Abstract
Russell inaugurated analytic philosophy new-style. His initial concern was with rationality. He took its display in science to be best and the clearest. He therefore had two initial aims: to render philosophy scientific and to prove that science is certain (since tradition equates reason with provability). This led him to efforts to improve logic. He argued that whereas every class is a subclass of itself, no class is a member of itself. He wanted to prove this. So he defined “normal” a class is normal if it is not a member of itself: x ε N ↔ ~(x ε x) and tried to prove that all classes are normal: N = V. This made him wonder: is the normal class itself normal? As we saw, the answer is, yes-and-no; which is absurd. Now absurdities were never problematic as everybody always pronounced them false. Russell’s paradox, like any paradox (or antinomy or puzzle), is demonstrably true and false. A language that contains a paradox has all its statements proven. Russell managed to block paradoxes but he allowed contradictions as well-formed. This was the birth of the first fully formal language.
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Notes
- 1.
Alam 199 , 135, 155. Landini 2007 describes all this in admirable detail. See also Agassi 2009 . In the 1959 preface of Popper’s magnum opus, he conceded to the analytic school what was Russell’s contribution and rejected the rest as due to misconceptions (alluding to Wittgenstein’s contribution, p. 17). See Glock 2008, 15 and 219. Glock places Popper in the analytic tradition, but while laudably mentioning his protest.
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A string of words is judged a pseudo-sentence in a language if its words are not placed in accord with the rules of the grammar of that language, or else if some of the words in it do not refer but should (TLP, §5.4733).
- 4.
This historical question invites a historical answer. Some writers have offered answers that include streamlined versions of Frege’s work. The streamlining requires of them to overlook the concerns that engaged the historical Frege. This is sad. Here is the place to stress that he worked as a mathematician. Berkeley’s critique of the calculus was conceptual; all effort to answer it were mathematical (Wisdom 1939). Frege and Peano approached the situation differently (Segre 1994) . Kreisel 1978 , 88 reports Wittgenstein’s reference to the “disastrous invasion of mathematics by mathematical logic”. This is strange, considering that Frege was the father of symbolic logic and the very first to perform this “invasion” and considering that Wittgenstein admired his work. This was “an exaggeration”; but since “his public performances which were always tense and often incoherent”, presumably Kreisel found it unsurprising. See also Rodych 2018 .
- 5.
Variance in notation alone need not alter content. This is true by definition. Rewording a problematic differential equation, however, may facilitate its solution. Hence, changing notations may help perform a task. Even the facilitation of reading a formula may help this way, The notations of Oliver Heaviside for both the equations for electric circuits and for electromagnetic equations were every helpful, as was the lovely simplifications that Einstein introduced into tensor calculus.
- 6.
Landini 2012 , Preface. Landini argues that simplifying Frege’s notation has raised many new problems for the reading of his ideas.
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More precisely, T is the set that contains all and only the true sentences in the language in use.
- 8.
TLP, §3.334: “The rules of logical syntax must go without saying, once we know how each sign signifies.” §5.132: “‘Laws of inference’ … are senseless and would be superfluous”. [Note: they would be superfluous; they are not superfluous tout court.] §6.1264: “(… one cannot express the modus ponens by means of a proposition.)” This is a truism: a rule is not a proposition; since early Wittgenstein limited meaning to propositions, his §6.1264 is trivial.
- 9.
Russell was never clear about the deduction theorem. His “Replies to Critics” (Russell 1944, 969) has a repetition of the theorem and a discussion of the value of any conditional statement. In the case of the dependence of our knowledge of the truth of the conditional on our knowledge of the truth of its antecedent or of the falsity of its consequence, Russell says there, the conditional is useless; it is useful in the case in which our knowledge of its truth is prior to such knowledge. He said this in oversight of conditionals that appear as hypotheticals – in the deduction theorem and in science. They are unknown yet most useful. More obvious is the reductio ad absurdum that here Russell overlooked (as inessential?). This also explains his tern “the contradiction” for his paradox (rather than “the proven contradiction”). See also Whitehead and Russell, 1911, I, 62.
- 10.
The criticism of Frege’s theory of denotation in Russell 1905, 484–5, is that it is hit by the paradox of empty terms (“the present king of France is bald”) and of referential opacity (“the king wanted to know who was the author of Waverley ”). Linsky 1967, 72–3 and Hintikka 1981, 170 have complained that Russell had not cleared the paradox of referential opacity, and that it does not involve definite descriptions. The two complaints may neutralize each other. Yet referential opacity remains an obstacle – not for PMia that is extensional, not for other purposes. Students of intensional logic may do better if the specify them.
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Quine 1943, conclusion. This is the program for developing a free logic, one with no existential import.
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Hintikka 1981. (His oversight of Linsky here is baffling.)
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“Use” is a social term. Modern logic and modern methodology are sociological. Traditional (Locke-style ) conceptual analysis is intentionally psychological: classical philosophy was individualist and demanded the reduction of sociology to psychology. Though Wittgenstein’s “use” is clearly sociological rather than psychological, many members of the analytic school prefer psychology to sociology.
- 15.
Popper 1959 , Preface to the English Translation is devoted to a thorough criticism of this idea.
- 16.
Moore noticed (Schilpp 1944, 214) that that “the Whale” differs from “Man” and “Horse” in its possession of a definite article. Russell dismissed this (op. cit., 690) as a minor irregularity.
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Limiting this discussion for unique entities leaves open the question, is it necessary to assume that a class is non-empty before naming it? This is a quaint variant of the problem of induction. For, the assumption that atoms exist is a corollary to the statement that atomic theory that describes them is a true explanation.
- 20.
Some comments seem obviously exaggerated, such as the refutation of Russell’s view that definite descriptions are unique. The alleged refutation is with the expression “the men who … .” This is an unintended insult. It stems from understanding Russell’s “On Denoting” as a theory of all the diverse usages of the definite article in familiar English rather than as a solution to a problem within the task of formalizing parts of a natural language into a language adequate for mathematics. Russell expressly referred to the definite article in the singular. (In some languages, singular and plural expressions may differ widely.)
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Bunge uses the universe of discourse freely to solve the problem at hand the easy way.
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This is a disclaimer: the theory is no complete account of the diverse acceptable usages of the definite article. Russell ends his paper by asserting a smaller claim as more general: the problem has no easy solution. All his commentators agree with him, but few pay him the courtesy of saying so explicitly.
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To render an opaque problematic inference invalid one needs to use a technique similar to the classical fallacy of four terms (quaternio terminorum).
- 25.
Russell 1905, 484: “one would suppose that ‘the King of France is bald’ ought to be nonsense; but it is not nonsense, since it is plainly false.” Here we see the importance of not calling a falsehood “nonsense”.
- 26.
Russell stressed repeatedly that propositions that contain expressions with no meanings may nevertheless possess clear and obvious meanings. Wittgenstein agreed, even though not clearly: TLP, §4.064 says, “Every proposition must already have sense; assertion cannot give it a sense, for what it asserts is the sense itself.” Nevertheless, Wittgenstein often spoke of the meanings of words, not of statements (“and then whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions”; TLP, 6.53). This is but a slip of his pen: he proudly presented his atheism as a part of modern logic. Popper 1963 , 373 has corrected Wittgenstein here: the theory of definite description allows describing the deity in many ways, all of them traditional. Putnam 2008 repeated the claim that Wittgenstein’s philosophy supports religion as a practice devoid of any doctrine.
- 27.
Russell 1905, 484–5: “A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science.” This then is a partial success of his plan to render philosophy scientific.
- 28.
Russell 1914, title to Chapter 2: Logic as the Essence of Philosophy.
- 29.
- 30.
Monk 1996, 241, 250–2272, 280, 289, 282: “No one except Wittgenstein understood it all”.
- 31.
This may be a mistake of mine: as commentators are reluctant to discuss this, I may be missing a point.
- 32.
Forster 2004 , 230. Engel 2012 , 48 denies that Wittgenstein’s Tractatus relates to the ideal language. “He was attempting, rather, to reveal the essential structure” of any language. As Forster refers to TLP, §§3.325 and 4.0031 here, it is hard to ascribe to Engel ignorance rather than defensiveness. Dummett 1991c , 83–5 argues for the impossibility of the ideal language, as all language is social.
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Agassi, J. (2018). Russell. In: Ludwig Wittgenstein’s Philosophical Investigations. Synthese Library, vol 401. Springer, Cham. https://doi.org/10.1007/978-3-030-00117-9_7
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