View source for Categorical proposition

{{short description|Statement regarding whether or not an item belongs to a category}}
{{refimprove|date=February 2013}}
In [[logic]], a '''categorical proposition''', or '''categorical statement''', is a [[proposition]] that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term'').<ref>{{cite book |last=Churchill |first=Robert Paul |title=Logic: An Introduction |year=1990 |publisher=St. Martin's Press |location=New York |isbn=0-312-02353-7 |oclc=21216829 |edition=2nd |page=143 |quote=A categorical statement is an assertion or a denial that all or some members of the subject class are included in the predicate class.}}</ref> The study of [[argument]]s using categorical statements (i.e., [[syllogism]]s) forms an important branch of [[deductive reasoning]] that began with the [[Ancient Greeks]].

The Ancient Greeks such as [[Aristotle]] identified four primary distinct types of categorical proposition and gave them standard forms (now often called ''A'', ''E'', ''I'', and ''O''). If, abstractly, the subject category is named ''S'' and the predicate category is named ''P'', the four standard forms are:
* All ''S'' are ''P''. (''A'' form)
* No ''S'' are ''P''. (''E'' form)
* Some ''S'' are ''P''. (''I'' form)
* Some ''S'' are not ''P''. (''O'' form)
A large number of sentences may be translated into one of these canonical forms while retaining all or most of the original meaning of the sentence. Greek investigations resulted in the so-called [[square of opposition]], which codifies the logical relations among the different forms; for example, that an ''A''-statement is contradictory to an ''O''-statement; that is to say, for example, if one believes "All apples are red fruits," one cannot simultaneously believe that "Some apples are not red fruits." Thus the relationships of the square of opposition may allow [[immediate inference]], whereby the truth or falsity of one of the forms may follow directly from the truth or falsity of a statement in another form.

Modern understanding of categorical propositions (originating with the mid-19th century work of [[George Boole]]) requires one to consider if the subject category may be empty. If so, this is called the ''hypothetical viewpoint'', in opposition to the ''existential viewpoint'' which requires the subject category to have at least one member. The existential viewpoint is a stronger stance than the hypothetical and, when it is appropriate to take, it allows one to deduce more results than otherwise could be made. The hypothetical viewpoint, being the weaker view, has the effect of removing some of the relations present in the traditional square of opposition.

Arguments consisting of three categorical propositions — two as premises and one as conclusion — are known as [[categorical syllogism]]s and were of paramount importance from the times of ancient Greek logicians through the Middle Ages. Although formal arguments using categorical syllogisms have largely given way to the increased expressive power of modern logic systems like the [[First-order logic|first-order predicate calculus]], they still retain practical value in addition to their historic and [[Pedagogy|pedagogical]] significance.

==Translating statements into standard form==

Sentences in [[natural language]] may be translated into standard forms. In each row of the following chart, ''S'' corresponds to the [[Subject (grammar)|subject]] of the example sentence, and ''P'' corresponds to the [[Predicate (grammar)|predicate]].

{| class="wikitable"
|-
! Name
! English Sentence
! Standard Form
|-
| align="center" | ''A''
| {{yes|All cats have four legs.}}
| {{yes|All S is P.}}
|-
| align="center" | ''E''
| {{yes|No cats have eight legs.}}
| {{yes|No S is P.}}
|-
| align="center" | ''I''
| {{yes|Some cats are orange.}}
| {{yes|Some S is P.}}
|-
| align="center" | ''O''
| {{yes|Some cats are not black.}}
| {{yes|Some S is not P.}}
|}

Note that "All ''S'' is not ''P''" (e.g., "All cats do not have eight legs") is not classified as an example of the standard forms. This is because the translation to natural language is ambiguous. In common speech, the sentence "All cats do not have eight legs" could be used informally to indicate either (1) "At least some, and perhaps all, cats do not have eight legs" or (2) "No cats have eight legs".

==Properties of categorical propositions==
Categorical propositions can be categorized into four types on the basis of their "quality" and "quantity", or their "distribution of terms". These four types have long been named ''A'', ''E'', ''I'', and ''O''. This is based on the Latin ''{{Lang |la | '''a'''ff'''i'''rmo}}'' (I affirm), referring to the affirmative propositions ''A'' and ''I'', and ''{{Lang |la | n'''e'''g'''o'''}}'' (I deny), referring to the negative propositions ''E'' and ''O''.<ref>{{cite book |last=Churchill |first=Robert Paul |title=Logic: An Introduction |year=1990 |publisher=St. Martin's Press |location=New York |isbn=0-312-02353-7 |oclc=21216829 |edition=2nd |page=144 |quote=During the Middle Ages, logicians gave the four categorical forms the special names of ''A'', ''E'', ''I'', and ''O''. These four letters came from the first two vowels in the Latin word '' 'a''ff''i''rmo' ('I affirm') and the vowels in the Latin 'n''e''g''o' '' ('I deny').}}</ref>

===Quantity and quality===
'''Quantity''' refers to the number of members of the subject class (A ''class'' is a collection or group of things designated by a term that is either subject or predicate in a categorical proposition.<ref>{{Cite web|date=2021-08-25|title=Dictionary|url=http://www.philosophypages.com/dy/c5.htm#class|url-status=live|website=Philosophy Pages|archive-url=https://web.archive.org/web/20010209090207/http://www.philosophypages.com:80/dy/c5.htm |archive-date=2001-02-09 }}</ref>) that are used in the proposition. If the proposition refers to all members of the subject class, it is ''universal''. If the proposition does not employ all members of the subject class, it is ''particular''. For instance, an ''I''-proposition ("Some ''S'' is ''P''") is particular since it only refers to some of the members of the subject class.

'''Quality''' It is described as whether the proposition affirms or denies the inclusion of a subject within the class of the predicate. The two possible qualities are called ''affirmative'' and ''negative''.<ref>{{cite book |last1=Copi |first1=Irving M. |authorlink1=Irving Copi |last2=Cohen |first2=Carl |authorlink2=Carl Cohen (professor) |title=Introduction to Logic |edition=11th |year=2002 |publisher=Prentice-Hall |location=Upper Saddle River, NJ |isbn=0-13-033735-8 |page=185 |quote=Every standard-form categorical proposition is said to have a '''quality''', either affirmative or negative.}}</ref>  For instance, an ''A''-proposition ("All ''S'' is ''P''") is affirmative since it states that the subject is contained within the predicate. On the other hand, an ''O''-proposition ("Some ''S'' is not ''P''") is negative since it excludes the subject from the predicate.

{| class="wikitable w"
|+The Four Aristotelian Propositions
|-
! Name
! Statement
! Quantity
! Quality
|-
| align="center" | ''A''
| All S is P.
| align="center" | universal
| align="center" | affirmative
|-
| align="center" | ''E''
| No S is P.
| align="center" | universal
| align="center" | negative
|-
| align="center" | ''I''
| Some S is P.
| align="center" | particular
| align="center" | affirmative
|-
| align="center" | ''O''
| Some S is not P.
| align="center" | particular
| align="center" | negative
|}

An important consideration is the definition of the word ''some''. In logic, ''some'' refers to "one or more", which is consistent with "all". Therefore, the statement "Some S is P" does not guarantee that the statement "Some S is not P" is also true.

===Distributivity===
The two terms (subject and predicate) in a categorical proposition may each be classified as '''distributed''' or '''undistributed'''. If all members of the term's class are affected by the proposition, that class is ''distributed''; otherwise it is ''undistributed''. Every proposition therefore has one of four possible ''distribution of terms''.

Each of the four canonical forms will be examined in turn regarding its distribution of terms. Although not developed here, [[Venn diagram]]s are sometimes helpful when trying to understand the distribution of terms for the four forms.

====''A'' form (otherwise known as Universal Affirmative)====
An ''A''-proposition distributes the subject to the predicate, but not the reverse. Consider the following categorical proposition: "All dogs are mammals".  All dogs are indeed mammals, but it would be false to say all mammals are dogs. Since all dogs are included in the class of mammals, "dogs" is said to be distributed to "mammals". Since all mammals are not necessarily dogs, "mammals" is undistributed to "dogs".

====''E'' form (otherwise known as Universal Negative)====
An ''E''-proposition distributes bidirectionally between the subject and predicate. From the categorical proposition "No beetles are mammals", we can infer that no mammals are beetles. Since all beetles are defined not to be mammals, and all mammals are defined not to be beetles, both classes are distributed.

The [[empty set]] is a particular case of subject and predicate class distribution.

====''I'' form (otherwise known as Particular Affirmative)====
Both terms in an ''I''-proposition are undistributed. For example, "Some Americans are conservatives". Neither term can be entirely distributed to the other. From this proposition, it is not possible to say that all Americans are conservatives or that all conservatives are Americans. Note the ambiguity in the statement: It could either mean that "Some Americans (or other) are conservatives" (''[[De dicto and de re#Context of modality|de dicto]]''), or it could mean that "Some Americans (in particular, Albert and Bob) are conservatives" (''[[De dicto and de re#Context of modality|de re]]'').

====''O'' form (otherwise known as Particular Negative)====
In an ''O''-proposition, only the predicate is distributed. Consider the following: "Some politicians are not corrupt". Since not all politicians are defined by this rule, the subject is undistributed. The predicate, though, is distributed because all the members of "corrupt people" will not match the group of people defined as "some politicians". Since the rule applies to every member of the corrupt people group, namely, "All corrupt people are not some politicians", the predicate is distributed.

The distribution of the predicate in an ''O''-proposition is often confusing due to its ambiguity. When a statement such as "Some politicians are not corrupt" is said to distribute the "corrupt people" group to "some politicians", the information seems of little value, since the group "some politicians" is not defined; This is the ''[[De dicto and de re#Context of modality|de dicto]]'' interpretation of the [[intensional statement]] (<math>\Box \exists{x} [Pl_{x}\land \neg C_x]</math>), or "Some politicians (or other) are not corrupt". But if, as an example, this group of "some politicians" were defined to contain a [[singular term|single person]], Albert, the relationship becomes clearer; This is the ''[[De dicto and de re#Context of modality|de re]]'' interpretation of the intensional statement (<math>\exists{x} \Box [Pl_{x}\land \neg C_x]</math>), or "Some politicians (in particular) are not corrupt". The statement would then mean that, of every entry listed in the corrupt people group, not one of them will be Albert: "All corrupt people are not Albert". This is a definition that applies to every member of the "corrupt people" group, and is, therefore, distributed.

====Summary====
In short, for the subject to be distributed, the statement must be universal (e.g., "all", "no"). For the predicate to be distributed, the statement must be negative (e.g., "no", "not").{{sfn|Damer|2008|p=82}}

{| class="wikitable"
|-
! rowspan="2" | Name
! rowspan="2" | Statement
! colspan="2" | Distribution
|-
! Subject
! Predicate
|-
| align="center" | ''A''
| All S is P.
| {{yes|distributed}}
| {{no2|undistributed}}
|-
| align="center" | ''E''
| No S is P.
| {{yes|distributed}}
| {{yes|distributed}}
|-
| align="center" | ''I''
| Some S is P.
| {{no2|undistributed}}
| {{no2|undistributed}}
|-
| align="center" | ''O''
| Some S is not P.
| {{no2|undistributed}}
| {{yes|distributed}}
|}

====Criticism====
[[Peter Geach]] and others have criticized the use of distribution to determine the validity of an argument.<ref name="SEP Medieval">{{cite encyclopedia |last=Lagerlund |first=Henrik |date=January 21, 2010 |title=Medieval Theories of the Syllogism |encyclopedia=Stanford Encyclopedia of Philosophy |url=http://plato.stanford.edu/entries/medieval-syllogism |accessdate=2010-12-10}}</ref><ref name=Murphree>{{cite journal |last=Murphree |first=Wallace A. |date=Summer 1994 |title=The Irrelevance of Distribution for the Syllogism |journal=Notre Dame Journal of Formal Logic |volume=35 |number=3 |pages=433–449 |doi=10.1305/ndjfl/1040511349 |doi-access=free }}</ref>

It has been suggested that statements of the form "Some A are not B" would be less problematic if stated as "Not every A is B,"{{sfn|Geach|1980|pp=62–64}} which is perhaps a closer translation to [[Aristotle]]'s original form for this type of statement.<ref name="SEP Square">{{cite encyclopedia |title=The Traditional Square of Opposition |url=http://plato.stanford.edu/entries/square/ |date=2006-10-01 |accessdate=2010-12-10 |encyclopedia=Stanford Encyclopedia of Philosophy |last=Parsons |first=Terence}}</ref>

Another criticism is that there is a little step from "All corrupt people are not some politicians" to "All corrupt people are not politicians" (whether meaning "No corrupt people are politicians" or "Not all corrupt people are politicians", which are different from the original "Some politicians are not corrupt"), or to "Every corrupt person is not some politician" (also different).

==Operations on categorical statements==
There are several operations (e.g., conversion, obversion, and contraposition) that can be performed on a categorical statement to change it into another. The new statement may or may not be equivalent to the original. [In the following tables that illustrate such operations, at each row, boxes are green if statements in one green box are equivalent to statements in another green box, boxes are red if statements in one red box are inequivalent to statements in another red box. Statements in a yellow box means that these are implied or valid by the statement in the left-most box when the condition stated in the same yellow box is satisfied.]

Some operations require the notion of the ''class complement''. This refers to every [[Domain of discourse|element under consideration]] which is ''not'' an element of the class. Class complements are very similar to [[Complement (set theory)|set complements]]. The class complement of a set P will be called "non-P".

===Conversion===
{{main|Converse (logic)|l1=Logical conversion}}
The simplest operation is '''conversion''' where the subject and predicate terms are interchanged. Note that this is not same to the [[Converse (logic)|implicational converse]] in the modern logic where a [[Material conditional|material implication]] statement <math>P \rightarrow Q</math> is converted (conversion) to another material implication statement <math>Q \rightarrow P</math>. Both conversions are equivalent only for A type categorical statements.
{| class="wikitable"
|-
! Name
! Statement
! Converse / Obverted Converse
! Subaltern / Obverted Subaltern / Condition of Validity
! Converse ''per accidens'' / Obverted Converse ''per accidens'' / Condition of Validity
|-
| align="center" | ''A''
| {{no2|All S is P.}}
| {{no2|All P is S.<br>No P is non-S.}}
| {{partial|Some S is P.<br>Some S is not non-P.<br>(''if S exists'')}}
| {{partial|Some P is S.<br>Some P is not non-S.<br>(''if S exists'')}}
|-
| align="center" | ''E''
| {{yes|No S is P.}}
| {{yes|No P is S.<br>All P is non-S.}}
| {{partial|Some S is not P.<br>Some S is non-P.<br>(''if S exists'')}}
| {{partial|Some P is not S.<br>Some P is non-S.<br>(''if P exists'')}}
|-
| align="center" | ''I''
| {{yes|Some S is P.}}
| {{yes|Some P is S.<br>Some P is not non-S.}}
|rowspan=2 colspan=3 {{n/a}}
|-
| align="center" | ''O''
| {{no2|Some S is not P.}}
| {{no2|Some P is not S.<br>Some P is non-S.}}
|}
From a statement in ''E'' or ''I'' form, it is valid to conclude its converse (as they are equivalent). This is not the case for the ''A'' and ''O'' forms.

===Obversion===
{{main|Obversion}}
Obversion changes the ''quality'' (that is the affirmativity or negativity) of the statement and the predicate term.<ref>{{cite book |last1=Hausman |first1=Alan |last2=Kahane |first2=Howard |authorlink2=Howard Kahane |last3=Tidman |first3=Paul |title=Logic and Philosophy: A Modern Introduction |year=2010 |publisher=Thomson Wadsworth/Cengage learning |location=Australia |isbn=9780495601586 |page=[https://books.google.com/books?id=EH3l1EW6kKgC&pg=PA326 326] |url=http://www.cengagebrain.com/shop/en/US/storefront/US?cmd=CLHeaderSearch&fieldValue=9780495601586 |edition=11th |accessdate=26 February 2013 |quote=In the process of '''obversion''', we change the quality of a proposition (from affirmative to negative or from negative to affirmative), and then replace its predicate with the negation or '''complement''' of the predicate.}}</ref> For example, by obversion, a universal affirmative statement become a universal negative statement with the predicate term that is the class complement of the predicate term of the original universal affirmative statement. In [[Square of opposition#Summary|the modern forms of the four categorical statements]], the negation of the statement corresponding to a predicate term P, <math>\neg Px</math>, is interpreted as a predicate term 'non-P' in each categorical statement in obversion. The equality of <math>Px = \neg (\neg Px)</math> can be used to obvert affirmative categorical statements.

{| class="wikitable"
|-
! Name
! Statement
! Obverse (obverted)
|-
| align="center" | ''A''
| {{yes|All S is P.}}
| {{yes|No S is non-P.}}
|-
| align="center" | ''E''
| {{yes|No S is P.}}
| {{yes|All S is non-P.}}
|-
| align="center" | ''I''
| {{yes|Some S is P.}}
| {{yes|Some S is not non-P.}}
|-
| align="center" | ''O''
| {{yes|Some S is not P.}}
| {{yes|Some S is non-P.}}
|}

Categorical statements are logically equivalent to their obverse. As such, a Venn diagram illustrating any one of the forms would be identical to the Venn diagram illustrating its obverse.

===Contraposition===
{{main|Contraposition (traditional logic)}}Contraposition is the process of simultaneous interchange and [[negation]] of the subject and predicate of a categorical statement. It is also equivalent to converting (applying conversion) the obvert (the outcome of obversion) of a categorical statement. Note that this contraposition in the traditional logic is not same to [[contraposition]] (also called transposition) in the modern logic stating that [[Material conditional|material implication]] statements <math>P \rightarrow Q</math> and <math>\neg Q \rightarrow \neg P</math> are logically equivalent. Both contrapositions are equivalent only for A type categorical statements.
{| class="wikitable"
|-
! Name
! Statement
! Contrapositive / Obverted Contrapositive
! Contrapositive ''per accidens'' / Obverted Contrapositive ''per accidens'' / Condition of Validity
|-
| align="center" | ''A''
| {{yes|All S is P.}}
| {{yes|All non-P is non-S.<br>No non-P is S.}}
| {{partial|Some non-P is non-S.<br>Some non-P is not S.<br>(''if non-P exists'')}}
|-
| align="center" | ''E''
| {{no2|No S is P.}}
| {{no2|No non-P is non-S.<br>All non-P is S.}}
| {{partial|Some non-P is not non-S.<br>Some non-P is S.<br>(''if S exists'')}}
|-
| align="center" | ''I''
| {{no2|Some S is P.}}
| {{no2|Some non-P is non-S.<br>Some non-P is not S.}}
|rowspan=2 colspan=2 {{n/a}}
|-
| align="center" | ''O''
| {{yes|Some S is not P.}}
| {{yes|Some non-P is not non-S.<br>Some non-P is S.}}
|}

==Treatment in first-order logic==
First-order logic is a much more expressive logic than that given by categorical propositions.
In first order logic, the four forms can be expressed as:
* ''A'' form: <math>\forall{x} [S_{x}\rightarrow P_x]\equiv \forall{x} [\neg S_{x}\lor P_x]</math>
* ''E'' form: <math>\forall{x} [S_{x}\rightarrow \neg P_x]\equiv \forall{x} [\neg S_{x}\lor \neg P_x]</math>
* ''I'' form: <math>\exists{x} [S_{x}\land P_x]</math>
* ''O'' form: <math>\exists{x} [S_{x}\land\neg P_x]</math>

==See also==
* [[Square of opposition]]
* [[Term logic]]

==Notes==
{{Reflist}}

==References==
* {{cite book |last1=Copi |first1=Irving M. |authorlink1=Irving Copi |last2=Cohen |first2=Carl |authorlink2=Carl Cohen (professor) |title=Introduction to Logic |publisher=Prentice Hall |year=2009 |isbn=978-0-13-136419-6}}
* {{cite book |last=Damer |first=T. Edward |authorlink=T. Edward Damer |title=Attacking Faulty Reasoning |year=2008 |isbn=978-0-495-09506-4 |publisher=Cengage Learning}}
* {{cite book |last=Geach |first=Peter |authorlink=Peter Geach |title=Logic Matters |year=1980 |isbn=978-0-520-03847-9 |publisher=University of California Press}}
* {{cite book |last=Baum |first=Robert |authorlink=Robert Baum |title=Logic |year=1989 |isbn=0-03-014078-1 |publisher=Holt, Rinehart and Winston, Inc.}}

==External links==
* [http://changingminds.org/disciplines/argument/making_argument/categorical_propositions.htm ChangingMinds.org: Categorical propositions]
* [https://github.com/jeffreycwitt/catlogic Catlogic: An open source computer script written in Ruby to construct, investigate, and compute categorical propositions and syllogisms]
{{Aristotelian logic}}
[[Category:Term logic]]
[[Category:Ancient Greek logic]]
[[Category:Propositions]]