Argumentum a fortiori


Argumentum a fortiori is a form of argumentation that draws upon existing confidence in a proposition to argue in favour 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, although he writes it is "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 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, 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.

Examples

  • If a person is dead, then one can, with equal or greater certainty, argue a fortiori that the person is not breathing. "Being dead" trumps other arguments that might be made to show that the person is dead, such as "he is no longer breathing"; therefore, "he is no longer breathing" is an extrapolation from his being dead and is a derivation of this strong argument.
  • If it is known that a person is dead on a certain date, it may be inferred a fortiori that he is exempted from the suspect list for a murder that took place on a later date, viz. "Allen died on 2 April, therefore, a fortiori, Allen did not murder Joe on 3 April".
  • If driving 10 km over the speed limit is punishable by a fine of $50, it can be inferred a fortiori that driving 20 km over the speed limit is also punishable by a fine of at least $50.
  • If a teacher refuses to add 5 points to a student's grade because the student does not deserve an additional 5 points, it can be inferred a fortiori that the teacher will also refuse to raise the student's grade by 10 points.
  • If married couples are forbidden from sharing a room, for example in a hotel, it can be inferred a fortiori that unmarried couples will also be unable to share a room.

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:
  1. For any set A, there does not exist a function mapping A onto its powerset P.
  2. There cannot exist a one-to-one correspondence between A and P.
Because bijections are a special case of onto functions, it automatically follows that if holds, then will also hold. Therefore, any proof of also suffices as a proof of. Thus, 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:
  • From general to particular
  • From greater to smaller
  • From the whole to the part
  • From stronger to weaker

''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" – works in two ways:
  • "who may more, all the more so may less" and relates to the statutory provisions that permit to do something
  • "who is ordered more, all the more so, is ordered less" and relates to the statutory provisions that order to do something
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 to the regulated case than this case is similar to itself.