Part II, Lesson 2

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Part II, Lesson 2

• Divisions of the Proposition

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Divisions of the Proposition

• There are at least five ways to divide the
  proposition according to types. Some are
  mentioned here, but will not be studied until
  later.

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1. According to their Unity

• Our minds can conceive different types of unity
  that can be expressed in the formation of a
  proposition.
• This unity can be absolute or relative.

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• A proposition has absolute unity when it
  signifies that the mind says one thing of another
  without qualification.

A tiger is a mammal.
Example
Here, we are saying one thing, mammal, of
another, tiger.
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• Since in this example we are signifying one act
  of predication, the resulting proposition has
  absolute unity. What matters is not the
  complexity of the subject or predicate, but
  whether one act of predication is intended.

Example
All human beings are capable of learning Logic.
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• This simple type of proposition is known as the
  categorical proposition.

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• A proposition has relative unity if it express
  the union of two or more categorical propositions
  by means of certain characteristic connective
  expressions.

Example
If man is rational, then he is capable of using
tools.
This type of proposition does not have absolute
unity, since combines various categorical
propositions with relative unity rather than
expressing a simple act of predication.
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• This type of proposition is known as the compound
  or hypothetical proposition.
• It has different parts than the categorical
  proposition, whose parts are subject, predicate
  and copula.
• The parts of the compound proposition are two
  categorical propositions united by some type of
  connection.

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• The categorical proposition uses a verb to
  signify the union of predicate and subject.
• The hypothetical proposition uses a non-verbal
  copula. Other examples are if then,
• either or, and, etc.

Examples
If the light is green, then you may cross the
intersection.
Either you sink or you swim.
Today is Monday, and we have a test.
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2. According to Quality

• This manner of dividing propositions applies only
  to categorical propositions.
• This division of propositions gives us
  affirmative and negative propositions.

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• A proposition is affirmative if it expresses
  union of predicate with subject. An affirmative
  proposition is a sign of the intellectual act of
  composition.

Example
The tiger is a quadruped.
This proposition expresses the union of the
predicate with the subject by means of the
verb-copula is. It affirms quadruped of
tiger.
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• A proposition is negative if it expresses the
  separation of the predicate from the subject.

Example
A tiger is not a lion.
This proposition denies lion of the subject
tiger.
In this case, it does so by using a negative verb
is not.
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• Yet it isnt quite as simple as looking at the
  verb to decide whether a proposition is negative,
  because there are alternate ways to express
  separation between subject and predicate.

No tiger is a lion.
Example
This proposition is also negative, because the
intellect is denying lion of tiger.
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• In this case, although the no appears with the
  subject instead of with the verb, its function is
  to express a denial.
• To decide whether a proposition is affirmative or
  negative, then, we need to ask whether the
  predicate is being affirmed or denied of the
  subject.

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• It must also be remembered that this
  consideration of the affirmative or negative
  quality of a proposition only applies in the case
  of categorical propositions, since all compound
  propositions will necessarily be affirmative.
  The categorical propositions of which they are
  formed may be negative, but the copulas of
  compound propositions are always connective.

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• Also, there may be more than one verb even in a
  categorical proposition, but only one of them can
  function as the copula.

Example
People who are not friendly are unsociable.
From this we can see the importance of
properly distinguishing the subject, the
predicate, and the verb.
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• People who are not friendly are unsociable.

Although there are two verbs here, are not and
are, only one is the copula.
Also, although the subject is negative, as is
the predicate, this does not make the
proposition negative, since the negative
predicate is being affirmed of the negative
subject.
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• Non-citizens are ineligible to vote.

Non-citizens are not eligible to vote.
The first proposition is affirmative, while
the second is negative.
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3. According to Quantity

• The quantity of a proposition refers to the
  quantity of its subject. The quantity the
  subject has is the quantity of the proposition as
  a whole.
• In Part One, we distinguished between singular
  and universal. What is singular is not
  predicable of many.

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• So a singular proposition is one whose subject is
  singular, of which we affirm or deny some
  predicate.

Example
John is tall.
or
This man is a thief.
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• When the subject of a proposition is universal,
  there are three ways we can predicate something
  of this universal

• We can predicate something of a universal
• subject universally, when we want to
• express that the predicate belongs to the
• subject and to all its singular members.

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Example
All tigers are mammals.
We understand by this that mammal can be said
of tiger as a universal nature, and also of
each and every singular tiger.
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• Negative propositions can also be universal. In
  this case, we wish to express that a predicate
  can be denied of the universal nature and also of
  all its singular members.

No tiger is a lion.
We universally deny lion of the subject tiger
and of all singular tigers.
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• 2. We can also predicate something of a
  universal subject particularly. The predicate is
  said of the subject, but is meant to be limited
  to only some of the singulars.

Example
Some tigers are ferocious.
or
Some tigers are not ferocious.
The subject is understood to stand for more than
one individual, without specifying exactly which
ones. Note that to indicate this, the subject
tiger (a universal) is modified by the
inclusion of some to indicate we are talking
about more than one.
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• 3. Finally, we can predicate something of a
  universal subject without indicating anything
  about its quantity, that is, indefinitely.

The tiger is a carnivorous animal.
Example
It is not clear whether the predicate is meant
to be understood as applying to all things called
tigers, or only of some.
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• Words such as all, none, no, some,
  this, every, half of, most, any, etc.,
  serve to indicate the quantity of the subject,
  and thus the quantity of the proposition.

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• To sum up, there are four possible quantities a
  proposition may have
• Universal
• Particular
• Indefinite
• Singular

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• We can analyze all categorical propositions
  according to both quantity and quality.

Example
Every tiger is ferocious.
This proposition is universal and affirmative. We
can refer to it as universal affirmative.
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• What happens if we add a not to this
  proposition?

Every tiger is ferocious.
Not every tiger is ferocious.
The first is universal affirmative.
But the second is particular negative. The
not has done two things it changes the
quality of the proposition from affirmative to
negative, and it alters the quantity from
universal to particular.
It means the same as Some tigers are not
ferocious, not No tigers are ferocious.
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• The analysis of propositions requires careful
  attention to exactly what is being said. This
  cannot be a merely mechanical or overly hasty
  examination.

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Tips for determining the quantity of propositions

• One form that causes difficulties is the sort

Not all exams are difficult.
It would be wrong to hastily conclude that this
is a universal proposition merely because the
word all appears in it. It does not mean that
No exam is difficult, but rather some exams
are not difficult.
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4. According to their Matter

• The Predicables can help us understand the
  division of propositions according to their
  matter. If a predicate is said of a subject as a
  genus, species, specific difference, or property,
  then this predicate has necessary matter with
  relation to this subject, because it expresses
  something that has to do with its essence.

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• Example

Man is capable of speech.
capable of speech is a
property
of the subject man,
so this proposition has necessary matter.
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• When the predicate is an accident of the subject,
  the proposition has contingent matter.

has contingent matter
Tigers are carnivores.
since carnivore is an accident of tiger.
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• If we attempt to use a predicate that is not a
  predicable with regard to that subject, the
  proposition has impossible matter.

Example
The tiger is a lion.
Lion is not a predicable with relation to
tiger and cannot properly be predicated of it.
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5. According to their Mode

• A proposition can be simple or modal.
• In a course on Elementary Logic, we will limit
  ourselves to simple propositions, only mentioning
  the existence of modal propositions.

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• A modal proposition contains some sort of
  modifier that indicates in what mode the
  predicate belongs to the subject.
• There are four modes necessary, impossible,
  possible, and contingent.

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• A modal proposition consists of a categorical
  proposition with the addition of an indicator of
  its mode.
• For example, It is necessary that man have a
  sense of humor. The simple proposition man has
  a sense of humor has been modified to indicate
  its mode.

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• The indicators of the other modes are phrases
  such as it is possible that . . ., it is
  impossible that . . . or it is contingent that
  . . .