This essay is part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results in modern logic and their significance for philosophical and systematic theology.
Most theologians who have studied logic have encountered quantifiers and perhaps heard that Georg Cantor proved that there are different sizes of infinity. What is easier to miss is just how revolutionary these developments were. In the final decades of the nineteenth century, logic ceased to be primarily a theory of propositions of the Aristotelian sort—“All men are mortal,” “Some Greeks are philosophers”—and acquired the resources needed to describe indefinitely complicated structures of objects and their relations. At roughly the same time, mathematics learned that the infinite was not a single undifferentiated beyond, but possessed an articulated internal structure.
Both developments matter for theology. Neither proves anything theological. But each profoundly enlarges the conceptual space in which theological claims can be formulated and assessed.
1. Frege, Peirce, and the Revolution in Quantification
The decisive breakthrough associated with Gottlob Frege’s Begriffsschrift of 1879 was not simply the invention of some new logical symbols. Frege supplied a new analysis of the logical structure of propositions. Traditional syllogistic logic had treated propositions largely in subject-predicate form. Frege instead analyzed propositions in terms of functions, arguments, variables, and quantification.
Consider a simple statement:
Every human being is mortal.
In contemporary notation we write
Read aloud, this says: for every object , if is human, then is mortal.
The statement does not name some peculiar object called “every human being.” Rather, it says that anything whatsoever in the relevant domain that is human is also mortal. The logical form becomes visible only when we distinguish the predicates—being human and being mortal—from the variable over which the quantifier ranges.
The real power of quantification appears when quantifiers are nested. Compare
with
The first reads: for every , there is some such that bears relation to .
The second reads: there is some single such that every bears relation to that .
The difference is only the order and scope of the quantifiers, but the difference in meaning can be enormous. “Everyone loves someone” does not entail “There is someone whom everyone loves.”
Once this machinery is available, logic can represent patterns of dependence that traditional syllogistic logic could scarcely express. Mathematics, science, metaphysics, and theology suddenly possess a far more exact language for describing structures.
Frege was not alone in bringing about this transformation. Charles Sanders Peirce and his collaborators, especially Oscar Howard Mitchell, independently developed powerful systems of quantification during roughly the same period, and Peirce’s work on the logic of relations deserves particular notice. Aristotelian logic had been especially comfortable with one-place predicates—“is human,” “is mortal,” “is wise.” Peirce emphasized relations among two, three, or more objects. Thus
can be read simply as bears relation to , while a three-place relation
says that , , and stand in some specified three-place relation. We might represent “ gives to ,” for example, by such a structure.
This sounds elementary to us precisely because the revolution succeeded.
For theology the significance is immediate. Christian theological vocabulary is saturated with relations: the Father begets the Son; the Son is begotten of the Father; the Spirit proceeds; God creates the world; God justifies the sinner; Christ assumes a human nature; believers participate in Christ; promise is addressed to hearer. Such claims cannot adequately be represented merely by assigning properties to isolated objects. Their logical articulation requires relations, often asymmetric relations, and sometimes relations whose formal properties themselves become doctrinally significant.
Consider merely the difference between saying that each divine person is God and saying that the divine persons stand in particular relations to one another. The first set of claims concerns predication; the second concerns relational structure. No piece of predicate logic solves the doctrine of the Trinity. But modern quantificational and relational logic allows theologians to see with far greater precision what kinds of claims they are actually making and where apparently similar formulations differ logically.
There is another consequence that may be even more important. Quantification makes explicit the distinction between what is true of some object and what is true of every object. Theology constantly moves among existential, universal, and uniqueness claims: there is a God; everything other than God depends upon God; there is exactly one God; every human being is a creature; some human beings believe; every justified sinner stands in a particular relation to Christ. The logical differences among these claims are not merely stylistic. They determine what follows from them.
The theological payoff, then, is not that Frege or Peirce secretly supplied Christian doctrine with its proper metaphysics. It is rather that modern logic makes possible a level of structural clarity that theological argument badly needs. Quantifier scope, relational order, dependence, uniqueness, and identity can all be made explicit. Much theological disagreement that appears initially to concern “concepts” turns out, upon examination, also to concern logical form.
2. Cantor and the Discovery of Different Infinities
The second revolution came from Georg Cantor. Before Cantor, philosophers and mathematicians had certainly discussed infinity, but infinity was commonly treated as though it were a single notion: the indefinite, the unbounded, or that which simply exceeds every finite magnitude. Cantor showed that infinite collections can themselves differ in size.
The key idea is deceptively simple. Two sets have the same cardinality when their members can be paired one-to-one: every member of the first set is paired with exactly one member of the second, and none is left over. For finite sets this seems trivial. A set containing five books has the same cardinality as a set containing five chairs because each book can be paired with exactly one chair.
Cantor applied the same criterion to infinite sets.
Consider the natural numbers:
and the even numbers:
At first the even numbers seem to form a smaller collection, since they constitute only part of the natural numbers. Yet every natural number can be paired with exactly one even number, namely : 1 with 2, 2 with 4, 3 with 6, and so on without end. Thus the natural numbers and the even numbers have the same cardinality.
Infinity therefore behaves differently from finite magnitude. A proper part of an infinite set can have the same number of members as the whole set.
Cantor’s deeper result was more startling. Not all infinite sets have the same cardinality. His diagonal argument establishes that the real numbers cannot be paired one-to-one with the natural numbers. In ordinary language: there are strictly more real numbers than natural numbers, even though both collections are infinite.
Cantor proved something still stronger. Given any set , form its power set, written
which simply means the set of all subsets of . Cantor’s theorem tells us that
Read in words: the set of all subsets of is always strictly larger than itself.
This means that there can be no greatest cardinal number. Begin with any infinity whatsoever, and one can specify a still greater infinity.
Cantor’s discovery forced philosophy to reconsider what it meant by “the infinite.” There is no single mathematical infinity. There is an ordered hierarchy of infinite cardinalities.
The theological temptation at this point is obvious and should generally be resisted. Cantor’s transfinite mathematics does not provide a mathematical model of divine infinity, and God’s infinity should not simply be identified with some enormously large cardinal number. Indeed, since Cantor proved that there is no greatest cardinality, saying that God possesses “the largest mathematical infinity” would not even make mathematical sense.
Precisely here, however, Cantor becomes theologically useful, for the mathematics teaches theologians to distinguish different senses of infinity rather than allowing the word infinite to do undisciplined work. Mathematical infinity concerns the cardinality or ordering of mathematical structures. Divine infinity, by contrast, has traditionally concerned the absence of creaturely limitation in God. To say that God is infinite is not ordinarily to say that God has infinitely many parts, occupies infinitely many locations, or instantiates some transfinite cardinality.
Cantor therefore offers theology a conceptual warning: do not infer from the single word “infinite” that two uses of the term belong to the same logical or ontological category.
At the same time, Cantor expands our imagination. Human reason can rigorously distinguish structures beyond every finite magnitude without thereby collapsing into incoherence. The infinite need not simply mean “something too large for us to think.” Certain infinities can be precisely defined, compared, and ordered.
This has interesting consequences for classical theological discussions of divine knowledge. Suppose we say that God knows every natural number, every real number, every subset of the natural numbers, and every mathematical truth. The point is not that divine knowledge thereby acquires some particular cardinality. Rather, Cantorian mathematics demonstrates how quickly apparently simple talk of “all things” becomes structurally complicated. A theology of omniscience must therefore take seriously the logical character of the totalities over which its claims range.
Cantor also helps us resist an old rhetorical maneuver in theology: invoking infinity whenever conceptual analysis becomes difficult. “God is infinite” cannot function as a license for contradiction. Cantor’s work is a striking historical demonstration that infinity and rigor are not opposites. Modern mathematics became more rigorous precisely by refusing to leave infinity unanalyzed.
Why These Two Revolutions Belong Together
Frege, Peirce, and Cantor transformed different areas of mathematics and logic, but the two revolutions belong together. Fregean and Peircean logic gave us vastly more powerful ways of saying what is true of objects and structures; Cantor revealed that the domains over which our variables range may themselves possess unsuspected structure.
Together they prepared the ground for twentieth-century model theory. Once we possess quantifiers ranging over a domain, predicates and relations interpreted over that domain, and a mathematically disciplined conception of infinite structures, questions that would previously have been nearly impossible to formulate become natural: Which sentences are true in a structure? Can two different structures satisfy exactly the same sentences? Must an infinite theory have models of different cardinalities? Can a formal language uniquely characterize the structure it is intended to describe?
Those questions will eventually lead us to Löwenheim, Skolem, Gödel, Tarski, compactness, and some of the most philosophically unsettling results of modern logic.
For theology, however, the first lesson is already substantial. Modern logic teaches us that form matters. Who quantifies over what, which relations hold between which objects, what depends upon what, and what sort of infinity is being invoked are not technical details added after theology has finished its real work. They are part of determining what our theological claims actually say.
Bibliographical Note
For Frege, the obvious starting point is Gottlob Frege, Begriffsschrift: A Formula Language, Modeled upon That of Arithmetic, for Pure Thought (1879), conveniently available in translation in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Harvard University Press, 1967). Van Heijenoort’s volume remains especially valuable because it allows readers to encounter many of the foundational texts of modern logic themselves.
The parallel American development should not be overlooked. See Oscar Howard Mitchell, “On a New Algebra of Logic” (1883), and Charles S. Peirce, “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885). For the importance of Peirce and his circle in the development of quantification and the logic of relations, see Geraldine Brady, From Peirce to Skolem: A Neglected Chapter in the History of Logic (North-Holland, 2000).
For Cantor, the classic sources include his 1874 paper establishing the non-denumerability of the real numbers and his 1891 diagonal argument, “On an Elementary Question of the Theory of Manifolds.” Both are widely reprinted and translated. A particularly accessible philosophical introduction remains Michael Hallett, Cantorian Set Theory and Limitation of Size (Oxford University Press, 1984). For the broader historical setting, see Joseph W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton University Press, 1979).
