Report On Logic_immediate Inferences.pptx

IMMEDIATE INFERENCES

STANDARD FORM FORM

PROPOSITION

EXAMPLE

A

All S is P

All lawyers are smart

E

No S is P

No lawyers are smart

I

Some S is P

Some lawyers are smart

O

Some S is not P

Some lawyers are not smart

Note: A and E propositions are universal propositions. (All, No) I and O propositions are particular propositions. (Some, Some are not) A and I propositions are affirmative propositions. (All, Some) E and O propositions are negative propositions. (No, Some are not)

A. CONVERSION A valid form of immediate inference for some but not all types of propositions. To form the converse of the proposition, the subject and predicate terms are simply interchanged. Thus, applied to the proposition “No circles are squares,” conversion yields “No squares are circle,” which is called the “converse” of the original proposition.

The original proposition is called the convertend.

I. CONVERSION FORMS

PROPOSITION

CONVERSE

VALIDITY

A

All S is P

All P is S

invalid

E

No S is P

No P is S

valid

I

Some S is P

Some P is S

valid

O

Some S is not P

Some P is not S

invalid

…more valid conversions • “No men are angels” converts to “No angels are men” • “Some women are writers” converts to “Some writer are women”

O Propositions in Conversion • The conversion of an O proposition is not valid. The O proposition, “Some animals are not dogs” is plainly true; its converse is the proposition, “Some dogs are not animals,” which is plainly false. An O proposition and its converse are not logically equivalent.

A Proposition in Conversion • The A proposition presents a special problem. The converse of A proposition does not follow from its convertend. From “All doges are animals” we certainly may not infer that “All animals are dogs.” The A proposition says something about all members of the subject class (dogs); the I proposition makes a more limited claim, about only some of the members of that class. • If we are given the A proposition, “All dogs are animals,” we first infer that “Some dogs are animals” by subalternation, and from that subaltern we can by conversion validly infer that “Some animals are dogs.” Hence, by combination of subalternation and conversion , we advance validly from “All S is P” to “Some P is S.” This pattern of inference, called conversion by limitation or (conversion per accidens), proceeds by interchanging subject and predicate terms and changing the quantity of the proposition from universal to particular.

CLASSES & CLASSESCOMPLEMENTS CLASS : The collection of all things that belong to a given class. CLASS-DEFINING CHARACTERISTICS: The collection of all objects that have a certain common attribute. The class of all human is the collection of all things that have the characteristic of being human; its class-defining characteristic is the attribute of being human.

COMPLEMENTARY CLASS or COMPLEMENT • The collection of all things that do not belong to the original class. • The complement of the class of all people is the class of all things that are not people. The class-defining characteristic of that complementary is the (negative) attribute of not being a person. The complement of the class of all people contains no people, but it contains everything else: shoes, ships, cabbages, books –but no kings, because kings are people. • To designate the complement of all persons , we add the word “non” to the subject, thus “class of nonpersons.” • The complement of the class designated by the term S is then designated by the term non-S.

• One must be careful not to mistake contrary terms for complementary terms. “Cowards” and “hero” are contraries because no person can be both a coward and a hero. Do no identify “cowards” with “nonhereos” because they do not necessarily provide the same meaning. • Likewise, the complement of the term “winner” is not ”loser” but “nonwinner,” for although not everything, or even everyone, is either a winner or a loser, absolutely everything is either winner or nonwinner.

B. OBVERSION • A valid form of immediate inference for every standard-form categorical proposition. To obvert a proposition we change its quality (from affirmative to negative, or from negative to affirmative) and replace the predicate term with its complement. Thus, applied to the proposition, “All dogs are mammals,” obversion yields “No dogs are nonmammals,” which is called obverse of the original proposition. The original proposition is called the obvertend.

OBVERSION FORM

PROPOSITION

OBVERSE

VALIDITY

A

All S is P

No S is non-P

valid

E

No S is P

All S is non-P

valid

I

Some S is P

Some S is not non-P

valid

O

Some S is not P

Some S is non-P

valid

• The A proposition, “All residents are voters,” has its obverse “No residents are nonvoters.” • The E proposition, “No umpires are partisans,” has its obverse “All umpires are nonpartisans.” • The I proposition, “Some metals are conductors,” has its obverse, “Some metals are non nonconductors.” • The O proposition, “Some nations were non belligerents,” has the obverse, “Some nations were nonbelligerent.”

C. CONTRAPOSITION • A valid form of immediate inference for some, but not for all types of propositions. To form the contrapositive of a given proposition, its subject terms is replaced by the complement of its predicate terms, and its predicate term is replaced by the complement of its subject term. Thus the contrapositive of the proposition “All humans are mammals” is the proposition “All nonmammals are nonhumans.” • It is a plainly valid form of immediate inference when applied to A propositions. It is also a valid form of immediate inference when applied to O propositions, although its conclusion may be awkward to express.

CONTRAPOSITION

• • • •

FORM

PROPOSITION

CONTRAPOSITIVE

VALIDITY

A

All S is P

All non-P is non-S

valid

E

No S is P

No non-P is non-S

invalid

I

Some S is P

Some non-P is non-S

invalid

O

Some S is not P

Some non-P is not non-S

valid

The contrapositive of the A propositions, “All members are voters,” is “All nonvoters are nonmembers.” The contrapositive of the O propositions, “Some students are not idealist,” is “Some nonidealist are not nonstudents.” For I propositions, however, contraposition is not a valid form of inference. The true I proposition, “Some citizens are nonlegislators,” has its contrapositive the false proposition, “Some legislators are noncitizens.” In E propositions, the contrapositives does not follow validly from the original, if we begin with the true proposition, “No wrestlers are weakling,” we get, as its contrapositive, “No nonweaklings are nonwrestlers.”

CONTRAPOSITION BY LIMITATION • It is when we infer an O proposition from an valid E proposition (e.g., we infer “Some non-P is not non-S” from “No S is P”), has the same peculiarity as conversion by limitation, on which it depends. Because a particular proposition is inferred from a universal proposition, the resulting contrapositive cannot have the same meaning and cannot be logically equivalent to the original premise. • On the other hand, the contrapositive of an A proposition is an A proposition, and the contrapositive of ab O proposition is an O proposition, and in each of these cases the contrapositive and the premise of which it is derived are equivalent.

APPLICATION 1. PREMISE: Some books are thought provoking materials.

3. PREMISE : All delegates to the convention are tough negotiators.

Converse: Some thought-provoking materials are books. (valid)

Converse: All tough negotiators are delegates to the convention. (invalid)

Obverse: Some books are not non-thought provoking Obverse: No delegates to the convention are not materials. (valid) tough negotiators. (valid) Contrapositive: Some non-thought-provoking materials are non-books. (invalid)

Contrapositive: All people who are not tough negotiators are not delegates to the convention. (valid)

2. PREMISE : No children are pranksters.

Converse: No pranksters are children. (valid)

4. PREMISE: Some scientists are not pragmatic people.

Obverse: All children are non-pranksters. (valid)

Converse: Some pragmatic people are not scientists. (invalid)

Contrapositive: No non-pranksters are non-children. Obverse: Some scientists are non-pragmatic people. (invalid) (valid) Contrapositive: Some non-pragmatic people are not non scientists. (valid)