# 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` holds true,
then `x > 2` also holds true.

We can consider `x > 3` as the stronger of the two statements, as it restricts the possible values for `x` more.

## Other names

• 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”.

### 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.