The document discusses different types of hypothetical and logical propositions including: - Conditional propositions expressed with "if-then" statements. - Disjunctive propositions expressed with "either-or" statements. - Conjunctive propositions that express two alternatives cannot be true simultaneously. - Complex statements that contain logical components like hypothetical and negative propositions. It also covers propositional logic concepts like negation, material implication, disjunction, conjunction, and material equivalence. Truth tables are provided to define the truth conditions for each logical operator.

1. THE HYPOTHETICAL PROPOSITIONS (Aristotelian Classification)

2. Conditional Proposition The conditional is the “if - then” proposition. It expresses a relation in virtue of which one judgment or proposition necessarily flows from the other. Example. If you are a Catholic, then you are a Christian. Parts: Antecedent If it rains Consequent then the ground is wet.

3. Antecedent: condition, cause, reason Consequent: conditioned, effect, result If – then , unless when Not all if –then statements are conditional. Example: If you need money then go to the bank . Both antecedent and consequent must be statements There is a necessary or logical connection between the antecedent and consequent. Terms of conditional:

4. Disjunctive Proposition The disjunctive is the “either - or” statement It expresses that two things cannot be false at the same time; one at least must be true. Examples: A person is either a male or female. You are either for or against Christ. Parts are called disjuncts

5. Proper Disjunctive - when the component parts are mutually exclusive or contradictories, they cannot be both true . Pass or fail Dead or alive Today is either Monday or Tuesday. A parent is either a father or a mother. Improper Disjunctive - when the component parts are not mutually exclusive. Doctor or engineer Red or blue I will visit my friend either on Monday or Tuesday. The parent who will attend the meeting is either the father or the mother.

6. Conjunctive Proposition The conjunctive proposition expresses that two alternatives cannot be true simultaneously. Examples: One cannot be in Davao and in Cebu at the same time. You cannot inhale and exhale at the same time. Parts are called conjuncts

7. Two-part conjunctive – the conjuncts are limited to two alternatives only. More than two-part conjunctive – the conjuncts are not limited to two alternatives. The suspect cannot be guilty and not guilty. A parent cannot be a father and a mother at the same time. One cannot study and play at the same time. One cannot study and play at the same time The president cannot be a conservative and liberal at the same time.

9. COMPLEX AND HYPOTHETICAL PROPOSITIONS (Modern Logic Classification )

10. Simple and Complex Statements Simple statement - does not contain any other statement as its component. Examples: John is a student . Manila is a city. Complex or compound statement contains logical components. ( hypothetical and negative propositions in the Aristotelian logic are considered complex.)

11. John is a student and John is honest. It is not the case that Marie is the best student in the class. Either the rebels surrender or they die in battle. If there is a weather disturbance, then classes will be suspended. For statements to be compound, it is necessary that the components are statements in themselves and that they are sensical on their own. The man who killed Ninoy was a murderer. Not a complex statement Additional examples: :

12. The truth-value of a complex/compound statement depends of the truth-value of its components. The truth value of the statement: Manila is a city and AMV is a college. depends on the truth value of the components: . A compound statement is a truth-functional statement. A truth-function is an expression whose truth-value (truth or falsity) is completely determined by the truth values of its component statements. Manila is a city AMV is a college.

13. Additional Examples: "The sun is shining" "the boys are playing." * It is not the case that Marie is the most beautiful girl in the world. The truth value of this proposition is determined by the truth-value of the logical component: * The sun is shining and the boys are playing . is a function of its component simple propositions: Marie is the most beautiful girl in the world .

14. Negation The negation of a statement is formed by using "not“ in the original statement or by prefixing the phrases "it is false that.." "it is not the case that.." Examples: John is not the president of the class. It is not the case that the Philippines is the poorest country in Asia. It is false that life is tragic.

15. The symbol for negation is the curl . Examples: Students are responsible. - ~ S - S It is not the case that students are responsible. The definition of negation may be presented in this truth table: p ~p --------- T F F T The curl denies or contradicts the statement it preceeds. '~' This is a simple statement the negative statement If p is true then ~p is false If p is false then ~p is true

16. Material Implication Material implication is an if then statement. If the Philippines is a democratic country, then Vatican is a state. Antecedent Consequent Parts: Example: - If.. - then..

17. Material Implication This is another type of implication where there is no real or causal connection between the antecedent and the consequent. If man could be have wings, then our ancestors are pigs. symbol of material implication - the horse shoe - ﬤ Material implication is an emphatic way of denying the antecedent . Example:

18. material implication is true if it is not the case that the antecedent of the statement is true and its consequence is false. The only it can be false is if the antecedent is true and the consequent is false If Manila is a city then UST is a university. M ﬤ U Philosophy is not myth then science is empirical . ~ P ﬤ S The horse shoe is a truth functional connective

19. Material implication may be defined by this truth table: p q p ﬤ q T T T T F F F T T F F T _____________ Five is an odd number ﬤ one is an even number . This is true (T) This is false (F) T F thus the material implication is false (F) Application:

20. Disjunction The disjunctive is the “ either - or ” statement. Two things cannot be false at the same time one at least must be true. The disjunction of two statements is formed by inserting the word " or " between them. Examples: An A proposition is affirmative or O is negative. The antecedent is condition or consequent is cause.

21. The symbol for disjunction (inclusive) is the wedge The wedge is a truth-functional connective. It connotes that a disjunctive statement is true if at least one of the components it connects is true. I either run or I get hit. Examples: p v q Logic is not fiction or arts is not magic. ~p v ~q v

22. Disjunction may be defined by this truth table: p q p v q T T T T F T F T T F F F _____________ Five is an odd number v one is an even number . This is true (T) This is false (F) T F thus the disjunction is true (T) Application:

23. Conjunction The conjunctive is formed by placing the word “ and ” between two statements. The conjunction in Aristotelian is not a conjunction in modern logic. Examples: An I proposition is affirmative and E is negative. Three is a odd number and six is divisible by two. PNB is a bank and SM is a department.

24. The symbol for conjunction is the dot. The dot is a truth-functional connective. It connotes that a conjunctive statement is true if both components it connects are true. PNB is a bank and SM is a mall. Examples: p . q Politics is not showbiz and painting is an art. ~p . q ( . ) p . ~q Snakes are reptiles and birds are not wild.

25. Conjunction may be defined by this truth table: p q p . q T T T T F F F T F F F F _____________ Five is an odd number . one is an even number . This is true (T) This is false (F) T F thus the conjunction is false (F) Application:

26. Material Equivalence Material equivalence is the bi-conditional statement The phrase “ if and only if ” is placed between two statements. Examples: UST is a university if and only if a university is a big school. Ten is an even number if and only if an even number is divisible by two.

27. The symbol for material equivalence is The ≡ is a truth-functional connective. It connotes that a materially equivalent statement is true if both components are true or both are false. PNB is a bank if and only if and SM is a mall. Examples: p ≡ q Politics is showbiz if and only if painting is not an art. p ≡ ~q ( ≡ ) ~p ≡ q Metal is not air if and only if lightning is electrical.

28. Material equivalence may be defined by this truth table: p q p ≡ q T T T T F F F T F F F T _____________ Five is an odd number ≡ one is an even number . This is true (T) This is false (F) T F thus the material equivalence is false (F) Application:

29. ~ (If critical thinking is important then Logic is required in college) or (St. Thomas is a philosopher and St. Dominic is the founder of the Order of Preachers.) T T ﬤ v T . T ( T ﬤ T ) v ( T . T) ~ ~ (T) v T F v T T Application