Principia Mathematica 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 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...
(The entire section is 1,394 words.)