Abstract
This chapter highlights Russell’s original conception of logical atomism as a new and boldly scientific method in philosophy that makes Principia’s mathematical logic its essence and applies the logical forms uniquely revealed by Principia to undermine the metaphysician’s arguments for the ineliminability of abstract particulars and their specialized kinds of necessity. Philosophy, Russell explained, is the science of the possible. Wittgenstein expressed agreement in his Notes on Logic and in his Tractatus, proclaiming that the only necessity is logical necessity. I argue that the pristine form of logical atomism is not found in the 1918 lectures, but in Russell’s 1911 “Analytic Realism” and it finds its canonical articulation in his 1914 Our Knowledge of the External World As a Field for Scientific Method in Philosophy. The logical atomism lectures reflect Russell in flux, with his acquaintance epistemology at an impasse. Russell could not imagine how his multiple-relation theory, even amended by acquaintance with logical forms, could find the truth-bearers for general and molecular belief. Russell eventually turned, not to Wittgenstein, but to neutral monism and a tenuous alliance with behaviorism to get over the impasse. I offer hope of returning to Russell’s original acquaintance epistemology and its multiple-relation theory. I argue that modern so-called rival logics, relevant and modal, can be studied within Principia’s logic just as are rival geometries. Principia’s logic is an impredicative (simple) type theory and it studies relational structures by studying the way relations, exemplified or not, order their fields. Rival logics and non-logical necessities are simply distinct structures. Thus, in spite of its hostility to modern non-logical necessities (especially metaphysical necessity de re), Russell’s logical atomism remains the last and best hope for a genuinely scientific philosophy.
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Notes
- 1.
I use the expression “cpLogic ” to characterize the thesis that logic embodies the impredicative comprehension of functions which makes it a genuine informative synthetic a priori science.
- 2.
See Wittgenstein “Letter of 29 November 1913”, in Wittgenstein (1914), p. 123.
- 3.
As I interpret Principia, the syntax is that of simple impredicative type theory. The exclamation sign (!) serves only to distinguish genuine bindable object-language predicate variables (e.g., φ!, ψ!, f!, g!) from schematic letters (φ, ψ, f, g) for wffs. Followers of Church’s interpretation hold that Principia’s grammar codes ramification into its object-language, so that letters (φ, ψ, f, g) are for object-language bindable non-predicative variables. This interpretation is quite unhistorical and generates insuperable difficulties for an interpretation of Russell’s Theory of Knowledge. For all of Russell’s trials and tribulations, there is not one mention of ramification in any of his works on the epistemology of Principia’s logic. Ramification, as I see it, concerned Whitehead and Russell’s failed nominalistic semantics for Principia’s bindable predicate variables. It was never part of its object-language grammar.
- 4.
See Fara and Williamson (2005).
- 5.
The following are the basic axioms, where “o” designates a basis:
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Idempotence: (x)Rxxx
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Identity: (x)Roxx
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Monotony: Roab ∧ Rbcd .⊃. Racd
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Star1: Rabc ⊃ Rac*b*
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Inheritance: Roab ∧ Aa .⊃. Ab
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Star2: b** = b
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Commutation: Rabc ⊃ Rbac
If one adds the axiom (x)Roox, the system emulates the classical quantifier-free calculus.
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References
Works by Other Authors
Cocchiarella, Nino B. (1975). “On the Primary and Secondary Semantics of Logical Necessity.” Journal of Philosophical Logic, Vol. 4: 13–27.
Fara, Michael and Timothy Williamson (2005). “Counterparts and Actuality.” Mind Vol. 114: 1–30.
Landini, Gregory (2007). Wittgenstein’s Apprenticeship with Russell. Cambridge: Cambridge University Press.
Lewis, David (1968). “Counterpart Theory and Quantified Modal Logic.” Journal of Philosophy, 65: 113–126.
Sheffer, H. (1913). “A Set of Five Independent Postulates for Boolean Algebras with Application Logical Constants.” Transactions of the American Mathematical Society, Vol. 14: 481–488.
Wittgenstein, Ludwig (1914). Notebooks 1914–1916, eds. G. H. von Wright and G. E. M. Anscombe, second edition 1979. Chicago: The University of Chicago Press.
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Landini, G. (2018). Logical Atomism’s Necessity. In: Elkind, L., Landini, G. (eds) The Philosophy of Logical Atomism. History of Analytic Philosophy. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-94364-0_2
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