logic instructional material module 1
TRANSCRIPT
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LOGIC: THE
SCIENCE AND ART
OF CRITICAL
ANALYSIS (IN INTERACTIVE FORMAT)
SIMEON C. BERNADOS, JR 2010
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INTRODUCTION
That man is a rational being, and a creation higher than the rest of the animals in the
animal kingdom are just some of the common notions about him. However, when it comes to
his physical prowess, he is weaker than the rest of the animals. He does not have the strength of
an elephant, the bravery of a lion, keenness of the eyes of a cat to see in the dark, and the
wings of birds to migrate. When he is still a baby, he is entirely helpless, and his helplessness is
shown through his dependence to the other older members of the family for survival.
For instance, in the first chapter of Genesis, man was given the dominion over the rest of
all creatures (v.28). He is the apex of creation for which all things in the world are at his disposal
(v. 28-29). In other words, he is created to be special.
Man is special because he has the ability to think and reason. He is endowed with
intellect. With his intellect, he discovered the way to science and developed himself through the
technology he discovered. With his intellect, man thinks.
Everyday, we are surrounded with many issues affecting our lives. These issues span
many topics ranging from economics, politics, and religion to education. Because of these
issues, our ideas are molded or may be influenced by the prevailing opinions in society.
However, what may prevail may not be right. We have to weigh the evidences, the form the
arguments as presented and the worthiness of the ideas advanced. In other words, these ideas
have to be evaluated. Nothing should be accepted unless it has passed the rigid mental
evaluation.
Arguments are presented in so many ways. There can be direct presentation while some
are implied in the course of discussion. The trained mind can easily recognize the arguments
and can judge their validity.
As experienced by the authors in teaching this course, students usually comment on the
relevance of the study of Logic in their chosen career. The most common notion in the mind of
students is that logic is only for students intending to major in Philosophy or planning to take up
Law. Although this comment is real, nevertheless the comment demonstrated that Logic really
needs to be studied. Students are proposing something thus, they are arguing or wanting to
discuss an idea.
The need to discuss an idea only ensures the necessity to study Logic. How does one
know that his/her discussions are strongly supported? While it can be claimed that the study of
Logic can be done outside the classroom and logical analysis is not a monopoly of those who
have studied Logic, yet the formal study of the discipline can greatly help the learners evaluate
the quality of the arguments presented.
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To demonstrate the discussion above, let us take a look at the two statements below:
1. Romantic love is a natural part of human experience, and is therefore found in all societies, in
close connection with marriage.*
2. In all societies, some people will be unhappy or depressed; therefore, rates of suicide will tend
to be the same throughout the world.*
Both statements are false. As in the case of the first statement, romantic love is not a
requirement for marriage. For some early Filipino families or even among the contemporary
ones, pre-arranged marriages are practiced; the proliferation of the mail-order bride is just an
example of the absence of romantic love on marriages.
For the second statement, on the other hand, human emotion is universal i.e. each
society has its own way of expressing grief, bliss, anger, hatred, and love, yet the degree to
bear pain, disillusionment, discouragement, and despair is not equal or to be at the same rate
with other culture. Thus, it cannot be said that the rates of suicide tend to be at the same rate
throughout the world.
The two examples we have discussed above demonstrate that not only the study of
Logic is important but it is also vital for man’s existence. Much more, the study of Logic is
essential for the development of the human mind. If man cannot think clearly and reason
succinctly and with precision, how is he different from other species in the animal kingdom?
Without the intellect, man would be devoid of morality or even in the sense of it; without the
intellect, man would just be a plain animal. With his intellect, precisely man is specifically called
an homo sapiens.
At the end of this course, the students are expected to:
� define Logic and its importance in human affairs
� evaluate issues using the logical method
� identify arguments and be able to assess their validity, and finally
� be able to assess his own thinking
METHODOLOGY
This course is designed to be in modular format covering the following modules, to wit:
MODULE 1 – PROBING THE TRUTH OF ARGUMENTS IN THE PROPOSITION
MODULE 2 - PROBING THE TRUTH OF ARGUMENTS IN REASONING
MODULE 3 - PROBING THE TRUTH OF ARGUMENTS THROUGH THE MATERIAL FALLACIES
* Giddens, A. 1989. “Sociology: Problems and Perspectives”. In: Sociology. Cambridge and Oxford: Polity. p.14
* ibid.
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Aside from its being in modular form, the course is meant to be in the interactive mode in
which discussion - activity processes will be integrated in the text for the students to perform. At
the end of every topic, an evaluation of students’ learning is undertaken.
MODULE 1. PROBING THE TRUTH OF ARGUMENTS IN THE PROPOSITION
Everyday, we heard arguments that oftentimes shape our opinions in certain matters
concerning our utmost attention. Either we are in favor or against an issue. We react to them,
debated on them, and if possible spread our position about them. Being a talking being, we
cannot help but always engage ourselves in some forms of discussion.
When we discuss ideas, we use arguments. Aside from using argument, we also believe
in the truth of our beliefs. Simply put, arguments are either true or false.
As defined in dictionary, an argument is 1) a reason put forward, 2) a chain of reasoning,
and 3) a discussion or a debate. These definitions carry some implications to wit:
P1. An arguer must prove his own assertion. As the Latin proverb goes, “Asserentis est
probare” --- He who asserts must prove his own assertion. One of the common practices
in proving an assertion is to use the opinion or ideas of somebody to prove one’s
contention. For instance, we are faced with the question of proving God’s existence.
Instead to prove our belief, we use the Bible to prove our point. This method may not be
wrong, but it is not appropriate. The writers of the Bible have proven the existence of God
through revelation, direct intervention in man’s activities, through the personality of Jesus
Christ, and His passion, death and resurrection. Yet, the question remains unanswered:
How would you prove that God exists.
P2. Reasons imply reasons. This implication only demonstrates the limitation of the human
mind. Since the human mind cannot know everything at once, man has to proceed to
higher steps in order to know the truth. In doing so, the arguer has to establish himself a
chain of reasoning. For instance, in an effort to prove that God does exists , man has
establish reason after reason, proof after proof just to demonstrate what he/she believes
in. Reasoning is only man’s tool. As Leo Tolstoy said: “Man has received direct from God
only the instrument wherewith to know himself and to know his relation to the universe- he
has no other- and that instrument is reasoning. Hence in this light, Alexander Hamilton
said: “Man is reasoning rather than a reasonable animal.”
P3. An argument is a discussion, an exchange of ideas. Since it is an exchange of ideas,
communication is not a one-way process.
Follow Up
1. What must be proven by an arguer?
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P1. _________________________________________________________________________________.
2. State in your own words the principle stated in P1.
_____________________________________________________________________________________.
3. What does Alexander Hamilton say regarding reasoning?
P2. _________________________________________________________________________________.
4. State in your own terms the principle stated in P2.
_____________________________________________________________________________________.
5. What does P3 say about arguments?
_____________________________________________________________________________________.
6. Is communication a one-way process? Yes or No and why?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
In philosophy, arguments are not just mere verbal squabbles. They are the judgments of
two ideas compared together to examine whether the ideas in question are compatible or
incompatible (Bittle, 1950). When one say “The apple is red”, he compares the two ideas,
“apple” and “redness”.
P4. In the comparison of ideas, the mind must follow three essential factors. First, the
mind must have an understanding of the ideas in question. What is a melamine? Is it a drug?
How about cyanide? Unless the mind knows the ideas in question, it cannot make judgment.
P5. Knowledge of ideas is a fundamental requirement in its analysis, for the absence of it
will lead to misrepresentation. The knowledge of ideas may consists of its definition and
difference with other ideas, its possible extension, differentiating notes or characteristics and
other variables that will make an idea stand out. When an idea is predicated to another idea
e.g. “Virtue is good”, we are proposing an identical relationship between two different ideas,
“virtue” and “good”.
P6. Secondly, the mind must compare the two ideas in question. The comprehension of
each idea, their identity and non-identity, must be recognized first by the mind before it
pronounces their compatibility or incompatibility.
P7. Lastly, the mind must express mentally the agreement or disagreement of the ideas in
question. The mental pronouncement is precisely the essence of judgment (Bittle, 1950).
Follow Up
7. As stated in P4, what is the first factor essential in the comparison of ideas?
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_____________________________________________________________________________________.
8. What is the fundamental requirement in the analysis of ideas? What does it consist of?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
9. What is the second factor in the comparison of ideas? Read P6 as your reference.
_____________________________________________________________________________________
_____________________________________________________________________________________
10. What is the third factor essential in the comparison of ideas? Read P7 for your reference.
_____________________________________________________________________________________
_____________________________________________________________________________________
Arguments are always found in conversation, since man is a communicative animal.
Thus, it is worth noting to discuss the acceptable arguments from the non-acceptable ones.
Recognizing acceptable arguments from the non-acceptable ones is just a matter of thinking,
analysis, and evaluation. An individual has to reflect on matters to be decided on. The term to
reflect is the basis of these three processes namely, thinking, analysis and evaluation.
Furthermore, the knowledge of ideas is an essential procedure in establishing the
acceptability of arguments. For example, “Cyclops are four-eyed monsters” will lead us to the
conclusion that the argument is not acceptable, for the simple reason that the Cyclops are one
-eyed monsters. Without my knowledge that Cyclops are one-eyed monsters, I will be led to
believe that they are four-eyed.
Various definitions of Logic
P8. Logic is the science and the art of right thinking. Unlike the physical sciences, logic
does not concern itself with reality but only with the operation of thinking itself. Logic is not a part
of philosophy but it is a study for preparation towards the study of philosophy. For this reason, the
ancients especially Aristotle called logic as Organon (McCall, 1952).
P9. Furthermore, logic is the science of principles, laws, and methods which the mind of
man in its thinking must follow for the accurate and secure attainment of truth (Bittle, 1950). The
principles such as, the Principle of Causality, the Principle of Identity, the Principle of
Contradiction, the Principle of Contrariety, etc. are developed by philosophy through the years
of study. Moreover, Bittle further explained that the “science of logic is not a parade ground for
mental gymnastics where the mind can disport itself in quibbles and subtle distinction in order to
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squirm out some disagreeable conclusion of fact or theory; rather, logic has the purpose to assist
the mind honestly in discovering and attaining truth wherever it can be found” (Bittle,1950). The
definition does not imply that only students of logic can distinguish between valid reasoning from
invalid ones (Copi, 1970). Rather, the study of logic can facilitate in the distinction between the
valid and invalid reasoning.
Follow Up
11. What is Logic (P8 )and what is its main concern?
_____________________________________________________________________________________
_____________________________________________________________________________________
12. How does Aristotle call Logic?
_____________________________________________________________________________________
13. How does Bittle define Logic in P9.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
14. P9. Enumerate the principles mentioned by Bittle in P9.
a.___________________________________________________________________________________
b. __________________________________________________________________________________
d. __________________________________________________________________________________
15. According to Bittle, what are the purposes of Logic?
The purpose of logic is to assist the mind in ________________ and _______________ truth.
Logic is both a science and an art. It is a science for it has a body of knowledge which
has been verified and systematized. A mere collection of facts does not make a science. On the
other hand, logic as an art, aims to develop the mastery and skill in reasoning. It does not only
provide the knowledge or tools; it also seeks to develop skills through constant use and training
as well as the facility to use such tools. Art, too, is developed in this manner. Hence, in this unit
the learners are expected to do the exercises found in every end of the unit. In this manner,
learners will gain mastery. Through constant practice, the study of logic tends to increase
proficiency in reasoning.
Formal and Material Logic: Distinguished
P10. Thinking and /or reasoning, which is the object of any inquiry, can be viewed from
two standpoints: MATTER and FORM. The former refers to the thought-object of reasoning
(Material Logic) while the latter (Formal Logic) refers more to the rules in attaining the truth.
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P11. Material Logic, then, refers to the trustworthiness, truth, and certainty of the
statements involved in thinking, particularly of the conclusions arrived at by reasoning; whereas,
the Formal Logic refers to the structure and the form of the argument regardless of whether or
not the elements of this reasoning conform with nature.
Example # 1:
Every cat is an animal;
Every tiger is an animal;
Therefore, every tiger is cat.
The contents of the propositions, materially taken, are true. It is true that cats and tigers
are animals. Much more, it is also true that every tiger is cat considering that the cats and tigers
are of the same feline family. But it does not follow that both cat and tiger are the same animal.
A horse is an animal too, yet it is neither a tiger nor a cat. Obviously, we cannot argue:
Example #2
Every cat is animal
Every pig is animal
Therefore, every pig is cat.
Example # 3:
Every animal is vertebrate;
Every cabbage is an animal;
Ergo, every cabbage is vertebrate.
Based on the third example, it is clear that both the minor premise (“Every cabbage is
animal”) and the conclusion (“Every cabbage is vertebrate”) are false statements, but formally
taken, the whole syllogism is valid. If we only assume that all animals are vertebrates and all
cabbages are animals is to conclude that all cabbages are vertebrates. It is very clear in the
example that the conclusion does follow from the premises.
P12. In this case, it is clear that the formal logic is concerned only on the form or the
structure of reasoning rather than on the correctness of the facts. Material logic, on the other
hand, concerns itself with the facts of reasoning rather than its form, as reflected in Example # 1.
As a conclusion, it is not enough to pursue that an argument must be materially correct but
formally invalid nor formally valid but materially incorrect, for the arguments would lack
foundation. Any argument therefore, must be valid and true: valid, i.e. in accordance with the
rules; true, i.e. the statements composing the argument conform with facts or reality. In other
words, an argument to serve its purpose must show validity and truth.
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Name:____________________________________________________ Course/Year/ Section:__________
Class Schedule:___________________________ Date:___________ Score:____________
Activity 1.
What do you think of the arguments below? Do you agree or disagree on them. Explain
briefly your answer and write your explanation on the space provided.
1. I believe that there is God because the Bible says so.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
2. Doctors say that exercise is good; therefore, we must all exercise.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
3. TV advertisements endorse Biogesic as safe and potent drug for headaches. Then, Biogesic is
good for me.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
4. Pedro’s mother is a witch. I am sure, Pedro is also a witch. After all, no fruit tree will be bear
fruit not of its own kind.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
5. I believe that nothing is impossible with God. Therefore, God can tell a lie.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
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Lesson 1. THE TRUTHS IN PROPOSITIONS: THE DETERMINATE AND INDETERMINATE TRUTHS
Learning Outcome
After this lesson, the students will be able to:
� define the terms associated with this lesson
� differentiate the determinate truths from the indeterminate truths
To facilitate the discussion for this lesson, answer the short test below.
Let’s Do It
Activity 1. Among the sentences given below, identify those which can be answered by either
True or False. Mark those sentences with a check mark on the space provided.
_______1. No men are plants.
_______2. Some soldiers are gays.
_______3. A signage in the library: DO NOT SHELVE BACK THE BOOKS.
_______4. Please, open the door.
_______5. What is your name?
_______6. This is rose is red.
_______7. It is not necessary for saints to be martyrs.
_______8. All men are mortal beings.
_______9. Some plants are not vines.
_______ 10. Anybody who passes this door must bring a black book.
What we learn from the activity is that not all sentences can be answered by the
Truth/False method. Of all the sentences we learned from the English grammar classes, only the
declarative sentences can be answered through the Truth/False method. Other sentence types
such as exclamatory, interrogative, exhortatory (requests/ commands/ instructions) are in no
way can be answered by the true /false method.
Follow Up
1. Based on the items, list down on the space provided the non-declarative sentences?
a._____________________________________________________________________________________
b._____________________________________________________________________________________
c._____________________________________________________________________________________
d._____________________________________________________________________________________
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2. Write five (5) examples of declarative sentences.
a._____________________________________________________________________________________
b._____________________________________________________________________________________
c._____________________________________________________________________________________
d._____________________________________________________________________________________
e._____________________________________________________________________________________
By definition, a statement can be considered determinately truth or false if its value is
known immediately. This means that the knower immediately understands or has immediate
knowledge about the truth/falsity of the sentence in question. This immediate knowledge
becomes the knower’s basis for accepting the truth of an argument. For instance,
Corazon C. Aquino is the first woman president of the Philippines. Those who are familiar with the contemporary history of the Philippine political affairs can
immediately infer that the sentence above is true. In this sense, the argument has a determinate
value of true.
On the other hand, arguments whose truth-falsity value is not immediately known are
said to have indeterminate value. Usually, arguments having indeterminate value are expressed
in the future tense or in any manner which connotes mediacy. For example:
Water will boil at 100o C.
Planets may collide with each other.
In summary, sentences that are answerable with either true or false are called arguments
and/or propositions. Hence, the other types of sentences e.g. interrogative, exclamatory, and
imperative sentences are not considered as arguments or propositions, for they can neither be
true or false.
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Name:_____________________________________________________ Course/Year/ Section:__________
Class Schedule:________________________________ Date:_________________ Score: _____________
Activity 3.
Classify the truths-falsity value of the propositions as either Determinate or Indeterminate.
By using the X mark, write your answers on the space provided.
PROPOSITIONS TRUTH-FALSITY VALUE
DETERMINATE INDETERMINATE
1. Ferdinand Magellan conquered Philippines on March 16, 1521.
2. With the coming of the Spaniards to the Philippines, it signaled the
exposition of the country to globalization.
3. Computers will later on invade all aspects of our lives from simple
business functions to space travels.
4. Acts of non-violence are good devices to use for social and political
changes.
5. It is necessary for politicians to enact laws for the betterment of the
country.
6. With the clamor of the youth and for the sake of being relevant,
Congress will soon sponsor a bill making Andres Bonifacio as the
national hero to replace Rizal.
7. The president will soon step down and will turn over the
government to the military junta.
8. Temptations are good indicators of a person’s spiritual strength.
9. Filipinos are still politically immature for they still adhere to
turncoating as the rule of their game.
10. With the value of liberalism and individualism, the family is now
threatened with extreme humanism.
11. The economic performance of a country will be conditioned the
people’s mobility.
12. With the effects brought about the works of Dan Brown, the
church will be divided as either pro or against Dan Brown.
13. Progress is one of our common aspirations as a people.
14. An absolute being, to perfect itself, must exist.
15. To classify human beings, the concept of race is not an
acceptable device.
16. Man is always doomed to peril and nobody can refute this truth.
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Lesson 2. CLASSIFICATION OF CATEGORICAL PROPOSITIONS: THE SIMPLE CATEGORICAL AND THE
MODAL PROPOSITIONS
Learning Outcome
After this lesson, the learners are expected to:
� define the different concepts related to this lesson
� recognize the different types of propositions
� typify propositions
Discussions, discourses, homilies, lectures and other communication avenues use some
forms of generalizations, conclusions or inferences. Some propositions or arguments are stated in
the (a) categorical sense, (b) others in the hypothetical way while (c) some are in the manner of
their existence. These classes of assertions correspond to the three classes of propositions, to wit:
a) categorical, b) hypothetical and c) modal. These three classes we are going to discuss in this
lesson.
Read the text below taken from the book of James 2: 14-26
“What good is it, my brothers, if a man claims to have faith but has no deeds? Can such
faith save him? Suppose a brother or sister is without clothes and daily food. If one of you says
to him, ‘Go, I wish you well, keep warm and well fed,’ but does nothing about his physical
needs, what good is it? In the same way, faith, by itself, if it is not accompanied by action, is
dead. x x x x x. As the body without the Spirit is dead, so faith without deeds is dead.”
A. THE CATEGORICAL PROPOSITIONS
When we say that “All men are created by God”, or that “No men are immortal beings”,
these assertions advanced a truth that is neither qualified nor limited to something. The assertions
did not imply a condition that could modify the truth-falsity value of the said propositions. The
same when I say “Some men are created by God” or “Some men are not immortal”. This is
what we mean by categorical propositions.
S1. By definition, propositions when not compounded with other statements are called
categorical propositions. Furthermore, they have subject and predicate terms, and assert that
either some or all of the classes in the subject are either included or excluded from the classes
designated by the predicate term (Hinacay, and Hinacay, 2004:52). For example,
All students in the SUCs are government scholars.
No imported cars are tax exempt.
Some grapes sold in the Philippines are seedless fruits.
Some dogs are not imported animals.
B. The Types of Categorical Proposition
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B.1. Propositions According to Quantity
S2. Categorical proposition can be further classified into a) Quantity and b) Quality. The
former is taken in consideration of its extension, either Universally extended or Particularly
extended. By the quantity of the subject terms, propositions in questions are accordingly
designated as Universal propositions, Particular propositions, Singular propositions and Indefinite
propositions. To illustrate the types according to quantity, take time to study the pattern or
structure of the propositions below:
Universal Proposition
Every man is mortal.
All men are fallible.
No dog is fish. (All dogs is not fish.)
Particular Proposition
Some men are selfish beings.
Some men are not cowards.
Singular Proposition
This man is a liar.
Paul is an apostle of Jesus Christ.
Indefinite Proposition
Men are selfish
Beauty is truth.
S3. Excluding the indefinite propositions, most of the examples above have quantifiers
such as “all, every, some, and this”. The quantifiers designate the extent of distribution of the
subject term, i.e. whether the subject terms are wholly or partly distributed or extended. The
indefinite propositions, since they do not have quantifiers, are oftentimes designated as universal
propositions. The designation as universal propositions is mainly on the reason of intent of the
argument. For instance, the statement, “He, who looks at a woman with lust commits adultery in
his heart” although singular, but by reason of intent, is designated as a universal proposition. The
examples below
1. Man is a rational animal.
2. Virtue is desirable.
are universal propositions. But
1. Peter is an apostle of Jesus Christ.
2. Charity is a virtue.
are particular propositions.
B.2. Propositions According to Quality
S4. Propositions in this classification are evaluated according to the quality of the copula
whether the subject term is either affirmed or denied of its predicate. In other words, the quality
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of the copula is either AFFIRMATIVE or NEGATIVE. The diagram below shows the structure of
an/a affirmative/ negative proposition.
S is P
Affirmative copula
S is not P
Negative Copula
Examples
a. Some politicians are lawyers.
b. Some lawyers are not lawyers.
c. No men are angles (All men are not angels).
d. All triangles are not equilateral plane figures.
Based from this discussion, it can be deduced that the copula determines the quality of
the proposition. In this way, propositions can either be AFFIRMATIVE or NEGATIVE.
S6. Based on our discussion, we can come up with four (4) types of propositions, to wit:
universal affirmative proposition, universal negative proposition, particular affirmative
proposition, and particular negative proposition. As shorthand, the following letters are used to
symbolize the proposition we have mentioned: A is used for the universal proposition, E, for
universal negative, I, for the particular affirmative, O, for particular negative propositions. These
letters come from the words AffIrmo (I affirm ) and nEgO (I deny).
Examples:
Universal Affirmative (A)
1. All angels are spiritual beings.
2. Every being is a substance.
3. Men are mortal.
Universal Negative (E)
1. No angels are men.
2. No students are professionals.
Particular Affirmative (I)
1. Some athletes are women.
2. A few men are intelligent.
3. Saul is a king.
Particular Negative (O)
1. Some cars are not expensive.
2. A few books are not worth reading materials.
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3. Paul is not a communist.
Follow Up
Q1. Based on your understanding of S1, define briefly the term “categorical propositions.”
_______________________________________________________________________________________
Q2. What are the two main parts of a categorical proposition?
a .__________________________________________________________________________________
b. _________________________________________________________________________________
Q3. Give four (4) examples of categorical propositions. Refer to the examples above for your
reference.
a .___________________________________________________________________________________
b. ___________________________________________________________________________________
c .___________________________________________________________________________________
d. ___________________________________________________________________________________
Q4. Based on S2, how do we know that a proposition is either particular or universal?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Q5. Give one (2) examples of Universal and Particular propositions.
a. ________________________________________________________________________(Universal)
b. ________________________________________________________________________(Universal)
c. ________________________________________________________________________(Particular)
d. ________________________________________________________________________(Particular)
Q6. With the statements below, underline the quantifiers. If a statement has no quantifier, write X
on the space provided.
a. All Doberman dogs are ferocious animals. __________
b. Some knives are rusty. __________
c. No traditional politicians are statesmen. __________
d. Some students are not government scholars. __________
e. All newspapers are tabloid papers. __________
f. Honesty is a virtue. __________
Q7. What part of the proposition that is indicative of its quality?
________________________________________________________________________________________
Q8. Give your own examples of two (2) affirmative and negative propositions.
a. (affirmative)_______________________________________________________________________
b. (affirmative)_______________________________________________________________________
c. (negative)_________________________________________________________________________
d. (negative)_________________________________________________________________________
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Q9. Enumerate the four types of propositions.
a. _________________________________________________________________________________
b. _________________________________________________________________________________
c. _________________________________________________________________________________
d. _________________________________________________________________________________
Q10. Give two (2) examples each of the four types of propositions.
a. (A)________________________________________________________________________________
b. (A) ________________________________________________________________________________
c. (E) ________________________________________________________________________________
d. (E) ________________________________________________________________________________
e. (I) ________________________________________________________________________________
f. (I) ________________________________________________________________________________
g. (O) ________________________________________________________________________________
h. (O) ________________________________________________________________________________
THE MODAL PROPOSITIONS
The modal categorical propositions are actually special types of propositions. The modal
propositions is a composite single sentence in which the copula is so manifested as to express
the manner or mode in which the predicate terms belongs to the subject terms. The
qualification does not affect the subject and the predicate but the copula itself, namely it
stated whether the objective connection between the subject and predicate as expressed by
the copula is necessary, impossible, possible or contingent or not necessary (Bachjhuber, 1954).
In this point, we can come up with four types of modal propositions:
1. S7. necessary mode: The necessary proposition is expressed with the use of the verb “must”,
“have to”, “is necessary” and other verbs which imply necessity. For instance,
Man must be rational animals
It is necessary for man to be rational animals.
Men have to be rational animals.
Some brand new cars must be expensive.
All brand new cars must be expensive.
A brand new car must be expensive.
A brand new Toyota must be expensive.
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In the second set of the examples given above, you can notice that there is a change of
the quantity of the subject term from particular to universal quantity. With the use of the verb
listed above, we can infer that these propositions still belong to the necessary mode regardless
of the quantity of the subject terms.
2. S8. impossible mode: The impossible mode states that the predicate cannot and does not
belong to the subject. In other words, there is no way for the predicate to be included or part of
the extension of the subject terms. Thus, the impossible mode is expressed in the verbs “cannot”,
“can never”, “is impossible”, etc. For instance,
It is impossible for square to be round.
Squares cannot be triangles.
Round can never be circular.
You cannot pass my course without studying.
3. S9. possible mode: The possible proposition enunciates that the predicate is not actually
found in the subject but, it might be such as
It is possible for brand new cars to be expensive
Freshmen students can be accelerated to higher levels.
Women can be presidents.
4. S10. contingent mode: The contingent proposition states that the predicate belongs to the
subject, but it need not be. It is expressed in the verb “is not necessary”, “need not”, etc. For
instance,
It is not necessary for Filipinos to render military service to the state.
Students need not pay their tuition fees before taking the final exams.
It is contingent for politicians to be lawyers.
Politicians need not be lawyers.
However, it is very important for us to note that modal propositions presuppose a former
judgment; thus, the statement “It is impossible for God to be unjust” presuppose a former
judgment that “God is just”.
S11. With reference to the classification of propositions according to quantity and quality,
the necessary mode is classified as A proposition; impossible mode, E; possible mode, I, and
finally, contingent mode, O.
Necessary Mode (A)
1. An absolute being must exist.
2. It is necessary for cats to pur.
Impossible Mode (E)
1. Angels cannot be mortals.
18
2. It is impossible for squares to be circulars.
Possible Mode (I)
1. This car can run 200 miles per hour.
2. It is possible for believer to become saints.
Contingent Mode (O)
1. It is not necessary for God to be a creator just to become a God.
2. It is contingent for tables to have four legs.
Again, it has to be emphasized that only the verb determines the classification of the
modal propositions into their respective classes. Hence, regardless the number or quantity of the
subject term, the type of the verb used overrule the quantity of the subject term
Follow Up
Q11. Define necessary mode and give at least two (2) examples of modal categorical
propositions.
Definition:___________________________________________________________________________
____________________________________________________________________________
Examples:
1:____________________________________________________________________________________
_____________________________________________________________________________________
2:____________________________________________________________________________________
______________________________________________________________________________________
Q12. What is an impossible mode and what does it imply?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Q13. Give at least two (2) propositions belonging to the impossible mode.
1:___________________________________________________________________________________
____________________________________________________________________________________
2:___________________________________________________________________________________
____________________________________________________________________________________
Q14. What is an possible mode and what does it imply?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Q15. Give at least two (2) propositions belonging to the possible mode.
19
1:___________________________________________________________________________________
____________________________________________________________________________________
2:___________________________________________________________________________________
____________________________________________________________________________________
Q16. What is a contingent mode and what does it imply?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Q17. Give at least two (2) propositions belonging to the contingent mode.
1:___________________________________________________________________________________
____________________________________________________________________________________
2:___________________________________________________________________________________
____________________________________________________________________________________
Q18. True or False. The quantity of the subject term is the sole determinant on the classification
of the modal propositions.
Answer: ____________________________
20
Name:_____________________________________________________ Course/Year/ Section:__________
Class Schedule:_________________________________ Date:___________ Score:______________
Activity 4.
Classify the propositions below into their proper class. Write A, E, I, O on the space
provided
1. Some saints are martyrs. ________________
2. All patriots welcome martyrdom. ________________
3. Few good men oppose tyranny. ________________
4. No skeptics believe that truth exists. ________________
5. Some men are not morally sensitive. ________________
6. Books are good sources of information. ________________
7. All computers are educational tools. ________________
8. This nation is bereft of moral integrity. ________________
9. No politicians are lawyers. ________________
10. Some lawyers are not politicians. ________________
11. Some politicians are lawyers. ________________
12. All trees are fruit-bearing plants. ________________
13. Some trees are not fruit-bearing plants. ________________
14. Some animals are vertebrate. ________________
15. No animals are vertebrate. ________________
16. Negative propositions are either E or O propositions. ________________
17. All buildings are skyscrapers. ________________
18. Some tables are not four-legged.. ________________
19. No web pages are educational sites. ________________
20. Some web pages are artistically created. ________________
21. Some technologies are Filipino inventions. ________________
22. Some technologies are not Filipino inventions. ________________
23. No Filipina women are domestic helpers abroad. ________________
24. Some OFWs are Filipinos. ________________
25. All OFWs are the new heroes of the country. ________________
21
Lesson 3: THE RULES OF IMMEDIATE INFERENCE
Learning Outcome
After this lesson, the learners are expected to:
� answer the activities in this lesson
� apply the rules of logical inference
The lessons we have learned from the previous lesson (Lesson 2) have shown the types of
propositions, e.g. A, E, I, and O. if we are going to compare each one of them retaining their
subject and predicate terms differing only on the quality of the copula -- either affirmative or
negative -- a relation can be deduced and an inference can be derived from such relation. For
instance, if we assume the proposition “All athletes are players” to be true, what can we say
about the following proposition. Let your answers be either True or False.
a. Some athletes are players. ________
b. No athletes are players. ________
c. Some athletes are not players. ________
If you have answered False for item letters B and C and T for item letter A, then you got it
right. Thus, if we assume to be true that “All athletes are players”, then it follows that “Some
athletes are players”. This method of inference accepts that the truth of the universal
proposition is also the truth of the particular proposition. In other words, whatever is held
universally is also held particularly. In illustrating this concept, if all my students are present, then I
can say that Juan, being my student, is also present.
In order to know the truth value of propositions, it is imperative for us to study the
principles covering the logical oppositional relation. By opposition here, we mean the
difference between the quality and quantity of the propositions although the subject and
predicate terms are similar. These principles are: the Principle of Contradiction, the Principle of
Contrary, the Principle of Subcontrary, and the Principle of Subalternation.
Principle of Contradiction
S7. This principle can be simply stated that nothing can be true and at the same time
false. In other words, if the first proposition is true, the second proposition is false. On the other
hand, if one of the propositions is false, the other proposition is true. In other words, the
contradicting proposition outrightly negates the other proposition for it is impossible for
propositions to be both true and false at the same time.
S8. In connection with this principle, we have the principle of the Excluded Third. This
principle demands that there is no middle ground between the two contradictories. Hence, a
22
proposition cannot be true at the same time false. The relation between A and O, E and I falls
under the principle of contradiction
Example:
All men are mortal beings. (A)
Some men are not mortal beings. (O)
No men are mortal beings. (E)
Some men are mortal beings. (I)
All textbooks are classroom materials. (A)
Some textbooks are not classroom materials. (O)
No birds are two-legged fowl. (E)
Some birds are two-legged (I)
The Principle of Contrariety
S8. This principle states that the relation between two universal propositions wherein the
first proposition is assumed to be true, the other universal proposition is assumed to be false. For
instance, the proposition “All animals are mammals” is assumed to be true, then the value of the
proposition “No animals are mammals” necessarily is false.
On the other hand, the relation between the two universal propositions wherein the
value of the first proposition is assumed to be false, the value of the second universal proposition
becomes indeterminate. Just for the sake of illustration, let us examine the statement “Alice is
older than Betty”. If the statement “Alice is older than Betty” is true, the statement “Betty is older
than Alice” is false. On the other hand, if the statement “Alice is older than Betty” is false, does
it necessarily follow that “Betty is older than Alice’? Good judgment requires that it is not so. The
statement “Betty is older than Alice’ becomes false if both Alice and Betty are of the same age.
Example:
All idolaters are sinful people .(A) -- T
No idolaters are sinful people. (E) -- F
If A is true, then E becomes false.
All metals melt under high pressure (A). -- F
No metals melt under high pressure (E) -- Doubtful
If proposition A is assumed to be false, the proposition E becomes doubtful.
23
Let’s Practice
Instruction: By using the principles of logical opposition (Principle of Contradiction and Principle
of Contrariety), state the truth-falsity of value of the remaining proposition?
1. All stars are heavenly bodies - T
a. Some stars are not heavenly bodies. ___________
b. No stars are heavenly bodies. ___________
2. No plants are perennial trees. -- T
a. All plants are perennial trees. ___________
b. Some plants are perennial trees. ___________
3. Some imported cars are luxury cars. T
a. No imported cars are luxury cars. ___________
4. Some dogs are not imported animals. T
a. All dogs are imported animals. ___________
The Principle of Subcontrariety
S9. The principle of subcontrariety involves the relation between the two particular
propositions e.g. I and O proposition. The principles states that if one of the particular
proposition is false, necessarily the remaining proposition become false. Hence, if I is false, then
O becomes true. Thus, if we argue that “Some dogs are imported animals” is false, then the
proposition “Some dogs are not imported animals” is true. What if one of the particular
propositions is true? If one of the particular propositions, e. g., I is true, the O proposition
becomes doubtful or indeterminate. For instance
Some Filipinos are overseas workers. (I)
Some Filipinos are not overseas workers. (O)
If we assume I to be true, then O becomes indeterminate or doubtful.
We have to noted, however, that the validity of the subcontraries cannot be claimed if
the propositions in question materially contain necessary truths. The truth of this principle can be
applied validly to propositions containing only contingent truths. Thus, the propositions,
Some squares are circles. -- I
Some squares are not circles -- (O)
is not tenable for the propositions to be compared contain necessary truths. This is the violation
of the existential import for the propositions contain contradictions in themselves. Thus if we deny
the I proposition, then we have to admit the O proposition.
24
The Principle the Subalternation
S10. The principle of Subalternation states the truth of the universal propositions is also the
truth of their subsequent particular propositions for the whole cannot be considered as such
unless their whole possible extension would be part of the whole. I cannot infer that all of my
students are present in the class if one of them is absent. By simple enumeration, everybody
during my roll call, everybody is present thus I concluded that all my students are present. This is
called inference by simple enumeration.
S11. The other application of this principle is inference by class. The inference by class
does not involve enumeration rather taking the possible extension as a class or category of
being. For instance, the proposition “All men are rational animals” indicates that the class of
men is a class of rational animals. I do not have to go through individual investigation or
enumeration just to know whether all men are rational animals. The fact that I am a man,
therefore I am a rational animal.
Example:
All carabaos are black (A) -- T.
Some carabaos are black (I) -- T.
From the example, it is clearly evident that the truth of the universal proposition is also the
truth of the particular propositions. Thus, if A is true, then I is also true; if E is true, then O is also
true. However, it has to be noted that the truth of the particular proposition cannot be the truth
of the universal propositions. For instance,
Some carabaos are black (I) -- T.
All carabaos are black (A) -- ?
Analyzing the example above, if the proposition “Some carabaos are black” is true, to
infer that “All carabaos are black” does not necessarily follow.
S12. The other side of the principle of Subalternation concerns the falsity values. The
principle states that the falsity of the universal proposition does not constitute the falsity of the
particular proposition rather the falsity of the particular proposition makes the universal
proposition false.
Some students are diligent -- (I) - F
All students are diligent -- (A) - F
Follow Up
Q19. Define the Principle of Contradiction.
_____________________________________________________________________________________
25
Q20. What are propositions whose relation fell under the principle of contradiction?
_____________________________________________________________________________________
Q21. Restate in your terms the principle of the Excluded Third.
_____________________________________________________________________________________
Q22. State in your own words, the principle of Contrariety.
______________________________________________________________
______________________________________________________________
Q23. In your own words, simplify the principle of subcontrariety.
a. ___________________________________________________________________________________
b. ___________________________________________________________________________________
Q24. Enumerate the propositions covered by the principle of Subalternation.
a. ___________________________________________________________________________________
b. ___________________________________________________________________________________
Q25. State in your own words, the principle of Subalternation.
a. ___________________________________________________________________________________
b. ___________________________________________________________________________________
Q26. Enumerate the propositions covered by the principle of subcontrariety.
a. ___________________________________________________________________________________
b. ___________________________________________________________________________________
Q27. Evaluate whether the inference is Valid or Invalid. Put your answer in the space provided
and explain very briefly.
a. If Pedro is absent from the class is true, then to conclude that all students are present
becomes false.
Answer: _____________________________________________________________________________
Reason: _____________________________________________________________________________
________________________________________________________
b. All Filipino presidents must be natural born citizens. Therefore, Gloria Macapagal-
Arroyo is a natural born citizen.
Answer: _____________________________________________________________________________
Reason: _____________________________________________________________________________
_____________________________________________________________________________
c. I am certain that all soldiers are trained warriors. Hence it will false to argue that some
soldiers are not trained warriors.
Answer: _____________________________________________________________________________
Reason: _____________________________________________________________________________
_____________________________________________________________________________
26
d. All judges are liars therefore, some judges are not liars.
Answer: _____________________________________________________________________________
Reason: _____________________________________________________________________________
_____________________________________________________________________________
Q28. Complete the diagram below and give the truth-falsity value of each proposition if we
assume to the first to be true.
A E - T
I O
27
Name: _____________________________________________________ Course/Year/ Section:__________
Class Schedule:________________________________________ Date:___________
Activity 5.
Change the given propositions into their opposites and identify the truth value of their
corresponding relation.
1. No terrorists are patriots. (True) T/F
1.1. Contradiction:________________________________________ _____
1.2. Contrary:___________________________________________ _____
1.3. Subalternation.:______________________________________ _____
2. All mayoralty candidates are professionals. (False)
2.1. Contradiction:________________________________________ _____
2.2. Contrary:___________________________________________ _____
2.3. Subalternation.:______________________________________ _____
3. Some quotations are words of wisdom. (False)
3.1. Contradiction:________________________________________ _____
3.2. Subcontrary:________________________________________ _____
3.3. Superalternation.:____________________________________ _____
4. No Christians are Muslims. (True)
4.1. Contradiction:________________________________________ _____
4.2. Contrary:___________________________________________ _____
4.3. Subalternation.:______________________________________ _____
5. Some cabinet members are career professionals. (True)
5.1. Contradiction:_______________________________________ _____
5.2. Subcontrary:________________________________________ _____
5.3. Superalternation.:____________________________________ _____
6. Some families are convention delegates. (False)
6.1. Contradiction:________________________________________ _____
6.2. Subcontrary:________________________________________ _____
6.3. Superalternation.:____________________________________ _____
7. No animal rights are human rights. (True)
7.1. Contradiction:________________________________________ _____
7.2. Contrary:___________________________________________ _____
7.3. Subalternation.:______________________________________ _____
8 .Some professionals cannot be illiterates (False)
8.1. Contradiction:________________________________________ _____
8.2. Subcontrary:________________________________________ _____
8.3. Superalternation.:____________________________________ _____
9. All hospitals must be medical institutions. (True)
9.1. Contradiction:________________________________________ _____
9.2. Contrary:___________________________________________ _____
9.3. Subalternation.:_______________________________________ _____
28
10. Some candidates need not be professionals. (False)
10.1. Contradiction:________________________________________ _____
10.2. Subcontrary:_________________________________________ _____
10.3. Superalternation:_____________________________________ _____
B. Evaluate whether the inference is valid or invalid. Put your answer in the space provided.
1. It is certain that no text messages are censored communications. Hence, if we conclude that
there are some text messages being censored is doubtful.
Answer: __________________________________________________________________________________
2. It is true that some voters are poll watchers. But to say that all voters must be poll watchers is
false.
Answer: __________________________________________________________________________________
3. It is certain that oil producing countries are rich countries. Therefore, Kuwait is rich.
Answer: __________________________________________________________________________________
4. If all medicines are therapeutic preparation is true, then BIOGESIC is a medicine.
Answer: __________________________________________________________________________________
5. It is certain that some tourist sites are historical sites. But to say that no tourist sites are historical
sites is false.
Answer: __________________________________________________________________________________
6. That all saints must be virtuous is true. Whether some saints need not be virtuous is doubtful.
Answer: __________________________________________________________________________________
7. It is true that some philanthropists are generous. Whether all philanthropists must be generous
is doubtful.
Answer: __________________________________________________________________________________
8. It is true that all brand new cars are expensive. Whether brand new cars must be expensive is
doubtful.
Answer: __________________________________________________________________________________
9. It is certain that some politicians are lawyers. Therefore, all politicians are lawyers.
Answer: __________________________________________________________________________________
10. It is certain that all presidents are citizens. Hence it is false to conclude that some presidents
need not be citizens.
Answer: __________________________________________________________________________________
29
LESSON 4: THE EDUCTIVE REASONING
Learning Outcome:
After this lesson, the learners are expected to learn:
a. define the different types of eductive arguments
b. form the eductive equivalent of the simple categorical propositions
The four types of simple categorical propositions (A, E, I, O) that we study also implied an
equivalent relation or an immediate inference. Immediate inference is a process of deriving a
conclusion or assumption from an original argument whereby retaining the truth value of the
original proposition. What is implied by the statement: “All men are bipeds”?
This proposition “All men are bipeds” implies that those who do not belong to the class of
bipeds are not men. Thus we can deduce that “No men are no-bipedss”. If we assume as true
that “All men are bipeds”, then it follows to assume that “No men are non-bipeds”.
S1. This type of argument or propositional derivation is called eduction. By definition,
eduction is a process of immediate inference, whereby, from any proposition taken as true, we
derive others implied in it, though differing from the first in subject or predicate or both (Bittle,
1950:155). There are three main forms of eduction: obversion, conversion, and contraposition.
OBVERSION
Obversion (L., Ob, before, towards, and vertere, tu turn) is a process of immediate
inference in which the inferred judgment, while retaining the original subject, has far its
predicate the complementary class of the original predicate. By complementary class, we
mean the excluded members or assertions of the predicate class. For the sake of illustration, in
the proposition, “All men are mortal beings”, the predicate term “mortal beings” has a
complementary class of “non-mortal beings”. In the same way, the proposition, “Some scholars
are genius”, has a complementary class of “non-genius”. To simplify the procedure, the
complementary class of any term is the affixing of the term “non” to any term in question.
IN Obversion, the process of eduction involves two changes. These changes occur in the
(1) quality of the proposition (but not in its quantity) and the (2) status of the predicate term
(Barker, 1989, as cited by Hinacay and Hinacay, 2004:99). In obversion, the original proposition is
called obvertend while the derived proposition is the obverse (Bittle, 1950:156). Following the
procedures outline above, we can now form the obverse of the following proposition:
1. If the obvertend is an A proposition, its obverse will be the E proposition with the predicate of
the obvertend substituted with the complementary class of its predicate.
30
2. If the obvertend is an E proposition, its obverse will be the A proposition with the predicate of
the obvertend substituted with the complementary class of its predicate.
3. If the obvertend is an I proposition, its obverse will be the O proposition with the predicate of
the obvertend substituted with the complementary class of its predicate.
4. If the obvertend is an O proposition, its obverse will be the I proposition with the predicate of
the obvertend substituted with the complementary class of its predicate.
OBVERTEND OBVERSE Example
OBVERTEND OBVERSE
All S is P No S is non P. All angels are spiritual
beings.
No angels are non
spiritual beings.
No S is P. All S is non P. No angels are carnal
beings.
All angels are non carnal
beings.
Some S is P. Some S is not non P. Some students are
scholars.
Some students are not
non scholars.
Some S is not P. Some S is non P. Some plants are not vines. Some plants are non
vines.
Further Examples:
Obvertend: No men are non-bipeds.
Obverse: All men are bipeds.
Obvertend: All angels are non-mortal beings.
Obverse: No angels are mortal beings.
Obvertend: Some brand new cars are non-expensive.
Obverse: Some brand new cars are not expensive
Obvertend: Some homicides are not non-murders.
Obverse: Some homicides are murders.
Let’s Practice:
Instruction: Form the obverse of the propositions below. Write your answers on the space
provided.
1. All men are rational animals.
____________________________________________________________________________________________
2. No umpires are prejudiced.
____________________________________________________________________________________________
31
3. Some saints are martyrs.
____________________________________________________________________________________________
4. No fish is a mammal.
____________________________________________________________________________________________
5. Some professors are not scholars.
____________________________________________________________________________________________
CONVERSION
Conversion (L., convertere, to turn) is a process of immediate inference, in which the
inferred judgment takes the subject of the original proposition (convertend) for its predicate and
the predicate of the convertend as its subject (Bittle, 1950:158). The derived proposition is called
converse. In simple terms, conversion is a simple switching of the subject and predicate terms.
Thus, the proposition “Some new graduates are employed professionals” becomes “Some new
employed professionals are new graduates”.
There are three rules to observe in making the conversion.
1. Interchange the subject and predicate terms. This rule implies that the quality of the
proposition and the quantity of the terms should be left unchanged.
Example:
Convertend: No carabaos are pigs.
Converse: No pigs are carabaos.
2. Retain the quality of the proposition. If the convertend is affirmative, then the converse must
also be affirmative. If the convertend is negative, then the converse must also be negative.
Example:
Convertend: Some students are athletes.
Converse: Some athletes are students.
3. Do not extend any term. The quantity of the term must not be affected in the process of
conversion. If a term, either subject term or predicate term, is particular, then in conversion, it
must retain its quantity. Concerning the quantity of the terms, as a general rule, the predicate
term of all affirmative propositions is always particular while the predicate term of all negative
proposition is always universal.
Example:
Convertend: No rubies are diamonds (In this case, the predicate is universal).
Converse: No diamonds are rubies.
32
Convertend: Some Filipinos are millionaires. (In this case, the predicate is particular).
Converse: Some millionaires are Filipinos.
Convertend: All plants are living beings.
Converse: Some living beings are plants. (Conversion by limitation)
Convertend: All men are rational animals.
Converse: All rational animals are men. (Full conversion).
Convertend: Some plants are not trees.
Converse: Some trees are not plants.
It is clear that the conversion of O proposition is generally invalid.
Let’s Practice
Instruction: Form the converse of the following convertend below. Write your answers on the
space provided.
1. All men are rational animals.
2. No judges are partisan.
3. Some saints are martyrs.
4. No fish are mammals.
5. Some students are not athletes.
33
CONTRAPOSITION
Contraposition is the process of eduction which combines conversion and obversion. It is
formed by two steps:
1. Application of the conversion process.
2. Substitution of the subject and predicate terms of the converse with their
complementary class.
The original proposition is referred to as contraponend while the equivalent proposition is
called the contrapositive.
Example:
Contraponend: All men are mortal beings.
Contrapositive: All non-mortal beings are non-men.
Or
All immortal beings are non-men.
Thus, if the proposition, “All men are mortal beings” is true, then its contrapositive “All
immortal beings are non-men” is true.
Example:
Contraponend: Some prisoners are not innocent individuals.
Contrapositive: Some non-innocent individuals are not non-prisoners.
Concerning the I propositions, the contrapositive is generally invalid. For the E proposition, its
contrapositive is an O proposition. This procedure is called as partial contraposition. Hence, if
we argue
No spiritual beings are moral being
Its contraposition will be:
Some non-mortal beings are not non-spiritual beings.
If we want to check our contrapositive propositions for its validity, the following procedure
can be adopted:
34
Original Proposition
Step 1. Form the obverse
Step 2. Form the converse of Step 1.
Step 3. Form the obverse of Step 2. This is the contrapositive.
For example:
Original Proposition: All men are mortal beings.
Step 1. Obverse: No men are n on-mortal beings.
Step 2. Converse: No non-mortal beings are men.
Step 3. Obverse: All non-mortal beings are non-men. (This is the contrapositive of the original
proposition “All men are mortal beings”).
Let’s Practice:
Instruction: Form the contrapositive of the proposition below. Write your answers on the space
provided.
1. All men are rational animals.
_________________________________________________________________________________
2. Some saints are martyrs.
_________________________________________________________________________________
3. No umpires are partisans.
_________________________________________________________________________________
4. Some books are not educational materials.
_________________________________________________________________________________
5. No fish is a mammal.
_________________________________________________________________________________
6. No elements are living.
_________________________________________________________________________________
7. Every plant is an organism.
_________________________________________________________________________________
8. Some professors are scholars.
_________________________________________________________________________________
9. Some athletes are drug-dependent individuals.
_________________________________________________________________________________
10. Every professor must be a scholar.
_________________________________________________________________________________
35
Name: _____________________________________________________ Course/Year/ Section:__________
Class Schedule:________________________________________ Date:___________
Activity 6. By applying the rules of immediate inference, what can be said about the validity of
the inferences?
1.
All Filipino lawyers are members of the Philippine Bar. Therefore,
no Filipino lawyers are non-members of the Philippine Bar.
2.
All presidents are natural-born citizens. It follows then that all non
natural-born citizens are non-presidents.
3.
It is certain that every dog is a member of a canine family.
Therefore, Rottweilers are dogs.
4.
It is true that some research findings are questionable. Whether
there are some research findings that are not questionable.
5.
No judges are biased is true. Therefore, all judges are non-
biased.
6.
No typewriters are high-end technology is true. Therefore, it is
also true to infer that all typewriters are non high-end technology.
7.
All Sony VAIO Notebooks are high-end products. Therefore, all
high-end products are Sony VAIO Notebooks.
8.
It is certain that some mortal beings are men. Whether all men
are mortal beings is doubtful.
9.
It is acceptable that no carabaos are pigs. It further follows that
no pigs are carabaos.
10.
It is true that no perishable things are spiritual beings. Therefore,
all non spiritual beings are perishable things.