In logic, categorical propositions , or categorical statements , are propositions that affirm or deny that all or part members of one category (subject term ) include in another (predicate term ). The study of arguments using category statements (ie, syllogism) forms an important branch of deductive reasoning that began with Ancient Greece.
Ancient Greeks such as Aristotle identified the four main major types of categorical propositions and standardized ones (now often called A , E , I , and O ). If, in abstract, the subject category is named S and a predicate category named P , four standard shapes are:
- All S are P . (Form A )
- Not S is P . (Form E )
- Some S are P . ( I form)
- Some S are not P . (Form O )
Surprisingly, a large number of sentences can be translated into one of these canonical forms while retaining all or most of the original meaning of the sentence. The Greek inquiry produced what was called a square of opposition, which codified the logical connection between the various forms; for example, that the statement A is contrary to O -statement; that is, for example, if someone believes "All apples are red fruit," one can not simultaneously believe that "Some apples are not red fruit." Thus the quadratic squad of opposition may permit a direct conclusion, in which the truth or falseness of one form may follow directly from the truth or falsity of the statement in another.
The modern understanding of the categorical proposition (originating from George Boole's 19th century work) requires one to consider whether the subject category may be empty. If so, this is called a hypothetical point of view, contrary to an existential viewpoint that requires that the subject category has at least one member. The existential viewpoint is a stronger attitude than hypothetical and, when appropriate to take, allows one to deduce more results than can otherwise be made. The hypothetical view, as a weaker view, has the effect of eliminating some of the relationships that exist in the traditional square of opposition.
The argument consisting of three categorical propositions - two as premises and one as a conclusion - is known as categorical syllogism and very important from the days of ancient Greek logic to the Middle Ages. Although the formal argument using categorical syllogism has largely given way to the enhancement of the expressive forces of modern logic systems such as first-order predicate calculus, they still retain a practical value in addition to their historical and pedagogical significance.
Video Categorical proposition
Translate the statement into standard form
Sentences in natural language can be translated into standard form. In each line of the following chart, S matches the subject of the sample sentence, and P matches the predicate.
Note that "All S is not P " (eg, "All cats do not have eight legs") are not classified as examples of standard shapes. This is because translation to natural language is unclear. In a general speech, the phrase "All cats do not have eight legs" can be used informally to show either (1) "At least some, and perhaps all, cats do not have eight legs" or (2) "No cat has eight legs" feet ".
Maps Categorical proposition
Properties of categorical propositions
Category propositions can be categorized into four types based on their "quality" and "quantity", or "term distribution". 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 proposition A and I , and n e g o (I refuse), referring to the negative proposition E and O .
Number and quality
Quantity refers to the number of subject class members used in the proposition. If the proposition refers to all members of the subject class, it is universal . If the proposition does not use all members of the subject class, it is special . For example, a I -proposition ("Some S is P ") is special because it refers only to some members of the class subject.
Quality describes whether the proposition confirms or rejects the inclusion of a subject in a predicate class. Two qualities that may be called affirmative and negative . For example, an A -proposition ("All S is P ") is affirmative because it states that the subject is in predicate. On the other hand, O -proposition ("Some S is not P ") is negative because it does not include the subject of the predicate.
An important consideration is the definition of the word some . In logic, some refers to "one or more", which can mean "all". Therefore, the statement "Some S is P" does not guarantee that the statement "Some S not P" is also true.
Distributivity
Both terms (subject and predicate) in each categorical proposition can be classified as distributed or not shared . If all members of the term class are influenced by the proposition, the class is distributed ; otherwise not shared . Therefore, each proposition has one of four possible distribution terms .
Each of the four canonical forms will be examined in turn regarding the distribution of its terms. Although not developed here, Venn diagrams are sometimes helpful when trying to understand the distribution of terms for four forms.
Forms
An A -proposition distributes subject to predicate, but not vice versa. Consider the following categorical proposition: "All dogs are mammals". All dogs are mammals, but it is wrong to say all mammals are dogs. Since all dogs belong to the mammalian class, "dogs" are said to be distributed to "mammals". Since all mammals are not always dogs, "mammals" are not distributed to "dogs".
E form
An E -proposition distributes both directions between subject and predicate. From the categorical proposition "No beetle is a mammal", we can conclude that no mammals become beetles. Since all the beetles are defined not to be a mammal, and all mammals are defined not to be beetles, both classes are distributed.
I form
Both terms in I -proposition are not distributed. For example, "Some Americans are conservative". Both terms can not be completely distributed to others. From this proposition, it is impossible to say that all Americans are conservative or all conservatives are Americans.
O form
In 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 not distributed. The predicate is distributed because all members of the "corrupt person" will not match the group of people defined as "some politicians". Because the rules apply to every member of a corrupt group of people, that is, "All corrupt people are not politicians", the predicate is distributed.
The predicate distribution in O -proposition is often confusing because of its ambiguity. When a statement such as "Some politicians are not corrupt" is said to distribute the group of "corrupt people" to "some politicians", the information appears to be of little value, since the "multiple politicians" group is not defined. But if, for example, this group of "politicians" is defined to contain one person, Albert, the relationship becomes clearer. The statement would then mean that, of every entry listed in a group of corrupt people, none of them would be Albert: "All corrupt people are not Albert". This is the definition that applies to every member of the group of "corrupt people", and is therefore distributed.
Summary
In short, for a distributed subject, the statement must be universal (eg, "all", "no"). For the predicate to be distributed, the statement must be negative (eg, "no", "no").
Criticism
Peter Geach and others criticized the use of distribution to determine the validity of an argument. It has been suggested that the statements of the form "Several A not B" will become less problematic if expressed as "Not every A is B," which may be a translation closer to Aristotle's original form for this type of statement.
Operation on category statement
There are several operations (eg, conversions, obversion, and contraposition) that can be done on a category statement to convert it to another. The new statement may or may not be equivalent to the original. [In the following table illustrating such an operation, lines with equivalent statements should be marked in green, while those with unequal statements should be marked in red.]
Some operations require the supplementary idea of ​​the class. This refers to any element under consideration that is not is an element of the class. Class complement is very similar to the complement circuit. The complementary class of the set P will be called "non-P".
Conversions
The simplest operation is the conversion in which the subject and the predicate term are interchangeable.
From a statement in the form of E or I , it is legitimate to conclude otherwise. This is not the case for forms A and O .
Obversion
Obversion changes the quality of (ie affirmative or negativity) of the statement and predicate term. For example, a universal affirmative statement would be a universal negative statement.
Categorical statements are logically equivalent to their fronts. Thus, the Venn diagram illustrating one of the forms would be identical to the Venn diagram illustrating the front.
Contraposition
See also
- Opposition square
- The term logic
Note
References
- Copi, Irving M.; Cohen, Carl (2009). Introduction to Logic . Prentice Hall. ISBN 978-0-13-136419-6.
- Damer, T. Edward (2008). Striking Reasoning Failed . Learning Cengage. ISBN: 978-0-495-09506-4.
- Geach, Peter (1980). Logic Problem . University of California Press. ISBN 978-0-520-03847-9. Baum, Robert (1989). Logic . Holt, Rinehart and Winston, Inc. ISBNÃ, 0-03-014078-1.
External links
- ChangingMinds.org: categorical proposition
- Katlogik: Open source computer scripts written in Ruby to compile, investigate, and compute categorical propositions and syllabologies
Source of the article : Wikipedia
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