Particular Affirmative
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logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, a categorical proposition, or categorical statement, is a
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
s using categorical statements (i.e.,
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
s) forms an important branch of
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
that began with the
Ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cultu ...
. The Ancient Greeks such as
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
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, \forall _\rightarrow P_xequiv \forall neg S_\lor P_x/math>) *No ''S'' are ''P''. (''E'' form, \forall _\rightarrow \neg P_xequiv \forall neg S_\lor \neg P_x/math>) *Some ''S'' are ''P''. (''I'' form, \exists _\land P_x/math>) *Some ''S'' are not ''P''. (''O'' form, \exists _\land\neg P_x/math>) Surprisingly, 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 In term logic (a branch of philosophical logic), the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate ''On Interpre ...
, 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 An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" ...
, 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 George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ire ...
) 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 A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
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 predicate calculus First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quan ...
, they still retain practical value in addition to their historic and
pedagogical Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political and Developmental psychology, psychological development of le ...
significance.


Translating statements into standard form

Sentences in
natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
may be translated into standard forms. In each row of the following chart, ''S'' corresponds to the subject of the example sentence, and ''P'' corresponds to the
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
. 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 ' (I affirm), referring to the affirmative propositions ''A'' and ''I'', and ' (I deny), referring to the negative propositions ''E'' and ''O''.


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.) 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''. 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. 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 A Venn diagram is a widely used diagram style that shows the logical relation between set (mathematics), sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple ...
s are sometimes helpful when trying to understand the distribution of terms for the four forms.


''A'' form

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

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 In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is a particular case of subject and predicate class distribution.


''I'' form

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 ''De dicto'' and ''de re'' are two phrases used to mark a distinction in intensional statements, associated with the intensional operators in many such statements. The distinction is used regularly in metaphysics and in philosophy of language. T ...
''), or it could mean that "Some Americans (in particular, Albert and Bob) are conservatives" (''
de re ''De dicto'' and ''de re'' are two phrases used to mark a distinction in intensional statements, associated with the intensional operators in many such statements. The distinction is used regularly in metaphysics and in philosophy of language. T ...
'').


''O'' form

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 ''De dicto'' and ''de re'' are two phrases used to mark a distinction in intensional statements, associated with the intensional operators in many such statements. The distinction is used regularly in metaphysics and in philosophy of language. T ...
'' interpretation of the
intensional statement In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase, or another symbol. In the case of a word, the word's definition often implies an intension. For instance, the intensions of ...
(\Box \exists l_\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
single person In legal definitions for interpersonal status, a single person refers to a person who is not in committed relationships, or is not part of a civil union. In common usage, the term 'single' is often used to refer to someone who is not involved in ...
, Albert, the relationship becomes clearer; This is the ''
de re ''De dicto'' and ''de re'' are two phrases used to mark a distinction in intensional statements, associated with the intensional operators in many such statements. The distinction is used regularly in metaphysics and in philosophy of language. T ...
'' interpretation of the intensional statement (\exists \Box l_\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").


Criticism

Peter Geach Peter Thomas Geach (29 March 1916 – 21 December 2013) was a British philosopher who was Professor of Logic at the University of Leeds. His areas of interest were philosophical logic, ethics, history of philosophy, philosophy of religion and t ...
and others have criticized the use of distribution to determine the validity of an argument. 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," which is perhaps a closer translation to
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
's original form for this type of statement.


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 set complements. The class complement of a set P will be called "non-P".


Conversion

The simplest operation is conversion where the subject and predicate terms are interchanged. Note that this is not same to the implicational converse in the modern logic where a material implication statement P \rightarrow Q is converted (conversion) to another material implication statement Q \rightarrow P. The both conversions are equivalent only for A type categorical statements. 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

Obversion changes the ''quality'' (that is the affirmativity or negativity) of the statement and the predicate term. 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 the modern forms of the four categorical statements, the negation of the statement corresponding to a predicate term P, \neg Px, is interpreted as a predicate term 'non-P' in each categorical statement in obversion. The equality of Px = \neg (\neg Px) can be used to obvert affirmative categorical statements. 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

Contraposition is the process of simultaneous interchange and
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
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 In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as Proof by contrapositive, proof by contraposition. The cont ...
(also called transposition) in the modern logic stating that material implication statements P \rightarrow Q and \neg Q \rightarrow \neg P are logically equivalent. The both contrapositions are equivalent only for A type categorical statements.


See also

*
Square of opposition In term logic (a branch of philosophical logic), the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate ''On Interpre ...
*
Term logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, th ...


Notes


References

* * * *


External links


ChangingMinds.org: Categorical propositionsCatlogic: An open source computer script written in Ruby to construct, investigate, and compute categorical propositions and syllogisms
{{Aristotelian logic Term logic Propositions