Begriffsschrift

Begriffsschrift (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in 1879. It introduced a formal system that changed how people think about reasoning and ideas.

The book is usually called concept writing or concept notation. Its full title describes it as "a formula language, modeled on that of arithmetic, for pure thought." Frege wanted to create a clear way to show logical steps, much like how numbers help with math.

Frege used his new logical system in his work on the foundations of mathematics for many years after publishing the book. Begriffsschrift is considered the first work in analytical philosophy, a field that later philosophers like Bertrand Russell helped to grow.

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 chose nine key statements as starting points, called axioms. He explained that these statements are clearly true based on their meaning.

These nine statements cover different parts of logic:

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

All the other statements in his work were built using these nine starting points and some basic rules for moving 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

The work 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 that could show parts of math and everyday language.

Some symbols from Begriffsschrift are still used today. For example, the "turnstile" symbol ⊢, which comes from Frege's ideas, is still used in logic. Also, the sign for "no" (¬) was used first by Frege and later brought back by others.

Famous thinkers like Ludwig Wittgenstein and others respected Frege's work and used his ideas in their own writing.

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.