Equivalence of Propositions

1. Equivalence of Propositions – the formulation of a new proposition, with the same meaning as the original, by interchanging the subject and predicate terms of the latter and/or by the use or removal of negatives.

1. conversion – interchanging the subject and predicate terms of the original but leaving its quality unchanged.

convertend original proposition	S is P
converse resultant proposition	Ps is Sp

1.a. simple done by simply interchanging subject and predicate terms because they have the same quantity. Note that the quantity of the converse should be the same as the quantity of the convertend.

E – E	No dog is a cat – No cat is a dog.
I - I	Some pictures are clear prints – Some clear print are pictures.

1.b. partial – the unioversal quantity of the convertend becomes particular in the converse.

A – I

All horses are animals – Some animals are horses.

It is advisable to reduce a proposition to its logical form before attempting conversion. Simple conversion is applied to a singular proposition whose predicate is also singular.

2. obversion – formulating a new proposition [obverse] by retaining the subject term and quantity of the original proposition [obvertend], changing its quality, and using as predicate term the contradictory of the original predicate term. The procedure for obversion is as follows:

1. Retain the subject quality of the obvertend.

2. Change the quality.

3. As new predicate, use the contradictory of the predicate of the obvertend.

A – E	All me are sinful.	-	All me are not sinless.
I - O	Some house are white.	-	Some houses are not non-white.
E – A	No person is indispensable.	-	All persons are dispensable.
O - I	Some jewels are not rare.	-	Some jewels are non-rare.

3. contraposition – the formulation of a new proposition [contraposit] whose subject is the contradictory of the predicate of the original proposition [contraponend]. There are two types of contraposition and the corresponding procedures are as follows:

Type 1:

1. Obvert

2. Convert the obverse.

[contraponend] A - E [contraposit]

E - I

O - I

A - E
Every dog is an animal.	Every dog is not a non-animal.	Every non-animal is a dog.
E - I
No dog is a cat.	All dogs are non-cats.	Some non-cats are dogs.
O - I
Some men are not voters.	Some men are non-voters.	Some non-voters are men.

Type 2:

1. Obvert.

2. Convert the obverse.

3. Then obvert the converse of the obverse.

[contraponend] A - A [contraposit]

E - O

O - I

A - A
Every man is mortal.	Every man is not immortal.	Every immortal is not a man.	Every immortal is a non-man.
E - O
No dog is a cat.	All dogs are non-cats.	Some non-cats are dogs.	Some non-cats are not non-dogs.
O - I
Some men are not voters.	Some men are non-voters.	Some non-voters are men.	Some non-voters are not non—men.

Opposition of Proposition

1. Opposition of Proposition – the difference as to quantity or quality or both of two propositions having the same subject term and predicate term.

1. contradictory – difference of two propositions as to quantity and quality such that one is necessarily the negation of the other. Both cannot be true but both cannot be false.

If one is true, the other is false and vice-versa

A vs O

E vs I

2. contrary – difference as to quality of two universal propositions. Both cannot be true but both can be false.

If one is true, the other is false; if one is false the other is

doubtful.

A vs E

3. subcontrary – difference of two particular propositions as to quality. Both can be true but both cannot be false.

If one is false the othet is true; if one is true the other is

doubtful.

I vs O

4. subalternate – difference as to quantity of two propositions having the same quality, between a universal proposition (superaltern) and its corresponding particular (subaltern).

If the superaltern is true, the subaltern is also true but not

vice-versa. If the subaltern is false, the superaltern is also

false but not vice-versa.

A vs I

E vs O

A being true	E is false	I is true	O is false
E being true	A is false	I is false	O is true
I being true	E is false	A is doubtful	O is doubtful
O being true	A is false	E is doubtful	I is doubtful
A being false	O is true	E is doubtful	I is doubtful
E being false	I is true	A is doubtful	O is doubtful
I being false	A is false	E is true	O is true
O being false	A is true	E is false	I is true

Classification of Proposition

1. Classification of Propositions – There are four kinds of categorical propositions:

1. those that assert that the whole subject class included in the predicate class;
2. those that assert that the whole subject class is excluded from the predicate class;
3. those that assert that part of the subject class is included in the predicate; and,
4. those that assert that part of the subject class is excluded from the predicate class.

A	Universal-Affirmative	All S are P	All politicians are honorable men.
E	Universal-Negative	All S are not P	All politicians are not honorable men.
I	Particular-Affirmative	Some S are P	Some politicians are honorable men.
O	Particular-Negative	Some S are not P	Some politicians are not honorable men.

Form A	All S are P asserts that the predicate is affirmed of all members of S, the subject category.
Form E	No S are P asserts that the predicate is denied of all members of S.
Form I	Some S are P asserts that there is at least one member of S and the predicate is affirmed of it.
Form O	Some S are not P asserts that there is at least one of S and the predicate is denied of it.

All S are P	Every member of the S class is a member of the P class; that is, the S class is included in the P class.
No S are P	No member of the S class is a member of the P class; that is, the S class is excluded from the P class.
Some S are P	At least one member of the S class is a member of the P class.
Some S are not P	At least one member of the S class is not a member of the P class.

Standard Categorical Format

1. Standard Categorical Format – translating ordinary language statements into categorical form. While few statements in ordinary written and oral expression are categorical propositions in standard form, many of them can be translated into standard form propositions. The necessity for this translation is twofold: the first is that the operations and inference pertinent to standard form categorical propositions become applicable to these statements, and the second is that such statements, once translated are completely clear and unambiguous as to their meaning. The simple rule for this translation is to understand the meaning of the given statement and then reformulate it in a new statement that has a quantifier, subject term, copula, and predicate term.

a. terms without nouns – The subject and predicate terms of a categorical proposition must contain either a noun or noun substitute that serves to denote the class indicated by the term. Nouns and noun substitute denote classes, while adjectives connote attributes. If a term consists of only an adjective, a noun or noun substitute should be introduced to make the term genuinely denotative.

Some rose are red.	Some roses are red flowers.
All tigers are carnivorous.	All tigers are carnivorous animals.

b. nonstandard verbs – statements in ordinary usage often incorporate other forms of the verb to be. Such statements may be translated according to the following examples:

Some persons who go to college will become educated.	Some persons who go to college are persons who will be educated.
Some dogs would rather bark than bite.	Some dogs are animals that would rather bark than bite.

1. singular proposition – A singular proposition makes an assertion about a specific person, place, thing, or time. Singular propositions are typically translated into universals by means of a parameter, a phrase that when introduced into a statement affects the form but not the meaning. Some parameters that may be used to translate singular propositions are: persons identical to, place identical to, things identical to, times identical to.

Gloria Macapagal-Arroyo is the president of this country.	All persons identical to Gloria Macapagal-Arroyo are the president of this country.
There is a radio in the back bedroom.	All places identical to the back bedroom are places where there is a radio.

2. adverbs and pronouns – when a statement contains a spatial adverb such as where, wherever, anywhere, everywhere, or nowhere, or a temporal adverb such as when, whenever, anytime, always, or never, it may be translated in terms of places or times, respectively. a statement containing pronouns such as who, whoever, anyone, what, whatever, or anything may be translated in terms of persons or things, respectively.

He is always clean shaven.	All times are times he is clean shaven.
Whoever works hard will succeed.	All persons who work hard are persons who will succeed.

3. unexpressed quantifiers – many statements in ordinary usage have quantifiers that are implied but not expressed. In introducing the quantifiers one must be guided by the most probable meaning of the statement.

Emeralds are green gems.	All emeralds are green gems.
Children live next door.	Some children live next door.

4. nonstandard quantifiers – sometimes the quantity of the statement is indicated by a word or words other than the three quantifiers that are allowed.

A few soldiers are heroes.	Some soldiers are heroes.
Anyone who voted is a citizen.	All voters are citizens.

5. conditional statements – when the antecedent and consequent of a conditional statement talk about the same thing, the statement can usually be translated into categorical form.

If it is a mouse, then it is a mammal.	All mice are mammals.
If an animal has four legs, then it is not a bird.	No four-legged animals are birds.

In translating conditional statements it is sometimes useful to employ a rule called transposition. According to this rule, the antecedent and consequent of a conditional statement may switch places if both are negated. For example, the statement If something is not valuable, then it is not scarce is logically equivalent to If something is scarce, then it is valuable. This is then translated as All scarce things are valuable things.

The word unless means if…not. For example, the statement A car will not run unless there is gas in the tank means A car will not run if there is not gas in the tank which means If there is not gas in the tank, then a car will not run. by transposition, this means, If a car runs, then there is gas in the tank which is translated as All cars that run are cars with gas in the tank.

6. exclusive propositions – propositions that involve the words only, none but, none except, and no… except are exclusive propositions. Efforts to translate them into categorical propositions frequently lead to confusion of the subject term with the predicate term. Such confusion can be avoided if the statement is phrased as a conditional statement first, then as a categorical statement. For example, the statement Only executives can use the silver elevator is equivalent to If a person can use the silver elevator, he is an executive. The correct categorical proposition is All persons who can use the silver elevator are executives.

Only elected officials will attend the convention.

Only elected officials will attend the convention.

When only and none but occur in the middle of a statement, the statement must first be restructured so that the term preceded by only or none but occurs first. Then the statement can be translated as those above. For example, the statement Executives can use only the silver elevator is equivalent to Only the silver elevator can be used by executives. This, in turn, is equivalent to If an elevator can be used by executives, which is translated: All elevators that can be used by executives are elevators identical to the silver elevator.

He owns only the shirt on his back.

All things owned by him are things identical to the shirt on his back.

Statements beginning with the words the only are translated differently from those beginning with only. For example, the statement The only cars that are available are Chevrolets means that If a car is available, then it is a Chevrolet. This, in turn, is translated as All cars that are available are Chevrolets. In other words, the only, when it occurs at the beginning of a statement, can simply be replaced with all, and the order of the term is not reversed in the translation. When the only occurs in the middle of a statement, the statement must be restructured so that it occurs at the beginning. For example, Romances are the only books he sells is equivalent to The only books he sells are romances. This is then translated as All books that he sells are romances.

Accountants are the only ones who will be hired.

All those who will be hired are accountants.

7. exceptive propositions – propositions of the form All excepts S are P and All but S are P are exceptive propositions. They must be translated not as single categorical propositions but as pairs of conjoined categorical propositions. Statements that include the phrase none except, on the other hand, are exclusive and not exceptive propositions.

All except students are invited.

No students are invited persons + All non-students are invited persons.

Categorical Proposition

1. Logic originated with the Greek philosophers. As far back as seven hundred years BC, they already had an understanding of good argument and used reasoning to draw abstract conclusions. This can be gleaned from fragments of their writings. But it was Aristotle who first developed a logical system. His comprehensive theory includes precise definitions, principles of reasoning, a system for exposing arguments, and rules for evaluating reasoning. The Aristotelian system is called categorical logic. It is also known as deductive logical system. It is a method for analyzing deductive argument which Aristotle called syllogisms. One of the requirements of categorical logic is that statement follows a certain form. They have to be categorical statements and must assert a relationship between one group of things and another.

Categories are classes or group of things. The words and phrases used to refer or name categories are called terms.

Statements in categorical logic assert a relationship between one category, the subject category, and another, the predicate category. The statement “all men are good” is thus to be interpreted as meaning that “all those belonging to the category of man belong to the category of good”. For Aristotle all statements can be expressed in terms of categories and the assertion of a relationship between categories. And all reasoning in categorical logic is reasoning about categories.

ORDINARY STATEMENT	CATEGORICAL STATEMENT
Some dogs bark.	At least one member of the class of dogs is a member of the class of animals that bark.
All spiders have eight legs.	All members of the class of spiders are members of the class of things having eight legs.
No bankers favor the withholding of interest on savings account.	No member of the class of bankers are members of the class of person favoring withholding interest on savings account.
Some of the people are not for the death penalty.	At least one member of the class of people is not a member of the class of those favoring the death penalty.

A categorical statement is also called a proposition. Its distinctive character is that it asserts that something is the case or that something is not the case. Its assertion has a truth-value, that is, it may either be true or false.

a. components and structure – A categorical proposition follows the standard subject-copula-predicate form. The subject is introduced be a quantifier which specifies how much of the subject class is included in or excluded from the predicate class. The copula is expressed in the third person, present tense, of the verb to be.

All students majoring in philosophy are required to take Metaphysics and Epistemology.

quantifier: all

subject term: students majoring in philosophy

copula: are

predicate term: required to take Metaphysics and Epistemology

b. quantity of the proposition – refers to the extension of the subject. A proposition could be either universal [if it refers to all the members of the class designated by the subject term] or particular [if it refers to some members of the class designated by the subject term].

c. quality of the proposition – could be either affirmative [if the predicate is asserted of the subject – AffIrmo] or negative [if the predicate is denied of the subject – nEgO].

d. distribution in the predicate – the predicate of an affirmative proposition is undistributed [does not refer to all the members of the class designated] while the predicate of the negative proposition is distributed [refers to all the members of the class designated].

Meaning and Definition

Over the years philosophers have held various conflicting views about the purpose of definition. For most logicians today, however, definitions are intended exclusively to explicate the meaning of words. A definition therefore is a group of words that assign a meaning to some word or group of words. It consists of two parts:

DEFINITION
definiendum word to be defined	definiens words that do the defining

1.Purposes of Definition

a. to increase vocabulary – There are situations when a deliberate explanation of the meaning of terms is required. To explain the meaning of a term is to give a definition to it. To understand what is being said, it is necessary to find out what the words mean and here a definition is required.

b. to eliminate ambiguity – it is not always clear what sense of a given word is intended. To resolve ambiguity a definition is required to specify the different meanings of the ambiguous word or phrase.

c. to reduce vagueness – Sometimes the limits of the applicability of the meaning of a term are not clear. There is a need for clarification and this is accomplished by giving a definition of the term that will permit a decision as to its applicability in a given situation where it was previously doubtful.

d. to explain theoretically – Technical terms carry a meaning of their own. To define is to elaborate the meaning that a term carries in a particular discipline like science among others.

e. to influence attitudes – There is a way to influence attitudes or stir the emotions and this is done by defining a term with a certain stant of meaning.

2.Classification of Definitions

a. stipulative definition – assigns meaning to new symbols.

b. lexical definition – reports the established usage of a term.

c. precising definition – helps to decide borderline cases of ambiguous meanings.

d. theoretical definition – attempts to formulate an adequate characterization of the objects to which it is applied.

e. persuasive definition – functions expressively to influence attitudes.

3.Techniques of Definition

DENOTATIVE	CONNOTATIVE
definition by giving examples or describing the objects being defined.	definition by genus and specific difference

1. Denotative Techniques – extensional – to define a term by indicating

the members of the class to which the definiendum refers.

a.1. by pointing them – demonstrative or ostensive – ex. A chair

means this, a table that; cafeteria here, infirmary there.

a.2. by naming them individually – enumerative – ex. Actor means

Gregory Peck, Tom Cruise, Antonio Banderas, Mel Gibson

a.3. by naming the individuals in group – subclass – ex. Vertebrate

means mammals, fishes, amphibians, birds; fiction means a poem,

novel, drama.

2. Connotative Techniques – intensional – to define a term by indicating the qualities or attributes that the world connotes.

1.by giving an equivalent term – synonymous – Physician means

doctor; Intentional means willful means deliberate.

To define by synonym requires that the definiens has the same intentional meaning as the definiendum. Also, the definiens must be simpler, more common or understandable in order to shed light on the definiendum. The following violate this last requirement: Concealed means clandestine or sob rosa. Esoteric means abstruse.

2.by specifying certain experimental procedures that determine

whether or not the word applies to a certain thing

- operational – ex. A solution is an acid if and only if litmus paper

turns red when placed in contact with it. One substance is

harder than another if and only if one scratches the other when

the two are rubbed together.

3.by identifying the genus and the specific difference. Genus

refers to a relatively large class to which the definiendum

belongs. Specific difference refers to the attribute which

distinguishes the definiendum from other members of the genus.

- ex. Ice is frozen water: Water is the genus and frozen is

the specific difference.

4.Rules for Defining

Rule 1	A definition should convey the essential meaning of the word being defined. Man is a rational animal / Man is being who smiles.
Rule 2	A definition should be neither too broad now too narrow. In some instances the context in which the definiens is used must be indicated. A school is a public institution of learning / A fish is an animal.
Rule 3	A definition must not be circular. The word to be defined should not be contained in the definition. A circle is a circular object. Silence means the state of being silent.
Rule 4	A definition should not be negative when it can be affirmative. Peace means harmony / Peace means the absence of war.
Rule 5	A definition should not be expressed in figurative, obscure, vague, or ambiguous language. A window is an orifice in an edifice to admit certain emissions of light energy. Faith is the substance of things hoped for, the guarantee of things unseen.
Rule 6	A definition should avoid affective terminology. Communism is the brilliant invention of Karl Marx and other foolish political visionaries in which the national wealth is supposed to be held in common by the people.