THE FATE OF ANALYSIS, #8–Russell & The Limits of Unlimited Logicism.

By Robert Hanna



II. Classical Analytic Philosophy

II.1 What Classical Analytic Philosophy Is: Two Basic Theses

II.2 What Classical Analytic Philosophy Officially Isn’t: Its Conflicted Anti-Kantianism

II.3 Classical Analytic Philosophy Characterized in Simple, Subtler, and Subtlest Ways

II.4 Three Kinds of Analysis: Decompositional, Transformative, and Conceptual

II.5 Frege, The First Founding Father of Classical Analytic Philosophy

II.6 Frege’s Project of (Transformative or Reductive) Analysis

II.7 Frege’s Dead End

II.8 Frege’s Semantics of Sense and Reference, aka Meaning

II.9 Some Biggish Problems For Frege’s Semantics

II.10 Husserl, Logic, and Logical Psychologism, aka LP

II.11 What LP is, and its Three Cardinal Sins

II.12 Husserl’s Three Basic Arguments Against LP

II.13 Has Husserl Begged the Question Against LP? Enter The Logocentric Predicament, and a Husserlian Way Out

II. Moore, Brentano, Husserl, Judgment, Anti-Idealism, and Meinong’s World

III.1 G.E. Moore, the Second Founding Father of Classical Analytic Philosophy

III.2 Brentano on Phenomenology, Mental Phenomena, and Intentionality

III.3 Husserl on Phenomenology and Intentionality

III.4 Moore and the Nature of Judgment

III.5 Moore and the Refutation of Idealism

III.6 Meinong’s World

IV. Russell, Unlimited Logicism, Acquaintance, and Description

IV.1 Russell Beyond Brentano, Husserl, Moore, and Meinong

IV.2 Russell and Mathematical Logic versus Kant

IV.3 Russell’s Unlimited Logicist Project

IV.4 Pursued by Logical Furies: Russell’s Paradox Again

IV.5 Russell’s ‘Fido’-Fido Theory of Meaning

IV.6 Knowledge-by-Acquaintance and Knowledge-by-Description

IV.7 Russell’s Theory of Descriptions

IV.8 Russell’s Multiple-Relation Theory of Judgment

IV.9 Russellian Analysis, Early Wittgenstein, and Impredicativity Again

IV.10 Russell and The Philosophy of Logical Atomism

V. Wittgenstein and the Tractatus 1: The Title, and Propositions 1–2.063

V.1 A Brief Synopsis of the Tractatus

V.2 The Tractatus in Context

V.3 The Basic Structure of the Tractatus: A Simple Picture

V.4 Tractarian Ontology

V.5 Reconstructing Wittgenstein’s Reasoning

V.6 What Are the Objects or Things?

V.7 The Role of Logic in Tractarian Ontology

V.8 Colorless Objects/Things

V.9 Tractarian Ontology, Necessity, and Contingency

V.10 Some Initial Worries, and Some Possible Wittgensteinian Counter-Moves

VI. Wittgenstein and the Tractatus 2: Propositions 2.013–5.55

VI.1 What is Logical Space? What is Real Space?

VI.2 Atomic Facts Necessarily Are in Manifest or Phenomenal Space, But Objects or Things Themselves Necessarily Aren’t in Manifest or Phenomenal Space

VI.3 Logical Space is Essentially More Comprehensive than Manifest or Phenomenal Space

VI.4 Why There Can’t/Kant Be a Non-Logical World

VI.5 A Worry About Wittgenstein’s Conception of Logic: Non-Classical Logics

VI.6 What is a Tractarian Proposition?

VI.7 Naming Objects or Things, and Picturing Atomic Facts

VI.8 Signs, Symbols, Sense, Truth, and Judgment

VI.9 Propositions Again

VI.10 Language and Thought

VII. Wittgenstein and the Tractatus 3: Propositions 4–5.61

VII.1 The Logocentric Predicament, Version 3.0: Justifying Deduction

VII.2 The Logical Form of Deduction

VII.3 Logic Must Take Care of Itself

VII.4 Tautologies and Contradictions

VII.5 What is Logic?

VII.6 Logic is the A Priori Essence of Language

VII.7 Logic is the A Priori Essence of Thought

VII.8 Logic is the A Priori Essence of the World

VIII. Wittgenstein and the Tractatus 4: Propositions 5.62–7

VIII.1 Tractarian Solipsism and Tractarian Realism

VIII.2 Tractarian Solipsism

VIII.3 Tractarian Realism

VIII.4 Is the Tractatus’s Point an Ethical One?

VIII.5 The Meaning of Life

VIII.6 Three Basic Worries About the Tractatus

VIII.7 Natural Science and the Worry About the Simplicity of the Objects or Things

VIII.8 Natural Science and the Worry About the Logical Independence of Atomic Facts

VIII.9 Tractarian Mysticism and the Worry About Metaphilosophy

IX. Carnap, The Vienna Circle, Logical Empiricism, and The Great Divide

IX.1 Carnap Before and After the Tractatus

IX.2 Carnap, The Vienna Circle, and The Elimination of Metaphysics

IX.3 The Verifiability Principle and Its Fate

IX.4 The Davos Conference and The Great Divide

X. Wittgenstein and the Investigations 1: Preface, and §§1–27

X.1 From the Tractatus to the Investigations

X.2 The Thesis That Meaning Is Use

X.3 A Map of the Investigations

X.4 The Critique of Pure Reference: What the Builders Did

XI. Wittgenstein and the Investigations 2: §§28–242

XI.1 The Picture Theory and the Vices of Simplicity

XI.2 Wittgenstein’s Argument Against The Picture Theory: A Rational Reconstruction

XI.3 Understanding and Rule-Following

XI.4 Wittgenstein’s Rule-Following Paradox: The Basic Rationale

XI.5 Wittgenstein’s Rule-Following Paradox: A Rational Reconstruction

XI.6 Kripkenstein’s Rule-Following Paradox: Why Read Kripke Too?

XI.7 Kripkenstein’s Rule-Following Paradox: A Rational Reconstruction

XI.8 How to Solve The Paradox: Wittgenstein’s Way and Kripkenstein’s Way

XI.8.1 Wittgenstein and The Rule-Following Paradox: A Rational Reconstruction

XI.8.2 Kripkenstein and The Rule-Following Paradox: A Rational Reconstruction

XII. Wittgenstein and the Investigations 3: §§242–315

XII.1 What is a Private Language?

XII.2 The Private Language Argument: A Rational Reconstruction

XII.3 Is Wittgenstein a Behaviorist? No.

XII.4 Wittgenstein on Meanings, Sensations, and Human Mindedness: A Rational Reconstruction

XIII. Wittgenstein and the Investigations 4: §§316–693 & 174e-232e

XIII.1 Linguistic Phenomenology

XIII.2 Two Kinds of Seeing

XIII.3 Experiencing the Meaning of a Word

XIII.4 The Critique of Logical Analysis, and Logic-As-Grammar

XIV. Coda: Wittgenstein and Kantianism

XIV.1 World-Conformity 1: Kant, Transcendental Idealism, and Empirical Realism

XIV.2 World-Conformity 2: Wittgenstein, Transcendental Solipsism, and Pure Realism

XIV.3 World-Conformity 3: To Forms of Life

XIV.4 The Critique of Self-Alienated Philosophy 1: Kant’s Critical Metaphilosophy

XIV.5 The Critique of Self-Alienated Philosophy 2: Wittgensteinian Analysis as Critique

XV. From Quine to Kripke and Analytic Metaphysics: The Adventures of the Analytic-Synthetic Distinction

XV.1 Two Urban Legends of Post-Empiricism

XV.2 A Very Brief History of The Analytic-Synthetic Distinction

XV.3 Why the Analytic-Synthetic Distinction Really Matters

XV.4 Quine’s Critique of the Analytic-Synthetic Distinction, and a Meta-Critique

XV.5 Three Dogmas of Post-Quineanism

XVI. Analytic Philosophy and The Ash-Heap of History

XVI.1 Husserl’s Crisis and Our Crisis

XVI.2 Why Hasn’t Post-Classical Analytic Philosophy Produced Any Important Ideas in the Last Thirty-Five Years?

XVI.3 On Irad Kimhi’s Thinking and Being, Or, It’s The End Of Analytic Philosophy As We Know It (And I Feel Fine)

XVI.4 Thinking Inside and Outside the Fly-Bottle: The New Poverty of Philosophy and Its Second Copernican Revolution


But you can also read or download a .pdf version of the complete book HERE.


IV. Russell, Unlimited Logicism, Acquaintance, and Description

Bertrand Russell (1872–1970)

IV.1 Russell Beyond Brentano, Husserl, Moore, and Meinong

In such theories [as Meinong’s], it seems to me, there is a failure of that feeling for reality which ought to be preserved even in the most abstract studies. Logic, I should maintain, must no more admit a unicorn than zoology can; for logic is concerned with the real world just as truly as zoology, though with its most abstract and general features.[i]

Russell, like Moore, began his philosophical career, first, as a neo-Hegelian, and then, second, as a psychologistic neo-Kantian, in an early treatise on the nature of geometry, An Essay on the Foundations of Geometry (1897), which was also based, like Moore’s early essays, on his Trinity fellowship dissertation.

The basic point of the Essay was to determine what could be preserved of Kant‘s Euclid-oriented theories of space and geometry after the discovery and development of non-Euclidean geometries.

Again like Moore, Russell had been supervised by Ward, but also and above all by Whitehead, who discovered and nurtured Russell’s logical and mathematical brilliance.

At the same time there was also a significant neo-Hegelian element in Russell‘s early thought, inspired by his close study of Bradley‘s Logic and discussions with another Trinity man and fellow Apostle, the imposingly-named Scottish neo-Hegelian metaphysician John McTaggart Ellis McTaggart.[ii]

Despite being a close friend of Russell, Moore wrote a sternly critical review of the Essay that was comparable in its both its philosophical content and its impact on Russell to Frege’s review of Husserl‘s Philosophy of Arithmetic, in that it accused Russell of committing the “Kantian fallacy” of grounding a priori modal claims on psychological facts.[iii]

Moore‘s “friendly” criticism seems to have almost instantly liberated Russell from his neo-Kantian and neo-Hegelian beliefs.

At the same time, however, Russell retained a serious philosophical interest in Husserl’s early phenomenology in his 1900 Logical Investigations, construed as a robustly anti-psychologistic and realistic doctrine.

Indeed, Russell took a copy of the second (1913) edition of Logical Investigations with him to prison in 1918 for the purposes of re-reading and reviewing it, and more generally, as Andreas Vrahimis puts it,

[Russell] seems to have thought highly of the book, which he would later praise as being “a monumental work” which he sees as part of “a revolt against German idealism … from a severely technical standpoint” and which he places alongside the work of Frege, Moore, and himself.[iv]

His encounter with early Husserl’s anti-psychologism and phenomenological realism , combined with the close, critical study of Meinong’s writings, led Russell to a radically realistic Moorean see-through epistemology, and to a correspondingly rich looking-glass ontology of concrete and abstract real individuals, although as we have seen he also prudently deployed his logician‘s “feeling for reality” and stopped short of accepting the being or existence, in any sense, of Meinongian impossibilia.[v]

One might well wonder, of course, why accepting the being or existence of Redness, Hotness, Goodness, Negation, Conjunction, and Disjunction is perfectly acceptable for Russell, while postulating unicorns, Hamlet, and round squares is rationally abominable.

By what criterion does the logician’s “feeling for reality” justifiably distinguish between the good Meinongian objects and the bad ones?

We’ll come back to that dangerous thought again shortly.

IV.2 Russell and Mathematical Logic versus Kant

By 1903 Russell had produced the fairly massive Principles of Mathematics, and then by 1910, in collaboration with Whitehead, the even more massive Principia Mathematica.

Above all however, on the collective basis of his intellectual encounters with Kant, Bradley, Boole, Frege, Peano, Whitehead, Moore, and Meinong, Russell developed a fundamental conception of mathematical logic (sometimes also called “pure logic” or “symbolic logic”).

Mathematical logic, as Russell understood it, is the non-psychological, universal, necessary, and a priori science of deductive consequence, expressed in a bivalent propositional and polyadic predicate calculus with identity as well as quantification over an infinity of individuals, properties, and various kinds of functions.

Mathematical logic in this heavy-duty sense has some correspondingly heavy-duty metaphysical implications.

But most importantly for our purposes here, Russell‘s logic expresses the direct avoidance of Kant‘s appeal to intuition in the constitution of mathematical propositions and reasoning:

[T]he Kantian view . . . asserted that mathematical reasoning is not strictly formal, but always uses intuitions, i.e. the à priori knowledge of space and time. Thanks to the progress of Symbolic Logic, especially as treated by Professor Peano, this part of the Kantian philosophy is now capable of a final and irrevocable refutation.[vii]

The result of all these influences, together with Russell‘s manic creative intellectual drive in the period from 1900 to 1913 (when he had a head-on collision with a Wittgensteinian juggernaut — of which, more below), was a refinement of Moore’s conception of philosophical analysis, now based on a radically platonistic, atomistic, and above all logicistic realism, according to which

(i) not merely arithmetic, but literally all of pure mathematics, including algebra, geometry, etc., etc., explanatorily and ontologically reduces to mathematical logic, which is the full-strength thesis of unlimited logicism (for more details, see directly below),

(ii) propositions literally contain both the simple concrete particulars (instantaneous sense-data) and also the simple abstract universals (properties or relations) that populate the mind-independently real world,[viii]

(iii) propositions also literally contain the logical constants (Negation, Conjunction, Disjunction, etc.) that express the purely logical form of propositions, and

(iv) not only the simple concrete particulars and the simple abstract universals, but also the logical constants are known directly and individually by cognitive acts of self-evident and infallible acquaintance.[ix]

IV.3 Russell’s Unlimited Logicist Project

Unlike Frege, Russell didn’t think of logic as the science of the most general laws of truth but instead as the maximally general science of deductive consequence, or of drawing conclusions from premises in a necessarily truth-preserving way.

Russell wasn’t as concerned as Frege with finding true premises, much less finding necessarily true premises: it’s the idea of necessary truth-preservation with respect to any subject-matter that matters for him, so the basic premises might in principle turn out to be false or purely suppositional.

The crucial point is that Russell’s mathematical logic, unlike other sciences, does not contain a body of fundamental general truths or axioms.

On the contrary, for mathematical logic, any axioms you postulate don‘t really matter, and the theorems derived from those axioms also don‘t really matter: what matters are the essentially logical logical relations between the axioms you happen to choose and the theorems you deductively derive.

Or in other words, for Russell mathematical logic doesn’t “say” or “state” anything about the world: rather it only expresses a set of special relations between propositions that, in turn, “say” or “state” things about the world.

IV.4 Pursued by Logical Furies: Russell’s Paradox Again

Russell, like Frege, had hoped to reduce numbers to classes or sets of equinumerous sets; but he was bedevilled by the discovery of his paradox of classes or sets, which persistently pursued his work in “mathematical philosophy” like a swarm of avenging logical Furies.

Sets are uniquely determined by their membership, and the membership of a set is specified by a conceptual description of what will count as a member of that set.

Some sets aren’t members of themselves: e.g., the set of all dogs isn’t a dog, so it isn’t a member of itself.

Other sets, by contrast, are members of themselves: e.g., the set of all non-dogs is a non-dog, so it is a member of itself.

This distinction then allows the formation of a set K whose membership is specified by the conceptual description that it contains all and only those sets that are not members of themselves.

Russell‘s paradox then follows immediately if one asks about the logical status of K: is K part of its own membership, or not?

Obviously, K is a member of itself if and only if K is not member of itself: paradox!

This in turn exemplifies a general point about paradoxes, namely that they express propositions that are not merely contradictory (both true and false) but in fact hyper-contradictory (true if and only if false).

Having discovered the set-theoretic paradox, and promptly having informed Frege about it, to the latter’s tragic dismay, Russell then adopted a series of strategies to try to avoid it, including

(i) the proto-eliminativist so-called no-class theory: there are no real classes or sets but instead only propositional functions, i.e., mappings from objects to propositions,[x]

(ii) the proto-stipulationist vicious circle principle: no totality of things can be constructed such that it increases its size only by including itself, aka “impredicativity,”[xi] and

(iii) the proto-structuralist theory of types: there’s a hierarchy of propositional functions such that the collections of objects formed by a given propositional function always occurs one level lower than that function and therefore cannot belong to that collection.[xi]

The problem with (i) is that it hand-waves the problem away and begs the question at issue, by merely banishing the problematic entities, classes or sets, into the realm of philosophical bad dreams.

The problem with (ii) is that it engages in logical overkill and also begs the question at issue, by merely banishing all logical constructions of the same general form (i.e., impredicativity) as the specific ones that lead to paradox, into the same realm of philosophical bad dreams.

And the problem with (iii) is that it’s formally possible to construct an exact analogue of the paradox of classes or sets by using Russellian propositions.[xiii]

So, sadly, Russell never actually solved the set-theoretic paradox.

Russell was also much exercised by a similar, yet also interestingly different, paradox in the foundations of the theory of meaning: reflexivity or self-reference can also lead to paradox whenever there is a sentence that says of itself that it is false, e.g. —

The only boldface italicized sentence in this blog post is false.

Let’s call this sentence ‘BERTIE’.

What is BERTIE’s logical status?

Or more precisely, is BERTIE true or false?

If BERTIE is true, then BERTIE is false; but if BERTIE is false then BERTIE is true; so BERTIE is a sentence that is true if and only if it is false: so, in short, BERTIE is a hyper-contradiction or paradox, although a logical Fury of a slightly different color than Russell’s set-theoretic paradox and others formally like it.

This differently-colored logical Fury and others formally like it are usually called semantic paradoxes.

Russell’s informal solution to the semantic paradoxes — mentioned, e.g., in Russell’s Introduction to the first English translation of Wittgenstein’s Tractatus in 1922 — is to postulate a hierarchy of languages, each of which contains the “semantic predicates” (e.g., ‘is true’, ‘is false’, ‘refers to’, etc.) of the language directly beneath it in the hierarchy, so that a sentence can never literally say of itself that it is true or false.

This is known as the meta–linguistic solution to the semantic paradoxes, because a language L2 that refers to another language L1 is said to be a “meta-language” of L1.

This basic strategy for avoiding the semantic paradoxes was later formally and famously developed by the brilliant Polish logician, mathematician, semanticist, and all-around-big-shot classical Analytic philosopher Alfred Tarski.[xiv]


[ii] See, e.g., Hylton, Russell, Idealism, and the Emergence of Analytic Philosophy (Oxford: Clarendon/Oxford Univ. Press, 1990), part I; and N. Griffin, Russell’s Idealist Apprenticeship (Oxford: Clarendon/Oxford Univ. Press, 1991).

[iii] G.E. Moore, “Review of B.A.W Russell, Essay on the Foundations of Geometry,” Mind 8 (1899): 397–405.

[iv] A. Vrahimis, “Legacies of German Idealism: From The Great War to the Analytic/Continental Divide,” Parrhesia 24 (2015): 83–106, at p. 94.

[v] See, e.g., Russell, Introduction to Mathematical Philosophy, pp. 169–170.

[vi] See, e.g., R. Monk, Bertrand Russell: The Spirit of Solitude (London: Jonathan Cape, 1996), part I.

[vii] B. Russell, Principles of Mathematics (2nd edn., New York: W.W. Norton, 1996), p. 4.

[viii] See, e.g., B. Russell, “Knowledge by Acquaintance and Knowledge by Description,” in B. Russell, Mysticism and Logic (Totowa, NJ: Barnes and Noble, 1981), 152–167.

[ix] See, e.g., Russell, The Problems of Philosophy (Indianapolis, IN: Hackett, 1995), ch. V.

[x] See Russell, Principles of Mathematics, pp. 79–80.

[xi] See B. Russell, “Mathematical Logic as Based on the Theory of Types,” in Russell, Logic and Knowledge, pp. 59–102; and A.N. Whitehead and B. Russell, Principia Mathematica to *56 (Cambridge: Cambridge Univ. Press, 1962), pp. 37–65.

[xii] Ibid.

[xiii] See M. Potter, Reason’s Nearest Kin: Philosophies of Arithmetic from Kant to Carnap (Oxford: Clarendon/Oxford Univ. Press, 2000), ch. 5, esp. section 5.5.

[xiv] See A. Tarski, “The Semantic Conception of Truth and the Foundations of Semantics,” Philosophy and Phenomenological Research, 4 (1943), 342–60; and A. Tarski, “The Concept of Truth in Formalized Languages,” in A. Tarski, Logic, Semantics, and Metamathematics (Oxford: Oxford University Press, 1956), pp. 152–278.


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