Categorical proposition: Difference between revisions
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In [[logic]], a '''categorical proposition''', or '''categorical statement''', is a [[proposition]] that asserts or denies members of one category (the ''subject term'') as belonging to 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. |
In [[logic]], a '''categorical proposition''', or '''categorical statement''', is a [[proposition]] that asserts or denies members of one category (the ''subject term'') as belonging to 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. |
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Modern understanding of categorical propositions (originating with the mid-19<sup>th</sup> 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 has the effect of removing some of the relations present in the traditional square of opposition. |
Modern understanding of categorical propositions (originating with the mid-19<sup>th</sup> 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 has the effect of removing some of the relations present in the traditional square of opposition. |
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Arguments consisting of exactly 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 predicate calculus]], they still retain practical value in addition to their historic and pedagogical significance. |
Arguments consisting of exactly 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 predicate calculus]], they still retain practical value in addition to their historic and pedagogical significance. |
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==Translating statements into standard form== |
==Translating statements into standard form== |
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==Properties of categorical propositions== |
==Properties of categorical propositions== |
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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 '''''a'''ff'''i'''rmo'' (I affirm), referring to the affirmative propositions ''A'' and ''I'', and ''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 ' |
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 '''''a'''ff'''i'''rmo'' (I affirm), referring to the affirmative propositions ''A'' and ''I'', and ''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> |
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===Quality and quantity=== |
===Quality and quantity=== |
Revision as of 06:44, 14 February 2013
This article needs additional citations for verification. (February 2013) |
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies members of one category (the subject term) as belonging to another (the predicate term).[1] The study of arguments using categorical statements (i.e., syllogisms) 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 surprisingly 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. Such relations 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 has the effect of removing some of the relations present in the traditional square of opposition.
Arguments consisting of exactly three categorical propositions — two as premises and one as conclusion — are known as categorical syllogisms 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, they still retain practical value in addition to their historic and pedagogical significance.
Translating statements into standard form
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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 affirmo (I affirm), referring to the affirmative propositions A and I, and nego (I deny), referring to the negative propositions E and O.[2]
Quality and quantity
Quality refers to whether the proposition affirms or denies the inclusion of a subject within the class of the predicate. The two qualities are affirmative and negative. For instance, the A-proposition ("All S are P") is affirmative since it states that the subject is contained within the predicate. On the other hand, the O-proposition ("Some S are not P") is negative since it excludes the subject from the predicate.
Quantity refers to the amount of members of the subject class 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, the I-proposition ("Some S are P") is particular since it only refers to some of the members of the subject class.
Name | Statement | Quantity | Quality |
---|---|---|---|
A | All S are P. | universal | affirmative |
E | No S are P. | universal | negative |
I | Some S are P. | particular | affirmative |
O | Some S are not P. | particular | negative |
An important consideration is the definition of the word some. In logic, some refers to "one or more", which could mean "all". Therefore, the statement "Some S are P" does not guarantee that the statement "Some S are not P" is also true.
Distribution of terms
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 diagrams 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.
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.
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 like "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. But if, as an example, this group of "some politicians" were defined to contain a single person, Albert, the relationship becomes more clear. The statement would then mean, 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").[3]
Name | Statement | Distribution | |
---|---|---|---|
Subject | Predicate | ||
A | All S are P. | distributed | undistributed |
E | No S are P. | distributed | distributed |
I | Some S are P. | undistributed | undistributed |
O | Some S are not P. | undistributed | distributed |
Criticism
Peter Geach and others have criticized the use of distribution to determine the validity of an argument.[4][5] 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,"[6] which is perhaps a closer translation to Aristotle's original form for this type of statement.[7]
Operations on categorical statements
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There are several operations that can be performed on a categorical statement that change it into another. These may or may not be equivalent to the original.
Conversion
The simplest operation is conversion where the subject and predicate terms are interchanged.
Name | Statement | Converse |
---|---|---|
A | All S are P. | All P are S. |
E | No S are P. | No P are S. |
I | Some S are P. | Some P are S. |
O | Some S are not P. | Some P are not S. |
From a statement in E or I form, it is valid to conclude its converse. This is not the case for the A and O forms.
Obversion
Contraposition
See also
Notes
- ^ Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 143. ISBN 0-312-02353-7. OCLC 21216829.
A categorical statement is an assertion or a denial that all or some members of the subject class are included in the predicate class.
- ^ Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 144. ISBN 0-312-02353-7. OCLC 21216829.
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 affirmo' ('I affirm') and the vowels in the Latin 'nego ('I deny').
- ^ Damer 2008, p. 82.
- ^ Lagerlund, Henrik (2010-01-21). "Medieval Theories of the Syllogism". Stanford Encyclopedia of Philosophy. Retrieved 2010-12-10.
- ^ Murphree, Wallace A. (Summer 1994). "The Irrelevance of Distribution for the Syllogism". Notre Dame Journal of Formal Logic. 35 (3).
- ^ Geach 1980, pp. 62–64.
- ^ Parsons, Terence (2006-10-01). "The Traditional Square of Opposition". Stanford Encyclopedia of Philosophy. Retrieved 2010-12-10.
References
- Copi, Irving M.; Cohen, Carl (2009). Introduction to Logic. Prentice Hall. ISBN 978-0-13-136419-6.
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(help) - Damer, T. Edward (2008). Attacking Faulty Reasoning. Cengage Learning. ISBN 978-0-495-09506-4.
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ignored (help) - Geach, Peter (1980). Logic Matters. University of California Press. ISBN 978-0-520-03847-9.
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