Begriffsschrift

Begriffsschrift

Begriffsschrift (German for, roughly, "concept-writing") is a book about logic by Gottlob Frege. It was published in 1879.

The book introduced a new formal system that changed how people think about reasoning and ideas. It is often called concept writing or concept notation. Its full title describes it as "a formula language, modeled on that of arithmetic, for pure thought."

Frege used this new logical system in his work on the foundations of mathematics for many years. Begriffsschrift is seen as the first work in analytical philosophy, a field later helped by philosophers like Bertrand Russell.

Notation and the system

This book shows one of the first ways to use special symbols to stand for ideas in logic. It uses a system where each sentence or idea can only be true or false. It also lets us talk about relationships between things.

In the first part of the book, the writer explains basic ideas like saying something is true, saying if one thing happens then another does, and saying something is not true.

The writer uses a special way to write these ideas that looks different from normal math. This way of writing helps show how ideas are connected without needing extra symbols like parentheses.

The book shows how ideas can be broken down into smaller parts, like building blocks. For example, one idea in the book can be written in a special shape that shows how the smaller parts fit together.

Basic concept	Frege's notation	The diagram shows (in modern notation)	Modern notation
Judging	⊢ A , ⊩ A {\displaystyle \vdash A,\Vdash A}		p ( A ) = 1 , {\displaystyle p(A)=1,} p ( A ) = i {\displaystyle p(A)=i} ⊢ A , ⊩ A {\displaystyle \vdash A,\Vdash A}
Negation		basic	¬ A {\displaystyle \neg A}
Material conditional		basic	B → A {\displaystyle B\to A}
Logical conjunction		¬ ( B → ¬ A ) {\displaystyle \lnot (B\to \lnot A)}	A ∧ B {\displaystyle A\land B}
Logical disjunction		¬ B → A {\displaystyle \lnot B\to A}	A ∨ B {\displaystyle A\lor B}
Universal quantification		basic	∀ x F ( x ) {\displaystyle \forall x\,F(x)}
Existential quantification		¬ ∀ x ¬ F ( x ) {\displaystyle \lnot \forall x\,\lnot F(x)}	∃ x F ( x ) {\displaystyle \exists x\,F(x)}
Material equivalence	A ≡ B {\displaystyle A\equiv B}		A ↔ B {\displaystyle A\leftrightarrow B}
Identity	A ≡ B {\displaystyle A\equiv B}		A = B {\displaystyle A=B}

The calculus in Frege's work

In the second chapter, Frege picked nine key statements to start with, called axioms. He said these statements are clearly true because of what they mean.

These nine statements cover different parts of logic:

• Some are about if-then statements.
• Others are about not or opposite statements.
• A few are about when things are the same.
• And one is about statements that apply to every item in a group.

All the other statements in his work were made using these nine starting points and some basic rules for changing from one statement to another.

Frege used these ideas later in his work on the foundations of numbers. For example, he showed that if we think of numbers in a certain way, we can explain what natural numbers are.

Influence on other works

Begriffsschrift had a big impact on later studies in logic. Many later works in formal logic depend on it because it was the first to use a special kind of logic.

Some symbols from Begriffsschrift are still used today. For example, the "turnstile" symbol ⊢ comes from Frege's ideas and is still used in logic. Famous thinkers like Ludwig Wittgenstein respected Frege's work and used his ideas.

Editions

Here are some ways you can find the book Begriffsschrift:

• Frege, Gottlob wrote the original version in German in 1879.
• There are translations available in English, such as "Concept Script" and "Conceptual notation and related articles."

These versions help people read and understand Frege's important work.