# Definition:Logical Argument

*This page is about Logical Argument. For other uses, see Argument.*

## Definition

A **logical argument** (or just **argument**) is a process of creating a new statement from one or more existing statements.

An **argument** proceeds from a set of premises to a conclusion, by means of logical implication, via a procedure called logical inference.

An **argument** may have more than one premise, but only one conclusion.

While statements may be classified as either **true** or **false**, an **argument** may be classified as either valid or invalid.

Loosely speaking, a valid argument is one that leads unshakeably from true statements to other true statements, whereas an invalid argument is one that can lead you to, for example, a false statement from one which is true.

Thus:

- An argument may be valid, even though its premises are false.
- An argument may be invalid, even though its premises are true.
- An argument may be invalid
*and*its premises false.

To be sure of the truth of a conclusion, it is necessary to make sure both that the premises are true *and* that the **argument** is valid.

However, while you may not actually know whether a statement is true or not, you can investigate the consequences of it being either true or false, and see what effect that has on the truth value of the proposition(s) of which it is a part. That, in short, is what the process of **logical argument** consists of.

An **argument** may be described symbolically by means of sequents, which specify the flow of an **argument**.

### Finitary Argument

A **finitary argument** is a logical argument which starts with a finite number of axioms, and can be translated into a finite number of statements.

## Sources

- 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): Chapter $1$: The Propositional Calculus $1$: $1$ The Nature of Logic - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $1$ Introduction: Logic and Language: $1.2$: The Nature of Argument - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (next): $\S \text{I}: 1$: The Logic of Statements $(1)$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**argument**:**3.** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**argument**:**2.** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Definition $1.1.3$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**argument**:**2.**