# Argumentum a Fortiori

Lat.: “argument from the stronger [argument]”: a principle of logic and law that a stronger argument supports a weaker one.

For example, in mathematics:

If`x > 3`

holdstrue,

then`x > 2`

also holdstrue.

We can consider `x > 3`

as the *stronger* of the two statements, as it restricts the possible values for `x`

more.

## Other names

Traditionally, we distinguish between:

*Argumentum a maiore ad minus*(“argument from the greater to the smaller”)*Argumentum a minori ad maius*(“argument from the smaller to the greater”)

However, this distinction is not relevant to the topic of this website.

## Description

The principle “*argumentum a fortiori* describes the fact that a *weaker* argument can be derived from a *stronger* one.

### Distributivity

A good example is the principle of ”distributivity“ which applies, among other things, to categorical statements. It describes, under which circumstances a category term can also referr to its parts or sub-categories. Thus, in a universal statement such as “all *dogs* are *mammals*”, the *subject* (here: “dogs”) is distributed and thus also refers to, for example, *poodles*, *greyhounds*, *watchdogs*, or even the *our neighbor’s dachshund*, etc. Thus, from the *strong* statement “all dogs are mammals” *weaker* statements such as ”*our neighbor’s dachshund* is a *mammal* “ can be derived.

### Universal and existential propositions

Another example concerns the relation of universal to existential propositions. For example, from the *strong* universal statement “all *dogs* are *mammals*”, we can derive the *weaker* “there exist *dogs*, which are *mammals*”.

**Note:** this derivation is only possible if we actually *know* that the subject of the statement *exists*, that is, that it does not refer to an empty extension. For more information, see ☞ Existential Fallacy.

### Syllogisms

This conclusion from all-phrases to existence-phrases also leads to the fact that for all syllogism-forms, where an all-phrase is in the conclusion, there is also a weaker variant, where only an existence-phrase is concluded. For example, the *weaker* variant of the Modus Barbara ◈ is the Modus Barbari ◈.

Here, too, it is important to note that the conclusion to an existential proposition is accompanied by an existential introduction, which has to be proved as a secondary condition.