Fundamental logical statements, as they are used e.g. in syllogisms.

There are four different types of categorical statements:

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.

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.

However, it should not be forgotten that these verbs can also be affected by equivocation. For more information and an example, see Equivocation § Copula and conjunctions.

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.

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.

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 so-called generic generalisations can be both universal as well as existential statements, and should therefore be avoided.

Also a statement like: “Socrates is an Athenian” does not at first glance seem to fit into the scheme of universal or existential propositions. It is not immediately clear whether this should be understood as “there exists at least one person with the name ‘Socrates’ …”, or rather as “all persons from the group [with exactly one member] that is named ‘Socrates’ …“.

In fact, both interpretations are possible. However, since the second form (as a universal statement) is the stronger form, it is usually chosen.

Thus the well-known example of a syllogism could also be formulated (somewhat awkwardly) as follows:

All humans are mortal.
All members of the group, which consists only of the well-known philosopher named ‘Socrates’, are humans.
Therefore: All members of the group consisting only of the well-known philosopher named ‘Socrates’ are mortal.

It is certainly easy to see why the shorter form is usually preferred in such cases.

A syllogism that contains such a singular statement is sometimes referred to as a “quasi-syllogism”.

As explained above, universal statements do not imply existence. This means that a propositions like the following is true even if (or rather: precisely because) there are no unicorns:

All unicorns are immortal.

Existential statement, on the other hand, at least presuppose existence, as in:

Some rectangles are squares. (equivalent to: “There exist rectangles which are squares”)

In the case of positive existential statements (as in this example here), the statement implies the existence of examples from both the subject and the predicate extension. With negative existential statements, this only applies to the subject extension.

“Distributivity” refers to the property of a term to refer to the total extension of all objects denoted as “distributed”, as opposed to undistributed terms which refer only to a subset.

For example in the following statement:

All dogs are mammals.

This is a statement about – quite literally – all dogs. This means that we can replace the term “dogs” with any subset of this group. Valid derived statements include: “all dachshunds are mammals”, “all golden retrievers are mammals“ or even “my neighbour’s dog is a mammal”. We call such a term “distributed”.

In contrast, the statement only refers to a part of the mammals (namely the part of the dogs). Therefore, we cannot make any statements about subsets without further information. We can not, for example, say “all dogs are marsupials”, even though marsupials are a subgroup of mammals. In this situation, the term “mammals” is undistributed.

For more information, see the article on Distributivity.

• Existential quantification
• Syllogism
• Universal quantification

• Categorical proposition on Wikipedia