Being a Compendium of Philosophical Paradoxes,
Logical Antinomies & Dialectical Tensions
Vol. I · First Edition · MMXXVI
Preface
On the Nature of Contradiction
The Chinese compound 矛盾 — máodùn, "spear-shield" —
derives from a parable preserved in the Han Feizi. A merchant sold both an
impenetrable shield and an irresistible spear; asked what would happen were one to
strike the other, he could not answer.
†
Han Fei, Han Feizi, ch. 36 ("Nan Yi"), c. 233 BCE.
The parable furnishes the modern Mandarin term for contradiction.
This volume gathers contradictions of three kinds: the logical, in which a
proposition entails its own denial; the semantic, in which the
referential machinery of language collapses upon itself; and the dialectical,
in which opposition is the engine of becoming rather than its obstacle.
Chapter I
The Liar & His Sentence
“All Cretans are liars,” said the Cretan. — Epimenides, fr. 1
Consider the simplest of antinomies. A sentence asserts only its own
falsity: This sentence is false. Should we suppose it true, the
content it conveys obliges us to call it false; should we suppose it
false, that very condition is what it asserts, and so we must call it
true.
‡
The earliest formulation is attributed to Eubulides of Miletus
(4th c. BCE), who reportedly devised seven such puzzles.
The bivalent law — tertium non datur — admits no third
option, and yet here neither option may stand.
Alfred Tarski's response, in his 1936 monograph on the concept of truth,
was to forbid the predicate true from applying to sentences in
its own language; truth-of-L is a notion residing only in a meta-language
L′. The hierarchy thus erected is infinite, and the liar is dispelled
only by the prohibition of self-reference.1
The semantic conception of truth implies nothing regarding the
conditions under which a sentence like “snow is white” can
be asserted; it implies only that, whenever we assert or reject this
sentence, we must be ready to assert or reject the correlated
sentence “the sentence ‘snow is white’ is true.”
— A. Tarski, The Concept of Truth in Formalized Languages, §1
Chapter II
Russell & the Set of All Sets
“The barber shaves all those, and only those, who do not shave themselves.” — B. Russell, 1901
Bertrand Russell, while reading Frege's Grundgesetze in the spring
of 1901, formed the set R of all sets that are not members of
themselves.
§
Russell's letter to Frege, 16 June 1902.
Frege's reply: “Your discovery… has rocked the ground on which
I meant to build arithmetic.”
Is R a member of itself? If yes, then by its defining condition
it is not. If not, then it satisfies the condition and therefore is.
The remedy proposed in Principia Mathematica was the theory of
logical types: a set of objects of type n belongs to type
n+1, and any predicate that purports to range over all
types is forbidden as ill-formed.2
The cost was a pyramidal universe; the gain was the avoidance of paradox.
Chapter III
Zeno & the Arrow's Flight
“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” — Aristotle, Physics VI:9
Zeno of Elea, defending his master Parmenides' doctrine of immutable
Being, framed motion as a contradiction. To traverse any distance,
one must first traverse half of it, and before that a quarter, and
before that an eighth, ad infinitum.
¶
The four arguments are preserved chiefly in Aristotle's
Physics VI:9 and Simplicius' commentary.
Yet motion is observed daily. Either the senses lie or the reason
misleads.
The modern resolution invokes the convergent geometric series
½ + ¼ + ⅛ + … = 1. The sum of
infinitely many durations is itself finite. The contradiction
dissolves once one concedes that infinity, properly handled, need
not balloon.3
Chapter IV
Hegel & the Dialectical Engine
“Contradiction is the very moving principle of the world.” — G.W.F. Hegel, Logic §119
For Hegel, contradiction was not a defect to be expelled but the
inner pulse of all becoming. A determinate concept —
thesis — gives rise to its negation —
antithesis — whose tension is preserved and surpassed
in a higher synthesis.
‖
The German Aufhebung bears the trebled sense of
cancelling, preserving, and lifting up.
Being, devoid of all determination, is indistinguishable from
Nothing; and from this very identity arises Becoming.
What for the formal logician is a syntactic catastrophe is, for the
dialectician, the engine by which spirit unfolds itself in history.
Marx, inverting the schema while keeping its motor, would later
ascribe the contradictions to the productive relations of material
life.4
Chapter V
Gödel & the Unprovable Truth
“This statement is not provable in the system S.” — K. Gödel, 1931 (paraphrase)
Kurt Gödel demonstrated that any consistent formal system rich
enough to express elementary arithmetic must contain a sentence
G which asserts of itself that it is not provable within
the system.
❦
K. Gödel, Über formal unentscheidbare Sätze,
1931. The numbering of formulae is the article's chief technical
invention.
If the system proves G, the system is inconsistent; if the
system is consistent, G is true but unprovable.
Self-reference, banished by Tarski and circumscribed by Russell, is
here re-admitted by means of arithmetisation: every formula receives
a unique natural number, and the predicate provable becomes
an arithmetical one. Truth, in arithmetic, exceeds proof.5
Apparatus
Footnotes & Apparatus
Tarski, A. Der Wahrheitsbegriff in den formalisierten Sprachen.Studia Philosophica, vol. 1, 1936, pp. 261–405.
Whitehead, A.N. & Russell, B. Principia Mathematica.
Cambridge: Cambridge University Press, 1910–1913, 3 vols.
Salmon, W. (ed.). Zeno's Paradoxes.
Indianapolis: Hackett, 2001 [1970].
Hegel, G.W.F. Wissenschaft der Logik.
Nürnberg: Schrag, 1812–1816, 2 vols.
Gödel, K. On Formally Undecidable Propositions of Principia
Mathematica and Related Systems. Transl. B. Meltzer.
Edinburgh: Oliver & Boyd, 1962.
Colophon
Set in EB Garamond, after the late-sixteenth-century types of
Claude Garamont, with sidenotes in Lato. Composed for the
screen in the year MMXXVI. The ornament below marks the end of the
volume.