Logic midterm notes

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CHAPTER II

JUDGMENT and PROPOSITION

Form is a revelation of essence.
As the drop becomes the ocean, so the soul is deified, losing her
Name and work, but not her essence.
You must break the outside to let out the inside; to get at
Kernel means breaking the shell. Even so to find nature herself all
Her likeness has to be shattered.
(Anonymous)

Chapter Outline
1. Judgment Defined
a) Elements of Judgment
two known ideas
comparison of these two ideas
mental pronouncement of
identity/non-identity between two ideas
2. Proposition Defined
a) Parts of Proposition
subject (S)
predicate (P)
copula (c)
3. Predicables
species
genus
differentia
property
accident

Suggested Learning Activities

1. Spotting the difference between a mere sentence and a judgment/proposition.
2. Imaginary ‘anatomy’ of the body.
3. Mind game: What’s more essential: a) character or fame? b) Beauty or brain? c) money or
person?
4. ‘Measuring depth of knowledge’ game.

The Proposition as Expression of Judgment

Judgment is the second mental process or operation that essentially figures in the
act of reasoning or thinking. A judgment is an act of the mind pronouncing an agreement
or disagreement between two ideas. Three things are required in the making of judgment,
namely: 1. two known ideas, their 2. comparison, and 3. the act of the intellect
pronouncing their identity or non-identity. Here is how a judgment exemplified. “A
computer is a machine.” This sentence is one written example of a judgment. The „computer‟
is the subject term while „machine‟ is the predicate term. These two terms then represent the
two ideas that the mind is comparing. The „is‟ in the same sentence represents the act of
pronouncement executed by the mind on the two ideas „computer‟ and „machine‟. In this
intellectual pronouncement, the mind is engaged in processing whether or not the two ideas are

in agreement or disagreement and/or identical or non-identical with each other. The example
demonstrates that the mind specifically makes a pronouncement of agreement since the verb
„is‟ indicates that the mind recognizes the identity between the “computer‟ and „machine‟. Had
the verb in the sentence been an “is not‟, it would have implied that the mind makes a
pronouncement of disagreement between the two ideas.
Truth or, for that matter, falsity is then contained not in „idea‟ but in judgment. In other
words, what is expressed by a judgment is either a truth or a falsity. If the judgment conforms to
the reality of the thing about which it is made, then it is true. If things are not as the judgment
asserts them to be, then the judgment is false.
It is not therefore difficult to figure out that a judgment is always expressed in a sentence,
and a declarative sentence at that. An interrogative sentence, or an exclamatory, or a request or
a command, cannot constitute a judgment per se because in these kinds of sentences the mind
does not make a pronouncement of agreement or disagreement between two ideas or term.
For instance, in this sentence, “Is Derrick Rose an NBA MVP?” no judgment is made by the
mind because it (the mind) is not engaged in making a pronouncement of agreement or
disagreement between the ideas „Derrick Rose‟ and „MVP‟. In asking question, the mind has
not yet arrived to a definitive judgment whether one idea is affirmed or denied of another idea.
Hence, no determination of truth or falsity is reached yet by the mind in an interrogative
sentence. The same can also be said in exclamatory and imperative sentences. For examples:
“Oh my goodness!” and “Please, hurry up.” In both sentences, the mind does not make any
comparison between two ideas. Thus, no judgments per se have been made by the mind in
these two sentences.
Just as idea is externally expressed in either verbal or written word technically called term,
a judgment is also expressed in an oral or written sentence technically called proposition. A
proposition is a judgment expressed in a sentence. It can then be readily stated here that
all propositions are sentences but not all sentences are propositions. Since there are sentences
that express neither truth nor falsity (thus they don‟t contain judgment), then not all sentences
are propositions. For instance, this interrogative sentence, “What day is today?”, although a
sentence, is not however a proposition because it does not contain a judgment. It expresses
neither truth nor falsity because it does not imply a mental act of pronouncing agreement or
disagreement between two ideas. Every proposition consists of a subject (S), a predicate (P),
and the copula (c); subject and predicate are the matter, and the copula is the form of the
proposition. It is the copula which indicates whether one term is denied or affirmed of the
subject. For example, in the proposition, “Her car is expensive,” the copula „is‟ expresses
affirmation; in the proposition, “Meriam Defensor Santiago is not media-friendly,” the copula
„is not‟ expresses denial or negation. Since the copula expresses the present act of the mind in
its judgment, it is always expressed in the present tense of the indicative mood of the verb „to
be‟ even if the sentence refers to past or future events.
When a proposition is structured or constructed grammatically as “S c P,” it is said to be in
its logical form. Examples: “Bin Laden is a Muslim extremist” and “Angel Locsin is not
arrogant.” These propositions are in their logical form. However, not all sentences employ „to
be‟ always as their verbs. Examples: “The communist countries disintegrated,” and “Brazil will
win hopefully again in the next Football World Cup.” The verbs used in these sentences are not
from the infinitive „to be,‟ and neither are they in the present tense, but past and future tenses
respectively. In cases like the above, what is needed is simply to reduce the propositions to
their logical form. So, “The communist countries disintegrated,‟ can be reduced to its logical
form which is, ”The communist countries are the counties that disintegrated.”; “Brazil will win
hopefully again in the next Football World Cup,” becomes “Brazil is the team that will win
hopefully again in the next Football World Cup.” Basically, no truth or falsity value is altered in
the propositions reduced to their logical forms.

Putting the proposition in its logical form makes its meaning clear and minimizes a lot of
misinterpretation, misunderstanding or confusion arising from the vague formulation of the
proposition.

The Predicables

Predicables (or logical universals) are the different modes or ways in which a universal
(used as a predicate term in a proposition) can be predicated of its subject. Understanding the
functionality of predicables in the thinking process enhances depth of knowledge about realities.
In fact, predicables serve in determining that there can be different levels of depth in
understanding things. Thus they anchor the thinker‟s capacity to define or describe realities.
That there are only five predicables can be seen by considering the essential and accidental
connections that a predicate of a proposition has with its subject. There is an essential
connection between a subject and predicate, if the latter expresses something that is intrinsic to
the nature or essence of the subject; the connection is accidental or non-essential when the
predicate expresses something that is extrinsic to the essence or nature of the subject. Thus,
for instance, „rational‟, if predicated of „man‟ in the proposition “Man is rational,” indicates
essential connection between the two precisely because rationality constitutes a part of man‟s
essence or nature; „religious fundamentalist,‟ if predicated of the same as in this proposition
“Man is a religious fundamentalist,” indicates only an accidental connection between them since
being a religious fundamentalist is not a necessary part of man‟s nature or essence.
The following are the five predicables:
1. Species refers to the manner in which a universal idea when used as a predicate of a
proposition expresses the whole essence of its subject. Example: „Man is a rational animal.‟
Precisely, „rational animal,‟ as predicated of „man‟ expresses the whole essence of what man is
in the real order.
2. Genus refers to the manner in which a universal idea when used as a predicate of a
proposition expresses a part of the essence of its subject, that part which the subject has in
common with other species in the same class. Example: „Man is an animal‟; „animal‟ as
predicated of „man‟ in the proposition expresses only a part of the essence of what man is.
Aside from man, however, there are also many other species of which „animal‟ can be
predicated, such as: „Dog is an animal.‟ „Dolphin is an animal.‟
3. Differentia refers to the manner in which a universal idea when used as a predicate of a
proposition expresses a part of the essence of its subject, that part which distinguishes one
species from another under the same genus. So: „Man is rational‟ and „Brute is irrational” are
two propositions whose predicates „rational‟ and irrational‟ express a part of the essence of their
respective subjects thus making the latter different (differentia) from each other, albeit they
belong to the same genus of „animal.‟
4. Property refers to the manner in which a universal idea when used as a predicate of a
proposition expresses something that flows necessarily from the essence, though not of the
essence itself. Example: „Man is capable of writing.‟ The predicate of this proposition „capable
of writing‟ is not of and even a part of the essence of what man is because man would still
remain to be a man even if he does not write. Yet, from man‟s rational nature, the capacity for
writing flows necessarily. Otherwise, writing would not be possible.
5. Accident refers to the manner in which a universal idea when used as a predicate of a
proposition expresses something of its subject which is neither of its essence nor necessarily
connected with its essence, but is merely contingently connected with the essence. Example:
„Man lives in a condominium.‟ In this proposition, the predicate „lives in a condominium‟ is never
of and part of the essence of what man is; neither is it necessarily flowing from man‟s essence,
but it is only an accidental circumstance or condition of man. It can entirely be otherwise.

_______________________
CHAPTER III

TYPES OF PROPOSITIONS

For thousand of years man has lost touch with his original intelligence...
...He starts interfering in the course of nature with a mind that is centered and one-pointed,
and analyzes everything, and breaks it down into bits...
The moment you do that you lost contact with your original know-how...
...getting back to being able to trust our original intelligence,
...(is drawing) an entirely new course for the
development of civilization.
Alan Ginsberg

Chapter Outline
I. The General Types of Propositions
A. Propositions according to their Quality
1. Affirmative
2. Negative
B. Propositions according to their Quantity
1. Universal
2. Singular
3. Particular
4. Collective
C. General Propositions expressed in Logical Shorthand Symbols
1. A proposition or Universal affirmative
2. E proposition or Universal negative
3. I proposition or Particular affirmative
4. O proposition or Particular negative
D. Propositions according to Relation between subject(s) and predicate(s)
1. Analytic (essential, necessary, ‘a priori’)
2. Synthetic (accidental, contingent, ‘a posteriori’)
II. The Special Types of Propositions
A. Single Categorical Propositions
1. Simple
2. Composite
o Complex
o Modal
Necessary
Impossible
Possible
Contingent
B. Multiple Categorical Propositions C. Hypothetical Propositions
1. Overtly Multiple Categorical a) Conditional
a) Copulative b) Disjunctive
b) Adversative c) Conjunctive
c) Relative
d) Causal
e) Comparative
2. Covertly Multiple Categorical
a) Exclusive
b) Exceptive
c) Reduplicative
d) Specificative

1. Group power point presentation of ‘standardization’ models of consumer products. Elaboration of
the practical benefits of this ‘standardization’ model by each group.
2. Random samples of mnemonics to enhance the mental processes of memory and imagination.
3. Board contest: ANALYSIS AND SYNTHESIS

Knowing considerably well all possible appearances of propositions in their grammatical
constructions contributes to the clarity and keenness of thinking and reasoning. More so, the
mind will easily penetrate to the truth and falsity found in the judgment and proposition if it can
see through the intricacies of linguistic structures. Hence, the knowledge of the various types of
propositions can come in handy here. There are general and special types of propositions as
delineated below.
The General Types of Propositions
The general types are based on the quality, quantity, and relation of subject and predicate
found in proposition.
Quality of Proposition. From the standpoint of quality a proposition is either affirmative
or negative. The quality of a proposition affects the copula and makes the proposition either
affirmative or negative. In other words, the predicate is either affirmed or denied of the subject.
Examples: „Bioinformatics is an academic course taught in medical school,‟ and „The
architectural design of the CICCT building is modern.‟ In these instances, the copula „is‟ implies
that each of the predicates of both propositions is affirmed of the subject. In these other
propositions „The consultant‟s grasp of the human psychology of the entire institution is not
impressive,‟ and „The feasibility study of the graduating accountancy students is not that viable,‟
the copula implies that each of the predicates of both propositions is denied of the subject.
As a rule, in an affirmative proposition, the predicate is always used according to the whole
of its comprehension and a part of its extension; it is, therefore, always a particular term. In a
negative proposition the predicate is always used according to a part of its comprehension and
the whole of its extension; it is, therefore, always a universal term. This logical rule should be
borne in mind because it bears important relation to the other laws and methods of correct
thinking that will be treated later in the other chapters.
Quantity of Proposition. The quantity of a proposition affects the extension of the whole
judgment as a judgment. It expresses the number of individuals to whom the judgment or
proposition applies. Since the predicate is applied to the subject, the proposition will be true of
all the individuals contained in the extension of the subject. From the standpoint of quantity,
then, propositions will be 1. universal, 2. particular, 3. singular, and 4. collective, whichever
way the extension of the subject is taken.
1. A proposition is universal if the subject is a universal term applied distributively to
each and all of the class. Examples: „All normal cows are four-legged.‟ „Every voter is a
citizen.‟ The use of „every‟ to prefix a subject denoting universality can never be ambiguous.
However, there are cases that when „all‟ is prefixed to a subject, it would not necessarily
indicate universality of a proposition. Take this for instance: „All players filled up the locker
room.‟ This can be taken only collectively not distributively because to say, „Every (single)
player filled up a locker room,‟ does not make any real sense. So, here, we must look to the
meaning of the judgment or proposition.
2. A proposition is particular when the subject is a universal term used partly and
indeterminately. „Particularity‟ of a term is indicated by some words like „some‟ or „not all,‟
attached to the subject. Examples: „Some artists are very eccentric,‟ „Not all nursing students
are females.‟ But sometimes, a statement with the word „all‟ attached to the subject makes that
statement appear like a universal proposition at first glance. Yet, a closer scrutiny would reveal

it to be otherwise. Example: „All Boholanos are not handsome.‟ This statement seems on the
face of it to be universal. But this is certainly not the sense intended, because obviously there
are Boholanos who are evidently (according to ultimate aesthetic standard) handsome.
Therefore, “All Boholanos are not handsome,” in its real sense, implies, „Not all Boholanos are
handsome,‟ which conversely is the same as saying „Some Boholanos are not handsome.”
Hence, it makes it a particular proposition.
3. A proposition is singular when the subject applies to a single individual only. Such are
the statements: „Carlos Polestico Garcia was a true-blooded Talibon, Bohol native.‟ „That
engineering student is not attentive.‟ „The present Pope is a German.‟ Singular propositions
have the same value as universal propositions and are treated the same way because the
subject is taken according to the whole of its extension, which in this case, is one.
4. A proposition is collective when the subject is a collective term, applying to all taken
together as a class, but not to the individuals composing the class. Examples: „Team
Philippines emerged as the champion in the 23rd South East Asian Games (SEA Games).‟ „All
troops of the 7th Infantry Brigade of the Philippine Army surrounded the swampy territory of
the insurgents.‟ „The entire gallery of spectators witnesses the buffoonery of the senators.‟
Since a collective term represents many considered as one, it is taken according to the whole of
its extension and it, too, is treated as a universal.
Addendum: There is one kind of proposition which has no definite sign of quantity
attached to the subject. This is called indefinite proposition. Example: „Athletes are physically
gifted.‟ „Mga Cebuanos hawod ug kamao mopili kinsa ang angayan nga mamuno sa nasud.‟
„Teenagers are restless.‟ Such and similar propositions indicate no definite quantity. They
evidently mean „some‟ or „all‟ and are either particular or universal propositions. The sense of
the statement or the context in which they are used must give us the exact quantity.
All propositions, considered simultaneously according to their quality and quantity result to
four general propositions, namely: the universal affirmative, the universal negative, the
particular affirmative and the particular negative. They are deemed general propositions in
order to point out the fact that practically all propositions can be reduced to this general
classification. This classification will in turn provide a kind of standardization of proposition thus
enhancing the mind‟s capacity for better and orderly comprehension.
Why then there are only four general types of propositions? As far as quantity is concerned,
all judgments have the value of either universal or particular propositions because singular and
collective propositions are equivalent to universal propositions as elucidated beforehand. And,
by reason of quality, all propositions will either be affirmative or negative. Logicians express
them in a sort of logical shorthand symbols with the letter A standing for universal affirmative; E
for universal negative; I for particular affirmative; and O for particular negative. The letters A
and I are the letters A and I in the Latin word „AffIrmo‟ which means „I affirm.‟ Letters E and O
are derived from „nEgO,‟ meaning „I deny.‟ This therefore gives us the following scheme.

Universal affirmative (A) ---------- ex: All men are mortal.
or Every man is mortal.
Universal negative (E) ---------- ex: No man is a mechanical invention.
or All men are not mechanical inventions.
Particular affirmative (I) ---------- ex: Some men are altruistic.
Particular negative (O) ---------- ex: Some men are not narcissistic.

Relation between Subject and Predicate of Proposition. From the standpoint of the
relation between subject and predicate, propositions will either be necessary or contingent.
When the relation is a necessary matter, the essence or nature of the subject is always
contained in the comprehension of the predicate or vice versa; hence, an analysis of one will

reveal the other and thereby reveal the truth of the proposition. Examples: „Man is a rational
animal.‟ „A square is a quadrangle.‟ „Two and two are four.‟ „A fish is capable of breathing
underwater.‟‟ But when this relation is a contingent matter, neither the subject nor the predicate
is contained in the comprehension of the other, and an analysis of one will not reveal the other;
since this relation is based on a contingent fact, we can prove the truth of the proposition only
by experience. Examples: „Carbon Market is a stone‟s throw away from the university
campus.‟ „The Mt. Pinatubo eruption in 1991 effected a change in global temperature.‟ „The
super typhoon left a huge area of destruction.‟ The necessary proposition is, therefore,
analytic, while the contingent proposition is synthetic; the former is also called „a priori‟ and
the latter „a posteriori.‟

The Special Types of Propositions

One function of language is to convey thought and the truth or falsehood of this thought from
one mind to another. But the complexity of language wrought by myriad of words tends to
muddle the meaning of a judgment or proposition. The knowledge of these special types of
propositions will definitely facilitate in understanding better the thought-contents or judgments
implied in many different grammatical or propositional structures.
Before exposing all these special propositions, it will be helpful to understand first and
foremost the individual characteristic feature(s) of a single, a multiple, a categorical, and an
hypothetical proposition. A single proposition consists of one subject and one predicate.
Examples: „A cab is a vehicle.‟ „The football field is deserted.‟ „An Oversea-Filipino-Worker
(OFW) is a modern day martyr.‟ A multiple proposition consists of two or more propositions
united into one sentence. Examples: „Ringo and Paul were members of the Beatles.‟ „Robert
Jaworski was a professional basketball player and a senator.‟ „If an earthquake hits Cebu, many
dilapidated buildings will collapse.‟ A categorical proposition attributes (affirms or denies) the
predicate of its subject outright. In other words, the predicate is directly asserted to (either
affirmed or denied of) its subject in a categorical proposition. Example: „Corruption is
immoral.‟ „Sex per se is not evil.‟ A hypothetical proposition asserts the dependence of one
judgment on another. Example: “If Manny Pacquiao does not focus on his training, he cannot
remain a pound-for-pound boxing king for long.‟
Single Categorical Proposition. The categorical proposition, as has been stated, makes a
direct assertion (hence it is also called assertoric) of agreement or disagreement between
subject and predicate. The single categorical proposition contains but a single sentence in its
construction – one subject, one predicate, and the copula. If the subject, the predicate and the
copula are not modified in any way, i.e., no qualification whatsoever enters into each and all of
them (the subject, the predicate and copula), then it is simple categorical. In other words, the
subject and predicate of a single categorical proposition are simple terms (consisting only of one
word) and the copula is at the same time not modified. Example: „Love is madness.‟
In this example, the subject „Love‟ is unmodified or it is a simple term; the predicate „madness‟ is
also a simple term; and the copula „is‟ is not modified. But if a qualification enters into the
subject or predicate or copula, what results is a composite single categorical proposition.
Examples: „This man is a passionate revolutionary.‟ „That lady wearing signature jeans is an
ex-nun.‟ „The intramural games may not be held this semester.‟ In the first example, the
predicate „passionate revolutionary‟ is not a simple but a compound term; in the second, the
subject „lady wearing a signature jeans‟ is also a compound term; and in the third, the verb „may
not be held‟ indicates a modified copula. Hence, all three propositions are composite
propositions.
There are two kinds of these composite propositions, namely, the complex and the modal.
The complex proposition is a composite single sentence in which both the subject and the
predicate or either one is a complex term. That‟s why „Colonel Gringo Honasan is a soldier,‟ „He

is a renegade lawmaker too,‟ and „The Armed Forces of the Philippines is a graft-ridden
government establishment,‟ are all examples of complex (composite single categorical)
propositions. The modal proposition is a composite single sentence in which the copula is so
modified as to express the manner (mode) in which the predicate belongs to the subject.
The qualification affects the copula, that is, it states whether the objective connection between
subject and predicate expressed by the copula is necessary, impossible, possible, or
contingent.
The necessary modal proposition states that the predicate belongs to the subject and
must belong to it. Examples: „God is necessarily powerful.‟ „A tsunami must be extremely
huge.‟ In these examples, „is necessarily‟ and „must be‟ are modified copulas. The impossible
proposition states that the predicate does not and cannot belong to the subject. Examples:
„Man cannot live forever.‟ „It is impossible that the globe is flat.‟ The possible proposition
enunciates the fact that the predicate is not actually found in the subject, but it might be.
Such are: 'One Filipino basketball player will be able to play in an NBA team.‟ „The mountain
climbers could have asphyxiated.‟ The underlined parts of both propositions given above also
indicate modified copulas. The contingent proposition states that the predicate actually
belongs to the subject, but it need not. Examples: „It is not necessary that the audience
keeps on standing.‟ „Blood need not be shed to attain peace.‟ Again, both examples show
modified copulas.
To facilitate familiarization, here is a chart of the single categorical propositions:

Single Categorical
Propositions

Simple
Composite

Complex Modal

Necessary Possible

Contingent Impossible

Multiple Categorical Propositions. Multiple categorical propositions contain two or
more sentences in their very construction. Some of these are overtly, while the others are
covertly multiple; the latter are called „exponibles.‟

The overtly multiple categorical propositions are plainly composed of two or more
propositions. They are five in number: 1. copulative 2. adversative 3. relative 4. causal and
5. comparative.
The copulative proposition is a multiple categorical proposition which has two or more
subjects, or two or more predicates, or two or more subjects and predicates, combined into
one sentence. Examples: „SM and Gaisano Country Malls are supermarkets.‟ „The passenger
bus skidded and plunged into a ravine.‟ „Tony Parker and Boris Diaw are non-American NBA
players and Spurs team mates.‟ Each of these sentences can be resolved into as many single
propositions as there are different subjects and predicates. The truth of copulative categorical
proposition depends upon the truth of all the single sentences which compose it.
The adversative (discretive) proposition is a multiple categorical proposition which consists
of two propositions united in opposition to each other by conjunctions like, „although,‟
„yet,‟ „even if,‟ „but,‟ etc. Examples: „Bill Gates became so successful, even if he dropped out
of college.‟ „A winner stumbles countless of times, yet he never quits.‟ „The torturers might have
broken him physically, but not his spirit.‟
The relative proposition is a multiple categorical proposition which expresses a
relationship of time or place between two sentences. Examples: „The incidence of bombing
in Iraq diminished a bit after they have installed a government.‟ „Before the cock crowed three
times, Peter chickened out and disowned his master.‟ „The demagogues or the traditional
politicians squirmed in their seats during the entire period that the witnesses divulged their
shenanigans.‟
The causal proposition is a multiple categorical proposition which combines two
statements in such a way that the one is given as the reason or cause of the other. The
words „for‟ or „because‟ or „due to‟ or „since‟ are some words used to connect the statements.
Examples: „There is massive poverty and hunger in the Philippines due to the absence of
moral leadership.‟ „Because of intolerance and bigotry, terrorism of all kinds persists in the
world today.‟
The comparative proposition is a multiple categorical proposition which compares the
relation between a subject and a predicate with the same relation between another subject
and predicate, and expresses the degree of this relationship as being either less or equal or
greater. Examples: „Rafael Nadal is a better lawn tennis player than Roger Federer in clay
court.‟ „As you live, so you shall die.‟ „An Apple laptop is flashier in design than an Acer.‟
The covertly multiple categorical propositions have the appearance of single propositions,
although they are really multiple. Their composition lies concealed in some words and needs an
exposition to show their multiple characters; hence, they are styled „exponibles,‟ and the
sentences into which the general proposition can be resolved are called the „exponents.‟ The
„exponibles‟ are four in number: 1. exclusive, 2. exceptive, 3. reduplicative, and 4.
specificative.
The exclusive proposition is a multiple categorical proposition which contains some
particles of speech like „only,‟ „alone,‟ „solely,‟ „none but,‟ etc., indicating the exclusion of any
other predicate from this subject or any subject from this predicate. Examples: „Only you can
make my dreams come true.‟ „Not only is the Philippines not economically progressive.‟ The
affirmative exclusive exponible will be resolved by means of a copulative proposition, in which
one sentence is affirmative and the other negative; thus, „Only you can make my dreams come
true,‟ will become, „You can make my dreams come true and no one or nobody else can.‟ The
negative exclusive exponible will be resolved into two negative exponents; the sentence „Not
only is the Philippines not economically progressive,‟ will become „The Philippines is not
economically progressive and some other countries are not also economically progressive.‟
The exceptive exponible is a multiple categorical proposition which contains a particle of
speech like „except‟ or „save‟ to indicate that a portion of the extension of the predicate does
not apply to the subject, or vice versa. Examples: „All hostages except an old mother were not

released by the kidnappers.‟ „The entire class passed save those who dropped out.‟ The
resolution of such exponible should go like this: „An old mother was released by the kidnappers,
and all others were not.‟
The reduplicative and specificative propositions are multiple categorical propositions which
contain an expression which duplicates the subject or predicate, giving them special
emphasis. The expressions „as such‟ „as a or such an‟ and „such a‟ are some expressions
employed in reduplicative or specificative propositions. Example of reduplicative proposition: „A
hovercraft, as an amphibious vehicle, can operate both on land and on sea.‟ In the
reduplicative proposition, the duplicated subject or predicate, can be taken as the reason itself
or cause for the connection between the subject and predicate. Example of a specificative
proposition: „Niña, as a student, is a freshman.‟ The duplicated subject or predicate in the
specificative proposition merely implies a circumstantial condition of the connection between
subject and predicate. For facility, here is a schematic outline of all multiple categorical
propositions:

Multiple
Categorical

Overt Covert

Copulative Adversative Relative Causal Comparative Exclusive Exceptive Reduplicative Specificative

The Hypothetical Propositions. The hypothetical proposition differs from the categorical
(assertoric) proposition, because it does not declare an unqualified affirmation or denial, but
expresses the dependence of one affirmation or denial on another affirmation or denial. This
proposition has three distinct kinds: the 1. conditional, the 2. disjunctive, and the 3.
conjunctive.
The conditional proposition is a hypothetical proposition which expresses a relation in
virtue of which one proposition necessarily flows from the other because a definite condition is
verified or not verified. Example: „If the international benchmark of the price of oil continues to
rise, social unrest will occur in many third world countries.‟ The statement containing „if‟ is
called „antecedent’ proposition; and the other is the „consequent.‟ The truth of this
hypothetical proposition does not depend on the truth of the two statements taken individually
and separately, but on the relation between them.
The disjunctive proposition is a hypothetical proposition which contains an
„either – or‟ statement, indicating that the implied judgment cannot be true together nor false
together, but one must be true and the other (or others) false. Example: “A student can either
be a scholar or a truant.‟ There are two variations of disjunctive proposition, namely: perfect or
proper disjunction which implies that the parts should be such that neither can be true or false
together; example: „A body is either in motion or at rest.‟ An imperfect or improper disjunction

is taken in wider sense and implies then „at least one, but possibly some or all of the parts. . .‟
Example: Either St. Peter or St. Paul or St. John died in Rome.‟ In this wider sense, all are not
allowed to be false, but all may be true.
A conjunctive proposition is a hypothetical proposition which expresses a judgment that
two alternative assumptions are not and cannot be true simultaneously. Examples: „An ace
athlete cannot be a disciplined person and voracious eater at the same time.‟ „A person cannot
be a hero and a villain at the same time.‟
A working knowledge of the various types of categorical and hypothetical propositions will
enable us to unravel the maze of words and detect the truth of the judgment implied in them.

____________________________
CHAPTER IV

OPPOSITION OF PROPOSITIONS
____________________________
“Not to speak with a man who can be spoken with is to
lose a man. To speak to a man who cannot be spoken with is to waste
words. He who is truly wise never loses a man; he too, never wastes his words.”

Chapter Outline

I. An introductory note on Chapters IV and V

II. The Logical Oppositions of Propositions A, E, I, and O
1. Subalternation
2. Contradiction
3. Contrariety
4. Subcontrariety

III. The Laws of Logical Opposition
1. Laws of Subalternation
2. Laws of Contradiction
3. Laws of Contrariety
4. Laws of Subcontrariety

1. Connecting the dots.
2. Mind’s Power of “finishing other’s sentence.”
3. Figuring out “things that are better left unsaid”
4. Pick up lines games

Note: For easy understanding of chapters IV and V, it is beneficial to situate these lessons properly. Both the logical
oppositions of propositions (chap. IV) and eductions (chap. V) are two intellectual processes that are inferential by nature.
Inference is a process whereby the mind, in its act of thinking, draws out or derives a certain judgment from one judgment that
it (mind) already comprehends or considers. Specifically, logical oppositions of propositions and eductions are immediate
inferences in contrast to reasoning (proper) which can be called more fittingly as a mediate inference. The distinction between
immediate and mediate inferences will be made clearer in the treatment of both in their respective chapters.

The logical opposition of proposition refers to the relation which exists between
propositions having the same subject and the same predicate, but differing in quality, or
in quantity, or in both. There are four possible ways for a proposition to have the same subject
and the same predicate, but differing in quality, or in quantity or in both: a universal affirmative
(A), a universal negative (E), a particular affirmative (I), a particular negative (O). Examples:
(A) – „All men are learned.‟ (E) – „No men are learned.‟ (I) – „Some men are learned.‟ and (O) –
„Some men are not learned.‟ The four oppositional relations that can possibly exist among
these propositions are exemplified in the subjoined Square of Opposition.

All men are learned. No men are learned.

A contrariety E

subalternation contradiction subalternation

I subcontrariety O
Some men are learned. Some men are not learned.

This diagram illustrates the four relations resulting from four oppositions, namely: 1.
subalternation, 2. contradiction, 3. contrariety, and 4. subcontrariety.

Note on the Square of Opposition

Seen against the square above,
Subaltenation can be traced with
line starting from the topmost of both the right and left
corners of the square and downward to their lowest counterparts.
It shows then the relations between A to I and E to O or vice versa.
Contrariety is the line connecting A and E or vice versa.
Subconrariety is the line between I and O or vice versa.
Contradiction is the diagonal line inside the square connecting
A to O and E to I or vice versa.

Subalternation is the opposition between a universal and particular affirmative (A and I),
and between a universal and particular negative (E and O). Both propositions, the universal
and particular, are called „subalterns‟; the universal (A and E) being the „subalternant,‟ while
the particular (I and O) is the „subalternate.‟
Contradiction is the opposition existing between the universal affirmative (A) and a
particular negative (O), and between universal negative (E) and a particular affirmative (I).
Contrariety is the opposition existing between a universal affirmative (A) and a universal
negative (E).
Subcontrariety is the opposition existing between a particular affirmative (I) and a particular
negative (O).

Laws of Logical Opposition. Our mind, in its natural state, that is, even without yet a
formal training or orientation in scientific logic, actually follows already patterns in reasoning or
thinking. These various thinking patterns and structures are explicated as mental laws or
methods in a formal study we now familiarly know as logic. Some of these very basic laws
govern the various relations existing among the four general propositions. The most practical
benefit we can derive from the knowledge of these laws is that it makes us figure out easily
whether or not judgment or proposition and its possible implications can be true or false. These
are the following laws.
1. Law of Subalternation: A-I and E-O There are two rules for this relation: the first rule
states that the truth of the universal involves the truth of the particular, but the truth of the
particular does not involve the truth of the universal. In other words, if A is true, I must also be

true; and if E is true, O must also be true. If I is true, A need not be true, but is doubtful
(doubtful means that the proposition is either true or false); if O is true, E need not be true, but is
doubtful.
There are thus two sections to this first rule. The first section implies that it is always logical
to conclude from the truth of the universal to the truth of the particular. What is true of „all‟
individuals of a class must also be true of „some‟ of these individuals, because what is true of
the „whole‟ must be true of every „part‟ of the whole. If „All men are mortal,‟ then surely „Some
men are mortal.‟ And if „No men are pigs,‟ then „Some men are not pigs.‟ either. On the other
hand, if I is true, we cannot conclude, in virtue of the proposition as such, that A is also true; and
if O is true, we cannot validly conclude that E is also true; What is true of „some‟ need not be
true of „all‟, because what is true of a „part‟ of a class need not be true of the „whole‟ of the class.
Examples: „Because some men are left-handed,‟ it does not follow that „All men are left-
handed.‟ and because „Some men are not polyglot,‟ it does not follow that „No men are polyglot.‟
It might happen, of course, that what is true of „some‟ is also true of „all‟ and what is true of a
„part‟ is also true of the „whole.‟ Examples: Both „Some men are mortal‟ (I), and „Some men are
not pigs‟ (O) are true. So: „All men are mortal‟ (A), and „No men are pigs‟ (E) are also true.
We are, therefore, never warranted to conclude from the truth of the particular to the truth of
the universal. It may be so, but it need not be so. In virtue of this phase of subalternation law,
we cannot validly argue from „some‟ to „all‟ and from the „part‟ to the „whole.‟ Hence, the second
section of the first rule is established: The truth of the particular does not involve the truth of the
universal; the truth of the universal will always be doubtful.
The second rule of the law of Subalternation states: The falsity of the particular involves the
falsity of the universal; but the falsity of the universal does not involve the falsity of the
particular. Here we begin with the falsity of one of the subaltern propositions (I to A, O to E),
and the rule states: if I is false, A is also false; and if O is false, E is also false. But if A is false,
I need not be false; and if E is false, O need not be false.
There are thus two sections to this second rule. The first section implies that it is valid for the
mind to draw conclusion from the falsity of the particular proposition to the falsity of the
universal. The reason is clear. In order that something be true of „all‟, it must be true of every
individual that belongs to the „all‟; that something be true of the „whole,‟ it must be true of every
„part‟ contained in the „whole‟. How, then, can something be true of „all‟ if it is false of „some‟?
That would mean that „all‟ are, although „some‟ of the „all‟ are not; and it would follow that every
„part‟ of the „whole‟ is, although a „part‟ of the „whole‟ is not; the same „some‟ and the same „part‟
would then both „be‟ and „not be‟ something at the same time. Hence, if it is false that „Some
men are pigs,‟ it is all the more false to state that „All men are pigs‟; and if it is false that „Some
men are not mortal,‟ it is also false to say that „No men are mortal.‟ From the falsity, therefore,
of the propositions, I or O, we must conclude to the falsity of the respective universal
proposition, A or E.
The second section of this second rule reads: If A is false, I need not be false; and if E is
false, O need not be false. In order that a universal be true, every individual of the class and
every „part‟ of the „whole‟ must be included in the truth of the universal; hence, the universal will
be false, if not every individual of the universal and not every „part‟ of the „whole‟ is included in
the truth of the universal statement. This means that if a universal proposition is false, some of
its individuals must also be false, but some (of the others) may be true. But if „some‟ may be
true, even if the universal („all‟) is false, it is obvious that we cannot legitimately conclude from
the falsity of the universal to the falsity of the particular. Thus, if „All men are learned‟ (A), and
„No men are learned‟ (E) are both false, respectively, „Some men are learned‟ (I) and “Some
men are not learned‟ (O) are true. This proves definitely that that the falsity of the universal
does not involve the falsity of the particular: the particular may be true, even if the universal is
false. However, the particular may also be false. Examples: Both „All men are pigs‟ (A) and
„No men are mortal‟ (E) are false; so, „Some men are pigs‟ (I) and, „Some men are not mortal‟

(O) are also false. We thus see that, when the universal proposition is false, its respective
opposite (subaltern) particular proposition may be either true or false. The falsity of the
universal only entitles us to conclude that some of the individuals are false, leaving the matter
undecided whether the others are true of false. Hence, the falsity of the universal propositions
(A and E) does not involve the falsity of their respective opposite (subaltern) particular
propositions (I and O): the particular proposition may or may not be false together with the
universal; the falsity of the particular will always be in doubt, when the universal proposition is
false.
We thus see the truth of the Law of Subalternation: the truth of the universal involves the
truth of the particular, but the truth of the particular does not involve the truth of the universal;
the falsity of the particular involves the falsity of the universal, but the falsity of the universal
does not involve the falsity of the particular.
The Law of Contradiction. A-O and E-I. This law also has double phases, of which the
first rule is: contradictories cannot be true together. If A is true, O is false; if O is true, A is
false; if E is true, I is false; if I is true, E is false. In an affirmative universal (A) proposition, it is
asserted that the predicate is affirmed of each and every individual belonging to the subject: „All
men are mortal.‟ If this is true, then it must be false to deny this statement of „some‟; hence, the
statement that „Some men are not mortal‟ (O) cannot be true. In a negative universal (E)
proposition, it is asserted that the predicate must be denied of each and every individual
belonging to the subject: „No men are pigs.‟ If this statement is true, then it must be false to
say that „Some men are pigs‟ (I). What is true of all must be true of every one of the class; to
state at the same time that „all are‟ and „some are not,‟ and that „none more‟ and „some are,‟
would violate the Principle of Contradiction. Hence, if the universal affirmative (A) is true, the
particular negative (O) must be false; and if the universal negative (E) is true, the particular
affirmative (I) must be false.
The law works also the opposite way: if O is true, A is false; and if I is true, E is false. If it is
true that „Some men are not intelligent‟, it is certainly false that „All men are intelligent‟; and if
it‟s true that „Some men are mortal,‟ it must be false to assert that „No men are mortal.‟ And
thus the rule is established: Contradictories cannot be true together.
The second rule reads: Contradictories cannot be false together. If A is false, O is true;
if E is false, I is true; if O is false, A is true; if I is false, E is true. If it is false that „All men are
intelligent,‟ it must be true that „Some men are not intelligent.‟ (A-O). If it is false that „No men
are learned,‟ it must be true that „Some men are learned.‟ (E-I). If it is false that „Some men
are not mortal,‟ it must be true that „All men are mortal.‟ If it is false that „Some men are pigs,‟ it
must be true that „No men are pigs.‟
From the above, it will be clear that the contradictory pairs form a perfect opposition among
themselves: they can be neither true nor false together; one must be true and the other must be
false. There is no neutral middle ground between contradictories: a thing either is or is not; if
the one is true, the other must be false.
Law of Contrariety: A-E. The rules are: Contraries cannot be true together; contraries
may be false together. If A is true, E is false; if E is true, A is false. If A is false, E may be true
or false; if E is false, A may be true or false.
The first rule states that contraries cannot be true together; if one of the contraries is true,
the other contrary must be false. The correctness of this rule is easily demonstrated with the
help of the Law of Subalternation and the Law of Contradiction. Suppose the universal
affirmative proposition (A) is true: „All men are mortal.‟ According to the Law of Contradiction, if
a universal proposition is true, its contradictory proposition must be false. Therefore, it is false to
say that „Some men are not mortal.‟ (I). Now, according to the Law of Subalternation, „the
falsity of the particular involves the falsity of the universal.‟ So, from the falsity of „Some men
are not mortal,‟ (O) the statement, „No men are mortal‟ (E) is also false. Hence, if A is true, E is
false, i.e., confirming that contraries cannot be true together.

The same line of thinking applies if we begin with E, the universal negative, as true: „No
men are pigs‟ (E). Since this is true, its contradictory (I) must be false, namely, „Some men are
pigs.‟ And since this particular affirmative (I) is false, it also involves (according to the Law of
Subalternation) the falsity of the universal (A) “All men are pigs.‟ Therefore if E is true, A must
be false; or the rule: Contraries cannot be true together.
The second rule states that contraries may be false together; if one contrary is false, the
other contrary may also be false (though it need not be false, but may be true). Examples: If
„All men are intelligent‟ (A) is false, its contradictory, „Some men are not intelligent‟ (O), must be
true. But the Law of Subalternation states that „the truth of the particular proposition does not
involve the truth of the universal.‟ Hence, although it is true that „Some men are not intelligent‟
(O), we cannot conclude legitimately that „No men are intelligent‟ (E) is also true; E may be true
or false. Hence, both contraries may be false. Similarly, granting that „No professors are
learned‟ (E) is false, its contradictory, „Some professors are learned‟ (I) must be true. Again,
„the truth of the particular proposition does not involve the truth of the universal.‟ Therefore, the
truth of A is not established; it may be true to say „All professors are learned‟; but the statement
may also be false. Contraries then may be false together.
Law of Subcontrariety: I-O. The twofold rule here states: Both subcontraries cannot be false
together; but both subcontraries may be true together.
The first rule demands: If I is false, O is true; if O is false, I is true. Examples: If „Some men
are pigs‟ (I) is false, according to the Law of Contradiction, “No men are pigs‟ (E) must be true.
Now, if a universal proposition is true, its particular proposition (according to the Law of
Subalternation) is also true. So, it must be true to state that „Some men are not pigs‟ (O). We
can arrive at the same result in a different way. If (I) „Some men are pigs‟ is false, (A) „All men
are pigs‟ (according to the Law of Subalternation) is also false; but if A is false, (O) „Some men
are not pigs‟ (according to the Law of Contradiction) must be true. Hence if I is true, O must be
true.
We arrive at the same conclusion, if we begin with the falsity of O. If O is false (‟Some men
are not mortal.‟), its contradictory A („All men are mortal.‟) must, according to the Law of
Contradiction, be true. But if A („All men are mortal.‟) is true, I („Some men are mortal.‟) must,
according to the Law of Subalternation, also be true. Hence, if O is false, I must be true. We
thus see the first rule governing subcontrary propositions (I and O): Subcontraries cannot be
false together; at least one of the two must be true.
The second rule of the subcontraries (I and O) states that both may be true together: if I is
true, O may be true; if O is true, I may be true. Suppose I is true: „Some Filipinos are
Cebuanos.‟ The contradictory of this proposition, namely E, must be false, and we cannot
assert that „No Filipinos are Cebuanos.‟ But we know from the Law of Subalternation that „the
falsity of the universal does not involve the falsity of the particular.‟ Hence, even though E is
false, we cannot conclude to the falsity of O: O may be true, that is, „Some Filipinos are not
Cebuanos.‟ We can also begin with O as true. If O is true, that is, “Some men are not learned‟,
its contradictory A is false, and that is, „All men are learned.‟ Since, however, we cannot
conclude from the falsity of the universal to the falsity of its particular (Law of Subalternation), it
does not follow that I is also false: the I statement, that is, „Some men are learned‟ may be true.
We have thus established the twofold rule governing the subcontrary propositions I and O:
Both subcontraries cannot be false together; but both subcontraries may be true together.
The following are legitimate conclusions from the study of the different oppositional
relations of propositions (subalternation, contradiction, contrariety, and subcontrariety):
If A is true: then I is true, E is false, O is false.
If A is false: then O is true, E is doubtful, I is doubtful.
If E is true: then O is true, A is false, I is false.
If E is false: then I is true, A is doubtful, O is doubtful.
If I is true: then E is false, A is doubtful, O is doubtful.

If I is false: then O is true, A is false, E is true.
If O is true: then A is false, E is doubtful, I is doubtful.
If O is false: then I is true, E is false, A is true.
Logical Opposition of Modals. The treatment of modal propositions is similar to the
ordinary categorical propositions, but the logical opposition affects the mode itself. The
„necessary‟ mode resembles the A proposition; the „impossible mode, the E proposition; the
„possible‟ mode, the I proposition; the „contingent‟ mode, the O proposition. The logical
opposition of the modal propositions may affect only the mode, or it may affect both the mode
and the quantity of the propositions.
Conclusion. The method of concluding from the truth or falsity of one statement to the truth
or the falsity of another is called immediate inference. It is called „immediate‟, because we can
pass directly from the one to the other, without the necessity of adducing any other idea or
judgment as proof. The Square of Opposition, therefore, with its relations of subalternation,
contradiction, contrariety, and subcontrariety, will act as a powerful aid toward correct thinking.

Logic midterm notes

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