# Principia Mathematica Summary

## Summary

The Principia Mathematica of Bertrand Russell and Alfred North Whitehead is an attempt to analyze the roots of mathematics in the language of logic. The new notation of symbolic logic, which is derived from Guiseppe Peano and had its origins in the work of Gottlob Frege, is the vehicle for this task.

Russell and Whitehead begin with the functions of propositions, in which letters such as p or q stand for logical propositions such as “Socrates is a man” or “Socrates is mortal.” These functions include the following:

the contradictory function “(p · p),” read as “not (p and not p)” and meaning that it is false to assert both a statement and its denial

the logical sum “p v q,” read as “p or q” and meaning that exactly one of the two propositions is true

the logical product “p · q,” read as “p and q” and meaning that both of the propositions are true

the implicative function “p > q,” read as “p implies q” and meaning that if p is true, then q must also be true

the function “p v q,” read as “not p or q” and meaning that either p is false or q is true, but not both

The standard symbol is used for equivalence: “p q” (meaning that p and q are either both true or both false). Truth value is denoted by either “T” for true or “F” for false. A less customary symbol is the assertion sign “⊦,” as in “ ⊦ p > ⊦ q,” which means “we assert p; thus, we assert q.” In other words, if the first statement is asserted as true, then the second statement must be asserted as true also—the standard form of modus ponens in greatly abbreviated notation.

Russell and Whitehead’s notation follows Peano’s notation, using dots instead of parentheses or brackets. One primitive propositional rule, for example, is the law of tautology. In the notation of the Principia Mathematica, this formula is expressed as“ ⊦: p. ≡ .p · p,” or ’we assert that any statement is equivalent to the logical product of the same statement.’ In slightly more customary notation, this would read“ ⊦ [p ≡ (p · p)].” Since the dots stand for bracket notation and for the logical product, the notation is a bit intuitively challenging. Most of the Principia Mathematics’s two thousand pages are expressed in symbolic notation, with additional operators added chapter by chapter.

The Principia Mathematica seeks to avoid paradoxes by redefining sets and lists. Several of the paradoxes in need of resolution are general knowledge. For example, the statement “all generalizations are...

(The entire section is 1089 words.)

## Bibliography

Goldstein, Laurence. “The Indefinability of ’One.’” Journal of Philosophical Logic 31 (2002): 29-42. Attempts to show that the reduction of all mathematics to a set of logical statements does not work.

Hylton, Peter, ed. Propositions, Functions, and Analysis: Selected Essays on Russell’s Philosophy. New York: Oxford University Press, 2008. Collection of essays that shed light on Russell’s general philosophical stances; provides a philosophical context for the Principia Mathematica.

Kripke, Saul. “Russell’s Notion of Scope.” Mind 114 (October, 2005): 1005-1037. Addresses scope ambiguities in statements with subjects that represent the null set, such as “The present king of France is bald.”

Link, Godehard, ed. One Hundred Years of Russell’s Paradox. New York: Walter de Gruyter, 2004. A collection of essays and conference papers of the International Munich Centenary Conference in 2001. The contributions all focus on Russell’s paradox.

Monk, Ray, and Anthony Palmer, eds. Bertrand Russell and the Origins of Analytical Philosophy. London: Continuum International, 1996. This collection of essays focuses on the precursors to Russell in the traditions of analytical philosophy, specifically on Gottlob Frege’s contributions.

Priest, Graham. “The Structure of the Paradoxes of Self-Reference.” Mind 103 (January, 1994): 25. Reviews a selection of paradoxes and their solutions in slightly more technical language.

Proops, Ian. “Russell’s Reasons for Logicism.” Journal of the History of Philosophy 44, no. 2 (April, 2006): 267-292. Reviews historical aspects of Russell’s philosophical development. The text is accessibly written and avoids highly technical language.

Soames, Scott. “No Class: Russell on Contextual Definition and the Elimination of Sets.” Philosophical Studies 139 (2008): 213-218. Explains the intension and extension of propositional functions.

Sorel, Nancy Caldwell. “When Ludwig Wittgenstein Met Bertrand Russell.” The Independent, August 19, 1995, p. 42. Amusing anecdote of the meeting between Russell and Wittgenstein.

Stevens, Graham. The Russellian Origins of Analytical Philosophy: Bertrand Russell and the Unity of the Proposition. New York: Routledge, 2005. Focuses on the historical development of analytical philosophy.