logic

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Phi losophy 103: I ntroduction to Logic

Quantity, Quali ty, and Distribution of Standard Form Categor ical Propositi ons

Abstract: The most important properties of standard form categorical propositions are explained and illustrated.

I. Categorical propositions and classes.

A. The long range goal is to give a theory of deduction, i.e., to explain the relationship

between the premisses and conclusion of a valid argument and provide techniques for the

appraisal of deductive arguments. Hence, we will be distinguishing between valid and

invalid arguments.

1. A deductive argumentis defined as one whose premisses are claimed to provide

conclusive evidence for the truth of its conclusion.

2. A valid deductive argumentis one in which it is impossible for the premisses to

be true without the conclusion being true also.

B. Our study of deduction, for the present, will be about arguments stated in categorical

propositions, e.g.,

No honest people are persons who embroider the truth.

Some politicians are persons who embroider the truth.Some politicians are not honest people.

1. A categorical propositionis defined as any proposition that can be interpreted as

asserting a relation of inclusion or exclusion, complete or partial, between two

classes.

2. A classis defined as a collection of all objects which have some specified

characteristic in common. This is no more complicated than observing that the classof "lightbulbs" all have the common characteristic of "being a lightbulb."

Thus, we can have four class relations in the various kinds of categorical

propositions:

Utilizing the classes, "people" and "good beings":

a. complete inclusion>>>"All people are good beings."

b. complete exclusion>>>"No people are good beings."

c. partial inclusion>>>"Some people are good beings."

d. partial exclusion>>>"Some people are not good beings."

We can also describe these four kind of statements respectively as

a. universal affirmative

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b. universal negative

c. particular affirmative

d. particular negative

3. Often, it is convenient to look at the general form of the statements given above.To these forms, special names are given: A, E, I, and O.

A: All SisP.

E: No SisP.

I: Some SisP.

O: Some Sis notP.

...where SandPstand for the logical subject and the logical predicate of the

statement respectively.

4. A mnemonic device for the four kinds of statements is to remember Affirmo and

Nego.

5. Note, here, the logical subject differs from the grammatical subject of a statement.

For example, in the statement, "All (unfledged floithoisters)are (things apt to

become unflaggled)," the logical subject is everything between the "all" and

the "are," and the logical predicate is everything after the "are."

6. Also note that the word "some" is taken to mean "at least one." This meaning

differs somewhat from ordinary language.

7. A model statement, then, can be represented as

Quantifier[subject term]copula[predicate term].

II. Analysis of the Categorical Proposition: Quality, Quantity, and Distribution

A. The quantityof a categorical proposition is determined by whether or not it refers to

all members of its subject class (i.e.,universal orparticular). The question "How many?"is asking for quantity.

B. The quality of a categorical proposition is determined by whether the asserted class

relation is one of exclusion or inclusion (i.e., affirmative or negative).

C. Indicators of "how much" are called quantity indicators (quantifiers) and specifically

are "all," "no," and "some."

D. Indicators of affirmative and negative are quality indicators (qualifiers) and

specifically are "are," "are not," "is," "is not," and "no,"

Note that "no" is both a quantifier and a qualifier.

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E. Memorize the following table:

Name Form Quantity QualityDistribution

Subject Predicate

A

All S is P

universal

affirmative

distributed

undistributed

E No S is P universal negative distributed distributed

I Some S is P particular affirmative undistributed undistributed

O Some S is not P particular negative undistributed distributed

F. Distribution of a term.

1. A distributedterm is a term of a categorical proposition that is used with

reference to every member of a class. If the term is not being used to refer to eachand every member of the class, it is said to be undistributed.

2. Consider the following propositions:

A: All birds are winged creatures.

E: No birds are wingless creatures.

I: Some birds are black things.

O: Some birds are not black things.

Read the above statements and see how the following chart represents distribution.

Subject Predicate

A: refers to all birds does not refer to every member, e.g.,bats, flying fish.

E: refers to all birds by

indicating that they are not

part of the predicate class

refers to all wingless creatures by

indicating that they are not part of

the subject class

I: refers only to some birds refers only to some black things,viz.,those which are birds

O: refers only to some birds,not all of them

refers to all members of theclass!Viz., not one of them is in the

class referred to by "some birds"

3. For the predicate of the Oproposition, consider the following analogy. If we

know that there is a book not in a bookcase, then we know something about each and

every shelf in that bookcase-- the book is not on that shelf.

4. There are three ways to remember the distribution status of subject and predicatefor standard form categorical propositions:

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a. Memorize it.

b. Figure it out from an example (as was done above).

c. Remember the following rule:

The quantityof a standard form categorical proposition determines thedistribution of the subject (such that if the quantity is universal, the subject

is distributed and if the quantity is particular, the subject is undistributed),and ...

the qualityof a standard form categorical proposition determines the

distribution status of the predicate (such that if the quality is affirmative,

the predicate is undistributed, and if the quality is negative, the predicate is

distributed).

http://philosophy.lander.edu/logic/prop.html

Propositions are either standard form or nonstandard. If a proposition is notstandard form, it is classified as nonstandard. We first consider the four

standard form propositions, then discuss nonstandard propositions in the lastsection of this Study.

Standard Form Propositions

There are only four standard form propositions. Each consists of a subject anda predicate. In each form, the subject and the predicate are joined togetherby isor are, the copula. The relation between the subject and predicate isidentified by the use of:All, No, Some, orSome ... not....If aand bstand forthe subject and predicate terms, respectively, the four forms are: (1)All a is b,

(2)No a is b, (3)Some a is b, and (4)Some a is not b.

The A Form

The proposition "All men are mortal" asserts a relation of inclusion betweenthe class of men and the class of mortals. More plainly, it states that allmembers of the class men fall within the class mortal. The form of all suchpropositions isAll a is b, orA(ab)where astands for the subject term
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and bstands for the predicate term. Note that inApropositions, the subject isincluded in the predicate, but not the predicate in the subject. For example,from "All men are mortals" (true) it does not follow that all mortals are men(false).

The E Form

The proposition "No Christian is an atheist" asserts a relation of exclusionbetween two classes, Christians and atheists. No member of the classChristians is a member of the class atheists, and conversely, no atheist is aChristian. The classes ofEpropositions are mutually exclusive. The form isNoa is b, orE(ab), where astands for any subject, and bstands for any predicate.Thus, withEpropositions all members of one class are excluded from theother, and vice versa.

The I Form

The proposition "Some Americans are Calvinists" asserts a relation of partialinclusion between the class Americans and the class Calvinists. Something lessthan all members of the subject-class is included in the predicate-class, andconversely, some members of the class Calvinists are included in the classAmericans. The form of theIproposition isSome a is b, orI(ab), where, asbefore, astands for any subject, bfor any predicate. Ordinarily, somecanmean a few in number. In logic, the word can also mean as few as one or anynumber less than all.

The O Form

The proposition "Some men are not Christian" asserts a relation of partialexclusion between the two classes, men and Christians. Some men are entirelyexcluded from all of the class of Christians. Does it follow then that someChristians are not men? Perhaps some angels are Christian? No, the converseof an Oproposition does not follow from the original. Its form isSome a is notb, orO(ab). Remember, there is no converse for an Oproposition.

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The following chart serves as a summary of the foregoing descriptions of thefour forms. Do not be confused in that the letters aand bare used throughout,even when the propositions contain different subject matter. Recall that theletters, aand b, stand for any subject and any predicate, respectively. Indeed,we could have usedxand yor any other pair of letters to stand for subjectsand predicates.
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Chart 1.1: Four Forms

All men are mortal All a is b. A(ab)

No Christian is an atheist. No a is b. E(ab)

Some Americans are Calvinists. Some a is b. I(ab)

Some men are not Christian. Some a is not b. O(ab)

The source of the letters for the four forms is of historical interest.

From affirmo (I affirm), meaning affirmative in quality, wegetAandI;Eand Ocome from nego (I deny), meaning negative in quality.

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Formal Properties of the Forms

The four forms share three important properties: distribution, quantity,and qualitydefined just below.

Distribution

The formal properties, quality and quantity, ofA, E, I,and Oforms depend onthe definition of distribution. We distinguish a distributed term (subject orpredicate) from an undistributed term in this manner: A distributed term isone modified byAllorNo. When a term is modified by "some," it isundistributed. Using the subscripts "d" for distributed and "u" forundistributed, the four forms distribute their terms as indicated below inChart 1.2.

Chart 1.2: Distribution of Terms

Forms Subject Term Predicated Term

A All sdis pu Distributed Undistributed

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E No sdis pd. Distributed Distributed

I Some suis pu. Undistributed Undistributed

O Some suis not pd. Undistributed Distributed

Where, s = subject term; p = predicate term.

To recapitulate: With theAform, only the subject term is distributed; thepredicate is undistributed, since, as noted previously, all of the predicate is not

included in the subject. The E form distributes both subject and predicateterms, since No s is p; and No p is s. With the I form, somepart of the subjectterm class is included in somepart of the predicate term class; therefore, bothterms are undistributed. Last, in the O form, somepart of the subject termclass is excludedfrom allof the predicate term class (Some s is not p);therefore, only the predicate term is distributed, the subject term,undistributed.

Quality

Previously we indicated that theAandI letters came from the Latin affirmo,andEand Ofrom the Latin nego. Remembering the sources of the letters mayhelp to recall that theAandIforms are affirmative in quality;Eand O,negative in quality. An affirmative form is one that does not distribute itspredicate. TheAandI forms do not distribute the predicates; therefore, theyare affirmative in quality. A negative form is one that does distribute itspredicate. TheEand Oforms distribute the predicates; therefore, they arenegative in quality.

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Quantity

Each of the four forms is either universal or particular in quantity. If a formdistributes its subject term, it is universal in quantity. TheAandEforms areuniversal, since each distributes its subject term. On the other hand, a form isparticular in quantity if its subject term is undistributed. TheIandthe Oforms have undistributed subject terms; therefore, these are particular.

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Chart 1.3: Distribution, Quantity, and Quality

Forms Quantity Quality

All sdis pu universal affirmative

No sdis pd universal negative

Some suis pu. particular affirmative

Some suis not pd particular negative

Chart 1.3 may serve the student as a memory device for reinforcing howquantity and quality is determined by distribution of terms in standard formpropositions. The chart is no substitute for memorizing the definition ofdistribution and understanding what it means. The importance of distributionof terms cannot be overemphasized, for it not only serves as the basis fordefining the quality and quantity of the four forms, but is the basis for some ofthe rules that test the validity of deductive inference.

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Nonstandard Propositions

Only standard form propositions are candidates for the premises andconclusion for the type of argument (syllogism) discussed in Study Three.Most nonstandard propositions are easily translated to standard form. Otherswill require practice and careful attention to the meaning of the proposition inquestion. This may result in some awkward formulations of English. The goalis clarity of meaning, not elegant prose.

Use of Parameters

In the case of an English verb other than the present tense of the verb to be,change the verbs into predicate adjectives. For example, "All competentstudents know logic" becomes "All competent students are knowers-of-logic.

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When the language of the sentence contains clauses or prepositional phrasesas well as a verb other than the English copula, the use of parameters will helpmake the sense of the proposition clear. For example, "All persons-who-are-competent-students are persons-who-are-knowers-of-logic." Here theword,persons, appears in both the subject and predicate, and together with

hyphens assists in reading the proposition as anAproposition. The purpose isto make the sense of the proposition crystal clear.

More effort is required with two other classes of propositions: exclusive andexceptive propositions.

Exclusive Propositions

How can we make clear the sense of this exclusive proposition? "Only atheistswill be ejected." What does it mean? It means "All persons-who-are-ejected

are persons-who-are-atheists." Thus the sense of exclusive propositions(only x is y) is theA form, the result obtained when subject and predicate areinterchanged.

Exceptive Propositions

Exceptive propositions (all except x is y) are really two in one form. Forexample, "All except the soldiers gave up the fight" means (1) All persons whoare non-soldiers (civilians) are persons who gave up the fight; and (2) Noperson who is a soldier is a person who gave up the fight. Not to anticipate the

material of the next Study, let it be merely noted for now that neither one ofthese can be deduced from the other.

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Propositions lacking Quantifiers

Some logic books label propositions with proper names, singularpropositions. We make no distinction between singular and other universalpropositions. All propositions using proper names are either FormAor

FormE, depending on the quality. The nameSocrates, in "Socrates is mortal"is the entire subject class, which happens to have only one member. Anexample of anEform is "Socrates is not immortal," or, "No Socrates isimmortal." These are not the only propositions wherein "all" or "no" isimplied. Some propositions appear to name only some members of a class,when all members of a class are either included or excluded. Example:"Dinosaurs are extinct" does not mean that some are or some may not beextinct. The sense of the statement is that all dinosaurs are extinct. In other

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words, the "all" is implied, and when the context calls for "all," or "no" theresult is either anA Formor anE Form, depending on the quality of theoriginal. Also, an implied "some" proposition is translated as theI Formor OForm, depending on quality.

Logical versus Grammatical Subjects

The grammatical and logical subjects of some propositions sometimes need tobe distinguished, if one is to achieve the correct sense of a proposition. Anexample cited in a logic book is: "You always squirm out of an argument." Thegrammatical subject, "you," is not the logical subject. Rather, alwaysmeaning"every time you get into an argument" is the logical subject. The sense of theoriginal is "Alltimes-you-get-into-an-argument are times-you-squirm-out-of-it." (The statement may appear to be awkward, but the meaning isaccurately worded and that's what matters!)

Application of tests to determine the validity of inference depends on the clearsense of standard form propositions. However, the job of re-wordingnonstandard propositions into standard formA, E, I,and Ohas benefitsbeyond the requirements of deductive inference. Where testing for validity isnot an issue, rewording nonstandard into standard forms will avoidmisunderstandings, mistakes, and confusion. If you can't reword anonstandard proposition into standard form, you probably do not know whatit means. Therefore, it is essential that you develop translation skills to achieveclarity of thought and to avoid misunderstanding or mistakes in reasoning.

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Summation

Standard form propositions consists of subject and predicate terms joined bythe copula "is" or "are" and qualified by "All," "No," "Some," or "Some ... not...." These requirements yield four forms: (1)All a is b, (2)No a is b, (3)Somea is b, and (4)Some a is not bknown asA,E,I, and O forms, respectively.(The forms are also expressed as A(ab), E(ab), I(ab), and O(ab).) The formal

properties of distribution, quality, and quantity of the four standard formswere explained and illustrated. A distributed term is one modified by "all" or"no." If a term is modified by "some," it is undistributed. If a proposition'spredicate term is distributed, the proposition is said to be negative in quality;if the predicate of a proposition is not distributed, then it is affirmative inquality. This definition of quality distinguishes E(ab) and O(ab), bothnegative, from A(ab) and I(ab), both affirmative. If a proposition distributes

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its subject term, it is universal in quantity. On the other hand, if aproposition's subject term is undistributed, it is particular in quantity. By thisdefinition, we distinguish A(ab) and E(ab), both universal, from I(ab) andO(ab), both particular. Finally, some guidelines for translating nonstandardpropositions into standard form were described.

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Review

1 Of the four standard forms, which distribute their subject terms? Which do not distributetheir subject terms? What formal property is defined in each case?

2 Of the four standard forms, which distribute their predicate terms? Which do not distribute

their predicate terms? What formal property is defined in each case?

3 Which of the other three forms differ in both quantity and quality from A(ab)? From I(ab)?

4 What is the general formulation of exclusive propositions? What is the procedure fortransforming an exclusive proposition into standard form. Of the four standard forms,which distribute their subject terms? Which do not distribute their subject terms?

5 Compose some examples of exceptive propositions. Identify the two component sentencesembedded in each.

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Proposition:

Propositions are the material of our reasoning. A propositionlinks nouns, pronouns and

phrases to other words in a sentence. The word or phrase that the proposition introduces is

called the object of the proposition.A proposition is a judgment expressed in a language and

a judgment is a mental act in which two or more than two ideas are combined together.

Judgments have two types:
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1. Affirmative- Indians are laborious.

2. Negative- Indians are not dull.

A proposition usually indicates the temporal, spatial or logical relationship of its object to

the rest of the sentence as in the following examples:

The book is onthe table.

The book is beneaththe table.

The book is leaning againstthe table.

The book is besidethe table.

She held the book overthe table.

She read the book duringclass.

Components of Proposition:

Every proposition has three components called as term. Any word or word phrase, which is

used in a proposition as a subject or predicate, is called as term.:

1. Subject- About whom something asserts or denies.

2. Predicate-What assert or deny.

3. Copula- Conjunct both subject and predicate terms. Copula will be negative or

affirmative.

For example:

Sonia is a good orator.

S C P

Difference with Sentences:

1. Propositions are different from sentences. Sentences have many kinds like questions,

exclamations etc. But none of these can be asserted and denied. Truth and Falsity

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apply always to proposition, but not apply to questions or commands or

exclamations.

2. Propositions must be differentiated from sentences by means of what they are

asserted. Two different sentences consisting the same proposition.

3. Sentences are parts of some language, but propositions are not tied to any given

language i.e. It is raining, Barsat ho rahi hai, both consist the same content.

4. A sentence called as proposition when its both term (subject and predicate) are

nouns i.e. Ram is a man. Flower is beautiful is not a proposition because its

predicate is adjective.

5. Proposition is always in present tense. But sentences are expressed in all tenses.

6. Proposition explains quantity and quality but sentence does not explain it.

7. All propositions are sentences but not all sentences are propositions.

Types of Proposition

According to the relation of terms proposition has three types:

Categorical Proposition: It is a type of proposition which has no condition for their

assertion. Roshan is a student.

Conditional or Hypothetical Proposition: A type of compound proposition, it is false only

when the antecedent is true and the consequent is false.- If Ram will pass, then he will geta bicycle.

Disjunctive Proposition: A type of compound proposition; if true, at least one of the

component of propositions must be true.-Ram is honest or clever.

Categorical Proposition:

A categorical proposition is simply a statement about the relationship between categories. It

states whether one category or categorical term is fully contained with another, is partially

contained within another or is completely separate.

A dog is an animal.

Some dogs are friendly.

No dog is a cat.

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Propositions may have quality: either affirmative or negative.

They may also have quantity: such as a, some, most or all. The all quantity is also

described as being universaland other quantitiesparticular.

Aristotelian Four-fold Classification of Categorical Propositions:

Aristotle classified categorical proposition in four, based on Quality and Quantity

distribution:

Universal Affirmative All S is P. A Type PropositionAll men are mortal.

Universal Negative No S is P. E Type Proposition No men are immortal.

Particular Affirmative Some S is P. I Type PropositionSome men are wise.

Particular Negative Some S in not P. O Type PropositionSome men are not wise.

Distribution of Terms

Both subject and predicate of a proposition are called as term. A term is a word or group of

words which is either a subject or a predicate of a proposition.

A term is said to be distributed if it refers to all the members of a class. In the other words,

a term is distributed when it includes or excludes all the members of a class. If a term

includes or excludes only some members of a class, then it is undistributed.

In a categorical syllogism the distribution of terms depends on the quantifier:

A Type: In All A are B-propositions the subject (A) is distributed.

E Type: In No A are B-propositions both the subject (A) and the predicate (B) are

distributed.

I Type : In Some A are B-propositions neither the subject nor the predicate are

distributed.

O Type : In Some A are not B-propositions the predicate is distributed.

Form Type Quality QuantityDistribution

of X

Distribution

of Y

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All S is P A Affirmative Universal Distributed Undistributed

No S is P E Negative Universal Distributed Distributed

Some S is P I Affirmative Particular Undistributed Undistributed

Some S is

not PO Negative Particular Undistributed Distributed

Copi and Cohen state two rules about distribution of terms in valid syllogisms:

1. The middle term must be distributed at least in one premise.

2. If the major term or the minor term is distributed in the conclusion, then it must

be distributed in the premises.

Venn and Boolean Expression of Categorical Proposition:

The modern interpretation of categorical logic also permits a more convenient way of

assessing the truth-conditions of categorical propositions, by drawing Venn diagrams,

topological representations of the logical relationships among the classes designated by

categorical terms. The basic idea is fairly straightforward:

All S is P. Universal Affirmative A Type Proposition

Or S non-P = 0

No S is P. Universal Negative- E Type Proposition

Or S P = 0

Some S is P.-Particular Affirmative-I Type Proposition

Or S P 0

Some S is not P.-Particular Negative-O Type Proposition

Or S non-P 0

Denotation and Connotation of Terms

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Denotation denotes the objects, connotation connotes the characteristics. Denotation of a

term refers to the objects or things which possess the quality. Connotation refers to the set

of characteristics essentially possessed by every object denoted by the term. For example,

Man, Gita, Mohan, Kamal etc. Man means that possess morality and rationality.

Man= Denotation Morality and rationality = Connotation

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