Solving the Conjunction Problem of Russell's Principles of Mathematics


  • Gregory Landini University of Iowa



The quantification theory of propositions in Russell’s Principles of Mathematics has been the subject of an intensive study and in reconstruction has been found to be complete with respect to analogs of the truths of modern quantification theory. A difficulty arises in the reconstruction, however, because it presents universally quantified exportations of five of Russell’s axioms. This paper investigates whether a formal system can be found that is more faithful to Russell’s original prose. Russell offers axioms that are universally quantified implications that have antecedent clauses that are conjunctions. The presence of conjunctions as antecedent clauses seems to doom the theory from the onset, it will be found that there is no way to prove conjunctions so that, after universal instantiation, one can detach the needed antecedent clauses. Amalgamating two of Russell’s axioms, this paper overcomes the difficulty.


Byrd, Michael, 1989. “Russell, Logicism and the Choice of Logical Constants.” Notre Dame Journal of Formal Logic 30: 343–61.

Cocchiarella, Nino, 1980. “The Development of the Theory of Logical Types and the Notion of a Logical Subject in Russell’s Early Philosophy.” Synthese 45: 71–115.

Frege, Gottlob, 1897. “On Herr Peano’s Begriffsschrift and My Own,” translated by V. H. Dudman. Australasian Journal of Philosophy 47 (1969): 1–14. Reprinted in Collected Papers on Mathematics, Logic and Philosophy, edited by B. McGuinness, New York: Basil Blackwell, 1984, pp. 234–48.

———, 1980. Philosophical and Mathematical Correspondence, edited by B. McGuinness, translated by H. Kaal. Chicago: University of Chicago Press.

Griffin, Nicholas, 1980. “Russell on the Nature of Logic 1903–1913.” Synthese 45: 117–88.

Landini, Gregory, 1996. “Logic in Russell’s Principles of Mathematics.” Notre Dame Journal of Formal Logic 37: 554–84.

Peano, Giuseppe, 1889a. Arithmetices principia, nova methodo exposita. Turin: Bocca. Reprinted in J. van Heijenoort, ed., From Frege to Gödel, Cambridge, MA: Harvard University Press, 1967, pp. 83–97.

———, 1889b. Formulaire de mathématiques. Turin: Bocca.

Russell, Bertrand, 1903. The Principles of Mathematics. Cambridge: Cambridge University Press.

———, 1906. “The Theory of Implication.” American Journal of Mathematics 28: 159–202.