LOGIC - Seminar In Problem Solving

Logic
• Logic is derived from the Greek word
‘LOGOS’ which means primarily the word
by which the inward thought is expresses
‘LOGIKE’ which means the work or what
is spoken (but coming to mean thought or
reason).

Logic: Definition
• The science and art of correct thinking. (Corazon Cruz
1995: 5)
• A practical philosophy of correct thinking. (Edgardo A.
Reyes, 1988: 1)
• The science of correct thinking, a systematized study of
the reasoning process for the purpose of helping us think
clearly, easily and correctly (Ramon B. Agapay, 1991: 2)
• The philosophical science that provides the student the
rational “tool” (organon) or instrumentality for pursuing
profitably the study of Philosophy. (Manual Pinon,1979:
1)

Logic: Definition
• Concerned with the quest pf knowledge and
truth, and is also a study of the validity or
correctness of our reasoning. (Mourant, 1963: 2)
• The study of the relationship between premises
and conclusions of the arguments. (Facione and
Scherer, 1978 : 60)
• Deals with arguments and inferences ; one of its
main purposes is to provide methods for
distinguishing those which are logically correct
from those which are not. (Wesley, 1963: 1)

Parts of Logic
• Reasoning- drawing a conclusion that was previous
unknown/ doubted/ unclear form judgments that are
known.
• Syllogism- a conclusion drawn from two premises which
have at least some part in the conclusion.
• Conclusion- the proposition or judgment whose validity
and/ or truth a syllogism seeks to establish.
• Premises- a judgment or proposition or reason given in
an argument that support or lead conclusion.

Subset of Logic
Informal Logic
• is the used in every
day reasoning and
argument analysis
Formal Logic
• it deals with deductive
reasoning and the
validity of the inferences
produced
Example:
Every cat is a mammal.
Some carnivores are cats.
Therefore, some
carnivores are mammals

Importance of Logic
• The study of Logic develops in the learner the
skills to reason out with order, validity, truth
and accuracy.
• The knowledge of logic helps to prevent us
from committing grave error in the acts of
thinking and reasoning.
• It is a necessary aid in evaluating and
understanding others studies.
• It is a tool in discerning validity and truth of
propositions and arguments.

Importance of Logic
• It prevents in making conclusions based on
false and biased assumptions.
• Logic contributes to the growth of individual
improving the quality of his life.
• Logic builds in the individual self-confidence,
provides a feeling of direction, and gives
assurance of being in control of one’s
situation.

Simple Apprehension
• A mental act in which the mind perceives
or notices something. This something
being perceived or noticed is what we call
a concept or an idea.

• Focuses on something that is being perceived or
noticed
Attention
• Notices the similarities and differences of the
characteristics
Comparison
• Singles out a characteristic or several characteristics
Abstraction
Forming an Idea

Properties of an Idea
Comprehension
• The property of a
term which is the
sum of
characteristic
notes of an idea
signified by a term
Extension
• The property of a
term by which
such term or
concept is applied
or extended to
other things

Rational Sentient Living Material Substance
Men
Sentient Living Material Substance
Animals Men
Living Material Substance
Plants Animals Men
Material Substance
Minerals Plants Animals Men
Substance
Spirits Minerals Plants Animals Men

Writing Instruments with Ink
Pen
Writing Instruments
Pen Chalk Pencils
Instrument
Compass Chalk Scissors Pen Pencils Etc.

Judgment?
• Is a mental act of affirming or denying the
relationship between two concepts

Prerequisites of Judgment
1. There must be at least two or more
concepts that exists.
2. In the act of comparing, the mind must
examine the similarities and differences
to verify the truth and falsity of the
concept.
3. The mind must lay down its acceptance
and rejection of the ideas.

Proposition
• A declarative sentence which expresses a
relation of affirmation or denial between
two terms
• Is the verbal or written expression of a
judgment

Elements of a Proposition
• Is the term or group of terms spoken of; being
talked about.; the one which is affirmed or denied
Subject
• Is the action the affirms or denies the subject
Predicate
• Links the subject to the verb; it also expresses
relationship of identity or diversity of terms
Copula

Example
• All Bicolanos are Filipinos. (Affirmative)
• Some criminals are not punished.
(Negative)

Truth or Falsity
• A tree is a plant.
• Igorots are Filipinos.
• The sun revolves around the earth.
• A dog is an irrational animal.
• Flowers are petals.
• A triangle has four sides.

Classification of Proposition
1. Quantity
• refers to the number of referents to
which the subject term is applied
2. Quality
• reveals the nature relationship between
the subject-term and the predicate-term

Quantity of Propositions
• Is one whose subject term stands for each and all
individuals to which it is applied
Universal Propositions
• Whose subject stands for a portion of a given
totality
Particular Propositions
• Is one whose subject is singular in concept
Singular Propositions

Quantity of Propositions: Example
• Every man is created by God.
• All cows are animals.
Universal Propositions
• Few students are bright.
• Some plants are edible.
Particular Propositions
• Baguio City is the summer capital of the Philippines.
• This guy is my friend.
Singular Propositions

Quality of Propositions
• Is one whose subject and predicate terms
are united by the copula and their
relationship is affirmed.
Affirmative Propositions
• Have subject and predicate terms that are
separated from each other due to a
negative copula.
Negative Propositions

Quality of Propositions: Examples
•Every man is a rational animal.
•All trees are with leaves.
Affirmative Propositions
•A person is not a dog.
•Not all that glitter are gold.
Negative Propositions

A
All S is P
E
All S is
not P
I
Some S
is P
O
Some S
is not P

• If one pair of proposition is true, the other is false.
• If one proposition is false, the other is true.
CONTRADICTORY
• If one of the opposed proposition is true, the other is
false.
• If one of them is false, the other is doubtful.
CONTRARY
• If one of the opposed proposition is false, the other is
true.
• If one of them is true, the other is doubtful.
SUBCONTRARY
• If the universal is true, the particular is also true but not
vice-versa.
• If the particular is false, the universal is also false but not
vice-versa.
SUBALTERN

Reasoning
• This is an act in which from the known
truth or certainty, the mind travels to
another truth. It is a mental process that
compares two similar propositions; and
out of these propositions, a conclusion is
drawn or formed.

Kinds of Reasoning
Deductive
• It is a reasoning process
that forms a conclusion
out of a generally
accepted fact – from
general or universal to
conclusion.
Inductive
• It is a kind of reasoning
that forms a conclusion
from a particular to a
universal or general
instance or fact – from
particular to general.

Kinds of Reasoning: Example
Deductive
All OFWs are suffering
from homesickness.
Pedro is an OFW.
Pedro is suffering from
homesickness.
Every good act is
rewarded.
Patience is a good act.
Patience is rewarded.
Inductive
Pedro is a man.
But all men are mortal.
Therefore, Pedro is a mortal.
Tigris is a river.
Euphrates is a river.
Nile is a river.
But, all rivers empty to the
sea.
Therefore, Tigris, Euphrates
and Nile empty to the sea.

Validating the Truth
1. The first two known truths which are
called premises should be both true.
2. The first two known truths or premises
must have a logical and close connection
so that the third proposition, the
conclusion or the new truth is the
necessary consequence of such logical
relationship.

Truth Table
• A truth table is a tool that helps
to analyze statements or
arguments in order to verify
whether or not they are logical, or
true.

Logic Operations
• AND ˄
• OR ˅
• NOT ~
• IF-THEN →
• IF AND ONLY IF ↔

AND Statements ˄
• These statements are true only when both p and
q are true.
• Example: “I will bring both a pen AND a pencil to
the tutoring session.”
p q p ˄ q
T T T
T F F
F T F
F F F
AND ˄

OR Statements ˅
• These statements are false only when both p and
q are false.
• Example: “I will bring a pen OR a pencil to the
tutoring session.”
p q p ˅ q
T T T
T F T
F T T
F F F
OR ˅

NOT Statements ~
• The “not” is simply the opposite or
complement of its original value.
p ~p
T F
F T
NOT ~

IF-THEN Statements →
• These statements are false only when p is true
and q is false.
• Example: “IF I am elected THEN taxes will go
down.”
p q p → q
T T T
T F F
F T T
F F T
IF-THEN →

IF AND ONLY IF Statements ↔
• These statements are true only when both p and
q have the same values.
• Example: “Taxes will go down IF AND ONLY IF I
am elected.”
p q p ↔ q
T T T
T F F
F T F
F F T
IF AND ONLY IF ↔

p q p ˄ q p ˅ q ~p p → q p ↔ q
T T T T F T T
T F F T F F F
F T F T T T F
F F F F T T T
Truth Table

Example: Truth Table
1.p˅~q
2.q˅~(~p˄q)

Tautology
•A tautology is a statement that
cannot possible be false, due to
its logical structure.

Truth Tables for Arguments
• A logical argument is made up of two parts: the premises
and the conclusion.
• Arguments are usually written in the following form:
• Example:
If it is cold, then my motorcycle will not start.
My motorcycle started.
It is not cold.

Logical Statement
If If it is cold, then my motorcycle will not start. "It is cold" = p
My motorcycle started. "It is not cold" = ~p
"My motorcycle will start" = q
"My motorcycle will not start" = ~q
It is not cold.
If If it is cold, then my motorcycle will not start. p→~q
My motorcycle started. q
It is not cold. ~p

[ Premise One ˄ Premise Two ] → Conclusion
[ (p → ~q) ˄ q] → ~p
Logical Statement

Example:
• Suppose
“ x > y “ is true.
“ ∫ f(x) dx=g(x) + C “ is false.
“Calvin Butterball has purple socks” is true.
Determine the truth value of the statement
( x > y → ∫ f(x) dx=g(x) + C ) → ~(Calvin Butterball
has purple socks)

Example:
( x > y → ∫ f(x) dx=g(x) + C ) → ~(Calvin Butterball
has purple socks)
• For simplicity, let
P = “ x > y ”
Q = “ ∫ f(x) dx=g(x) + C “
R = “ Calvin Butterball has purple socks “

Syllogism
•A formal argument in logic that
is formed by two statements
and a conclusion which must
be true if the two statements
are true.

Different Types of Syllogism
Categorical Syllogism
Hypothetical Syllogism
• Conditional Syllogism
• Disjuctive Syllogism
• Conjunctive Syllogism

•A categorical syllogism is an argument
consisting of exactly three categorical
propositions (two premises and a
conclusion) in which there appear a
total of exactly three categorical
terms, each of which is used exactly
twice.

Basic Propositions in Categorical
Syllogism
Major Premises
• The premise which
contains the major term.
Usually the first
proposition.
Minor Premises
• The premise which
contains the minor term.
Usually the second
proposition and it is
preceded by conjunction
BUT.

Basic terms in Categorical
Syllogism
• It is the PREDICATE in the conclusion and found in the major
premise. Usually designated by P.
Major Term
• It is the SUBJECT in the conclusion and found in the minor
premise. Usually designated by S.
Minor Term
• It provides the connection between 2 premises to form
conclusion. It is designated by M.
Middle Term

Rules Governing the Validity of
• Rule no.1
-there must only be three terms.
• Rule no.2
-conclusion will follow the weaker premise.
Example:
All wicked people will be punished.
But some people are wicked.
Therefore some people will be punished.

Rules Governing the Validity of
• Rule no.3
- if both premises are negative, no conclusion
follows.
Example: No mammals are fish.
No fish can fly
Therefore???
• Rule no.4
- if both premises are particular, then no
conclusion follows.
• Rule no.5
-If both premises are affirmative, then the
conclusion must also be affirmative.

Hypothetical Syllogism
• One wherein the major premise is a
hypothetical proposition and the minor
premise and conclusion are categorical.
1. Conditional Syllogism
2. Disjunctive Syllogism
3. Conjunctive Syllogism

Conditional Syllogism
• One whose major premise is a conditional proposition
and whose minor premise and conclusion. Consist of the
antecedent and consequent for the truth of the
hypothetical judgment lies in the truth of dependence
between the two clauses, the antecedent (cause) and the
consequent(effect).
Example:
If it will rain, then the grass will be wet.
It rains.
Therefore the grass is wet.

Valid Moods For Conditional
Syllogism
1. Moods
- Define as the classification of two premises and
conclusion.
2. Modus Ponens
- The truth of the antecedent implies the truth of the
consequent.
If A is B, then X is Y.
But A is B.
Therefore X is Y.
Example:
If Sharon dances, then she is moving.
But Sharon is dancing.
Therefore she is moving.

3. Modus Tollens
- The falsity of the antecedent implies the falsity of the
consequent.
If A is B, then X is Y.
But A is not B.
Therefore X is not Y.
Example:
If you are honest, then people will admire you.
But you are not honest.
Therefore people will not admire you.
Valid Moods For Conditional
Syllogism

Disjunctive Syllogism
• The major premise is a disjunctive propositions and the minor
premise and conclusion are categorical propositions. It is an
"either or" statement.
A is either B or C.
But A is B.
Therefore A is not B.
Example
Political candidates are either honest or corrupt.
Political candidates are corrupt.
Therefore they are not honest.

Valid mood for disjunctive
• Ponendo Tollens
- Positing Mood (Accept or Affirms)
- Minor premise affirms one of the
alternatives of the major premise and the
conclusion denies the other.
Example:
The criminal is either dead or alive.
But he is alive.
Therefore he is not dead.

• Tollendo Ponens
- Sublating Mood( Sublate or Negative)
- Minor premises deny the alternative of the
major premise and the conclusion affirms the
other.
Example:
May is either a liberated or a conservative
person.
But he is not liberated.
Therefore he is a conservative person.
Valid mood for disjunctive

Conjunctive Syllogism
• The major premise is a conjunctive
proposition (one that denies that the two
choices can be true at the same time) and
that the main premise and the conclusion are
categorical propositions.
Example
Our system of government cannot be
either presidential or parliamentary.
But it is presidential.
Therefore it is not parliamentary.

Valid mood for conjunctive
• Ponedo Tollens
- Positing one conjunct in the minor and sublating the
other in the conclusion.
Example:
I cannot be in Zambales and Manila at the same time
I am in Zambales (posited)
Therefore I am not in Manila. (Sublated)

A IS C
BUT B IS C
THEREFORE A IS B

All cats are animals.
All dogs are animals.
Therefore, all dogs
are cats.

Fallacies
•are the reasoning or arguments
which are valid but are actually
invalid; arguments, which you
know are correct but definitely
are not correct; or arguments that
seem to be true but are actually
false.

Classification Of Fallacies
• There are various ways of classifying
fallacies as there are different author. For
our purpose here, however, we follow the
classification by Aristotle, a Greek
philosopher.
Aristotle divides fallacies into the following
categories: (1) FALLACIES OF LANGUAGE and
(2) FALLACIES NOT OF LANGUAGE

Fallacies In Language
1. Fallacies In Equivocation
• the fallacy committed when some terms are
used in a premise but with different meanings.
• Example:
A ruler helps us to draw a straight line.
Datu Puti is a ruler.
Therefore Datu Puti helps us o draw a straight line.

2. Fallacy Of Amphiboly
• It arises from the ambiguous use not of a single
word but of a phrase or of a complete sentence.
• Example:
This woman her cousin loves.
For sale: Hyundai Car by a carboy with damaged
button

3. Fallacy Of Composition
• This fallacy lies on the fact that a group of words or
phrase is taken singly or a unit when they are
supposed to be taken separately.
• Example:
BISCAST Students are from different places
But Eric is a BISCAST Student
Therefore Eric is from different places.

4. Fallacy Of Division
• It is the opposite of the fallacy of composition. For
this fallacy is committed when words or phrases are
taken separately instead of using them jointly.
• Example:
BISCAST Students make up a good class
Ciano is a BISCAST Student
Therefore Ciano makes up a good class.

5. Fallacy Of Accent
• This arises from the use of word which changes
meaning when the accent of the word changes.
• Example:
Every invalid needs care and attention
But fallacy is invalid
Therefore a fallacy needs care and attention

6. Fallacy Of Figures Of Speech
• happens when the syllogism make use of sentence
structures having the same or from the similar form,
from which a conclusion is derived hastily.
• Example:
Insincerity is the antonym of sincerity
Dishonesty is the antonym of honesty
Therefore invaluable is the antonym of valuable.

Fallacies Not Of Language
1. Fallacy Of Accident
• It happens when what is essential or necessary o an
object is confused with what is merely accidental to
it.
• Example:
This watch is made in USA.
Therefore, this watch is excellent.

2. Fallacy Of False Cause
• It is an argument that attributes an effect or
result to an inadequate or false cause.
Oftentimes it is accompanied by superstition or
presumption.
• Example
The family became poor because they sweep the
floor in the evening.
He met an accident because its Friday the 13th.

3. Fallacy Of Consequent
• This fallacy takes the truth of the antecedent from
the truth or fact of the consequent or the falsity of
the antecedent from the falsity of the consequent.
• Example:
If the student is not diligent, he will not succeed.
But he did not succeed.
Therefore, he is not diligent.

4. Fallacy Of Begging The Question
• It consist in assuming as true what is still not proven. It
assumes an unproven statement which is the same as
the conclusion. Two types of this fallacy:
a. Not Proven
• The fallacy makes the assumption by employing
different words having different meanings.
• Example:
Men have rationality, because they can reason out.
To err is human, because man commits mistake.
Man is mortal because he dies.

b. Vicious Circle
• The fallacy consists of two propositions unproven yet,
trying to mutually prove one another.
• Example:
Man is imperfect because he is limited.
Man is limited because he imperfect.

5. Confusion Of Absolute Statement
• Is committed when one argues from the truth
of a general principle to the truth of specific
case. The special case may even be an
exception to the general law.
• Example:
To kill is morally criminal. (universal law)
But in self defense, one may kill. (specific case0
Therefore, self defense is morally criminal.

6. Confusion Of Qualified Statement
• The fallacy consists in concluding from the truth
of a proposition which is good only under
certain circumstances of time, place, or
condition to the truth of the same thing under
circumstances regardless of whatever the
circumstances are.
• Example:
Some Catholics are bad
But Mary and Joseph are Catholics
Therefore Mary and Joseph are bad.

LOGIC - Seminar In Problem Solving

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