Definition:Theorem/Logic

Definition

A theorem in logic is a statement which can be shown to be the conclusion of a logical argument which depends on no premises except axioms.

A sequent which denotes a theorem $\phi$ is written $\vdash \phi$, indicating that there are no premises.

In this context, $\vdash$ is read as:

It is a theorem that ...

Sources

• 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.1$: The Purpose of the Axiomatic Method
• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules
• 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $1$. Pure Mathematics: Games
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Definition $1.10$