Modern Logic

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Logic and Mathematics

By the turn of the twentieth century, philosophers had begun to devote careful attention to the foundations of logical and mathematical systems. For two millenia Aristotelian logic—with only minor scholastic modifications—had seemed a complete and final explanation of human reasoning. But the geometry of Euclid had also seemed secure, until Lobachevsky and Riemann showed that alternative conceptions were not only possible but useful in many applications. Similar efforts to re-think the structure of logic began late in the nineteenth century. John Stuart Mill tried to develop a comprehensive account of human thought that focussed on inductive rather than deductive reasoning. Even mathematical reasoning, he supposed, can be grounded on empirical observation. Many philosophers and mathematicians, however, took a different approach.

Mathematical Treatment of Logic

William Hamilton suggested that "quantifying the predicates" contained in traditional categorical propositions might permit algebraic interpretation of their content as explicit statements of identity. This view encouraged Augustus De Morgan to propose symbolic expression of the copula as a purely logical relation, whose formal features obtain in many distinct contexts. Thus, for example, De Morgan's Theorems (suitably interpreted) hold equally well for the intersection and union of sets as for logical conjunction and disjunction. De Morgan also explored Laplace's notion of probability as a degree of rational belief that may fall between perfect certainty of truth or falsity.

George Boole completed this transformation by explicitly interpreting categorical logic (as we now do) by reference to classes of things. The logical/set-theoretical/mathematical relations that hold among such classes can be expressed at least as well in a "Boolean algebra" as in traditional Aristotelean terms. What is more, as Leonhard Euler and John Venn showed, these relations can be represented perspicuously in purely topographical diagrams whose features model formal validity. All of these developments encouraged philosophers to examine the isomorphism of logic and mathematics more closely.

Frege: The Representation of Thought

The single most important figure in this process was Gottlob Frege, whose technical innovations helped to make it clear that logic and mathematics can be understood as inter-related parts of a single aspect of human thinking. Frege formalized the use of quantifiers in the symbolic representation of logical relations among classes of things and upon use of extensional equivalence (the one-to-one correspondence) as the basis for any adequate definition of number. With these tools, his Begriffsschrift (Concept-notation) (1879) was intended as a comprehensive method for expressing truths about the world.

But these achievements only focussed Frege's attention more clearly on fundamental issues about the meaning of terms and symbols. What are numbers, really?

not generalizations from experience, since they would be useful only for those objects we have already known;
not just the formal signs we employ, since that would render their application in the world uncertain; and
certainly not merely our ideas, since those can too easily differ from person to person.

Somehow, numbers (and other significant notions) must be entirely objective without linking them too directly to objects. Frege's solution involved an appeal to concepts as beings in their own right, independent both of the objects they represent and of the minds that have them.

In "Über Sinn und Bedeutung" ("On Sense and Reference") (1892), Frege developed this theory with respect to informative statements of identity such as "The morning star is the evening star." The status of such an assertion is a bit puzzling. Since "the morning star" and "the evening star" both refer in fact to the planet Venus, the statement reduces to the tautology that "Venus is Venus," which conveys no more significant content than does " 7 = 7 ." But since we experience the bright objects in the sky at distinct locations on different days, they do naturally come to have two names instead of one, and learning that they are one and the same seems genuinely informative. Frege explained such situations by distinguishing between the sense and the reference {Ger. Sinn und Bedeutung} of any linguistic unit of meaning. The sense of a word, phrase, or symbol is just the concept it expresses, while its refence is the object it represents.

Not only does this solution works fine for proper names, but it also has interesting results when generalized to apply to other units of meaning. According to Frege, every sentence has as its sense just the thought or judgment that it is ordinarily used to express. But each sentence must also have a reference, the real object to which it refers, and this in every instance Frege took to be either TRUE or FALSE. In this application, the theory accounts for cases of mistaken belief, establishing referential opacity while preserving a purely extensional notion of logical equivalence. The methods Frege employed in this paper had even greater influence than merely providing a respectable solution to a specific problem, since they served as an important model for the development of philosophical analysis generally.

Logical Foundations for Arithmetic

The culmination of the new approach to logic lay in its capacity to illuminate the nature of the mathematical reasoning. While the idealists sought to reveal the internal coherence of absolute reality and the pragmatists offered to account for human inquiry as a loose pattern of investigation, the new logicians hoped to show that the most significant relations among things could be understood as purely formal and external. Mathematicians like Richard Dedekind realized that on this basis it might be possible to establish mathematics firmly on logical grounds. Giuseppe Peano had demonstrated in 1889 that all of arithmetic could be reduced to an axiomatic system with a carefully restricted set of preliminary postulates. Frege promptly sought to express these postulates in the symbolic notation of his own invention. By 1913, Russell and Whitehead had completed the monumental Principia Mathematica (1913), taking three massive volumes to move from a few logical axioms through a definition of number to a proof that " 1 + 1 = 2 ." Although the work of Gödel (less than two decades later) made clear the inherent limitations of this approach, its significance for our understanding of logic and mathematics remains undimmed.

History of Philosophy

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