LOGIC

SIMPLE PROPOSITION

• A proposition is declarative in nature
• It is a statement that needs affirmation or denial – maybe TRUE or FALSE (truth value)

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CATEGORICAL PROPOSITION - Statement whose terms are joined by a verb copula

Types of Categorical Proposition:

1. Universal - the subject term taken in full extension.

Examples: All Triangles are polygons

No parallel lines meet at a point.

2. Particular - the subject term is taken only in some extension.

Examples: Most cats are afraid of water.

Some algebraic expressions are polynomials.

3. Singular - the subject term denotes a single person/thing.

Examples: A penguin is a bird.

A prime number has only two factors.

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Mood of Catgeorical Propositions:

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HYPOTHETICAL PROPOSITION

• It is a proposition that is compound
• Statement whose terms are joined by a nonverb copula.
• Commonly known as conditional statements
• Are mostly “if-then” statements

Example: If it is a nice day, then I wil go to the park.

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CONVERSE

• The converse of a conditional statement switches the hypothesis and the conclusion.
• Some conditional statements can’t have a converse

Example:

Conditional: If you can’t send a text message, then you are out of cell phone load.

Converse: If you are out of cell phone load, then you can’t send a text message.

Example:

Conditional: If 2 lines intersect to form right angles, then they are perpendicular.

Converse: If 2 lines are perpendicular, then they intersect to form right angles.

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State the converse of the following conditional statements:

1. If it is hot outside, then it must be sunny.

2. If there is smoke, then there is fire.

3. If you had an infection in your nose, throat and lungs then you have the flu.

4. If a number is divisible by 6, then it is divisible by 2 and 3.

5. If two angles have the same measure then they are congruent.

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INVERSE - To write the inverse of a conditional statement, simply negate the hypothesis and the conclusion

Example:

Conditional: If 2 lines intersect and form right angles, then they are perpendicular.

Inverse: If 2 lines intersect and didn’t form right angles, then they are not perpendicular.

Example:

Conditional: If you can’t send a text message, then you are out of cell phone load

Inverse: If you can send a text message, then you still have cell phone load.

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State the inverse of the following conditional statements:

1. If it is hot outside, then it must be sunny.

2. If there’s smoke, then there’s fire.

3. If you had an infection in your nose, throat and lungs then you have the flu.

4. If a number is divisible by 6, then it is divisible by 2 and 3.

5. If two angles have the same measure then they are congruent.

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CONTRAPOSITIVE - To write the contrapositive of a conditional statement, get its inverse first then interchange its hypothesis and conclusion.

Example:

Conditional: If 2 lines intersect and form right angles, then they are perpendicular.

Inverse: If 2 lines intersect and didn’t form right angles, then they are not perpendicular.

Contrapositive: If 2 lines are not perpendicular, then they don’t intersect to form right                          angles.

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State the contrapositive of the following conditional statements:

1. If it is hot outside, then it must be sunny.

2. If there’s smoke, then there’s fire.

3. If you had an infection in your nose, throat and lungs then you have the flu.

4. If a number is divisible by 6, then it is divisible by 2 and 3.

5. If two angles have the same measure then they are congruent.

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BICONDITIONAL STATEMENT

• the combination of a conditional statement and its converse (as long as both statements are true).
• statements are combined using the phrase “if and only if” (iff)

Example:

Conditional: If three points are collinear, then they lie on the same line.

Converse: If three points lie on the same line, then they are collinear.

Biconditional: Three points are collinear iff they lie on the same line.

Separate the biconditional into two conditional statements.

Biconditional: A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

Conditional: If a number is divisible by 3, then the sum of its digits is divisible by 3

Conditional: If the sum of a number’s digits is divisible by 3, then the number is divisible by 3.

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HYPOTHETICAL OR CATEGORICAL?

• If my cellphone’s battery is drained then I cannot turn it on.

• Illegal drugs are dangerous to your well being.

• Quadrilateral ABCD has equal sides therefore it is a square.

• A right triangle has a right angle.

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An argument is an assertion that a given set of statements called premises results in another statement called conclusions.

Example: Premise 1: Schools make use of textbooks.

**Premise 2:**Textbooks are made of paper.

Premise 3: Paper came from trees.

Premise 4: Trees give Oxygen.

Conclusion: Schools take away oxygen. Schools are killing us!!

Since an argument is a resulting statement, then it also has a truth value. If the premises are true and the conclusion is true, then it is considered valid

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PARTS OF A CATEGORICAL SYLLOGISM 

1. Minor Premise - contains the subject of the conclusion

2. Major Premise - contains the predicate of the solution.

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Example:

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DENOTE SOME TERMS

Let:

P - Major Term (Predicate of the Conclusion)

S - Minor Term (Subject of the Conclusion)

M - Middle Term (term not found in the conclusion)

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OPERATIONS ON PROPOSITIONS

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The fundamental elements of propositional logic are propositions statements that can be either true or false. A proposition is like a variable that can take two values, the val ue "true" and the value "false." Logical operators combine propositions to make other propositions, with the use of a truth-table you can determine whether the resulting pro position is true or false.

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CONJUNCTION -  this  is a proposition which is the result of combining two other propositions called conjuct with the connective word and

Rule: two statements are true only if both statements are true

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DISJUNCTION-  the result of combining two other proportions called disjunct with the connective word or

Rule: two statement is false only if both disjunctions are false

IMPLICATION -  This proposition, which is also called conditional proposition, is a proposition which is the result of combining a hypothesis or antecedent to a conclusion or a consequent in the form if…., then…

Rule: The implication is true in all cases, except when the antecedent is true and the consequent is false. In other words, a true hypothesis cannot imply a false conclusion.

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Example:

1. If 2 + 5 = 7 then (2) (5 )= 10

Solution:  2 +5 = 7 which is a true statement, and q is the consequent (2)(5)= 10 which is also a true statement. Then the implication p  q “if 2 +5 = 7 then (2)(5)= 10” is a true statement since both antecedent and consequent are true statements.

2. if 2 +5 = 7 then 2-5= 7

Solution: if p is the antecedent 2 + 5= 7 which is a true statement, and q is the consequent 2-5 = 7 which is a false statement, then the implication p  q “if 2 + 5 = 7 then 2-5= 7” is a false statement

3. If 2 + 7 = 5 then 7 -5 = 2

Solution: if p is the antecedent 2 + 7= 5 which is a false statement, and q is the consequent 7-5 = 2 which is a true statement, then the implication p  q “if 2 + 7 = 5 then 7 -5=2” is a true statement

4. if 2 + 7 = 5 then 2-5= 7

Solution: if p is the antecedent 2 + 7= 5 which is a false statement, and q is the consequent 2-5 =7 which is also a false statement, then the implication p  q “if 2 + 7 = 5 then 2 -5=7” is a true statement.

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EQUIVALENCE -  This proposition which is also called biconditional proposition, is a proposition which is the result of combining two propositions in the form…. if and only if…

Rule: The equivalence is true if both propositions are true or both are false

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Example:

5. A square is a polygon if and only if the square is a rectangle

Solution:  If p is the proposition “ A square is a polygon.” Which is a true statement a q i s the proposition “ The square is a rectangle” which is a true statement , then the eq uivalence p⇔q “A square is a polygon if and only if the square is a rectangle” is a true statement since both statements are true

6. A right triangle is a triangle if and only if the sum of the lengths of its sides is equal to the length of the hypotenuse

Solution: If p is the proposition “A right triangle is a triangle.” Which is a true statemen t a q is the proposition “The sum of the lengths of its sides is equal to the length of t he hypotenuse” which is a false statement, then the equivalence p⇔q “ A right triangle is a triangle if and only if the sum of the lengths of its sides is equal to the length of the hypotenuse” is a false statement

7. The diagonals of a square are parallel if and only if the square is a quadrilateral

Solution:  If p is the proposition “The diagonals of a square are parallel” Which is a false statement a q is the proposition “the square is quadrilateral” which is a true statement, then the equivalence p⇔q “The diagonals of a square are parallel if and only if the square is a quadrilateral” is a false statement

8. Parallel segments intersect at a point if and only if the segments have only one end point

Solution:   If p is the proposition “Parallel segment intersect at a point ” Which is a false statement a q is the proposition “the segments have only one end point” which is a false statement, then the equivalence p⇔q “ Parallel segments intersect at a point if and only if the segments have only one end point” is a true statement since both propositions are false stament

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EQUIVALENCE -  Simple sentence is true when the simple sentence is false and false when the simple sentence is true.

Rule: If a proportion is true, its negation is false, and if a proposition is false, its negatio is true.

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Example:

1. T - FSUU is a great University

~T - FSUU is NOT a great University

2. F - Philippines is part of the European Continent

~F - Philippines is NOT part of the European Continent

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ARGUMENT

• An assertion that a given set of statements called premises results in another statement called conclusions.

• Maybe inductive or deductive

• Since an argument is a resulting statement, then it also has a truth value. If the argument is a tautology, then it is considered valid; otherwise it is a fallacy

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FALLACY - A fallacy is an incorrect argument in logic which is a result of invalid reasoning. A fallacy is not synonymous with factual error. Some fallacies are popular beliefs, some are often persuasive, some are unintentionally created, and some are intentionally created for deception. Fallacies are either formal or informal.

1. Formal fallacies - are invalid arguments due to flaw in logical structure

2. Informal fallacies -  are arguments whose premises adequately support the validity of its conclusion.

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TYPES OF FALLACY

> AD POPULUM FALLACY

• the appeal to the popularity of a claim as a reason for accepting it.

Example:

> CIRCULAR REASONING

• committed when an arguer begins with a concept that is supposed to be the end of the argument.

Example: “A measurement is a quantity that expresses the measure of an object being measured.”

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> UNDISTRIBUTED MIDDLE

• happens when the middle term in a syllogism is not distributed as a subject and a predicate.

Example: All triangles are polygons.

All quadrilaterals are polygons.

Therefore, all triangles are quadrilaterals.

> INSUFFICIENT STATISTICS

• happens when the middle term in a syllogism is not distributed as a subject and a predicate.

Example: A class of 40 was subjected to an informal survey about the mayoralty candidate they will support in the election. The survey says that 21 of the students support Candidate A. The surveyor declares that the majority of the city residents support the candidacy of Candidate A.

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> QUATERNIO TERMINORUM

• a fallacy that used four, instead of three categorical terms.

Example: Every school has a principal.

Every principal has an interest.

Therefore, every school has an interest.

> DENYING THE ANTECEDENT

• Denying the antecedent of a conditional and then assuming that by doing so, a sufficient reason to deny the consequent can be claimed.

Example: If Ralph was a UP student, then he would know the UP’s official publication is The Collegian. But Ralph is not a UP student. Therefore, it is certain that Ralph does not know that the UP’s official publication is The Collegian.

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> FALLACY OF EQUIVOCATION

• committed when the ambiguity of a term or a phrase in an argument which has occurred at least twice is exploited of deliberately misused

Example: The end of life is death.

Happiness is the end of life.

Therefore, death is happiness

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> FALLACY OF AMPHIBOLY

• ambiguity due to syntax structure.

Example: The guidance counselor is asked to stop bullying

• This statement can be interpreted in at least two ways: The guidance counselor is bullying someone, or the guidance counselor is tasked to eradicate bullying.

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> FALLACY OF COMPOSITION

• committed when the characteristics of some are transferred to another.

Example:

Every member of the gymnastic team is an outstanding dance sports performer.

The gymnastic team is an outstanding team.

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> POST HOC ERGO PROPTER HOC

• (After this, therefore because of this) is a fallacy of causation which relates two events on the basis of temporal succession.

Example: Daniel failed most of his subjects in the 4th Grading Period because he did not submit his project in the 3rd Grading Period.

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> AD VERCUNDIAM

• involves using a pronouncement of a person taken to be an authority but is not really one as evidence

Example: Richard Dawkins, an evolutionary biologist and perhaps the foremost expert in the field, says that evolution is true. Therefore, it's true.

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> FALLACY OF FALSE DILEMMA

• limits unfairly an arguer to only two choices, as if the choice is on black or white

Example: Decide! Are you joining the field trip or not? You will still pay the contribution.

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> AD POPULUM

• involves using a popular opinion as evidence for a proposition.

Example: Survey shows that mathematics in the most dislike subject. So you must dislike mathematics too.

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> FALLACY OF FAULTY ANALOGY

• committed when analogies are used to support a conclusion using similarities of two things that are too remote.

Example:  I believe in the resurrection of Jesus Christ.

Zombies are resurrected entities.

Zombies are real!

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> AD HOC RESCUE

• committed when trying to rescue a troubled belief by citing different argument.

Example:  This soap will remove skin blemishes after 7 days of use.

Skin blemishes are still present after 7 days of use.

The soap is probably fake.

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> AD HOMINEM

• committed when ignoring the validity of an argument by attacking the personality of the arguer.

Example:  She is wrong when she claim that Girolamo Cardano was insane because she is only a Grade 10 student.

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> AVOIDING THE ISSUE

• occurs when an issue is answered by an argument not really a response to the issue.

Example:  Issue: Why is she appointed the leader of the group?

Answer: Why not she?

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