====== 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 =====

Traditionally, we distinguish between:

  * <i :la>Argumentum a maiore ad minus</i> (“argument from the greater to the smaller”)
  * <i :la>Argumentum a minori ad maius</i> (“argument from the smaller to the greater”)

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

===== Description =====

The principle "<i :la>argumentum a fortiori</i> describes the fact that a //weaker// argument can be derived from a //stronger// one.

==== Distributivity ====

A good example is the principle of "[[glossary:distributivity|distributivity]]" which applies, among other things, to [[glossary:categorical_statement|categorical statements]]. It describes, under which circumstances a category term can also referr to its parts or sub-categories. Thus, in a [[glossary:universal_quantification|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//".

<aside box>
**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 [[glossary:extension|extension]]. For more information, see <span maniculus "see:">[[logic:formal_fallacies:existential|Existential Fallacy]]</span>.
</aside>

==== 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 [[logic:inferences:modus_barbara:index|Modus Bar­bara]]<span noprint> [[app>#barbara|Show in Syllogism-Finder App]]</span> is the [[logic:inferences:modus_barbara:modus_barbari|Modus Bar­bari]]<span noprint> [[app>#barbari|Show in Syllogism-Finder App]]</span>.

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.