# Definition:Therefore

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## Definition

If statement $p$ logically implies statement $q$, then we may say:

**$p$, therefore $q$**.

The symbology:

- $p, q \vdash r$

means:

**Given as premises $p$ and $q$, we may validly conclude $r$**

So the symbol $\vdash$ is interpreted to mean **therefore**.

Thus, $p, q \vdash r$ reads as:

**$p$ and $q$, therefore $r$.**

A fallacy may be indicated by $p, q \not \vdash r$, which can be interpreted as:

**Given as premises $p$ and $q$, we may***not*validly conclude $r$.

## Also known as

The symbol $\vdash$ is sometimes called the **turnstile symbol** (or **gate post**), and is often (misleadingly) called the **assertion sign**.

Some older literature uses the symbol $\therefore$ but this is falling out of use.

In contrast to $\vdash$, which is a formal symbol used in proof writing, the $\therefore$ symbol is generally used as shorthand for "therefore," and as such is traditionally classified as a punctuation mark.

## Also see

- Definition:Because
- Definition:Logical Implication
- Definition:Logical Equivalence
- Definition:Conditional
- Definition:Biconditional

## 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): $\S 3.3$: The Construction and Application of Truth-Tables - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 1.2$: Conditionals and Negation - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 1$: The Logic of Statements $(1)$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*: Entry:**Turnstile** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**assertion sign**