Frege's Revolution: From Subject-Predicate to Function-Argument

1. The Ancient Paradigm and Its Limits

For over two millennia, Western logic was dominated by the framework inherited from Aristotle. At its heart was the analysis of a simple proposition into two parts: a subject and a predicate. In the sentence "Socrates is mortal," "Socrates" is the subject and "is mortal" is the predicate. This S-P structure seems intuitive and works well for a certain class of simple statements.

However, this grammatical division, when used as the basis for logic, has profound limitations. As logicians tried to analyze more complex sentences, the S-P model proved to be a straitjacket.

The key problem, which stumped logicians for centuries, is known as multiple generality. This occurs in sentences with more than one quantifier (like "all," "some," "every," "a").

To understand this problem, please watch the following video clip. Professor Michael Potter explains the ambiguity in the sentence "Every man respects some teacher," an ambiguity that traditional logic simply could not represent.

As you saw, the sentence has two possible logical forms:

1. ∀x (Man(x) → ∃y (Teacher(y) ∧ Respects(x,y)))
   For every man x, there exists some teacher y, such that x respects y.
2. ∃y (Teacher(y) ∧ ∀x (Man(x) → Respects(x,y)))
   There exists some teacher y, such that for every man x, x respects y.

The subject-predicate form provides no way to capture the different scopes of "every" and "some." Both Boole's and Aristotle's systems would try to treat "every man" as the subject, forcing the sentence into a structure that cannot handle the nested relationship between the two quantified terms. This is not just a minor technical issue; it's a complete failure to capture the logic of a vast range of human thought.

2. Frege's Revolution: Logic as Function and Argument

Frege's solution was to discard the subject-predicate analysis entirely. As a mathematician, he saw that a different structure, one already fundamental to mathematics, could be applied to language: the distinction between a function and an argument.

Your background in mathematics will make this move feel very natural. You are used to functions like $f(x)=x^2+1$. The function $f(\cdot )$ is an "incomplete" or "unsaturated" entity; it has a gap waiting to be filled. When you provide an argument, say the number 3, the function yields a value: $f(3)=10$.

Frege's brilliant move was to propose that this is how all propositions are structured.

Let's break down this new analysis:

• Argument: An expression that refers to a specific object. In logic, these are proper names, like "Socrates" or "Plataea".
• Function (or Concept): An expression with one or more gaps. When the gaps are filled with arguments (names), the expression becomes a complete sentence that expresses a thought. For Frege, the value yielded by a concept-function when supplied with an argument is a truth-value (either The True or The False).

Consider the sentence "John respects Mary."

• Traditional analysis: Subject = "John", Predicate = "respects Mary".
• Frege's analysis allows for multiple decompositions:
  1. We can remove "John" to get the function ( ) respects Mary. This is a one-place function (a concept) that is true of anyone who respects Mary.
  2. We can remove "Mary" to get John respects ( ). This is a different one-place function, true of anyone respected by John.
  3. Crucially, we can remove both "John" and "Mary" to get ( ) respects ( ). This is a two-place function (a relation).

This ability to analyze sentences into multi-place functions is what gives Frege's logic the power to handle relations, precisely the weakness of Boole's system.

This process of decomposition, using variables to mark the gaps, is explained brilliantly by Michael Potter in the following clip, where he discusses the most important paragraph in Frege's Begriffsschrift ("Concept-Script").

3. The Power of the New Analysis

With the function-argument structure in place, the problem of multiple generality dissolves. The relational predicate Respects(x, y) provides the core structure, and the quantifiers can then be applied to the variables x and y with different scopes, perfectly capturing the two distinct meanings we saw earlier.

But the philosophical impact goes even deeper. Frege's analysis shows that the grammatical form of a sentence is often a poor guide to its logical form.

As the reading highlights, consider these two sentences:

1. Socrates is mortal.
2. All Greeks are mortal.

Grammatically, they look similar. Both have a subject ("Socrates", "All Greeks") and a predicate ("is mortal"). But logically, Frege shows they are worlds apart:

1. Mortal(Socrates) — A function being applied to an object.
2. ∀x (Greek(x) → Mortal(x)) — A relation between two concepts (Greek and Mortal), bound by a universal quantifier.

In Frege's analysis, the grammatical subject "All Greeks" is not a logical unit at all; it is "analyzed away" into a more complex structure involving a quantifier, a variable, and two concepts linked by a conditional. This reveals a hidden logical depth that grammar conceals. Logic, for Frege, is not about the surface structure of language but the deep structure of the thoughts that language expresses.

This leap in expressive power is so great that it marks a fundamental difference in complexity. As Professor Potter mentioned (in a part of the video you didn't watch), Aristotelian logic is decidable: there is an algorithm that can determine whether any given syllogism is valid or not. In contrast, modern predicate logic, the system Frege invented, is undecidable. This result, proven later by Church and Turing, is a formal testament to the non-trivial power and richness of the logical system Frege created.

Conclusion

Frege's replacement of subject-predicate analysis with a function-argument structure was not merely a technical improvement. It was a complete paradigm shift that redefined the subject of logic.

Key Takeaways:

• The 2000-year-old subject-predicate analysis of propositions, rooted in grammar, was unable to handle statements involving relations and multiple quantifiers (the problem of multiple generality).
• Frege replaced it with a function-argument analysis borrowed from mathematics. Concepts are treated as functions that take objects as arguments and yield a truth-value.
• This new structure, especially its ability to handle multi-place functions (relations), combined with a system of quantifiers and variables, gave logic unprecedented expressive power.
• The revolutionary impact was to sever logic's dependence on grammatical form. It revealed a deeper logical structure where grammatical subjects are often "analyzed away," and it provided a formal language powerful enough to express the reasoning of mathematics and solve long-standing philosophical puzzles.

Preview of the Next Lesson:

Frege's new logic was powerful, but it also raised new, more subtle philosophical questions. If the meaning of a name is simply the object it refers to, how can a statement like "The morning star is the evening star" be informative? If they refer to the same object (Venus), shouldn't the statement be as trivial as "Venus is Venus"? To solve this, Frege introduced another celebrated and crucial distinction: that between Sense (Sinn) and Reference (Bedeutung). In our next lesson, we will delve into this distinction and its profound implications for the philosophy of language.