Argumentum a fortiori (literally "argument from the stronger [reason]") is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more certain than, the first.

Usage

American usage

In Garner's Modern American Usage, Garner says writers sometimes use a fortiori as an adjective as in "a usage to be resisted". He provides this example: "Clearly, if laws depend so heavily on public acquiescence, the case of conventions is an a fortiori [read even more compelling] one."

Jewish usage

A fortiori arguments are regularly used in Jewish law under the name kal va-chomer, literally "mild and severe", the mild case being the one we know about, while trying to infer about the more severe case.

Relation to ancient Indian logic

In ancient Indian logic (nyaya), the instrument of argumentation known as kaimutika or kaimutya nyaya is found to have a resemblance with a fortiori argument. Kaimutika has been derived from the words kim uta meaning "what is to be said of".

Islamic usage

In Islamic jurisprudence, a fortiori arguments are proved utilising the methods used in qiyas (reasoning by analogy).

Examples

In mathematics

Consider the case where there is a single necessary and sufficient condition required to satisfy some axiom. Given some theorem with an additional restriction imposed upon this axiom, an "a fortiori" proof will always hold. To demonstrate this, consider the following case: Because bijections are a special case of functions, it automatically follows that if (1) holds, then (2) will also hold. Therefore, any proof of (1) also suffices as a proof of (2). Thus, (2) is an "a fortiori" argument.

Types

A maiore ad minus

In logic, a maiore ad minus describes a simple and obvious inference from a claim about a stronger entity, greater quantity, or general class to one about a weaker entity, smaller quantity, or specific member of that class:

A minore ad maius

The reverse, less known and less frequently applicable argument is a minore ad maius, which denotes an inference from smaller to bigger.

In law

"Argumentum a maiori ad minus" (from the greater to the smaller) – works in two ways: An a fortiori argument is sometimes considered in terms of analogical reasoning – especially in its legal applications. Reasoning a fortiori posits not merely that a case regulated by precedential or statutory law and an unregulated case should be treated alike since these cases sufficiently resemble each other, but that the unregulated case deserves to be treated in the same way as the regulated case in a higher degree. The unregulated case is here more similar (analogues) to the regulated case than this case is similar (analogues) to itself.