====== Categorical statement (logic) ====== Either affirmative (positive) or negative, existential or universal statements, as they are used e.g. in [[glossary:syllogism|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 [[glossary:universal_quantification|universal statement]] (type "**E**"): > ~~There exist no S, which is P.~~

==== Minimal existence ==== There is no implication on the [[mathematics:statistics:glossary/prevalence|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 ====== * [[glossary:existential_quantification|Existential quantification]] * [[glossary:syllogism|Syllogism]] * [[glossary:universal_quantification|Universal quantification]] ===== More information ===== * [[wp>Categorical proposition]] on //Wikipedia// {{page>templates:banner#Short-BG-Logic&noheader&nofooter}}