Political Science Observations

Wittgenstein's Use of Formal Logic

From the logical foundations of the Tractatus to the language-games of the Philosophical Investigations

Ludwig Wittgenstein's relationship with formal logic underwent a radical transformation between his early and later work. For the early Wittgenstein, formal logic was the foundation of reality, thought, and language. For the later Wittgenstein, it became just one "language-game" among many, with no privileged status.

Early Wittgenstein

Tractatus Logico-Philosophicus

Propositional & Predicate Logic

For the early Wittgenstein, formal logic was not just a tool but the very foundation of reality. His primary tools were:

Propositional Logic: Dealing with simple propositions and connections using operators like AND (∧), OR (∨), NOT (¬), IF...THEN (→)
Predicate Logic: Introducing predicates, variables, and quantifiers (∀, ∃) to analyze proposition structure

Picture Theory of Meaning

Wittgenstein's revolutionary idea was that a proposition is a logical picture of a state of affairs. For this to work, the proposition and the fact must share the same logical form.

Proposition: "The book is on the table"

Logical form: On(book, table)

Truth-Functional Analysis

He argued that all complex propositions are truth-functions of elementary propositions. The truth of any complex statement is entirely determined by the truth of its simplest parts and the logical operators connecting them.

If P → Q, and P is true, then Q must be true

If R ∧ S is true, then both R and S must be true

Later Wittgenstein

Philosophical Investigations

Rejection of Logic as Foundation

The later Wittgenstein completely abandoned his earlier view. He came to see his obsession with formal logic as a profound mistake that misunderstood how language actually works.

Language-Games (Sprachspiel)

Wittgenstein argued that meaning is determined by use within specific, rule-governed activities or "games." The rules are not logical rules but social conventions.

Example: The word "check" means different things in chess, banking, and pattern design

Family Resemblance (Familienähnlichkeit)

He rejected the idea that all things called by a general term share a common logical essence. Instead, they are linked by overlapping similarities.

Example: Games share family resemblances but no single common essence

Meaning as Use

This slogan captures his later philosophy. Meaning comes from how words are used in language, not from logical relationships to the world.

Comparison: Wittgenstein's Shift in Perspective

Aspect	Early Wittgenstein (Tractatus)	Later Wittgenstein (Investigations)
View of Logic	The essence of language and world	One language-game among many
Primary Tool	Formal Logic (Propositional, Predicate)	Descriptive Analysis of Ordinary Language
Method	Logical Analysis (reduction to truth-functions)	Description of Use (observing language-games)
Goal	Uncover hidden logical form of language	Describe various uses of language in practice
Meaning	Logical relationship to world (Picture Theory)	Use in language (Meaning as Use)

Conclusion

While Truth-Functional Propositional and Predicate Logic was undoubtedly the workhorse of the Tractatus, the later Wittgenstein replaced it with an entirely different approach: a method of conceptual clarification based on observing the fluid, practical, and context-dependent nature of our actual language.

This radical shift represents one of the most dramatic transformations in the history of philosophy, as Wittgenstein moved from seeing logic as the foundation of everything to viewing it as just one tool among many in our linguistic toolbox.