be given a name; suppose a family Fα has the children with the names Annie, Barbie and Charles:

[ a, b, c ]_Fα

This Dedekind-Frege construction could be symbolized by a bracketing process similar to, but to be distinguished from, the symbolism of contemporary set theory { a, b, c }, i.e. [ ] with the elements that satisfy the proposition separated by commas (an index to label the collection-as-a-unity will not be used):

[a, b, c], [d], [ ], [e, f, g], [h, i], [j, k], [l, m, n, o, p], [q, r], [s], [t, u], [v, w, x, y, z]

For Russell, on the other hand, a class consists of only those elements that satisfy the proposition; the class itself is not a unitary object and exists only as a kind of useful fiction: “We have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic” (First edition of Principia Mathematica 1927:24).

Russell does not waver from this opinion in his 1919; observe the words "symbolic fictions":

“When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. . . . But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them . . ..” (1919:184)

And by the second edition of PM (1927) Russell would insist that “functions occur only through their values, . . . all functions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, loose even that shadowy being which they retain in *20” (p. xxxix). In other words, classes as a separate notion have vanished altogether. Given this disappearance of class-as-object, the only correct way to symbolize the above listing is to eliminate the brackets. But this is visually confusing, so a dashed vertical line at each end of the collection will be used to symbolize the collection-as-aggregate:

┊a, b, c┊, ┊d┊, ┊┊, ┊e, f, g┊, ┊h, i┊, ┊j, k┊, ┊l, m, n, o, p┊, ┊q, r┊, ┊s┊, ┊t, u┊, ┊v, w, x, y, z┊

Step 2: Collect “similar” classes into bundles (equivalence classes): These above collections can be put into “similarity”, i.e. one-one correspondence of the elements, and thereby create Russellian classes of classes or what Russell called “bundles”.

Take for example ┊h,i┊. Its terms h, i cannot be put into one-one correspondence with the terms of ┊a,b,c┊,┊d┊,┊┊,┊e,f,g┊, etc. But it can be put in correspondence with itself and with ┊j,k┊,┊q,r┊, and ┊t,u┊. These similar collections can be assembled into a “bundle” (equivalence class) as shown in the fourth row, below.

┊┊h,i┊, ┊j,k┊, ┊q,r┊, ┊t,u┊┊

The bundles (equivalence classes) are shown below. ┊ ┊a, b, c┊, ┊e, f, g┊ ┊ ┊ ┊d┊, ┊s┊ ┊ ┊ ┊┊ ┊ ┊ ┊h, i┊, ┊j, k┊, ┊q, r┊, ┊t, u┊ ┊ ┊ ┊ l, m, n, o, p┊, ┊v, w, x, y, z┊ ┊

Step 3: Define the null-class’’’: Notice that the third class-of-classes, ┊ ┊┊ ┊ , is special because it contains no elements, i.e. no elements satisfy the predicate that created this particular class/collection. Example: the predicate is:

“For all childnames: “childname is the name of a child in family F_ρ”.

This particular predicate cannot be satisfied because family F_ρ is childless. There are no terms (names) that satisfy this particular predicate. Remarkably, this class ┊┊ not only is empty, it does not exist at all (more or less for Russell the agnostic-about-class-existence; for Dedekind-Frege it does exist).

This peculiar non-existent entity is nicknamed the “null class” or the “empty class”. this is not the class of all null classes ┊ ┊┊ ┊: the class of all null classes is destined to become “0”; see below. Russell symbolized ┊ ┊ with Λ. In PM Russell says that “A class is said to exist when it has at least one member . . . the class which has no members is called the “null class” . . . “α is the null-class” is equivalent to "α does not exist". One is left uneasy: ‘’Does the null class itself “exist”?” This problem bedeviled Russell throughout his writing of 1903. After he discovered the paradox in Frege’s ‘’Begriffsschrift’’ he added Appendix A where through the analysis of the nature of the the null and unit classes, he he discoveds a need for a “doctrine of types”.

Step 4: Assign a "numeral" to each bundle: For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a “numeral”). These symbols are arbitrary. (The symbol ≡ means "is an abbreviation for" or "is a definition of"):

┊ ┊a, b, c┊, ┊e, f, g┊ ┊ ≡ ✖

┊ ┊d┊, ┊s┊ ┊ ≡ ■

┊ ┊┊ ┊ ≡ ♣

┊ ┊h, i┊, ┊j, k┊ ┊, ┊q, r┊, ┊t, u┊ ┊ ≡ ❥

┊ ┊ l, m, n, o, p┊, ┊v, w, x, y, z┊ ┊ ≡ ♦

Step 5: Define “0”: In order to "order" the bundles into the familiar number-line a starting point traditionally called “zero”, is required. Russell picks the empty or ‘’null’’ class of classes to do this. This null class-of-classes ┊ ┊┊ ┊ has been labelled "0" ≡ ♣.

Step 6: Define the notion of “successor”: Russell defines a new characteristic "hereditary", a property of certain classes with the ability to "inherit" a characteristic from another class (or class-of-classes) i.e. “A property is said to be " hereditary" in the natural-number series if, whenever it belongs to a number n, it also belongs to n+1, the successor of n.” (1903:21). He asserts that “the natural numbers are the posterity [the “children”, the inheritors of “successor”] of 0 with respect to the relation “the immediate predecessor of (which is the converse of “successor”) (p. 23).

Note Russell has used a few words here without definition, in particular “number series”, “number n”, and “successor”. What is a “successor?”

Step 7: Define the notion of "successor": Russell does not use the unit class-of-classes (e.g. ┊ ┊d┊, ┊s┊ ┊ ≡ ■ ) to construct the successor. The reason is that, in Russell's detailed analysis, a unit class ■ can become a entity in its own proposition, thereby causing the proposition to become "impredicative" and result in a "vicious circle". Rather, he states (confusingly): “We saw in Chapter II that a cardinal [natural] number is to be defined as a class of classes, and in Chapter III that the number 1 is to be defined as the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, unit classes must be defined so as not to assume that we know what is meant by one (1919:181).

He will define the successor by use of a single entity or "term" as follows:

“It remains to define "successor." Given any number n let α be a class which has ‘’n’’ members, and let x’’ be a term which is not a member of α. Then the class consisting of α with x added on will have +1 members. Thus we have the following definition :

the successor of the number of terms in the class α is the number of terms in the class consisting of α together with x where x is not any term belonging to the class.” (1919:23)

Russell’s definition requires a new “term” (name, thing) which is “added into” the collections inside the bundle. To keep things abstract this will be abbreviated by the name "Smiley" ≡ ☺ (on the assumption that no one has ever actually named their child "Smiley").

Step 8: Construct the successor of the null class: For example into the null class Λ [ ] stick the smiley face. From the previous, it is not obvious how to do this. The predicate:

“For all childnames: "childname is the name of a child in family F_α”.

has to be modified to creating a predicate that contains a term that is always true:

“For all childnames: "childname is the name of a child in family F_α *AND* Smiley";

In the case of the family with no children, “Smiley” is the only “term” that satisfies the predicate. Russell fretted over the use of the word *AND* here, as in “Barbie AND Smiley”, and called this kind of AND (symbolized below with *&* ) a “numerical conjunction” :

┊ ┊ *&* ☺ → ┊☺┊

By the relation of similarity, this new class can be put into the equivalence class (the unit class) defined by ■: ┊ ┊☺┊,┊d┊,┊s┊ ┊≡ ■

Step 9: For every equivalence class, create its successor: Note that the smiley-face symbol must be inserted into every collection in a particular equivalence-class bundle, i.e.: ❥ *&* ☺ ≡┊┊h,i┊, ┊j,k┊, ┊q,r┊, ┊t,u┊┊ *&* ☺ → ┊ ┊h, i,☺┊, ┊j, k,☺┊ ┊, ┊q, r,☺┊, ┊t, u,☺┊ ┊ ≡ ✖

And in a similar manner, by use of the abbreviations set up above, for each numeral its successor is created:

0

0 *&* ☺ = ■

■ *&* ☺ = ❥

❥ *&* ☺ = ✖

✖ *&* ☺ = ? [no symbol]

? *&* ☺ = ♦

♦ *&* ☺ = etc, etc

Step 10: Order the numbers: The process of creating a successor points to a relation " . . . is the successor of . . .", call it "S", between the various "numerals", for example ■ S 0, ❥ S ■, and so forth. “We must now consider the serial character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the ‘’class’’ of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later. (1901:31)

Russell applies to the notion of “ordering relation” three criteria: First, he defines the notion of “asymmetry” i.e. given the relation such as S (" . . . is the successor of . . . ") between two terms x, and y: x S y ≠ y S x. Second, he defines the notion of transitivity for three numerals x, y and z: if x S y and y S z then x S z. Third, he defines the notion of “connected”: “Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)

He concludes: ”. . . [natural] number m is said to be less than another number n when n possesses every hereditary property possessed by the successor of m. It is easy to see, and not difficult to prove, that the relation "less than," so defined, is asymmetrical, transitive, and connected, and has the [natural] numbers for its field [i.e. both domain and converse domain are the numbers].” (p. 35)

Criticism

‘’’The problem of a cardinal number being defined as the class defined by ALL properties of cardinal [natural] numbers’’’: Kleene 1952 points out that “we presuppose the totality of all properties of cardinal numbers as existing in logic, prior to the definition of the natural number sequence. Note that this definition is impredicative, because the property of being a natural number, which it defines, belongs to the totality of properties of cardinal numbers, which is presupposed in the definition” (p. 44).

‘’’The problem of presuming the notion of “iteration”’’’: Kleene points out that, “the logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the Intuitionistic view, an essential mathematical kernel is contained in the idea of iteration” (Kleene 1952:46)

Bernays 1930-1 in Mancosu:243 observes that this notion “two things” already presupposes something, even without the claim of existence of two things, and also without reference to a predicate, which applies to the two things; it means, simply, “a thing and one more thing. . . .With respect to this simple definition, the Number concept turns out to be an elementary ‘’structural concept” . . . the claim of the logicists that mathematics is purely logical knowledge turns out to be blurred and misleading upon closer observation of theoretical logic. . . . [one can extend the definition of “logical”] however, through this definition what is epistemologically essential is concealed, and what is peculiar to mathematics is overlooked”.

Hilbert 1931:266-7, like Bernays, detects “something extra-logical” in mathematics: “Besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea of the Kantian epistemology retains its significance: to ascertain the intuitive ‘’a priori’’ mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion this is essentially what happens in my investigations of the principles of mathematics. The ‘’a priori’’ is here nothing more and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: something is already given to us in advance in our faculty of representation: certain ‘’’extra-logical concrete objects’’’ that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed next to one another is immediately and intuitively given to us, along with the objects, as something that neither can be reduced to anything else, nor needs such a reduction.” (Hilbert 1931 in Mancosu 1998: 266, 267) .

In simpler words: the notion of “sequence” or “successor” is an ‘’a priori’’ notion that lies outside “symbolic logic”.

Hilbert dismissed logicism in just a few words: “It would be too great a digression . . . to discuss the many false paths that are today recognized as such; some tried to define the numbers purely logically; others simply took the usual number-theoretic modes of inference to be self-evident. On both paths they encountered obstacles that proved to be insuperable.” ((Hilbert 1931 in Mancoso 1998: 267) .

Mancosu states that Brouwer concluded that: “the classical laws or principles of logic are part of [the] perceived regularity [in the symbolic representation]; they are derived from the post factum record of mathematical constructions” (Mancosu 1998:9), “theoretical logic . . . [is] an empirical science and an application of mathematics” (Brouwer quoted by Mancosu 1998:9).

Goedel 1944: TBD

The unit class, impredicativity and the vicious circle principle

A benign impredicative definition: Suppose the local librarian wants to catalog (index) her collection into a single book (call it Ι for "index"). Her index must list ALL the books and their locations in the library. As it turns out, there are only three books, and these have titles Ά, β, and Γ. To form her index-book I, she goes out and buys a book of 200 blank pages and labels it "I". Now she has 4 books: I, Ά, β, and Γ. Her taks is not difficult. When completed, the contents of her index I is 4 pages, each with a unique title and unique location (each entry abbreviated as Title.Location_T):

I ← { I.L_I, Ά.L_Ά, β.L_β, Γ.L_Γ}.

This sort of definition of I was deemed by Poincaré to be "impredicative". He opined that only predicative definitions can be allowed in mathematics:

"a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it".

By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In the teeth of these, Russell would enunciate a strict prohibition -- his "vicious circle principle":

"No totality can contain members definable only in terms of this totality, or members involving or presupposing this totality" (vicious circle principle)" (Gödel 1944 appearing in Collected Works Vol. II 1990:125. This definition appears, verbatim, also in Kleene 1952:42.</ref>

A pernicious impredicativity: α = NOT-α: To create a pernicious paradox, apply input α to the simple function box F(x) with output ω = 1 - α. This is the [[Boolean logic|agebraic-logic equivalent of the symbolic-logical ω = NOT-α, for truth values 1 and 0 rather than "true" and "false". In either case, when input α = 0, output ω = 1, etc.

To make the function "impredicative", wrap around output ω to input α, i.e. identify (equate) the input with (to) the output (at either the output or input, it does not matter):

α = 1-α

Algbraically equation is satisfied only when α = 0.5. But logically, when only "truth values" 0 and 1 are permitted, then the equality cannot be satisfied. To see what is happening employ an illustrative crutch: assume (i) the propagation of starting α = α₀ and (ii) observe the propagation in discrete time-instants that proceed left to right in sequence across the page:

α₀ → F(x) → 1-α₀ → F(x) → (1 -(1-α_o)) → F(x) → (1-(1-(1-α_o))) → F(x) → ad nauseum

Start with α₀ = 0:

α₀ = 0 → F(x) → 1 → F(x) → 0 → F(x) → 1 → F(x) → ad nauseum

Observe that output ω oscillates between 0 and 1. If the "discrete time-instant" crutch (ii) is dropped, the function-box's output (and input) is both 1 and 0 simultaneously.

Fatal impredicativity in the definition of the unit class: The problem that bedevilled the logicists (and set theorists too, but with a different resolution) derives from the α = NOT-α paradox. Russell discovered in Frege's 1879 Begriffshrift that Frege had allowed a function to derive its input "functional" (value of its variable) not only from an object (thing, term), but from the function's own output as well.

As described above, Both Frege's and Russell's construction of natural numbers begins with the formation of equinumerous classes-of-classes (bundles), then with an assignment of a unique "numeral" to each bundle, and then placing the bundles into an order via a relation S that is assymetricc: x S y ≠ y S x. But Frege, unlike Russell, allowed the unit class (in the example above [[d],[s]] ) to be identified as a unit itself:

[[d], [s]] ≡ ■ ≡ 1

But, as the class ■ or 1 is a single object (unit) in its own right, then it too must be included in the class-of-unit-classes containing a single entity [■]. And this inclusion requires an "infinite regress" (as Godel called it) of increasing "type" and increasing content:

[[d], [s], [■]] ≡ ■

[[d], [s], [[d], [s], [■]]]] ≡ ■

[[d], [s], [[d], [s], [[[d], [s], [[d], [s], [■]]]]]]]] ≡ ■, ad nauseum

Russell would make this problem go away by declaring a class to be a "fiction" (more or less). By this he meant that the class would designate only the elements that satisfied the propositional function (e.g. d and s) and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a "unit". It is an assemblage e.g. d,s but it is not (in Russell's view) worthy of thing-hood:

"The class as many . . . is unobjectionable, but is many and not one. We may, if we choose, represent this by a single symbol: thus xεu will mean " x is one of the u 's." This must not be taken as a relation of two terms, :x and u, because u as the numerical conjunction is not a single term . . . Thus a class of classes will be many many's; its c<>nstituents will each be only many, and cannot therefore in any sense, one might suppose, be single constituents.[etc]" (1903:516).

This is fine, and dandy to boot -- it supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem -- a hierarcy of "types" of classes.

A solution to impredicativity: a hierachy of types

"The logical doctrine which is thus forced upon us is this: The subject of a proposition may be not a single term, but essentially many terms; this is the case with all propositions asserting numbers other than 0 and 1" (1903:516).

When Russell proclaimed classes are "fictions" he solved the problem of the "unit" class, but the overall problem, did not go away; rather, it arrived because of this definition of the null and unit classes: "It will now be necessary to distinguish (1) terms, (2) classes, (3) c1asses of classes, and so on ad infinitum; we shall have to hold that no member of one set is a member of any other set, and that x ε u requires that x should be of a set of a degree lower by one than the set to which u belongs. Thus x ε x will become a meaningless proposition; and in this way the contradiction is avoided" (1903:517)

This is Russell's "doctrine of types":

"Thus the final conclusion is, that the correct theory of classes is even more extensional than that of Chapter VI; that the class as many is the only object always defined by a propositional function, and that this is adequate for formal purposes" (1903:518).

Russell proposed, as a kind of working theory, that all impredicative functions could be made predicative. First of all, functions are to be classified by their "order", where functions of individuals are of order 1, functions of functions (classes of classes) are of order 2, and so forth. Next, he defined the "type" of a function's arguments to be their "range of significance",, i.e. what are those inputs α (indivduals? classes? both?) that plug into f(x) and yield a meaningful output ω? Note that this means that a "type" can be of mixed order, as the following example shows:

"Joe Dimaggio and the Yankees won the 1947 World Series".

This sentence can be decomposed into two: "x won the 1947 World Series" #&# "y won the 1947 World Series". The first sentence takes an individual "Joe Dimaggio" as its input, the other taking an aggregate "Yankees" as its input. Thus overall the sentence has a (mixed) type of 2, mixed as to order (1 and 2).

By "predicative" Russell meant that the function is of an order higher than the "type" of its variable(s). Thus a function (of order 2) that creates a class of classes can only entertain arguments for its variable(s) that are classes (type 1) and individuals (type 0), as these are lower types. Type 3 can only entertain types 2, 1 or 0, and so forth. (Functions come in various "orders" and the arguments/"objects" that plug into the functions come in various "types". Thus one has to be cautious about plugging in an argument of the wrong "type" into a function; the type has to be less than the )

If the function is to be "predicative" it must accept only arguments of lower orders (of a lower type). The axiom of reducibility is the assumption that any function of any "order" can be reduced to a predicative first-order function

A careful reading of the first edition leaves the impression that an n^th order predicative function need not be expressed "all the way down" as a huge "matrix" or aggregate of individual atomic propositions. "For in practice only the relative types of variables are relevant; thus the lowest type occurring in a given context may be called that of individuals" (p. 161). But the axiom of reducibility proposes that in theory a reduction "all the way down" is possible. By the 2nd edition of PM of 1927, though, Russell had given up on the axiom of reducibility and concluded he would indeed force any order of function "all the way down" to its elementary propositions, linked together with logical operators:

"All propositions, of whatever order, are derived from a matrix composed of elementary propositions combined by means of the stroke" (Appendix A, p. 385),

The "stroke" is Sheffer's inconvenient logical NAND that Russell adopted for this 2nd edition -- a single logical function that replaces logical OR and logical NOT.

The net result, though, was a collapse of his theory. Russell arrived this disheartening conclusion: that "the theory of ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealt with. . . .Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom." (PM 1927:xiv).

Gödel agreed that Russell's logicist project was stymied; he seems to disagree that even the integers survived:

"[In the second edition] The axiom of reducibility is dropped, and it is stated explicitly that all primitive predicates belong to the lowest type and that the only purpose of variables (and evidently also of constants) of higher orders and types is to make it possible to assert more complitcated truth-functions of atomic propositions" (Godel 1944 in Collected Works:134). Gödel asserts, however, that this procedure seems to presuppose arithmetic in some form or other (p. 134). He deduces that "one obtains integers of different orders" (p. 135), but the proof in Russell's Appendix B that "the integers of any order higher than 5 are the same as those of order 5" is "not conclusive" and "the question whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy [of the 2nd Edition?] must be considered as unsolved at the present time". Furthermore, it wouldn't matter anyway because propositional functions of order n (any n) must be described by finite combinations of symbols (cf all quotes and content derived from page 135).

In compensation, though, Gödel proposes that

Thus, in mathematical logic this colloquial sentence is not permitted:

"Joe Dimaggio and the Yankees won the 1941 World Series,"

The propositional function is " 'x' won the 1941 World Series". If it is to be evaluated then all the 'x' should be of the same type: "Joe Dimaggio" is an object of type 0 (a thing) and "the Yankees" are an object of type 1 (a collection of which Joe Dimaggio happens to be a "term", i.e. element) .

And this is the end of his analysis of Frege from "the more philosophical part of Frege's work" (1903:518).

Russell believed that, in its fictionalized form, the class of null classes contained nothing at all:

"0" ≡ ┊ ┊┊ ┊┊ ┊┊ ┊┊ ┊┊ . . .┊

And the unit class was yet different from this weird, null/empty not-thing, and the class of "duals" i.e. "2" was different yet. From this he concluded that logicistic mathematics required a "hierarchy of types", and that different types could not appear in the same propositional function.

Criticism: The question of whether or not "impredicativity" is to be disallowed in mathematics is not a trivial matter. Zermelo in his 1908 argued forcefully against Poincaré's stance that "a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it". Rather, he offered up the example of an impredicative definition "in analysis wherever the maximum or the minimum of a previously defined "completed" set of numbers Z is used for further inferences"; his example is the Cauchy proof of the fundamental theorm of algebra that "up to now ... has not occurred to anyone to regard this as something illogical" (Zermelo 1908 in van Heijenoort 167:191). Mancosu 1998 observes that Zermelo "insisted that the object being defined is not "created" through such a determination. As we shall see, this is exactly the crucial philosophical issue concerning impredicative definitions" (Mancosu 1998:69).

Another example, from Kleene 1952, is the impredicativity of the "least upper bound" in the theory of real numbers: ALL the numbers on the line between 0 and 1 must be defined in total (as an aggregate) before the l.u.b. can have definition itself (Kleene 1952:42-43).

A more accessible example of the "least upper bound" is to be found in Ramsey 1926: "Ramsey also criticised the vicious circle principle for its denial of harmless descriptions presupposing a totality, such as 'a man as the tallest in a group' (p. 41)". Carnap 1931 commented on this exact quote: "Consider, he said, the description 'the tallest man in this room'. Here we describe something in terms of a totality to which it itself belongs. Still no one thinks this description inadmissible since the person described already exists and it only singled out, not created, by the description. Ramsey believed that the same consideration applied to properties. The totality of properties alrady exists in itself . . . for these reasons Ramsey allows impredicative definitions (Carnap 1931, pp. 49-50) Mancosu opines that Ramsey's solution is derived from "a strong form of Platonism" and that Carnap called it "theological mathematics" (Mancosu 1998:76), but was sympathetic to the attempt. And Gödel 1944, also referencing Ramsey's 1926, agrees and states that: "If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence of totalities containing members which can be described (i.e. uniquily characterized)²⁰ only by reference to this totality²¹. [²⁰An object a is said to be described by a propositional function φ(x) if φ(x) is true for x = a and no other. ²¹Cf. Ramsey 1926.]

Impredicativity of the definition of the natural numbers: Remember that in the logicistic construction of the numbers, EVERY class formed by a "property" would be placed one-to-one to every other class in order to form "equinumerous" bundles. Kleene observes this, in a devastating example of logicistic impredicativity, as follows: "A finite cardinal (or natural number) can be defined as a cardinal number which possesses every property P such that (1) 0 has the property P and (2) n+1 has the property P whenever n hs the property P. Here we presuppose the totality of all properties of cardinal numbers as existing in logic, prior to the definition of the natural number sequence. Note that this definition is impredicative, because the property of being a natural number, which it defines, belongs to the totality of properties of cardinal numbers" (p. 44). He goes on to note that this latent impredicativity could be resolved with Russell's ordering

The only case when this "infinite regress" of library books will not occur is when there are no books B, i.e. no unit classes, to begin with, i.e. B is utterly null, i.e. non-existent, thus B+Λ = B (i.e. B will be empty after inclusion of the null book Λ Λ This would be exactly the approach Russell would take: Russell would declare the notion of a class "a mere fiction" (Godel 1944:128).

Suppose F(x) is the function, and x stands in for the unspecified thing that normally fills the empty place x; when x is assigned a specific "value" (such as "the girl in the last row on the far right" or "this rabbit") the function (such as "( . . .) is named Emily") becomes a proposition that can be evaluated as true or false. But suppose that the function itself can "plug into" the empty spot that normally receives the specific object:

"( "(. . .) is named Emily" ) is named Emily.

Normally, propositional functions are not named "Emily", so no matter what "x" were plugged into the form, the resulting proposition is false:

"( "x is named Emily" ) is named Emily" e.g.

"("The rabbit is named Emily)" is named Emily" --> false

"("The girl in the last row on the far left") is named Emily" --> false

Philosophy behind logicism

• impredicative definitions and the antinomies (Russell paradox, in particular)
• definition of "class"
• definition of "null class"
• definition of the "unit class"
• problem of "types"
• problem of "integer" (number)
• problem of "the infinite"

Goal: Precisify language for purposes of philosophic discourse. Thus impredicativity (silly paradoxes in language) is a reasonable problem to study. But attempts at the precisification result in a problem re defining a commonplace notion: that of "number", in particular, "the continuum" between 0 and 1. Why on earth would have a philosopher make the leap from precisifying language of discourse by use of "symbolic logic" to axiomatizing and "reducing" mathematics to logic??

The problem of logicism begins with, and ultimately at, with the Russell paradox. But the Russell paradox in its turn rests upon [epistomological assumptions] the notion of a definition of a "totality" that requires the notion of that totality in its definition

"According to Poincare . . . a definition is "predicative" and logically admissible only if it excludes all objects that are "dependent" upon the notion defined, that is, that can in any way be determined by it." (Zermelo 1908 in van Heijenoort 1967:190).

" . . .the vicious circle principle, which forbids a certain kind of "circularity" which is made responsibe for the paradoxes. The fallacy in these, so it is contended, consists in the circumstance that one defines (or tacitly assumes) totalities, whose existence would entail the existence of certain new elements of the same totality, namely elements definable only in the whole totality. This lead to a principle which says that "no totality can contain members definable only in terms of this totality, or members involving or presupposing this totality" (vicious circle principle)" (Goedel 1944:125 in Collected Works)

Suppose a library wants a book (call it "uber-book" U) that shall include all the books in the library, including the uber-book U itself? Is this a reasonable rquest? Suppose the library is empty of books. Then the uber-book U can contain itself (it will be empty). Suppose the library contains one book, call it B. The uber-book U necessarily must contain U = B. But now U must contain itself U as well as B the sole book in the library, so it will actually contain U <= U_B+B = BB. But now U must contain its augmented self (i.e. U = U_BB+B = BBB), ad infinitum in an infinite regress. Goedel observes that this will not occur if the "separate object" (U) is a "mere fiction" (Goedel 1944:128, also footnote 23). And this is exactly the tack that Russell in his 1903, 1912, and 1919 took to dodge the problem of the definition of a "class" -- although he claims to be "agnostic" about whether or not a "class" is "fictious" [cf 1919 and 1912].

But are impredicative definitions so bad? Zermelo 1908 argues in their favor; as does Goedel 1944 and Kleene 1952 argue that they're with us in mathematics, for better or worse. Zermelo argues that the numbers are "intuitive", not a construction from logical nuggets:

The point of view maintained here, that e are dealing with a productive science resting ultimately upon intuition, was recently urged in oppostion to Peano's "logistic", by Poincare, too, in a series of essays (1905, 1906, 1906a)" (p. 190 in van H)

"The shortest man in the room" requires us to have a knowledge of the totality M of all men in the room: M def "∀x: x is a man and x is in the room"; we see that the shortest man m_s is contained in the totality { m₁, m₂, . . ., m_s, m_n } and requires the totality, although it is plain that the totality M is not a man, it is a collection of men. Thus while "shortest man" requires the totality of men-in-the-room M in its definition it is not necessary to measure the height of M to find the shortest man in M. But what about this "construction"? Is it impredicative? Given a stick with a number of marks scribed across it, similar to a ruler, what is the longest length between scribe-marks? Intuitively or by experience, we know the answer: if we include the end-lengths its the total length of the stick. But a naive mechanical process would require all the possible measurements. But the 5th longest length? We need to know the totality of all the lengths, order them, and find the fifth longest. Thus the totality must be included in the construction necessary for the determination. [?]

Zermelo cites the set M determined by the totality of "θ-chains" "would have had to excluded from the definition of these chains, in my definition, which counts M' itself as a θ-chain would be "nonpredicative" and contain a vicious circle" (p. 190). Closer to the example of the longest length, or shortest man, he cites "in analysis wherever the maximum or the minimum of a previously defined "completed" set of numbers Z is used for further inferences. This happens, for example, in the well-known Cauchy proof of the fundamental theory of algebra" (Zermelo 1908:190-191).

“The number 0 is the number of terms in a class which has no members, i.e. in the class which we called the “null class." By the general definition of number, the number of terms in the null-class is the set of all classes similar to the null-class, i.e. (as is easily proved) the set consisting of the null-class all alone, i.e. the class whose only member is the null-class. (It is not identical with the null-class: it has one member, namely, the null-class, whereas the null-class itself has no members” (1919:23)

Philosophy as defined "schools of thought", the various "-isms", such as empiricism, realism, dualism, monism, idealism, -- question of whether or not "logicism" is to be included in one of these, or whether it should be restricted to a particular[epistomology] of mathematics, or what exactly.

A study of Logicism is made difficult because its foremost promulgator -- Bertrand Russell -- was (arguably) first a philosopher and secondarily a mathematician, whereas his forerunners Dedekind, Frege, and Peano were mathematicians first, and philosphers second (or mathematicians with a restricted intent e.g. to facilitate philosophical precision with "logic", or "symbolic logic", a Leibnizian calculus racinitor). So a confusion of "mathematical philosophy" with "the philosophy of mathematics" is an easy mistake to make, and teasing out the philosophy from the mathematics and vice versa is not straight-forward.

In the case of Russell [with all his various writings] one can derive with some certainty his [shifting philosophical background] versus the more-mathematically directed Logicism -- Dedekindian and Fregian]. These later two overtly expressed their desire to reduce the language of philosophical discoure to a precise "symbolic logic"; the problem remains what does "symbolic logic" entail? Russell expressed the same goal, too, backed [this intent] with a realist philosophy that included objective "truth and falsehood" as correlation of the mind's beliefs to an observation of a priori, external "facts" a priori knowledge, "knowledge of . Basically, by the 1920's Russell counted himself a neutral monist, one who believed that all physical and mental entities were united:

"My own belief -- for which the reasons will appear in subsequent lectures -- is that James is right in rejecting consciousness as an entity, and that the American realists are partly right, though not wholly, in considering that both mind and matter are composed of a neutral-stuff which, in isolation is neither mental nor material".[8]

G-G attributes to Russell a stance of "logical monism", "affirming the all-embracing generality of logic, in its bivalent form" (p. 437), although if one considers Russell a neutral monist, Symbolic Logic would be just another composition of the same neutral stuff that constitutes our universe.

In the executional sense, Logicism is a blend of the symbolic logic pioneered by Frege, and the formalist (axiomatic) arithmetic of Peano and Dedekind. The idea had been around since, at least, Liebniz. Russell summarized the idea many times:

"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age and when this fact has been

established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself" (1903:5),

"Pure logic, and pure mathematics (which is the same thing), aims at being true, in Leibnizian phraseology, in all possible worlds" (1919:155), "The two [mathematics and logic] are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians . . . and by mathematicians . . . The proof of their identity, is of course a matter of detail" (1919:156).

Russell identifies himself with the "rationalists" ("especially Descartes and Leibniz") as opposed to the "empricists" (who are best represented by the British philosophers Locke, Berkeley, and Hume" (1912:73). The rationalists "maintained that, in addition to what we know by experience, there are certain 'innate ideas' and 'innate principles', which we know independently of experience (1912:73); indeed, Russell counts the "logical principles" (laws of identity, contradiction and excluded middle) as some of this knowledge (1912:72-73).

"All pure mathematics is a priori. This seems to square with Dedekind 1887 where he states that "In speaking of arithmetic (algebra, analysis) as a part of logic, I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought." (1887:32: Preface to the First Edition The Nature and Meaning of Numbers").

Goedel 1944 begins with a summary of Russellian-logicism's history:

Mathematical logic . . . is a science prior to all others, which contains the ideas and principles underlying all sciences. It was in this second sense that mathematical logic was first conceived by Leibniz in his Characteristica universalis, of which it would have formed a central part. But it was almost two centuries after his death before his idea of a logical calculus . . . was put into effect (in some form at least, if not the one Leibniz had in mind) by Frege and Peano¹ (¹ Frege has doubtless the priority, since his first publication about the subject, which already contains all the essentials, appeared ten years before Peano's.) Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pear logic. Peano, on the other hand, was more interested in its applications within mathematics . . ." .

"It was in this line of thought of Frege and Peano that Russell's work set in . . . It was only in Principia mathematica that full usse was made of the new method for actually deriving large parts of mathematics from a very few logical concepts and axioms . . . In addition, the young science was enriched by a new instrument, the abstract theory of relations . . . In Principia not only Cantor's set theory but also oridinary arithmetic and the theory of meassurement are treated from this abstract relational standpoint.

With regards to the philosophy behind Russell's efforts, he states "what strikse one as surprising in this field is Russell's pronouncedly realistic attitude" although Gödel observes that for Russell classes or propositions "soon" turned into "logical fictions" meaning, per Gödel, "not necessarily . . . that these things do not exist, but only that we have no direct knowldege of them." (p. 121).

The "no-classes" theory: A Russellian "class" is not a "set" as understood in contemporary mathematics. The fundamental difference between a formal axiomatic theory such as that of Peano-Dedekind is the absence of a strong notion of a "class" as mathematical object in its own right. In Russellian notion a class is the null set and the ordered pair (the axiom of pairing), an identification between modus ponens (rule of detachment) and logical implication [confusion about substitution, identity and tautology.]

Absense of the null set: Russell in particular his 1903 observed that a consequence of his defintion of a "class"

"On questions discussed in these sections, I discovered errors after passing the sheet 4 for the press; these errors, of which the chief are the denial of the null-class, and the identification of a term with the class whose only member it is, are rectified in the Appendices." (page vi, entire section §73 "There are null class-concepts, but there is no null class", "A propositional function is said to be null when it is false for all values of x; and the class of x's satisfying the function is called the null-class,

eing in fact a class of no terms. Either the function or the class, following Peano, I shall denote by A" (p. 22), "rather I should regard them [failure of Peano's notion of null class] as proofs that Symbolic Logic ought to concern itself, as far as notation goes, with class-concepts rather than with classes", an intensional interpretation of "concept" allows for the existence of a null class (p. 72) because in extension, no entities at all that satisfy the proposition constitute nothing. "We agreed that the null class, which has no terms, is a fiction, though there are null class-concepts. It appeared throughout that, although any symbolic treatment must work largely with class-concepts and intension, classes and extension are logically more fundamental for the principles of Mathematics; and this may be regarded as our main general conclusion in the present chapter." (p. 81), "But in the present chapter we decided that it is necessary to distinguish a single term from the class whose only member it is, and that consequently the null-class may be admitted" (p. 106),p. 128 a derivation of the Peano axioms in terms of "similarity" or 1-to-1 correspondence, (1) 0 is the class of classes whose only member is the null-class. (2) A

number is the class of all classes similar to anyone of themselves, (3) definition of 1, etc..

Class existence: 1919 has the definition of 0 as "the class whose only member is the null class", keeping in mind that in extension the null class is nothingness. In other words one wonders what a "class" really is. Is it a basket that can be empty? Another very important notion is what is called the existence of a class-a word which must not be supposed to mean what existence means in philosophy. A class is said to exist when it has at least one term. A formal definition is as follows: a is an existent class when and only when any proposition is true provided " x is an a" always implies it whatever value we may give to x. It must be understood that the proposition implied must be a genuine proposition, not a propositional function of x. A class a exists when the logical sum of all propositions of the form "x is an a" is true, i.e. when not all such propositions are false." (p. 21)

The goal: derived from the quest to "precisify" philosophical debate, to eradicate [purge] from philosophy the confusion caused by the ambiguous language of discourse. To reduce all of mathematics to a symbolism that expresses in logical terms, in particular assertion of truth of propositions e.g. P, Q, etc, logical implication ( P → Q) and logical negation ( ~P meaning "not P"), the notion of generalization by use of a "predicate" applied to an indeterminate subject, (e.g. "One predicate always gives rise to a host of cognate notions; thus in addition to human and humanity, which only differ grammatically, we have man, a man, some man, any man,every man, all, all of which appear to be genuinely distinct one from another" (Russell 1903:45).

Results With resulting confusion between :"My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. . . . I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present ideography." (Frege 1879 in van Heijenoort 1967:5-6)

Frege would trace this [goal] to Bacon and Leibniz (p. 6). The goal was also expressed by Dedekind who claims to have, in the autum of 1858, "felt more keenly than ever before the lack of a really scientific foundation for arithmetic" and in his 1887 "In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result of the laws of thought." (Preface to the first edition, The Nature and Meaning of Numbers (p. 31). In the preface to the second edition (1893) he credits both Cantor (1878) and Balzano (1851) neither of whose work he was aware of with "the property which I have employed as the defintion of the infinite system" but he claims that niether had used "this property" to define the infinite "and upon this foundation to establish with rigours logic the science of numbers". Furthermore he references Frege 1884: "it contains, particularly from §79 on, points of very close contact with my paper, especially with my definition (44)" (p. 42)

Peano

Russell

"A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the philosophy of Dynamics. I was met by the difficulty that, when a particle is subject to several forces, no one of the component accelerations actually occurs, but only the resultant acceleration, of which they are not parts; this fact rendered illusory such causation of particulars by particulars as is I affirmed, at first sight, by the law of gravitation. It appeared also that the difficulty in regard to absolute motion is insoluble on a relational theory of space. From these two questions I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and thence, with a view to discovering the meaning of the word any, to Symbolic Logic." (1903: p. vi-vii).

He would reiterate this in his 1919 "commentary: It was Russell's beliefs that by using the new logic of his day, philosphers would be able to exhibit the underlying "logical form" of natural statement . . . [this] would help philosophers resolve problems of reference associated with the ambiguity and vagueness of natural language" (p. 6)

"we should arrive at a language in which everything formal belonged to sytax and not to vocabulary. In such a language we could express all the propositions of mathematics even if we did not know one single word of the language. The language of mathematical logic, if it were perfected, would be such a language." (p. 161)

"Because language is misleading, as well as because it is diffuse and inexact when applied to logic (for which it was never intented), logical symbolism is absolutely necessary to any exact or thorough treatment of our subject." (p. 165)

Frege, i.e. symbolic logic cf Russell 1903:532 index:

Logic, symbolic, 10-32; three parts of, 11; and mathematics, v, 5, 8, 106, 397, 429, 467

THE present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable

in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II.-VII. of this Volume, and will be established by strict symbolic reasoning in Volume II (page v, italics added).

This immediately creates a conundrum: Russell will have use symbols to discuss symbols. G-G 2000:452 observes that Sheffer 1926a was "puzzled that 'in order to give an account of logic; we must presuppose and employ logic' and saw no way out this "logocentric predicament". A point taken up by Wittgenstein 1918:

"3.33 From this observation we get a further view-- into Russell's Theory of Types. Russell's error is shown by the fact that in drawing up his symbolic rules he has to speak of the meaning of the signs. 3.332 No proposition can say anything about itself, because the propostional sign cannot be contained in itself (that is the "whole theory of types"). 3.333 A function cannot be its own argument, because the functional sign already contains the protoype of its own argument and it cannot contain itself.

"If for example , we suppose that the function F(fx) could be its own argument, then there would be a proposition "F(F(fx)), and in this the outer function F and the inner function F must have different meanings, [etc]."

According to G-G 2000:473 by 1922 Hilbert "stressed the importance of axiomatics and secure foundations for mathematics, and of distinguishing signs from their references . . . in contrast to signs lay the properties of arithmetic proper, such as the 'principle of complete induction ' . . . and to make the disnction still more important, he introduced into print the word 'a metamathematics ' (' Metamathematik '), which 'serves for the securing of that' mathematics" (G-G noting in footnote that at the same time the same word was introduced by Sir George Greenhill cf p. 473 footnote 29).

Furthermore, in his chapter 8.5.4 "Groping towards metalogic ", Bernstein's review 1926a of PM ended thus:

" 'As a mathematical system, the logic of propositions is amendable to the postulational treatment applicabe to any other branch of mathematics. As a language, this logic has all its symbols outside the system it expresses. This distinction between the propositional logic as a mathematical system and as a language must be made, if serious errors are to be avoided; this distinction the Principia does not make.' " (G-G 2000:454 quoting Bernstein 1926a). G-G goes on to note that other American authors were beginning to arrive at a similar conclusion."

Here Wittgenstein is proposing an utterly extensional description of a generalization e.g. "All dogs are mammals", to be reduced to infinite conjuctions of the form "dog1 is a mammal" & "dog2 is a mammal" & . . . "dogn is a mammal . . .", the sign and being what Russell would call a "numerical conjuction"

"As regards what is meant by the combination indicated by and, it is indistinguishable from what we before called a numerical conjunction." (1903:72).

By the second edition of PM (1925-1927) Russell would come to adopt Wittgenstein's suggestion, but in his 1903 he claimed that both extensional and intensional descriptions were required: extensional meaning that the indviduals or "terms" would be listed, intensional meaning that a generalization would be formed between "things" and a "concept", the "things" being those with proper names, and the "concpets" being all other words. He further divides the later (concepts) into adjectives (class coepts, predicates), and verbs (relations). Thus "classes" are connected with adjectives" (p. 52) and "propositional functions involve verbs" (p. 52). with unknown x and function such as "all dogs are mammals" [Every dog is a mammal].

Liebniz

"Mathematical logic . . . is a science prior to all others, which contains the ideas and principles underlying all sciences. It wa in this second sense that mathematical logic was first conceived by Leibniz in his Characteristica universalis, of which it would have formed a centeral part. But it was almost two centuries after his death before his idea of a logical calculus . . . was put into effect (in some form at least, if not the one Leibniz had in mind) by Freg and Peano¹ (¹ Frege has doubtless the priority, since his first publication about the subject, which already contains all the essentials, appeared ten years before Peano's.) Freg was chiefly interesting in the analysis of thought and used his calculus in the first place for deriving arithmetic from pear logic. Peano, on the other hand, was more interested in its applications within mathematics . . ." . "It was in this line of thought of Frege and Peano that Russell's work set in . . . It wa only in Principia mathematica that full usse was made of the new method for actually deriving large parts of mathematics from a very few logical concepts and axioms . . . In addition, the young science was enriched by a new instrument, the abstract theory of relations . . . In Principia not only Cantor's set theory but also oridinary arithmetic and the theory of meassurement are treated from this abstract relational standpoint.

However, in Gödel's opinion, Russell's effort is deeply flawed by "so greatly lacking in formal precision in the foundations (*1-*21 of Principia) that it represents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism . . . this is especially doubtful for the rule of substitution and of replacing defined symbols by their definiens.

But regards to the philosophy behind Russell's efforts, he states "what strinks one as surprising in this field is Russell's pronouncedly realistic attitude" although Gödel observes that for Russell classes or propositions "soon" turned into "logical fictions" meaning, per Gödel, "not necessarily . . . that these things do not exist, but only that we have no direct knowldege of them." (p. 121).

What exactly did Frege and Russell have to say on these matters?

Fregian logicism

Goedel 1944 observes that Russell's example "the author of Waverly" signifies Walter Scott, but in Frege's view, the substitution of "Scott" for x in " x is the author of Waverly signifies the same thing as "Scott is Scott"

"the author of Waverly = f

The propositional function "x is the author of Waverly can be written x is f

The only true substitution for x that yields a truth in this function is "Scott"

"Scott" is f, ie "Scott is [identical to] the author of Waverly

ergo the two sentences "Scott is Scott" is identical to "Scott is the author of Waverly.

Goedel observes that Frege draws this conclusion, that all true sentences have the same signification (aswell as all falseones). " 'The Ture' -- accorrding to Frege's view -- is analyzed by us in different ways in different propositions, 'the True' being the name he uses for the common signification of all true propositions.

"Closer examination, however, shows that this advantage of Russell's theory over Frege's subsists only as long as one interprets definitions as mere typographical abbreviations, not as introducing names for objects described by the definitions, a feature which is common to Frege and Russell." (p. 123-124)

Thus we might define a "pig" as a "stout four-legged mammal with short hair", and here the definition is introducing names for objects "legs", "mammal", and "hair". But this definition could denote (signify) any number of animals e.g. a Welsh Corgi dog. To precisify the definition the object "cloven hooves" could be added, but then there may be other possibilites.

Goedel concludes that Frege's conclusion re "the True" (e.g. "Scott is Scott") is "puzzling" and that he considers Russell's theory of descriptions an "evasion"; "that there is something behind it which is not yet completely understood." (p. 125)

Russellian logicism

In Russell 1903 Principles of Mathematics,

"About six years ago, I began an investigation into the philosophy of Dynamics. I was met by the difficulty that, when a particle is subject to several forces, no one of the component accelerations actually occurs, but only the resultant acceleration, of which they are not parts; this fact 1 rendered illusory such causation of particulars by particulars as is I affirmed, at first sight, by the law of gravitation. It appeared also that the difficulty in regard to absolute motion is insoluble on a relational theory of space. From these two questions I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and then, with a view to discovering the meaning of the word any, to Symbolic Logic." (Preface vi-vii)

the preface two [expressions of belief] are apparent -- Platonism (independence of any knowing mind), and a non-realist position with regards to propositions:

"On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not irreducible to adjectives of their terms or of the whole which these compose. . . Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour."

With regards to "truth", it must be found outside "the mental":

"Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour (p. viii)

"For·example, propositions are commonly regarded as (1) true or false, (2) mental. Holding, as I do, that what is true or false is not in general mental, I require a name for the true or false as such, and this name can scarcely be other than proposition. (p. viii)

"Pure logic, and pure mathematics (which is the same thing), aims at being true, in Leibnizian phraseology, in all possible worlds, not only in this higgledy-piggledy ob-lot of a world in which chance has imprisoned us." (Russell 1919/2005:155).

a priori knowledge:"Their truth is independent of the universe, . . . logical propositions are such as can be known a priori, without study of the actual world. . . without needing any appeal to experience (Russell 1919/2005:164)

Goedel observes "the analogy beween mathematics and a natural science is enlarged upon by Russell also in another respect", that of comparing the axioms of logic and mathematics with the laws of nature and logical evidence with sense perception, "so that . . . their justification lies (exactly as in physics) in the fact that they make it possible for these "sense perceptions" to be deduced" . . . the solution of certain arithmeticl problems requires the use of assumptions essentially transcending arithmetic, i.e. the domain of the kind of elementary indisputable evidence that may be most fittingly compared with sense perception

"Now, according to Russell, what corresponds to sentences in the outer world is facts . . . according to Russell's terminology and view, true sentences "indicate" facts and, correspondingly, false ones indicate nothing . . . the fact that Russell does not consider this whole question of the interpretation of descriptions as a matter of mere liguistic conventions, but rather as a questionof right and wrong, is another example of his realistic attitude . . ." (p. 123)

Goedel contrasts this with Frege's views . . . [see above]

Russellian reasoning: induction from observation probability etc (1968 The art of philosophizing p. 65, essays written during the 2nd world war). "law of animal habit" p. 47, "logic may be defined as the art of drawing inferences" (page 37). "There are two forms of logic, deductive and inductive" (p. 38), "In fact, pure mathematics and deductive logic are indistinguishable" (p. 41), strengthen inductive logic by means of "general laws" (p. 50) that have to be discovered: "general laws cannot be discovered unless they exist" p. 51, the role of mathematical deduction in prediction of hypotheses(p. 57, in particular p. 63).

Russellian a priori knowledge: From 1912 The Problems of Philosophy Chapter VII "On Our Knowledge of General Principles":

". . . even that part of our knowledge withi is logically independent of experience . . . is yet elicited and caused by experience. It is on occasion of particular experiences that we become aware of the general laws which their connexions exemplify. . . .For this reason [re babies not knowing everything that men know without experience] the word 'innate' would not bnow be employed to describe our knowledge of logical principles. The phrase 'a priori' is less objectionable . . . we shall nevertheless hold that some knowledge is a priori in the sense that the experience which makes us think of it does not suffice to prove it, but merrely so directs our attention that we see its truth without requiring any proof from experience.

"logic is a priori, namely in the sense that the truth of such knowldege can be neither proved nor disproved by experience. All pure mathematics is a priori, like logic. This was strenuously denied by the empriical philosphers . . . they maintained that by the repeated experience of seeing two things and two other things, and finding that altogether they made four things, we were led by induction to the conclusion that two things and two other things would always make four things together. . . . as soon as we are able to divest our thoughts of irrelevant particularity, we become able to see the general principle that two and two are four; any one instance is seen to be typical, and the examination of toher instances become unnecessary.¹ (¹ Cf. A. N. Whitehead, Introduction of Mathematics (Home University Library).)" (p. 77

Chapter VIII "How 'A Priori Knowldege is Possible": After an analysis of Kant's take on the a priori Russell concludes that that which is a priori knowable by the mind resides, not because of an a priori [prediliction, capability, tendency ] of the mind, but in the external-to-the mind objects themselves. Thus the mind's knowing "two objects plus two objects is equivalent to 4 objects" is a result of a priori with respecct to the objects and not the mind:

"It seems strange that we should apparently be able to know some truths in advance about particular things of which we have no experience; but it cannot be easily doubted that logic and arithmetic will apply to such things." (p. 85)

But no matter what we believe is the case about the facts of our experience, these facts must agree with logic and arithmetic (p. 87):

"The facts must always confirm to logic and arithmetic. To say that logic and arithmetic are contributed by us [our minds] does not account for this." p. 87

" . . . The Kantian view that time itself is a form imposed by the subject [a mind] upon phenomena [that mind's experience of time], so that our real Self is not in time and has tomorrow. But he will still have to suppose that the time order of phenomena is determined by characteristics of what is behind phenomena [? Does Russell mean the objects themseles, or the mental process of time ordering?], and this suffices for the substance of our argument." (p. 87)

Russell concludes that the law of contradiction derives from real objects, from "things" as opposed from the perceiving mind (p. 89): "Thus the law of contradiction is about things, and not merely about thoughts" (p. 89)

re qualities and relations . . .

Truth and falshood: in his chapter XII pp. 119ff and in PM (pages 441-47) Russell argues that, like the perception of objects, "truth and falsehood are properties of belief of a percepient and an external reality:

"This judgment of perception, considered as an actual occurrence, is a relation of four terms, namely a and b and R and the percipient. The perception [itself] is a relation of two terms, namely "a-in-the-relation-R-to-b," and the percipient. . . .

"hence a world of mere matter, since it would contain no beliefs or statements, would also contain no truth or falsehood" i.e. there must be a percipient, too. Nevertheless, there are "outside things" beyond the percipient and the percipient's beliefs, "we are driven back to correspondence with fact as constituting the nature of truth. "What makes a belief true is a fact, and this fact does not (exception in exceptional cases) in any way involve the mind of the person who has the belief" (p. 130).

InPM first edition, Russell asserts the notion of two "truth values" as "truth" and "falsehood", following the usage of Frege (p. 7). In the next paragraph, in consideration of "truth-functions", Russell ties this realist assertion of objective fact, to "mathematics is always concerned with extensions [objective reality] rather than intensions". By the second edition of PM, Russell limits this by stating "functions only occur through their values", and he restricts propositions to only "atoms" and "molecules" and identifies (equates) functions of functions to classes. Thus he has restricted the sizes of the matrices (the extensions of the functions and functions of functions) but this forces him to abandon the axiom of reducibility [cf p. xxxix]. This allows Russell to reduce his logic to a "first order propositional and predicate calculus] by use of propositional functions such as "x loves Socraties" to a listing of the x that either satisfy or do not satisfy this function, and when functions of functions are asserted, again by a listing of their members' truth values. He then "identifies" (equates) functions and classes (sets).

Collapse of Russellian logicism

Goedel 1944

Failure Derives from an unsuccessful attempt to solve the Russell paradox: Goedel 1944 asserts that this collapse occurred (in the 2nd edition of PM) when Russell adopted a "constructivist" stance with respect to impredicative definitions, i.e. Russell's adoption of a new axiom would not allow impredicative definitions because "functions can occur in propositions only 'through their values', i.e., extensionally" (p. 126). Russell flatly states this new axiom and its adverse consequences: "A function can only appear in a matrix through its values*. (*This assumption is fundamental in the following theiry. It has its difficulties . . . It takes the place (not quite adequately of the axiom of reducibility. It is discussed in Appendix C" (p. xxix).

Kleene 1952:42: states that "Russell (1906m 1910) enunciated the same explanation [as Poincare] in his vicious circle principle: No totality can contain members definable only in terms of this totality, or members involving or presupposing this totality [observe the three word-sets: "definable only in terms of", "involving", or "presupposing"]. Thus it might appear that we have a sufficient solution and adequate insight into the paradoxes, except for one circumstance: parts of mathematics we want to retain, particularly analysis, also contain impredicative definitions", and Kleene then offers as example the problem of the definition of the the least upper bound (pages 42-43) M as an element of C, defined in terms of the set of real numbers C and C - M.

In the introduction to the second edition of PM, Russell adopts the suggestions of Wittgenstein to "assume that functions of propositions are always truth functions, and that a function can only occur in a proposition through its values" (p, xiv).

Impredicative definitions, Russell's paradox: The goal is to dodge the Russell paradox that Russell found in Frege's 1879 Begriffsschrift. The paradox appears on page 23 , where Frege allows what is normally the "indeterminate" part of a propositional function i.e. the unknown x, to in fact be the function f(x) itself. This yields x = f(x). Then if substitution is permitted in the calculus one can produce an infinite regression as follows i.e. x = f(x) = f(f(x)) = f(f(f(x))) = ad infinitum if desired.

The paradox can be seen as follows: Suppose that f(x) = 1-x, but x in turn is the totality f(x), i.e. x = ∀x: f(x):

then f(x) = 1 - x = f(f(x)) = 1 - f(x) = 1 - (1 - x). this yields the paradoxical 0 = 1 when (1 - x) is subtracted from both sides of 1 - x = 1 - (1 - x). Or if you will: from 1 - x = x one derives 1 = 2x, i.e. 1 is even.

If, on the other hand, per Wittgenstein's proposal of the 2nd edition of PM the function f(x) can be the indeterminate part only when its values are substituted, then we have the case: f(x) = f( {values of f(x)} ).Tthis "new axiom" might be called the "extensional" axiom (cf Goedel 1944:126). For the finite assignment of x = { 0, 1 } then f(x) = 1 - x => { 1, 0 } when { 0, 1 } are substituted into the equation 1 - x. These can now be substituted into f(x) again (and again, and again) without a paradox resulting.

Goedel concludes that "there is no doubt that thee things are quite unobjectionable even from the constructive standpoint . . . provided that quantifiers are always restricted to definite orders" (boldface added, p. 127). He goes on to expand on Russell's "vicious cirecle principle", stating that it actually comes in three varieties (see the identical quote in Kleene 1952:xxx, above), (i) "definable only in terms of", (ii) "involving", (iii) "presupposing". Of these three, the first is the most pernicious:

"only this one makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of modern mathematics itself. It is demonstrable that the formalism of classical mathematics does not satisfy the vicious circle principle in its [definable-only-in-terms-of-] form, since the axioms imply the existence of real numbers definable in this formalism only by reference to all real numbers." (p. 127)

Indeed, Goedel wonders if his argument might be a proof that the vicious circle principle is false rather than classical mathematics is false. "first of all one may, on good grounds, deny that reference to a totality necessarily implies reference to all single all elements of it, or, in other words, that "all" means the same as an infinite logical conjunction . . . there are difficulties with this view; but there is no doubt that in this way the circularity of impredicative definitions disapperars . . . (p. 127).

For example, the index-book that contains the location of all the books in a library can indeed contain its own location as one of the listings. A library that contains all these books is not a book, it is a library. But a single grandbook that contains all the books in the library is also a book and it cannot contain itself because to do so would result in an infinite regress of grandbook+books-within-grandbook.

Thus "the construction of a thing [the grandbook] can certainly not be based on a totality of things [books + grandbook] to which the thing to be constructed [the grandbook] itself belongs" (p. 127).

Goedel proposes a philosophical viepoint to escape this conundrum: if the objects exist independent of our constructions, totalities can indeed contain members which can be described (uniquely) only by reference to the totality; he references Ramsey 1926 for this assertion (p. 128).

"Classes and concpets may, however, also be conceivd as real objects . . . existing independently of our definitions and constructions. ¶ It seems to me that the assumption of suchobjects is quite as legitimate as the assumption of physical bodies . . . they are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory system theory of our sense perceptions . . . Russell himself concludes in the last chapter of his book on Meaning and Truth [1940], though "with hesitation", that there exist "universals"" (p. 128) although these involve sense perceptions, not logical "concepts". In fact, Goedel asserts the word "concept" in the objective sense (i.e. concepts are as real as physical objects) (p. 128).

But he concludes that when restricted to the first type (i), the vicious circle principle applies only in the "constructivist (or nominalist) standpoint" (p. 128) . . . "Since the vicious circle princple, in its first form, does apply to constructed entities, impredicative definitions and the totality of all notions or classes or propostions are inadmissible in constructivistic logic."

However, what happens when the assignment of x is arbitrarily from an infinite set? This is where Russell's axiom of reducibility of the first edition of PM came into play. In the second edition he replaces it with the following restriction: "A function can appear in a matrix [truth table] only through its values" (p.xxix)

i.e. the subject or "argument") to be "indeterminate", i.e. a function. part of a proposition to be, not a "determinate" argument, x is not a rabbit", where "x are individuals John, Emily, Bill and Sarah. When an occurrence of one of these subjects is substituted for x the proposition can be evaluated for truth or falsity. But what happens when a function (f(y), say) is substituted for x? " f(y) is a function, a generalization, that can extend to an infinite number of instances(cf p. xv)." It is entirely possible that f(y) may evaluate to the class (collection) of all objects "rabbit" in the totality, so the whole totality x = f(y) = all rabbits, when substituted into the proposition " x is not a rabbit" yields the paradox " rabbit is not a rabbit"

If Wittgenstein's suggestion is adopted, then Russell concludes that "it eems that the theory of infinite Dedkindian and well-ordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. . . . Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom." (p. xiv) Since Russell has abandoned this, his entire logicistic edifice collapses. He concludes that "some logical axiom which is true will justify it [the theory of real numbers]. The axiom required may be more restricted than the axiom of reducibility, but, if so, it remains to be discovered."(xiv)

Definition of Logicism

¶, §

From Russell in his 1901[3?] specified two reasons for writing his 1903:

• First: "the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles" (Russell 1903:iv)
• Secondly: "the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task . . . The discussion of indefinables-which forms the chief part of philosophical logic -is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple." (Russell 1903:iv)

More explicitly in his 1919 he declares them identical:

". . . it has now become wholly impossible to draw a line beween the two [mathematics and logic]; the two are one. They differ as a boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians . . . and by mathematicians . . .. The proof of their identity is, of course, a matter of detail . . . [but] if there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the sucessive defintions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins" (Russell 1919/2005:156)

Origins of logicism

Seems to be two sources -- mathematics of the early-mid 1900s, philosophy. Russell 1903 credits Peano and Frege on the mathematical/logical side, his mentor A. N. Whitehead and his philosophy professor xxx

Leibniz calculus rationcitur

Peano

Frege

Grattan-Guinness 2000:303ff, "Section 6.5 Convoluting towards Logicism, 1900-1901":

6.5.1 logicism as generalized metageometry, January 1901.

"Generalizing this conception of metageometry heenvisioned logicism as the philosophy which defined all pure mathematics as hypotheitcial, and that the Peanist line between mathematics and logic did not exist. All mathematics, or at least those branches handled in this book, could be obtained from mathematical logic as an all-embraching implication, for this new category of 'pure mathematics'; the propostional and predicate calcuil (including relations) with quantification proved the means of deduction, while the set theory furnished the "sutff": terms or individuals, and classes or relations (of classes or relations...) of them. Maybe a trace memory of this origin of logicism came to him when he introduced the reprinting of the book many years later: 'I was originally led to emphasis this [implicational] form by the consideration of Geometry' (1937a, vii).

"Later on in Part 6 occurs a similar passage: 'And when it is realized that all mathematical ideas, except thos of Logic, can be defined, it is seen that there are no primitive propostions in mathematics except those of Logic' (1903a, 430). . . .

"The details of the logicisitic vision were still not clear, but Russell made his first puiblic statement of it in a popular essay 'On recent work on the principles of mathematics' . . .

[There's more good stuff here]

Grattan-Guinness 2000:542:

"In the following year Russell reprinted with Allen and Unwin two books which Cambridge University Press had published around the start of the century: the study of Leibniz, and The principles. To each reprint he added a new introduction, and the one for the latter volume revealed that his grasp on the philosophy of logicism was weaker than that on its technicaliaties. He began by stting that the book showed that 'mathematics and logic were identical' (1937a, v), whereas the inclusion thesis had clearly been argued (§6.5.1). Discussing in detail his implicational definition of logicism, he recorded its origins in geometry (p. vii)."

Russellian monism, Russellian logical monism

TBD. Russell is considered to be a "neutral monist".

Russell in his own words

Russell 1903 Principles of Mathematics, Introduction

Bertrand Russell, 1903, The Principles of Mathematics Vol. I, Cambridge: at the University Press, Cambridge, UK The boldface and boldface italics are mine!

page vi: PREFACE. THE present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II.-VII. of this Volume, and will be established by strict symbolic reasoning in Volume II.

The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken, in this volume, to defend its various parts, as occasion arose, against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established. The other object of this work, which occupies Part I, is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task, and I cannot flatter myself that I have done more than indicate a vast field of inquiry, and give a sample of the methods by which the inquiry may be conducted. The discussion of indefinables-which forms the chief part of philosophical logic-is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them; there is a process analogous to that which resulted in the discovery of Neptune, with the difference that the final stage-the search with a mental telescope for the entity which has been inferred-is often the most difficult part of the undertaking. In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions

vi Preface

requisite for the notion of class. And the contradiction discussed in Chapter x. proves that something is amiss, but what this is I have hitherto failed to discover.

[As a result he adds an Appendix where he proposes his type theory, this Appendix added at the laat second].

. . .

Professor Frege's work, which largely anticipates my own, was for the most part unknown to me when the printing of the present work began; I had seen his Gruwigeaetze der Arithmetik, but, owing to the great difficulty of his symbolism, I had failed to grasp its importance or to understand its contents. The only method, at so late a stage, of doing justice to his work, was to devote an Appendix to it; and in some points the views contained in the Appendix differ from those in Chapter VI., especially in §§ 71, 78, 74. On questions discussed in these sections, I discovered errors after passing the sheet.'4 for the press; these errors, of which the chief are the denial of the null-class, and the identification of a term with the class whose only member it is, are rectified in the Appendices. The subjects treated are so difficult that I feel little confidence in my present opinions, and regard any conclusions which may be advocated as essentially hypotheses.

A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the philosophy of Dynamics. I was met by the difficulty that, when a particle is subject to several forces,

Preface vii

no one of the component accelerations actually occurs, but only the resultant acceleration, of which they are not parts; this fact 1 rendered illusory such causation of particulars by particulars as is I affirmed, at first sight, by the law of gravitation. It appeared also that the difficulty in regard to absolute motion is insoluble on a relational theory of space. From these two questions I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and then, with a view to discovering the meaning of the word any, to Symbolic Logic. The final outcome, as regards the philosophy of Dynamics, is perhaps rather slender; the reason of this is, that almost all the problems of Dynamics appear to me empirical, and therefore outside the scope of such a work as the present. Many very interesting questions have had to be omitted, especially in Parts VI. and VII., as not relevant to my purpose, which, for fear of misunderstandings, it may be well to explain at this stage.

When actual objects are counted, or when Geometry and Dynamics are applied to actual space or actual matter, or when, in any other way, mathematical reasoning is applied to what exists, the reasoning employed has a form not dependent upon the objects to which it is applied being just those objects that they are, but only upon their having certain general properties. In pure mathematics, actual objects in the world of existence will never be in question, but only hypothetical objects having those general properties upon which depends whatever deduction is being considered; and these general properties will always be expressible in terms of the fundamental concepts which I have called logical constants. Thus when space or motion is spoken of in pure mathematics, it is not actual space or actual motion, as we know them in experience, that are spoken of, but any entity possessing those abstract general properties of space or motion that are employed in the reasonings of geometry or dynamics. The question whether these properties belong, as a matter of fact, to actual space or actual motion, is irrelevant to pure mathematics, and therefore to the present work, being, in my opinion, a purely empirical question, to be investigated in the laboratory or the observatory. Indirectly, it is true, the discussions connected with pure mathematics have a very important bearing upon such empirical questions, since mathematical space and motion are held by many, perhaps most, philosophers to be self-contradictory, and therefore necessarily different from actual space and motion, whereas, if the views advocated in the following pages be valid, no such self-contradictions are to be found in mathematical space and motion. But extra-mathematical considerations of this kind have been almost wholly excluded from the present work.

viii Preface

On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not Ireducible to adjectives of their terms or of the whole which these compose. Before learning these views from him, I found myself completely unable to construct any philosophy of arithmetic, whereas their acceptance brought about an immediate liberation from a large number of difficulties which I believe to be otherwise insuperable. The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. But I must leave it to my readers to judge how far the reasoning assumes these doctrines, and how far it supports them. Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour.

In Mathematics, my chief obligations, as is indeed evident, are to Georg Cantor and Professor Peano. If I had become acquainted sooner with the work of Professor Frege, I should have owed a great deal to him, but as it is I arrived independently at many results which he had already established. At every stage of my work, I have been assisted more than I can express by the suggestions, the criticisms, and the generous encouragement of Mr A. N. Whitehead; he also has kindly read my proofs, and greatly improved the final expression of a very large number of passages. Many useful hints owe also to Mr W. E. Johnson; and in the more philosophical parts of the book owe much to Mr G. E. Moore besides the general position which underlies the whole.

In the endeavour to cover so wide a field, it has been impossible to acquire an exhaustive knowledge of the literature. There are doubtless many important works with which I am unacquainted; but where the labour of thinking and writitig necessarily absorbs so much time, such ignorance, however regrettable, seems not wholly avoidable. Many words will be found, in the course of discussion, to be defined in senses apparently departing widely from common usage. Such departures, I must ask the reader to believe, are never wanton, but have been made with great reluctance. In philosophical matters, they have been necessitated mainly by two causes. First, it often happens that

Preface ix

two cognate notions are both to be considered, and that language has two names for the one, but none for the other. It is then highly convenient to distinguish between the two names commonly used as synonyms, keeping one for the usual, the other for the hitherto nameless sense. The other cause arises from philosophical disagreement with received views. Where two qualities are commonly supposed inseparably conjoined, but are here regarded as separable, the name which has applied to their combination will usually have to be restricted to one or other. For·example, propositions are commonly regarded as (1) true or false, (2) mental. Holding, as I do, that what is true or false is not in general mental, I require a name for the true or false as such, and this name can scarcely be other than proposition. In such a case, the departure from usage is in no degree arbitrary. As regards mathematical terms, the necessity for establishing the existence-theorem in each case i.e. the proof that there are entities of the kind in question-has led to many definitions which appear widely different from the notions usually attached to the terms in question. Instances of this are the definitions of cardinal, ordinal and complex numbers. In the two former of these, and in many other cases, the definition as a class, derived from the principle of abstraction, is mainly recommended by the fact that it leaves no doubt as to the existence-theorem. But in many instances of such apparent departure from usage, it may be doubted whether more has been done than to give precision to Ii notion which had hitherto been more or less vague.

For publishing a work containing so many unsolved difficulties, my apology is, that investigation revealed no near prospect of adequately resolving the contradiction discussed in Chapter x., or of acquiring a better insight into the nature of classes. The repeated discovery of errors in solutions which for a time had satisfied me caused these problems to appear such as would have been only concealed by any seemingly satisfactory theories which a slightly longer reflection might have produced ; it seemed better, therefore, merely to state the difficulties, than to wait until I had become persuaded of the truth of some almost certainly erroneous doctrine.

My thanks are due to the Syndics of the University Press, and to their Secretary, Mr R. T. Wright, for their kindness and courtesy in regard to the present volume. LoNDON, December, 1902.

Russell 1903 Principles of Mathematics, Classes

CHAPTER VI. CLASSES. 66. To bring clearly before the mind what is meant by class, and to distinguish this notion from all the notions to which it is allied, is one of the most difficult and important problems of mathematical philosophy. Apart from the fact that class is a very fundamental concept, the utmost care and nicety is required in this subject on account of the contradiction to be discussed in Chapter x. I must ask the reader, therefore, not to regard as idle pedantry the apparatus of somewhat subtle discriminations to be found in what follows.

It has been customary, in works on logic, to distinguish two standpoints, that of extension and that of intension. Philosophers have usually regarded the latter [intension] as more fundamental, while Mathematics has been held to deal specially with the former [extension]. M. Couturat, in his admirable work on Leibniz, states roundly that Symbolic Logic can only be built up from the standpoint of extension; and if there really were only these two points of view, his statement would be justified. But as a matter of fact, there are positions intermediate between pure intension and pure extension, and it is in these intermediate regions that Symbolic Logic has its lair. It is essential that the classes with which we are concerned should be composed of terms, and should not be predicates or concepts, for a class must be definite when its terms are given, but in general there will be many predicates which attach to the given terms and to no others. We cannot of course attempt an intensional definition of a class as the class of predicates attaching to the terms in question and to no others, for this would involve a vicious circle; hence the point of view of extension is to some extent unavoidable. On the other hand, if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential. It is owing to this con-

... La Logique de Leibniz, Paris, 1901, p. 387. _ 66-68 Classes 67

sideration that the theory of denoting is of such great importance. In the present chapter we have to specify the precise degree in which extension and intension respectively enter into the definition and employment of classes; and throughout the discussion, I must ask the reader to remember that whatever is said has to be applicable to infinite as well as to finite classes.

67. When an object is unambiguously denoted by a concept, I shall speak of the concept as a concept (or sometimes, loosely, as the concept) of the object in question. Thus it will be necessary to distinguish the concept of a class from a class-concept. We agreed to call man a class concept, but man does not, in its usual employment, denote anything. On the other hand, men and all men (which I shall regard as synonyms) do denote, and I shall contend that what they denote is the class composed of all men. Thus man is the class-concept, [all] men (the concept) is the concept of the class, and [all] men (the object denoted by the concept men) are the class. It is no doubt confusing, at first, to use class-concept and concept of a class in different senses; but so many distinctions are required that some stmining of language seems unavoidable. In the phraseology of the preceding chapter, we may say that a class is a numerical conjunction of terms. This is the thesis which is to be established.

68. In Chapter II we regarded classes as derived from assertions, i.e. as all the entities satisfying some assertion, whose form was left wholly vague. I shall discuss this view critically in the next chapter; for the present, we may confine ourselves to classes as they are derived from predicates, leaving open the question whether every assertion is equivalent to a predication. We may, then, imagine a kind of genesis of classes, through the successive stages indicated by the typical propositions "Socrates is human," "Socrates has humanity," "Socrates is a man," "Socrates is one among men." Of these propositions, the last only, we should say, explicitly contains the class as a constituent; but every subject-predicate proposition gives rise to the other three equivalent propositions, and thus every predicate (provided it can be sometimes truly predicated) gives rise to a class. This is the genesis of classes from the intensional standpoint.

On the other hand, when mathematicians deal with what they call a manifold, aggregate, Mengr:, ensemble, or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class. Here it is not predicates and denoting that are relevant, but terms connected by the word and, in the sense in which this word stands for a numerical conjunction. Thus Brown and Jones are a class, and Brown singly is a class. This is the extensional genesis of classes.

5-2 68 The Indefinables of Mathematics [CHAP. VI

69. The best formal treatment of classes in existence is that of Peano. But in this treatment a number of distinctions of great philosophical importance are overlooked. Peano, not I think quite consciously, identifies the class with the class-concept; thus the relation of an individual to its class is, for him, expressed by is a. For him, "2 is a number" is a proposition in which a term is said to belong to the class numher. Nevertheless, he identifies the equality of classes, which consists in their having the same terms, with identity--a proceeding which is quite illegitimate when the class is regarded as the class-concept. In order to perceive that man and featherless biped arenot identical, it is quite unnecessary to take a hen and deprive the poor bird of its feathers. Or, to take a less complex instance, it is plain that even prime is not identical with integer next after 1. Thus when we identify the class with the c1ass-concept, we must admit that two classes may be equal without being identical. Nevertheless, it is plain that when two class-conrepts are equal, some identity is involved, for we say that they have the same terms. Thus there is some object which is positively identical when two class-concepts are equal; and this object, it would seem, is more properly called the class. Neglecting the plucked hen, the class of featherless bipeds, every one would say, is the same as the class of men; the class of even primes is the same as the class of integers next after 1. Thus we must not identify the class with the class-concept, or regard "Socrates is a man" as expressing the relation of an individual to a class of which it is a member. This has two consequences (to be established presently) which prevent the philosophical acceptance of certain points in Peano's formalism.

The first consequence is, that there is no such thing as the null-class, though there are null class-concepts. The second is, that a class having only one term is to be identified, contrary to Peano's usage, with that one term. I should not propose, however, to alter his practice or his notation in consequence of either of these points; rather I should regard them as proofs that Symbolic Logic ought to concern itself, as far as notation goes, with class-concepts rather than with classes.

70. A class, we have seen, is neither a predicate nor a class concept, for different predicates and different class-concepts may correspond to the same class. A class also, in one sense at least, is distinct from the whole composed of its terms, for the latter is only and essentially one, while the former, where it has many terms, is, as we shall see later, the very kind of object of which many is to be asserted. The distinction of a class as many from a class as a whole is often made by language: space and points, time and instants, the army and the soldiers, the navy and the sailors, the Cabinet and the Cabinet Ministers, all illustrate the distinction. The notion of a whole, in the sense of aggregate

11- Neglecting Frege, who is discussed in the AppendiX. , 69-'11] Classes 69

which is here relevant, is, we shall find, not always applicable where the notion of the class as many applies (see Chapter x). In such cases, though terms may be said to belong to the class, the class must not be treated as itself a single logical subject-. But this case never arises where a class can be generated by a predicate. Thus we may for the present dismiss this complication from our minds. In a class as many, the component terms, though they have some kind of unity, have less than is required for a whole. They have, in fact, just so much unity as is required to make them many, and not enough to prevent them from remaining many. A further reason for distinguishing wholes from classes as many is that a class as one may be one of the terms of itself as many, as in "classes are one among classes" (the extensional equivalent of "class is a class-concept "), whereas a complex whole can never be one of its own constituents.

71. Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts.

I believe this distinction to be purely psychological: logically; the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal. Logically, therefore, extension and intension seem to be on a par. I will begin with the extensional view.

When a class is regarded as defined by the enumeration of its terms, it is more naturally called a collection. I shall for the moment adopt this name, as it will not prejudge the question whether the objects denoted by it are truly classes or not. By a collection I -mean what is conveyed by ".A. and B" or ".A. and Band C," or any other enumeration of definite terms. The collection is defined by the actual mention of the terms, and the terms are connected by and. It would seem that and represents a fundamental way of combining terms, and that just this way of combination is essential if anything is to result of which a number other than 1 can be asserted. Collections do not presuppose numbers, since they result simply from the terms together with and: they could only presuppose numbers in the particular case where the terms of the collection themselves presupposed numbers. There is a grammatical difficulty which, since no method exists of avoiding it, must be pointed out and allowed for. A collection, grammatically, is

• A plurality of terms is not the logical subject when a number is asserted of it: such propositions have not one subject, but many subjects. See end of § 74.

10 The Indefinables qf Mathematics [CHAPo VI

asingular, whereas A and B, A and B and C, etc. are essentially plural. This grammatical difficulty arises from the logical fact (to be discussed presently) that whatever is many in general forms a whole which is one; it is, therefore, not removable by a better choice of technical terms.

The notion of and was brought into prominence by Bolzano. In order to understand what infinity is, he says, we must go back to one of the simplest conceptions of our understanding, in order to reach an agreement concerning the word that we are to use to denote it. This is the conception which underlies the conjunction and, which, however, if it is to stand out as clearly as is required, in many cases, both by the purposes of mathematics and by those of philosophy, I believe to be best expressed by the words: 'A system (Inhegrijf) of certain things; or 'a whole consisting of certain parts.' But we must add that every arbitrary object A can be combined in a system with any others B, C, D, or (speaking still more correctly) already forms a system by itself [dagger], of which some more or less important truth can be enunciated, provided only that each of the presentations A, B, C, D, ... in fact represents a different object, or in so far as none of the propositions 'A is the same as B,' 'A is the same as C' 'A is the same as D' etc. is true. For if e.g. A is the same as B, then it is certainly unreasonable to speak of a system of the things A and B." .

The above passage, good as it is, neglects several distinctions which we have found necessary. First and foremost, it does not distinguish the many from the whole which they form. Secondly, it does not appear to observe that the method of enumeration is not practically applicable to infinite systems. Thirdly, and this is connected with the second point, it does not make any mention of intensional definition nor of the notion of a class. What we have to consider is the difference, if any, of a class from a collection on the one hand, and from the whole formed of the collection on the other.

But let us first examine further the notion of and. Anything of which a finite number other than 0 or 1 can be asserted would be commonly said to be many, and many, it might be said, are always of the form "A and B and C and ...." Here A, B, C, ... are each one and are all different. To say that A is one seems to amount to much the same as to say that A is not of the form "A1 and A2 and A3 and ...." To say that A, B, C, ... are all different seems to amount only to a condition as regards the symbols: it should be held that "A and A" is meaningless, so that diversity is implied by and, and need not be specially stated. A term A which is one may be regarded as a particular case of a

.,. Parado.1:ien deB UnendJichen, Leipzig, 1854 (2nd ed., Berlin, 1889), § 3. [dagger] i.e. the combination of A with B, C, D, '" already forms a system.

71J Classes 71

collection, namely as a collection of one term. Thus every collection which is many presupposes many collections which are each one: A and B presupposes A and presupposes B. Conversely some collections of one term presuppose many, namely those which are complex: thus "A differs from B" is one, but presupposes A and difference and B.

But there is not symmetry in this respect, for the ultimate presuppositions of anything are always simple terms. Every pair of terms, without exception, can be combined in the manner indicated by A and B, and if neither A nor B be many, then A and B are two. A and B may be any conceivable entities, any possible objects of thought, they may be points or numbers or true or false propositions or events or people, in short anything that can becounted. A teaspoon and the number 3, or a chimaera and a four dimensional space, are certainly two. Thus no restriction whatever is to be placed on A and B, except that neither is to be many. It should be observed that A and B need not exist, but must, like anything that can be mentioned, have Being. The distinction of Being and existence is important, and is well illustrated by the process of counting. What can be counted must be something, and must certainly be, though it need by no means be possessed of the further privilege of existence. Thus what we demand of the terms of our collection is merely that each should be an entity.

The question may now be asked: What is meant by A and B? Does this mean anything more than the juxtaposition of A with B? That is, does it contain any element over and above that of A and that of B? Is and a separate concept, which occurs besides A, B? To either answer there are objections. In the first place, and, we might suppose, cannot be a new concept, for if it were, it would have to be some kind of relation between A and B; A and B would then be a proposition, or at least a propositional concept, and would be one, not two. Moreover, if there are two concepts, there are two, and no third mediating concept seems necessary to make them two. Thus and would seem meaningless. But it is difficult to maintain this theory. To begin with, it seems rash to hold that any word is meaningless. When we use the word and, we do not seem to be uttering mere idle breath, but some idea seems to correspond to the word. Again some kind of combination seems to be implied by the fact that A and B are two, which is not true of either separately. When we say "A and B are yellow," we can replace the proposition by "A is yellow" and "B is yellow"; but this cannot be done for "A and B are two"; on the contrary, A is one and B is one.

Thus it seems best to regard and as expressing a definite unique kind of combination, not a relation, and not combining A and B into a whole, which would be one. This unique kind of combination will in future be called addition of individuals. It is important to observe that it applies to terms, and only applies to numbers in consequence of their being

72 The Indefinables qf Mathematics [CHAP. VI l• tenns. Thus for the pressent, 1 and 2 are two. and 1 and 1 is meaningless.

As regards what is meant by the combination indicated by and, it is indistinguishable from what we before called a numerical conjunction. That is, A and B is what is denoted by the concept of a class of which A and B are the only members. If u be a class-concept of which the propositions "A is a u" "B is a u" are true, but of which all other propositions of the same fonn are false, then " all u's" is the concept of a class whose only terms are A and B; this concept denotes the terms A, B combined in a certain way, and "A and B" are those tenns combined in just that way. Thus "A and B" are the class, but are distinct from the class-concept and from the concept of the class.

The notion of and, however, does not enter into the meaning of a class, for a single tenn is a class, although it is not a numerical conjunction. If u be a class-concept, and only one proposition of the form "x be a u" be true, then "all u's" is a concept denoting a single term, and this tenrm is the class of which "all u's" is a concept. Thus what seems essential to a class is not the notion of and, but the being denoted by some concept of a class. This brings us to the intensional view of classes.

72. We agreed in the preceding chapter that there are not different ways of denoting, but only different kinds of denoting concept'! and correspondi:ngly different kinds of denoted objects. We have discussed the kind of denoted object which constitutes a class; we have now to consider the kind of denoting concept. The consideration of classes which results from denoting concepts is more general than the extensional consideration, and that in two respects. In the first place it allow!" what the other practU:ally excludes, the admission of infinite classes; in the second place it introduces the null concept of a class. But, before discussing these matters, there is a purely logical point of some importance to be examined. If u be a class-concept, is the concept" all u's" analyzable into two constituents, all and u, or is it a new concept, defined by a certain relation to u, and no more complex than u itself? 'Ve may obsene, to begin with, that" all u's" is synonymous with" u's," at least according to a very common use of the plural. Our question is, then, as to the meaning of the plural. The word all has certainly some definite meaning, but it seems highly doubtful whether it means more than the indication of a relation. "All men" and "all numbers" have in common the fact that they both have a certain relation to a classconcept, namely to man and number respectively. But it is very difficult to isolate any further element of all-'Tle88 which both share, unless we take as this element the mere fact that both are concepts of classes. It would seem, then, that" all u's" is not validly analyzable into 011 71-73J Classes 73 and u, and that language, in this case 8.'l in some others, is a misleading guide. The same remark will apply to every, any, aome, a, and tke. It might perhaps be thought that a class ought to be considered, not merely 8.'l a numerical conjunction of terms, but 8.'l a numerical conjunction denoted by the concept of a class. This complication, however, would serve no useful purpose, except to preserve Peano's distinction between a single term and the class whose only term it isa distinction which is eMy to gr8.'lP when the class is identified with the class-concept, but which is inadmissible in our view of classes. It is evident that a numerical conjunction considered 8.'l denoted is either the same entity 8.'l when not so considered, or else is a complex of denoting together with the object denoted; and the object denoted is plainly what we mean by a class. With regard to infinite cl8.'lSe8, say the class of numbers, it is to be observed that the (:oncept all numbers, though not itself infinitely complex, yet denotes an infinitely. complex object. This is the inmost secret of our power to deal with infinity. An infinitely complex Iconcept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections, owing to the notion of denoting, call be manipulated without introducing any concepts of infinite complexity. Throughout the discussions of infin.~ty in later Parts of "the present work, this remark should be borne in mind: if it is forgotten, there is an air of magic which causes the .results obtained to seem doubtful. 73. Great difficulties are associated with the null-class, and generally with the idea of 1UJthing. It is plain that there is such a concept 8.'l nothing, and that in some sense nothing is something. In fact, the proposition "nothing is not nothing" is undoubtedly capable of an interpretation which makes it true-a point which gives rise to the contradictions discussed in Plato's SCYphist. In Symbolic Logic the ~JJJ'fl.s is the class which h8.'l no terms at all; and symbolically it is quite necessary to introouce some such notion. We have to consider whether the contradictions which naturally arise can be avoided. It is necessary to realize, in the first plac.-e, that a concept may denote although it does not denote anything. This occurs when there are propositions in which the said concept occurs, and which are not about the said concept, but all such propositions are false. Or rather, the above is a first step towards the explanation of a denoting concept which denotes nothing. It is not, however, an adequate explanation. Consider, for example, the proposition "chimaeras are animals" or ~'even primes other than 2 are numbers." These propositions appear to be true, and it would seem that they are not concerned with the denoting concepts, but with what these concepts denote; yet that is impossible, for the (:oncepts in question do not denote anything. The lndefinables of Mathematics [CHAP, VI Symbolic Logic says that these concepts denote the null-class, and that the propositions in question assert that the null-class is contained in certain other classes. But with the strictly extensional view of classes propounded above, a class which has no terms fails to be anything at all: what is merely and solely a collection of terms cannot subsist when all the terms are removed. Thus we must either find a different interpretation of classes, or else find a method of dispensing with the null-class. The above imperfect definition of a (:oncept which denotes, but does not denote anything, may be amended as follows. All denoting concepts, as we saw, are derived from class-concepts; and a is a classconcept when "x is an a" is a propositional function. The denoting concepts associated with a will not denote anything when and only when "x is an a" is false for all values of x. This is a complete definition of a denoting concept which does not denote anything; and in this case we shall say that a is a null class-concept, and that" all a's" is a null concept of a class. Thus for a system such as Peano's, in which what are called classes are really cla.'!S-concepts, technical difficulties need not arise; but for us a genuine logical problem remains. The proposition "chimaeras are animals" may be easily interpreted by means of formal implication, as meaning "x is a chimaera implies x is an animal for all values of x." But in dealing with "classes we have been assqming that propositions containing all or any or every, though equivalent to formal implications, were yet distinct from them, and involved ideas requiring independent treatment. Now in the case of chimaeras, it is easy to substitute the pure intensional view, according to which what is really stated is a relation of predicates: in the case in question the adjective animal is part of the definition of the adjective chimerical (if we allow ounrelves to use this word, contrary to usage, to denote the defining predicate of chimaeras). But here again it is fairly plain that we are dealing with a proposition which implies that chimaeras are animals, but is not the same proposition-indeed, in the present case, the implication is not even reciprocal. By a negation we can give a kind of extensional interpretation: nothing is denoted by a chimaera which is not denoted by an animal. But this is a very roundabout interpretation. On the whole, it seems most correct to reject the proposition altogether, while retaining the various other propositions that would be equivalent to it if there were chima.era.~. By symbolic logicians, who have experienced the utility of the nullclass, this will be felt as a reactionary view. But I am not at present discussing what should be done in the logical calculus, where the established practice appears to me the best, but what is the philosophical truth concerning the null-class. ·We shall say, then, that, of the bundle of normally equivalent interpretations of logical symbolic formulae, the class of interpretations considered in the present chapter, .. ~, . 73J Classes 75 which are dependent upon actual classes, fail where we are concerned with null class-concepts, on the ground that there is no actual null-class. Vie may now reconsider the proposition "nothing is not nothing"'a proposition plainly true, and yet, unless carefully handled, a source of apparently hopeless antinomies. Nothing is a denoting concept, which denotes nothing. The concept which denotes is of course not nothing, i.e. it is not denoted by itself. The proposition which looks so paradoxical means no more than this: Nothing, the denoting concept, is not nothing, i.e. is. not what itself denotes. But it by no means follows from this that there is an actual null-class: only the null class-concept and the null concept of a class are to be admitted. But now a new difficulty has to be met. The equality of classconcepts, like all relations which are reflexive, symmetrical, and transitive, indicates an underlying identity, i.e. it indicates that every class-concept has to some term a relation which all equal class-concepts also have to that term-the term in question being different for different sets of equal class-concepts, but the same for the various members of a single set of equal class-concepts. Now for all class-concepts which are not null, this term is found in the corresponding class; but where are we to find it for null class-concepts? To this question several answers may be given, any of which may be adopted. For we now know what a class is, and we may therefore adopt as our term the class of all hull class-concepts or of all null propositional functions. These are not nullclasses, but genuine classes, and to either of them all null class-concepts have the same relation. If we then wish to have an entity analogous to what is elsewhere to be called a class, out corresponding to null class-concepts, we shall be forced, wherever it is necessary (as in counting da.'lSeS) to introduce a term which is identical for equal class-concepts, to substitute everywhere the class of class-concepts equal to a given cla.'lS-concept for the cla."s corresponding to that class-concept. The class corresponding to the class-concept remains logically fundamental, but need not be actually employed in our symbolism. '1lle null-cla..'lS, in fact, is in some ways analogous to an irrational in Arithmetic: it cannot be interpreted on the same principles as other classes, and if we wish to give an analogous interpretation elsewhere, we must substitute for classes other more complicated entities-in the present case, certain correlated classes. The object of such a procedure will be mainly technical; but failure to understand the procedure will lead to inextricable difficulties in the interpretation of the symbolism. A very closely analogous procedure occurs constantly in Mathematics, for example with every generalization of number; and so far as I know, no single case in which it occurs has been rightly interpreted either by philosophers or by mathematicians. So many instances will meet us in the course of the present work that it is unnecessary to linger longer over the point at present. Only one possible misunderstanding must 76 The Indefinahles qf Mathematics [CHAP. VI be guarded against. No vicious circle is involved in the above account of the null-class; for the geneml notion of cla8a is first laid down, is found to involve what is called existence, is then symbolically, not philosophically, replaced by the notion of a class of equal class-concepts, and is found, in this new form, to be applicable to what corresponds to null class-concepts, since what corresponds is now a class which is not null. Between classes simpliciter and classes of equal class-concepts there is a one-one correlation, which breaks down in the sole case of the class of null class-concepts, to which no null-class corresponds; and this fact is the reason for the whole complimtion. 74. A question which is very fundamental in the philosophy of Arithmetic must now be discussed in a more or less preliminary fashion. Is a class which has many terms to be regarded as itself one or many? Taking the class as equivalent simply to the numerical conjunction "A and B and C and etc.,'" it seems plain that it is many; yet it is quite necessary that we should be able to count classes as one each, and we do habitually speak of a class. Thus classes would seem to be one in one sense and many in another. There is a certain temptation to identify the class as many and the class as one, e.g., all men and the human rru:e. Nevertheless, wherever a class consists of more than one term, it can be proved that no such identification is permissible. A concept of a class, if it denotes a class as one, is not the same as any concept of the class which it denotes. That is to say, claaaea of all rational ani11'U.l18, which denotes the human race as one term, is different from men, which denotes men, i.e. the human race as many. But if the human race were identical with men, it would follow that whatever denotes the one must denote the other, and the above difference woll1d be impossible. We might be tempted to infer that Peano's distinction, between a term and a class of which the said term is the only member, must be maintained, at least when the term in question is a class·: But it is more correct, I think, to infer an ultimate distinction between a class as many and a class as one, to hold that the many are only many, and are not also one. The class 8.'i one may be identified with the whole composed of the terms of the cla.'lS, i.e., in the case of men, the class as"one will be the human race. But can we now avoid the contradiction always to be feared, wh~re there is something that cannot be made a logical subject? I do not myself see any way of eliciting a precise contradiction in this case. In the case of concepts, we were dealing with what was plainly one entity; in the present case, we are dealing with a complex essentiall)' capable of analysis into units. In such a pl'opOliition as "A and B are two,'" there is no logical subject: the assertion is not about A, nor

• This conclusion is actually drawn by Frege from an analogous argument:

ArchivjUr '!lat. Phil. I, p. 444. See Appendix. '13-76] Classes '1'1 about B, nor about the whole composed of both, but strictly and only about A and B. Thus it would seem that assertions are not necessarily ahuut single subjects, but may be about many subjects; and this removes the contradiction which arose, in the case of concepts, from the impossibility of making assertions about them unless they were turned into subjects. This impossibility being here absent, the contradiction which was to be feared does not arise. 75. We may ask, as suggested by the above discussion, what is to be said of the objects denoted by a man, every man, aorTU: man, and any man. Are these objects one or many or neither?