Principia Mathematica by Bertrand Russell

First published: 1910-1913

Type of work: Philosophy

The Work:

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 false” is a generalization. Thus, if it is true, then it must be false (because it asserts that all statements in a class of which it is a member are false). Similarly, if one posits a town called Mayberry, in which all barbers are male and the barber only shaves people who do not shave themselves, one creates an infinite regression, because the barber should only shave himself if he does not shave himself, and if he does shave himself then he should not shave himself.

One more example that dates back to ancient Greece is the liar’s paradox: An Athenian comes to Sparta and reports “all Athenians are liars.” If the statement is true, the Athenian must be lying, so the statement must be false. If the statement is false, it is consistent with what it says about Athenians and thus should be true.

The solution in all of these cases is a reflection about the nature of sets and subsets. For example, my family is a set. My two sons are a subset of that set. Statements also make up sets. There is the set of generalizations, for example. The apparent paradox involving generalizations arises when the statement “all generalizations are false” is considered to be part of the set of generalizations. In fact, the statement is part of a descriptive superset that includes the set of generalizations, plus an additional statement describing that set. Ignoring the limits of sets creates logical contradictions.

The same error applies to the other paradoxes as well. The barber is part of a superset whose subset is “men from Mayberry”; thus, he is not targeted by the statement. The Athenian likewise must be considered external to the set of Athenians described by his statement before that statement can be evaluated. Sets in mathematics and geometry can generate similar paradoxes. Avoiding these problematic characteristics of sets, supersets, and subsets by way of the “axiom of reducibility” is the purpose of the Principia Mathematica. Russell and Whitehead assert the axiom of reducibility as a smaller assumption than the assumption that there are classes. The axiom asserts that a predicative function can always replace classes; in other words, any mathematical class reduces to a logical predication.

Another generally known problem that the Principia Mathematica addresses is called “existential presupposition.” Ordinary intuition indicates that if “John passed the examination” is true, then “John did not pass the examination” should be false—unless there are two different Johns. Where only one person is under consideration, one of the two statements should be true and the other should be false. If the world contained no people named “John” at all, problems would arise.

Russell and Whitehead explore this notion by speaking of kings. “The present king of England is bald” was a true statement in 1903. The subcontrary “The present king of England is not bald” was then false. However, “The present king of France is bald” and “the present king of France is not bald” would actually be both false statements, in 1903 as in the twenty-first century, because no present king of France exists. In other words, the subject of both propositions is the null set, a set without content. Set characteristics of the null set then produce a problem. The avoidance of classes altogether while becoming aware of ambiguities would avoid this problem.

The Principia Mathematica is not leisure reading. Russell himself is said to have quipped that he knew of only six people who had read it from cover to cover: three people from Poland and three from Texas. The notational formula employed by the work is difficult to follow without extensive notes and “cheat sheets.” The Principia Mathematica helped Bertrand Russell gain his reputation as the father of analytical philosophy, particularly in combination with his association with Ludwig Wittgenstein, who is associated with the Vienna Circle and logical positivism. The Principia Mathematica is still a treasure trove of ideas yet to be developed.

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.