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Categorical statement (logic)

Either affirmative (positive) or negative, existential or universal statements, as they are used e.g. in syllogisms.

Thus, there are four different types of categorical statements:

Other statement forms

Besides the formulations chosen here, there are various other ways of expressing the same concepts. However, some formulations have specific problems that need to be taken into account.

Using other verbs

In principle, statements using other verbs can be transformed into forms that use some for of “is”. For example:

All the pupils go to school.

is equivalent to:

All pupils are in the group of those who go to school.

These rephrased statements are usually a bit long-winded and clumsy, but they express the same idea. Using other verbs is therefore generally unproblematic and does not change the form of the statement.

Implicit or explicit existence

Existential statements generally imply – as the name suggests – the existence of something. That means that a statement of the form “some S are P” implies that S is not empty, i.e. that at least one S exists (note that existence is not automatically implied in universal statements).

This implication can be made explicit by chosing other formulations for such statements, e.g. as follows for type “I”:

There exist S which are P.

Likewise for type “O”:

There exist S, which are not P.

However, even without making it explicit, the existence is always implied in this type of statement.

Please note: the following is not a negative existential statement (type “O”), but a negative universal statement (type “E”):

There exist no S, which is P.

Minimal existence

There is no implication on the prevalence of the described phenomenon. For an existential statement to be true it is enough if a single example exists.

This can be made more explicit by using formulations like the following:

At least one S is P.
There exist one or more S which are P.

Also these are logically equivalent to the other forms listed on this page. Even if the word “some” is used, this does not imply that the statement refers to more than one subject.

Syntactic ambiguity

In addition to the alternative forms of statements mentioned above, there are also those that are expressly discouraged to use because they are ambiguous. This applies in particular (but not exclusively) to the following forms:

S are P.
S are not P.

These can be interpreted both as universal as well as existential statements, and should therefore be avoided.

See also

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