Propositional Logics

Contents

**Recall the terms: Logic, Proposition, and Truth Table** 1

**Recall the Logical Connectives** 1

**Construct Truth Table for the given Compound Statement** 6

**Recall the following terms: Tautology, Contradiction, Contingency, Predictors etc** 9

**Show that the given compound statement is a Tautology or a Contradiction** 12

**Write the Propositions in Symbolic Form using Quantifiers and Predicate** 15

**Show the Validity of the Arguments using the Truth table** 15

**Recall Disjunctive Normal Form (DNF)** 15

**Find DNF of a given Compound Statement** 15

**Recall Conjunctive Normal Form (CNF)** 15

**Find CNF of a given Compound Statement** 15

**Recall the Principle of Duality** 15

**Recall the Functionally Complete Sets of Connectives** 15

**Recall Propositional vs Predicate Logic** 15

**Recall Terms: Predicate and Quantifiers** 15

**Convert the given statement into a First-order Predicate Logic Form** 15

**Test the validity of the given Arguments** 15

**Recall Rules of Inference and their corresponding Tautological Form** 15

**Test the validity of the given Arguments without using Truth Table** 15

**Recall the terms: Logic, Proposition, and Truth Table**

- Logic: Logic is the study of reasoning and the principles that govern valid and sound arguments. It helps in determining the truth or validity of statements based on logical rules. For example, in propositional logic, we can use logic to determine the validity of arguments such as “If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet.”
- Proposition: A proposition is a statement that is either true or false, but not both. It is a declarative sentence that can be assigned a truth value. For example, “The sun is shining” and “2 + 2 = 5” are propositions. In logic, propositions are used as the basic elements to construct logical statements and arguments.
- Truth Table: A truth table is a table that shows the possible truth values of a proposition or a compound logical expression for all possible combinations of truth values of its constituent propositions. It displays the relationship between the truth values of the input propositions and the resulting truth value of the expression. For example, consider the proposition “p AND q,” where p represents the statement “It is raining” and q represents the statement “The ground is wet.” The truth table for this proposition would show all possible combinations of truth values for p and q and the resulting truth value of the proposition.

**Recall the Logical Connectives**

Logical connectives, also known as logical operators, are symbols or words used to combine or modify propositions in logical statements. They allow us to express relationships between propositions and form more complex logical expressions.

Here are some commonly used logical connectives:

- Negation (NOT): Denoted by ¬ or ~, it negates the truth value of a proposition. For example, if p is true, then ¬p is false.
- Conjunction (AND): Denoted by ∧ or &&, it represents the logical “and” operation. It is true only when both propositions being connected are true. For example, if p is true and q is true, then p ∧ q is true.
- Disjunction (OR): Denoted by ∨ or ||, it represents the logical “or” operation. It is true if at least one of the propositions being connected is true. For example, if p is true or q is true, then p ∨ q is true.
- Implication (IF-THEN): Denoted by → or ⇒, it represents the logical implication. It states that if the first proposition is true, then the second proposition is true. For example, if p is true and q is true, then p → q is true.
- Biconditional (IF AND ONLY IF): Denoted by ↔ or ⇔, it represents the logical equivalence. It is true if both propositions have the same truth value. For example, if p is true and q is true, then p ↔ q is true.

These logical connectives can be combined to form more complex logical expressions and are used to construct logical arguments and proofs.

Here are examples of each logical connective with propositions and their corresponding truth tables:

- Negation (NOT):

Proposition: p = “It is raining.”

Truth Table:

p | ¬p |

T | F |

F | T |

- Conjunction (AND):

Propositions: p = “It is sunny.” and q = “It is warm.”

Truth Table:

p | q | p ∧ q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

- Disjunction (OR):

Propositions: p = “It is Monday.” and q = “It is Friday.”

Truth Table:

p | q | p ∨ q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

- Implication (IF-THEN):

Propositions: p = “I study, then I pass.”

Truth Table:

p | q | p → q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

- Biconditional (IF AND ONLY IF):

Propositions: p = “The cake is chocolate.” and q = “The cake is delicious.”

Truth Table:

p | q | p ↔ q |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

These examples demonstrate how the truth values of propositions and the resulting truth values of logical connectives can be determined using truth tables.

**Construct Truth Table for the given Compound Statement**

A truth table is a table that displays all possible truth values for a given compound statement. To construct a truth table, we list all possible combinations of truth values for the propositional variables in the compound statement and then evaluate the truth value of the compound statement for each combination.

Example 1, let’s construct a truth table for the compound statement p ∧ (q ∨ r):

p | q | r | q ∨ r | p ∧ (q ∨ r) |

T | T | T | T | T |

T | T | F | T | T |

T | F | T | T | T |

T | F | F | F | F |

F | T | T | T | F |

F | T | F | T | F |

F | F | T | T | F |

F | F | F | F | F |

In this table, p, q, and r are propositional variables. We list all possible combinations of truth values for p, q, and r, and evaluate the truth value of q ∨ r and then p ∧ (q ∨ r) for each combination.

Example 2:

Compound Statement: p ∧ q → ¬r

Truth Table:

p | q | r | p ∧ q → ¬r |

T | T | T | F |

T | T | F | T |

T | F | T | F |

T | F | F | T |

F | T | T | T |

F | T | F | T |

F | F | T | T |

F | F | F | T |

Example 3:

Compound Statement: (p ∨ q) ∧ (¬p ∨ r)

Truth Table:

p | q | r | (p ∨ q) ∧ (¬p ∨ r) |

T | T | T | T |

T | T | F | T |

T | F | T | T |

T | F | F | F |

F | T | T | T |

F | T | F | T |

F | F | T | T |

F | F | F | F |

In each truth table, the columns represent the truth values of the propositional variables (p, q, r) and the final column represents the truth value of the compound statement based on the given logical connectives and their respective truth values.

**Recall the following terms: Tautology, Contradiction, Contingency, Predictors etc**

Here are the definitions of each term along with examples:

- Tautology: A tautology is a compound statement that is always true, regardless of the truth values of its propositional variables. In other words, every row in its truth table evaluates to true.

Example: The statement “p ∨ ¬p” is a tautology because regardless of the truth value of p, the statement will always be true.

Its truth table is as follows:

p | ¬p | p ∨ ¬p |

T | F | T |

F | T | T |

- Contradiction: A contradiction is a compound statement that is always false, regardless of the truth values of its propositional variables. In other words, every row in its truth table evaluates to false.

Example: The statement “p ∧ ¬p” is a contradiction because regardless of the truth value of p, the statement will always be false.

Its truth table is as follows:

p | ¬p | p ∧ ¬p |

T | F | F |

F | T | F |

- Contingency: A contingency is a compound statement that is neither a tautology nor a contradiction. Its truth value depends on the specific truth values assigned to its propositional variables.

Example: The statement “p → q” is a contingency because its truth value depends on the specific truth values of p and q.

Its truth table is as follows:

p | q | p → q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

- Predictors: In the context of statistics and data analysis, predictors refer to the independent variables or factors used to predict or explain the dependent variable or outcome.

Example: In a study examining the factors influencing students’ academic performance, predictors could include variables such as study hours, attendance, socioeconomic status, and prior academic achievement. These predictors are used to predict or explain the students’ academic performance as the dependent variable.

These terms are commonly used in logic and statistics to describe different aspects of statements, truth values, and data analysis.

**Show that the given compound statement is a Tautology or a Contradiction**

To determine whether a given compound statement is a tautology or a contradiction, we can use a truth table. A truth table is a table that lists all possible truth values of the individual components of a compound statement and the resulting truth value of the compound statement for each combination of truth values.

Here are five examples of compound statements along with their truth tables to determine whether they are tautologies or contradictions:

- Example: “p ∨ ¬p”

Truth table:

p | ¬p | p ∨ ¬p |

T | F | T |

F | T | T |

Explanation: The compound statement “p ∨ ¬p” is a tautology because regardless of the truth value of p, the statement is always true.

- Example: “p ∧ ¬p”

Truth table:

p | ¬p | p ∧ ¬p |

T | F | F |

F | T | F |

Explanation: The compound statement “p ∧ ¬p” is a contradiction because regardless of the truth value of p, the statement is always false.

- Example: “p → p”

Truth table:

p | p | p → p |

T | T | T |

F | F | T |

Explanation: The compound statement “p → p” is a tautology because the statement is always true. It states that if p is true, then p is true.

- Example: “(p ∧ q) ∨ (¬p ∧ q)”

Truth table:

p | q | (p ∧ q) ∨ (¬p ∧ q) |

T | T | T |

T | F | F |

F | T | T |

F | F | F |

Explanation: The compound statement “(p ∧ q) ∨ (¬p ∧ q)” is a contingency because its truth value depends on the specific truth values of p and q.

- Example: “(p ∨ q) ∧ (¬p ∨ q)”

Truth table:

p | q | (p ∨ q) ∧ (¬p ∨ q) |

T | T | T |

T | F | F |

F | T | T |

F | F | F |

Explanation: The compound statement “(p ∨ q) ∧ (¬p ∨ q)” is a contingency because its truth value depends on the specific truth values of p and q.

In these examples, we can observe that some compound statements are tautologies (always true), some are contradictions (always false), and some are contingencies (their truth value depends on the specific truth values of the propositional variables).

**Recall and apply the following Laws of Logic: i. Idempotent Law ii. Associative Law iii. Commutative Law iv. Distributive Law v. Implication Law vi. Identity Law vii. Absorption Law viii. Complement Inverse Law ix. De-Morgan’s Law**

i. Idempotent Law:

The Idempotent Law states that a logical operator applied to a proposition multiple times has the same effect as applying it once.

Example:

Let p be the proposition “It is raining.”

p ∧ p = p (p AND p = p)

p ∨ p = p (p OR p = p)

ii. Associative Law:

The Associative Law states that the grouping of propositions using a logical operator does not affect the truth value.

Example:

Let p, q, and r be propositions.

(p ∧ q) ∧ r = p ∧ (q ∧ r) ((p AND q) AND r = p AND (q AND r))

(p ∨ q) ∨ r = p ∨ (q ∨ r) ((p OR q) OR r = p OR (q OR r))

iii. Commutative Law:

The Commutative Law states that the order of propositions connected by a logical operator does not affect the truth value.

Example:

Let p and q be propositions.

p ∧ q = q ∧ p (p AND q = q AND p)

p ∨ q = q ∨ p (p OR q = q OR p)

iv. Distributive Law:

The Distributive Law relates the logical operators AND and OR, showing how they interact.

Example:

Let p, q, and r be propositions.

p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) (p AND (q OR r) = (p AND q) OR (p AND r))

p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r) (p OR (q AND r) = (p OR q) AND (p OR r))

v. Implication Law:

The Implication Law relates the logical operators AND, OR, and NOT, showing how they interact.

Example:

Let p and q be propositions.

p → q = ¬p ∨ q (p implies q = not p or q)

vi. Identity Law:

The Identity Law states that a proposition combined with a logical operator and an identity element results in the original proposition.

Example:

Let p be the proposition.

p ∧ T = p (p AND true = p)

p ∨ F = p (p OR false = p)

vii. Absorption Law:

The Absorption Law states that a proposition combined with a logical operator and its duplicate results in the original proposition.

Example:

Let p and q be propositions.

p ∨ (p ∧ q) = p (p OR (p AND q) = p)

viii. Complement Inverse Law:

The Complement Inverse Law states that the complement of the complement of a proposition is equal to the original proposition.

Example:

Let p be the proposition.

¬(¬p) = p (not (not p) = p)

ix. De Morgan’s Law:

De Morgan’s Law relates the logical operators AND and OR with negation.

Example:

Let p and q be propositions.

¬(p ∧ q) = ¬p ∨ ¬q (not (p AND q) = not p OR not q)

¬(p ∨ q) = ¬p ∧ ¬q (not (p OR q) = not p AND not q)

These examples demonstrate the application of each law of logic, showing how they manipulate propositions and their truth values.

**Write the Propositions in Symbolic Form using Quantifiers and Predicate**

Propositional statements can be written in symbolic form using quantifiers and predicates. Quantifiers are words that indicate the quantity of individuals in a statement, such as “all”, “some”, or “none”. Predicates are words that describe the characteristics or properties of the individuals in a statement.

Here are five examples of propositions written in symbolic form using quantifiers and predicates:

- “Every student in the class passed the exam.”

Symbolic form: ∀x (S(x) → P(x))

Where S(x) represents “x is a student in the class” and P(x) represents “x passed the exam.”

- “There exists a prime number greater than 100.”

Symbolic form: ∃x (P(x) ∧ G(x))

Where P(x) represents “x is a prime number” and G(x) represents “x is greater than 100.”

- “No dogs are allowed in the restaurant.”

Symbolic form: ¬∃x (D(x) ∧ R(x))

Where D(x) represents “x is a dog” and R(x) represents “x is allowed in the restaurant.”

- “For every real number x, there exists a real number y such that x + y = 0.”

Symbolic form: ∀x∈ℝ ∃y∈ℝ (x + y = 0)

Where ℝ represents the set of real numbers.

- “There exists a person who loves all animals.”

Symbolic form: ∃x (P(x) ∧ ∀y (A(y) → L(x, y)))

Where P(x) represents “x is a person,” A(y) represents “y is an animal,” and L(x, y) represents “x loves y.”

- “For every positive integer n, there exists a positive integer m such that m is a multiple of n.”

Symbolic form: ∀n∈Z+ ∃m∈Z+ (m = n*k)

Where Z+ represents the set of positive integers and k is an integer.

- “There exists a triangle such that all its angles are acute.”

Symbolic form: ∃T (A1(T) ∧ A2(T) ∧ A3(T))

Where A1(T), A2(T), and A3(T) represent predicates indicating that the angles of triangle T are acute.

- “For every real number x, there exists a real number y such that x*y = 1.”

Symbolic form: ∀x∈ℝ ∃y∈ℝ (x * y = 1)

Where ℝ represents the set of real numbers.

- “There exists a positive even integer that is a perfect square.”

Symbolic form: ∃n∈Z+ (E(n) ∧ PS(n))

Where E(n) represents “n is even” and PS(n) represents “n is a perfect square.”

- “For every student in the class, there exists a subject that the student enjoys.”

Symbolic form: ∀x (S(x) → ∃y (E(x, y)))

Where S(x) represents “x is a student in the class” and E(x, y) represents “x enjoys subject y.”

These examples illustrate how propositions can be expressed using quantifiers and predicates to make statements about different domains and their elements.

**Show the Validity of the Arguments using the Truth table**

To show the validity of an argument using truth tables, we need to construct a truth table for the premises and the conclusion of the argument, and verify that the conclusion is true in all cases where the premises are true.

Here are five examples of arguments and their validity demonstrated using truth tables:

- Argument: “If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet.”

Truth table:

P (Raining) | Q (Ground is wet) | P → Q (If it is raining, then the ground is wet) | P (It is raining) | Q (Conclusion: The ground is wet) | Valid? |

T | T | T | T | T | Valid |

T | F | F | T | F | Invalid |

F | T | T | F | T | Valid |

F | F | T | F | T | Valid |

Since the conclusion (Q) is true in all rows where the premises (P → Q and P) are true, the argument is valid.

- Argument: “All birds have feathers. Penguins are birds. Therefore, penguins have feathers.”

Truth table:

P (All birds have feathers) | Q (Penguins are birds) | R (Penguins have feathers) | P → Q (Premise 1) | Q (Premise 2) | R (Conclusion) | Valid? |

T | T | T | T | T | T | Valid |

T | F | T | T | F | T | Valid |

F | T | T | T | T | T | Valid |

F | F | T | T | F | T | Valid |

Since the conclusion (R) is true in all rows where the premises (P → Q and Q) are true, the argument is valid.

- Argument: “If it is sunny, then I will go for a walk. It is not sunny. Therefore, I will not go for a walk.”

Truth table:

P (It is sunny) | Q (I will go for a walk) | P → Q (If it is sunny, then I will go for a walk) | ¬P (It is not sunny) | ¬Q (Conclusion: I will not go for a walk) | Valid? |

T | T | T | F | F | Invalid |

T | F | F | F | T | Valid |

F | T | T | T | F | Valid |

F | F | T | T | T | Valid |

Since the conclusion (¬Q) is true in all rows where the premises (P → Q and ¬P) are true, the argument is valid.

- Argument: “All mammals are animals. Dogs are mammals. Therefore, dogs are animals.”

Truth table:

P (All mammals are animals) | Q (Dogs are mammals) | R (Dogs are animals) | P → Q (Premise 1) | Q (Premise 2) | R (Conclusion) | Valid? |

T | T | T | T | T | T | Valid |

T | F | F | F | F | F | Valid |

F | T | T | T | T | T | Valid |

F | F | T | T | F | T | Valid |

Since the conclusion (R) is true in all rows where the premises (P → Q and Q) are true, the argument is valid.

- Argument: “If it is Monday, then I have a meeting. I have a meeting. Therefore, it is Monday.”

Truth table:

P (It is Monday) | Q (I have a meeting) | P → Q (If it is Monday, then I have a meeting) | Q (Premise 2) | P (Conclusion: It is Monday) | Valid? |

T | T | T | T | T | Valid |

T | F | F | F | T | Valid |

F | T | T | T | F | Invalid |

F | F | T | F | F | Valid |

Since the conclusion (P) is true in all rows where the premises (P → Q and Q) are true, the argument is valid.

These examples demonstrate the validity of the arguments by showing that the conclusions are true in all rows where the premises are true.

**Recall Disjunctive Normal Form (DNF)**

DNF stands for Disjunctive Normal Form. It is a standard form used to represent logical formulas in propositional logic. In DNF, a logical formula is expressed as a disjunction (OR) of one or more conjunctions (AND) of literals, where a literal is either a variable or its negation.

Formally, a logical formula is in DNF if it has the following structure:

F = (L₁₁ ∧ L₁₂ ∧ … ∧ L₁ₖ) ∨ (L₂₁ ∧ L₂₂ ∧ … ∧ L₂ₘ) ∨ … ∨ (Lₙ₁ ∧ Lₙ₂ ∧ … ∧ Lₙₚ)

where F is the formula, each Lᵢⱼ is a literal, k, m, p are positive integers, and n is the total number of conjunctions in the formula.

The disjunction (∨) represents logical OR, which means at least one of the conjunctions must be true for the entire formula to be true. The conjunction (∧) represents logical AND, which means all the literals within a conjunction must be true for that conjunction to be true.

DNF provides a way to express complex logical formulas in a more structured and readable form. It is often used in logic simplification, theorem proving, and understanding the behavior of logical expressions.

To convert a logical formula into Disjunctive Normal Form (DNF), you can follow these steps:

- Eliminate implications and bi-implications using logical equivalences.
- Move negation inwards using De Morgan’s laws and double negation elimination.
- Distribute conjunction (∧) over disjunction (∨) to get the formula in DNF.

Here is a step-by-step explanation of the process:

Step 1: Eliminate implications and bi-implications

- Replace the implication (→) with its equivalent form using disjunction and negation: A → B is equivalent to ¬A ∨ B.
- Replace the bi-implication (↔) with its equivalent form using conjunction and implication: A ↔ B is equivalent to (A → B) ∧ (B → A).

Step 2: Move negation inwards

- Apply De Morgan’s laws to move negation inwards. For example, ¬(A ∧ B) becomes (¬A ∨ ¬B) and ¬(A ∨ B) becomes (¬A ∧ ¬B).
- Use double negation elimination to simplify expressions with double negations. For example, ¬¬A becomes A.

Step 3: Distribute conjunction over disjunction

- Apply the distributive property to distribute conjunction (∧) over disjunction (∨). For example, A ∧ (B ∨ C) becomes (A ∧ B) ∨ (A ∧ C).

By following these steps, you can transform a logical formula into Disjunctive Normal Form (DNF), which is a disjunction of one or more conjunctions of literals.

Note: It’s important to note that not all logical formulas can be converted into DNF. Some formulas may have complex structures or involve higher-order logic that cannot be represented in DNF.

Example 1:

Consider the logical expression: (A ∨ B) ∧ (C ∨ D) ∧ (E ∨ F)

The DNF form of this expression is:

(A ∧ C ∧ E) ∨ (A ∧ C ∧ F) ∨ (A ∧ D ∧ E) ∨ (A ∧ D ∧ F) ∨ (B ∧ C ∧ E) ∨ (B ∧ C ∧ F) ∨ (B ∧ D ∧ E) ∨ (B ∧ D ∧ F)

Example 2:

Consider the logical expression: (P ∨ ¬Q) ∧ (R ∨ S)

The DNF form of this expression is:

(P ∧ R) ∨ (P ∧ S) ∨ (¬Q ∧ R) ∨ (¬Q ∧ S)

Example 3:

Consider the logical expression: (X ∨ Y) ∧ (Z ∨ ¬W) ∧ (P ∨ Q)

The DNF form of this expression is:

(X ∧ Z ∧ P) ∨ (X ∧ Z ∧ Q) ∨ (X ∧ ¬W ∧ P) ∨ (X ∧ ¬W ∧ Q) ∨ (Y ∧ Z ∧ P) ∨ (Y ∧ Z ∧ Q) ∨ (Y ∧ ¬W ∧ P) ∨ (Y ∧ ¬W ∧ Q)

In each of these examples, the given logical expression is transformed into Disjunctive Normal Form by expressing it as a disjunction of conjunctions of literals.

**Find DNF of a given Compound Statement**

Here are five examples of finding the Disjunctive Normal Form (DNF) of compound statements, along with step-by-step explanations:

- Compound Statement: (P ∧ Q) → (R ∨ S)

Step 1: Apply the implication law: ¬(P ∧ Q) ∨ (R ∨ S)

Step 2: De Morgan’s law: (¬P ∨ ¬Q) ∨ (R ∨ S)

DNF: (¬P ∨ ¬Q ∨ R ∨ S)

- Compound Statement: (P → Q) ∧ (R ∨ ¬S)

Step 1: Apply the implication law: (¬P ∨ Q) ∧ (R ∨ ¬S)

DNF: (¬P ∧ R) ∨ (¬P ∧ ¬S) ∨ (Q ∧ R) ∨ (Q ∧ ¬S)

- Compound Statement: (P ∧ (Q → R)) → (S ∨ T)

Step 1: Apply the implication law: ¬(P ∧ (¬Q ∨ R)) ∨ (S ∨ T)

Step 2: De Morgan’s law: (¬P ∨ (Q ∧ ¬R)) ∨ (S ∨ T)

DNF: (¬P ∨ Q ∨ S ∨ T) ∨ (¬P ∨ ¬R ∨ S ∨ T)

- Compound Statement: (P ∨ (Q ∧ R)) → (¬S ∨ T)

Step 1: Apply the implication law: ¬(P ∨ (Q ∧ R)) ∨ (¬S ∨ T)

Step 2: De Morgan’s law: (¬P ∧ (¬Q ∨ ¬R)) ∨ (¬S ∨ T)

DNF: (¬P ∨ ¬Q ∨ ¬R ∨ ¬S ∨ T)

- Compound Statement: (P → (Q ∧ R)) → ((S ∨ T) → U)

Step 1: Apply the implication law: ¬(P → (Q ∧ R)) ∨ ¬((S ∨ T) → U)

Step 2: Apply the implication law again: ¬(¬P ∨ (Q ∧ R)) ∨ ¬(¬(S ∨ T) ∨ U)

Step 3: De Morgan’s law: (P ∧ ¬(Q ∧ R)) ∨ ((S ∨ T) ∧ ¬U)

DNF: (P ∨ (S ∨ T) ∨ ¬U) ∨ (P ∨ (S ∨ T) ∨ ¬U) ∨ (P ∨ (S ∨ T) ∨ ¬U) ∨ (P ∨ (S ∨ T) ∨ ¬U)

In each example, I followed the step-by-step process to transform the given compound statement into DNF, applying logical laws such as implication, De Morgan’s, and distribution. The resulting DNF is a disjunction (OR) of clauses, where each clause is a conjunction (AND) of literals.

**Recall Conjunctive Normal Form (CNF)**

Conjunctive Normal Form (CNF) is a standard form of representing logical formulas where a formula is expressed as a conjunction of one or more clauses, where each clause is a disjunction of literals. In CNF, the logical connectives used are only conjunction (∧) and disjunction (∨), and negation (¬) is applied only to literals.

To convert a logical formula into CNF, the following steps are typically followed:

- Eliminate implications and bi-implications using logical equivalences.
- Move negation inwards using De Morgan’s laws and double negation elimination.
- Distribute disjunction (∨) over conjunction (∧) to get the formula in CNF.

Here are a few examples of converting compound statements into CNF:

Example 1:

Compound Statement: (A ∧ B) ∨ (C → D)

CNF: (A ∨ C) ∧ (B ∨ C ∨ D)

Example 2:

Compound Statement: ¬(P ∧ Q) → R

CNF: (¬P ∨ ¬Q ∨ R)

Example 3:

Compound Statement: (X ∨ Y) ∧ (Z → W)

CNF: (X ∧ Z ∧ W) ∨ (Y ∧ Z ∧ W)

Example 4:

Compound Statement: (A → B) ∨ (C ↔ D)

CNF: (¬A ∨ B ∨ C) ∧ (¬A ∨ B ∨ ¬D) ∧ (A ∨ B ∨ C ∨ D) ∧ (A ∨ B ∨ C ∨ ¬D)

Example 5:

Compound Statement: (P ∧ ¬Q) ↔ (R ∨ S)

CNF: (P ∨ R ∨ S) ∧ (¬Q ∨ R ∨ S) ∧ (P ∨ ¬Q ∨ ¬R) ∧ (P ∨ ¬Q ∨ ¬S)

In each of these examples, the given compound statement is transformed into Conjunctive Normal Form (CNF) by expressing it as a conjunction of clauses, where each clause is a disjunction of literals.

**Find CNF of a given Compound Statement**

Here are five examples of finding the Conjunctive Normal Form (CNF) of compound statements, along with step-by-step explanations:

- Compound Statement: (P ∨ Q) → (R ∧ S)

Step 1: Apply the implication law: (¬(P ∨ Q)) ∨ (R ∧ S)

Step 2: De Morgan’s law: (¬P ∧ ¬Q) ∨ (R ∧ S)

Step 3: Distributive law: (¬P ∨ R) ∧ (¬P ∨ S) ∧ (¬Q ∨ R) ∧ (¬Q ∨ S)

CNF: (¬P ∨ R) ∧ (¬P ∨ S) ∧ (¬Q ∨ R) ∧ (¬Q ∨ S)

- Compound Statement: (P → Q) ∧ (R ∨ ¬S)

Step 1: Apply the implication law: (¬P ∨ Q) ∧ (R ∨ ¬S)

CNF: (¬P ∨ Q) ∧ (R ∨ ¬S)

- Compound Statement: (P ∧ (Q → R)) → (S ∨ T)

Step 1: Apply the implication law: (¬(P ∧ (¬Q ∨ R))) ∨ (S ∨ T)

Step 2: De Morgan’s law: (¬P ∨ (Q ∧ ¬R)) ∨ (S ∨ T)

CNF: (¬P ∨ Q ∨ S ∨ T) ∧ (¬P ∨ ¬R ∨ S ∨ T)

- Compound Statement: (P ∨ (Q ∧ R)) → (¬S ∨ T)

Step 1: Apply the implication law: (¬(P ∨ (Q ∧ R))) ∨ (¬S ∨ T)

Step 2: De Morgan’s law: (¬P ∧ (¬Q ∨ ¬R)) ∨ (¬S ∨ T)

CNF: (¬P ∨ ¬Q ∨ ¬S ∨ T) ∧ (¬P ∨ ¬R ∨ ¬S ∨ T)

- Compound Statement: (P → (Q ∧ R)) → ((S ∨ T) → U)

Step 1: Apply the implication law: (¬(P → (Q ∧ R))) ∨ (¬((S ∨ T) → U))

Step 2: Apply the implication law again: (¬(¬P ∨ (Q ∧ R))) ∨ (¬(¬(S ∨ T) ∨ U))

Step 3: De Morgan’s law: (P ∧ ¬(Q ∧ R)) ∨ ((S ∨ T) ∧ ¬U)

CNF: (P ∨ (S ∨ T) ∨ ¬U) ∧ (P ∨ (S ∨ T) ∨ ¬U) ∧ (P ∨ (S ∨ T) ∨ ¬U) ∧ (P ∨ (S ∨ T) ∨ ¬U)

In each example, I followed the step-by-step process to transform the given compound statement into CNF, applying logical laws such as implication, De Morgan’s, and distribution. The resulting CNF is a conjunction (AND) of clauses, where each clause is a disjunction (OR) of literals.

**Recall the Principle of Duality**

The Principle of Duality is a fundamental concept in Boolean algebra, which states that any theorem or algebraic expression remains valid if we interchange the logical operators AND (⋅) and OR (+), as well as replace the constants 0 and 1 with each other. In other words, if a statement is true in Boolean algebra, its dual statement obtained by replacing AND with OR and 0 with 1 will also be true.

The Principle of Duality can be summarized as follows:

- Interchange AND (⋅) with OR (+): If an expression is true, its dual expression obtained by replacing all AND operations with OR operations (and vice versa) will also be true.
- Replace 0 with 1 and 1 with 0: If an expression is true, its dual expression obtained by replacing all 0s with 1s and all 1s with 0s will also be true.

The Principle of Duality is useful in simplifying Boolean expressions and proving theorems. By using duality, we can derive the dual form of a given expression and apply various logical operations to simplify it.

For example, if we have the expression (A + B)⋅C, its dual form obtained by applying the Principle of Duality would be (A ⋅ B) + C.

The Principle of Duality is a powerful tool in Boolean algebra, allowing us to reason about expressions and theorems from multiple perspectives and find alternative representations.

**Recall Well Formed Formula**

A Well-Formed Formula (WFF) is a valid and syntactically correct expression in a formal language or logic system. It follows the defined rules and grammar of the language without any syntax errors. The concept of WFF is often associated with formal logic, where logical expressions are constructed using logical connectives, variables, and quantifiers.

In order for a formula to be considered well-formed, it must adhere to the following rules:

- Proper use of parentheses: Every opening parenthesis must have a matching closing parenthesis, and vice versa. This ensures that the formula is properly nested and unambiguous.
- Correct use of logical operators: The logical operators (such as AND, OR, NOT, IMPLIES) must be used in accordance with their defined semantics and syntax. For example, the NOT operator must be followed by a valid subformula.
- Proper use of variables: If variables are used in the formula, they must be defined and used consistently throughout the expression. Each variable should have a clear scope and meaning.
- Consistent quantifier usage: If quantifiers (such as FOR ALL or THERE EXISTS) are used, they must be used correctly and consistently. Quantified variables should be appropriately bound within the formula.

Overall, a well-formed formula ensures that the expression is valid and can be properly interpreted within the logical system. It is free from syntax errors and follows the established rules and grammar of the language.

Example of a well-formed formula:

((A AND B) IMPLIES C) OR (NOT D)

In this example, the formula is constructed using parentheses to define the grouping of subformulas. The logical operators (AND, IMPLIES, OR, NOT) are used correctly, and the variables (A, B, C, D) are properly defined. The formula follows the syntax rules and is considered well-formed.

**Recall the Functionally Complete Sets of Connectives**

Functionally complete sets of connectives are sets of logical connectives that can be used to express any logical function or truth table. In other words, using the connectives from a functionally complete set, we can construct logical expressions that can represent any logical operation.

There are several functionally complete sets of connectives, and two commonly used sets are:

- {AND, NOT}:
- The AND connective (∧) represents the logical conjunction or conjunction of two propositions. It outputs true only when both input propositions are true.
- The NOT connective (¬) represents the logical negation or negation of a proposition. It outputs the opposite truth value of the input proposition.

- Using these two connectives, we can express other logical connectives as follows:
- OR (∨): A OR B = NOT(NOT A AND NOT B)
- IMPLIES (→): A → B = NOT A OR B
- IF AND ONLY IF (↔): A ↔ B = (A → B) AND (B → A)

- {NAND}:
- The NAND connective (↑) represents the negation of the logical conjunction. It outputs true unless both input propositions are true.

- Using only the NAND connective, we can express other logical connectives as follows:
- OR (∨): A OR B = (A ↑ A) ↑ (B ↑ B)
- AND (∧): A AND B = (A ↑ B) ↑ (A ↑ B)
- NOT (¬): NOT A = A ↑ A

These sets of connectives are called functionally complete because they can be used to construct any truth table or logical operation. Any logical expression can be expressed using these connectives.

It’s worth noting that other sets of connectives, such as {OR, NOT}, {NOR}, and {XOR}, are also functionally complete and can be used to express any logical function.

Here are three examples of functionally complete sets of connectives and their usage:

Example 1: {AND, NOT}

- Propositional Function: p AND (q OR NOT r)

Using the functionally complete set {AND, NOT}, we can express the proposition as follows:

- p AND (q OR NOT r) = p AND (q OR (NOT r))

Example 2: {NAND}

- Propositional Function: NOT (p AND q) OR (r NAND s)

Using the functionally complete set {NAND}, we can express the proposition as follows:

- NOT (p AND q) OR (r NAND s) = (p NAND q) NAND (r NAND s)

Example 3: {OR, NOT}

- Propositional Function: (p OR q) AND (NOT r)

Using the functionally complete set {OR, NOT}, we can express the proposition as follows:

- (p OR q) AND (NOT r) = NOT (NOT (p OR q) OR (NOT r))

In each of these examples, we use the given functionally complete set of connectives to construct the logical expression that represents the given propositional function. The connectives from the set are used to combine the propositions and perform logical operations such as conjunction, disjunction, and negation.

**Recall Propositional vs Predicate Logic**

Propositional logic and predicate logic are two different branches of formal logic that deal with different aspects of logical reasoning:

Propositional Logic:

- Propositional logic, also known as sentential logic or propositional calculus, deals with propositions or statements that are either true or false.
- It focuses on the logical relationships between propositions and uses logical connectives such as AND, OR, NOT, IMPLIES, etc., to form compound propositions.
- Propositional logic does not involve variables or quantifiers and is concerned with the truth values and the logical relationships between propositions.

Predicate Logic:

- Predicate logic, also known as first-order logic or predicate calculus, extends propositional logic by introducing variables, quantifiers, and predicates.
- It allows for the representation of more complex statements involving variables, quantified statements, and relations between objects.
- Predicate logic uses quantifiers such as “forall” (∀) and “exists” (∃) to express general statements about objects and predicates to denote properties or relations.
- Predicate logic allows for more precise and detailed reasoning about the properties and relationships of objects.

In summary, propositional logic deals with simple statements and their logical relationships, while predicate logic extends propositional logic to handle variables, quantifiers, and more complex statements involving objects and relations. Predicate logic provides a more expressive language for logical reasoning and is commonly used in mathematics, computer science, and formal reasoning systems.

**Recall Terms: Predicate and Quantifiers**

In logic and mathematics, the terms “predicate” and “quantifiers” are fundamental concepts used to express statements and make generalizations about objects or variables within a given domain.

- Predicate:

A predicate is a statement or a function that takes one or more variables as input and produces a truth value (either true or false) as output. It is a way of expressing properties or characteristics of objects. Predicates are often denoted by uppercase letters, such as P(x), Q(x, y), or R(x, y, z), where x, y, z are variables representing elements from a specified domain.

For example:

- P(x): “x is a prime number.”
- Q(x, y): “x is greater than y.”
- R(x, y, z): “x is the parent of y, and y is the parent of z.”

Predicates can be true or false depending on the values assigned to the variables. When we substitute specific values for the variables, the predicate becomes a proposition (a statement that is either true or false).

- Quantifiers:

Quantifiers are used to make general statements about objects or variables in a given domain. They specify the quantity or extent to which a predicate holds true for a particular set of elements. The two most common quantifiers are:

- Universal Quantifier (∀): The symbol ∀ is used to indicate that a predicate is true for all elements in a specified domain. It asserts that a statement holds for every possible value of the variable.

For example:

∀x P(x) can be read as “For all x, x is a prime number.”

- Existential Quantifier (∃): The symbol ∃ is used to indicate that a predicate is true for at least one element in a specified domain. It asserts the existence of an element that satisfies the given predicate.

For example:

∃x P(x) can be read as “There exists an x such that x is a prime number.”

Quantifiers are often used in combination with predicates to express more complex statements and to establish generalizations about the elements in a domain. They play a crucial role in formal logic, mathematics, and computer science.

**Convert the given statement into a First-order Predicate Logic Form**

To convert the statement “Convert the given statement into a First-order Predicate Logic Form: Algorithm” into first-order predicate logic form, we need to identify the predicates and quantify the variables appropriately. However, the given statement does not involve any specific predicates or variables that can be expressed in first-order logic. It is a directive or command rather than a statement about objects or relationships within a domain.

First-order predicate logic is used to express statements or propositions that involve predicates and variables. Since the given statement does not contain any such propositions, it cannot be converted into first-order predicate logic form.

If you have a specific statement or proposition that you would like to convert into first-order predicate logic, please provide it, and I’ll be happy to assist you with its conversion.

To convert statements into first-order predicate logic form, we need to identify the predicates and quantify the variables appropriately. Here are five examples of statements and their conversions into first-order predicate logic:

- Statement: “All cats are mammals.”

Conversion: ∀x (Cat(x) → Mammal(x))

Predicate: Cat(x) – “x is a cat.”

Mammal(x) – “x is a mammal.”

Quantifier: ∀x – Universal quantifier, indicating the statement applies to all x (all elements in the domain).

- Statement: “There exists a prime number.”

Conversion: ∃x Prime(x)

Predicate: Prime(x) – “x is a prime number.”

Quantifier: ∃x – Existential quantifier, indicating the existence of at least one x that satisfies the predicate.

- Statement: “Every student loves at least one subject.”

Conversion: ∀x ∃y (Student(x) → Loves(x, y))

Predicates: Student(x) – “x is a student.”

Loves(x, y) – “x loves y.”

Quantifiers: ∀x – Universal quantifier, indicating the statement applies to all x (all students).

∃y – Existential quantifier, indicating that each student loves at least one y (subject).

- Statement: “Some birds can fly.”

Conversion: ∃x (Bird(x) ∧ CanFly(x))

Predicates: Bird(x) – “x is a bird.”

CanFly(x) – “x can fly.”

Quantifier: ∃x – Existential quantifier, indicating the existence of at least one x that satisfies the predicate.

- Statement: “No dogs are reptiles.”

Conversion: ∀x (Dog(x) → ¬Reptile(x))

Predicates: Dog(x) – “x is a dog.”

Reptile(x) – “x is a reptile.”

Quantifier: ∀x – Universal quantifier, indicating the statement applies to all x (all elements in the domain).

¬ – Negation symbol, indicating the absence of the predicate.

In these examples, I have assumed the existence of predicates such as Cat(x), Mammal(x), Prime(x), Student(x), Loves(x, y), Bird(x), CanFly(x), Dog(x), and Reptile(x). The choice of predicates depends on the specific domain and context in which these statements are being evaluated.

**Recall the following Rules of Inference: Modus Ponens, Modus Tollens, Hypothetical Syllogism, and Disjunctive Syllogism**

Here are the definitions of the rules of inference:

- Modus Ponens:

Modus Ponens is a valid rule of inference that allows you to infer a conclusion from a conditional statement and its antecedent. The rule can be stated as follows:

If you have a premise of the form “If P, then Q” (P → Q) and another premise that is simply P, then you can validly conclude Q.

Example:

Premise 1: If it is raining, then the ground is wet. (R → W)

Premise 2: It is raining. (R)

Conclusion: Therefore, the ground is wet. (W)

- Modus Tollens:

Modus Tollens is another valid rule of inference that allows you to infer a conclusion from a conditional statement and its negation. The rule can be stated as follows:

If you have a premise of the form “If P, then Q” (P → Q) and another premise that is the negation of the consequent (¬Q), then you can validly conclude the negation of the antecedent (¬P).

Example:

Premise 1: If it is raining, then the ground is wet. (R → W)

Premise 2: The ground is not wet. (¬W)

Conclusion: Therefore, it is not raining. (¬R)

- Hypothetical Syllogism:

Hypothetical Syllogism is a valid rule of inference that allows you to infer a conclusion from two conditional statements. The rule can be stated as follows:

If you have premises of the form “If P, then Q” (P → Q) and “If Q, then R” (Q → R), then you can validly conclude “If P, then R” (P → R).

Example:

Premise 1: If it is raining, then the ground is wet. (R → W)

Premise 2: If the ground is wet, then the grass is green. (W → G)

Conclusion: Therefore, if it is raining, then the grass is green. (R → G)

- Disjunctive Syllogism:

Disjunctive Syllogism is a valid rule of inference that allows you to infer a conclusion from a disjunction and the negation of one of its disjuncts. The rule can be stated as follows:

If you have a premise of the form “P or Q” (P ∨ Q) and another premise that is the negation of one of the disjuncts (¬P or ¬Q), then you can validly conclude the other disjunct.

Example:

Premise 1: It is either raining or sunny. (R ∨ S)

Premise 2: It is not raining. (¬R)

Conclusion: Therefore, it is sunny. (S)

These rules of inference are commonly used in deductive reasoning to derive valid conclusions from given premises. They provide a structured and reliable way to make logical deductions.

**Test the validity of the given Arguments**

To test the validity of the given arguments, we need to examine whether the conclusion logically follows from the premises. Here are five examples of arguments along with their validity assessments:

- Argument:

Premise 1: If it is raining, then the ground is wet.

Premise 2: The ground is wet.

Conclusion: Therefore, it is raining.

Validity: This argument is not valid. The argument commits the fallacy of affirming the consequent. While the premises establish a conditional relationship between rain and wet ground, the conclusion assumes that rain is the only possible cause of a wet ground, which is not necessarily true.

- Argument:

Premise 1: All mammals are warm-blooded.

Premise 2: A whale is warm-blooded.

Conclusion: Therefore, a whale is a mammal.

Validity: This argument is valid. It follows the pattern of modus ponens, where the premises establish a conditional relationship between mammals and being warm-blooded. Given that a whale is warm-blooded, the conclusion logically follows that it must be a mammal.

- Argument:

Premise 1: If it is a cat, then it is a mammal.

Premise 2: It is a mammal.

Conclusion: Therefore, it is a cat.

Validity: This argument is not valid. Similar to the first example, it commits the fallacy of affirming the consequent. While the premises establish a conditional relationship between cats and mammals, the conclusion assumes that being a mammal implies being a cat, which is not necessarily true.

- Argument:

Premise 1: All dogs have four legs.

Premise 2: Max is a dog.

Conclusion: Therefore, Max has four legs.

Validity: This argument is valid. It follows the pattern of modus ponens, where the premises establish a conditional relationship between dogs and having four legs. Given that Max is identified as a dog, the conclusion logically follows that Max must have four legs.

- Argument:

Premise 1: If it is a square, then it has four equal sides.

Premise 2: The figure has four equal sides.

Conclusion: Therefore, the figure is a square.

Validity: This argument is valid. It follows the pattern of modus ponens, where the premises establish a conditional relationship between squares and having four equal sides. Given that the figure in question has four equal sides, the conclusion logically follows that the figure must be a square.

Remember that validity is concerned with the logical structure of an argument, regardless of the truth or falsity of the premises or conclusion.

Here are five examples of arguments along with their symbolic representations, followed by an assessment of their validity:

- Argument:

Premise 1: (P → Q)

Premise 2: P

Conclusion: Q

Validity: This argument is valid. It follows the modus ponens rule of inference, where the premises establish a conditional relationship between P and Q. Given that P is true, the conclusion Q logically follows.

- Argument:

Premise 1: (P → Q)

Premise 2: ¬Q

Conclusion: ¬P

Validity: This argument is valid. It follows the modus tollens rule of inference, where the premises establish a conditional relationship between P and Q. Given that Q is false (denoted by the negation ¬Q), the conclusion ¬P logically follows.

- Argument:

Premise 1: (P → Q)

Premise 2: (Q → R)

Conclusion: (P → R)

Validity: This argument is valid. It follows the hypothetical syllogism rule of inference, where the premises establish conditional relationships between P and Q, and between Q and R. The conclusion logically follows that if P implies Q, and Q implies R, then P implies R.

- Argument:

Premise 1: (P ∨ Q)

Premise 2: ¬P

Conclusion: Q

Validity: This argument is valid. It follows the disjunctive syllogism rule of inference, where the premises establish a disjunction between P and Q, and the negation of P. Given that one disjunct is false (¬P), the conclusion Q logically follows as the remaining true disjunct.

- Argument:

Premise 1: (P ∧ Q)

Conclusion: P

Validity: This argument is valid. It follows a simple logical rule where the premise states that both P and Q are true (P ∧ Q). From this premise, we can validly conclude that P must be true since it is one of the conjuncts.

In each of these examples, the validity of the arguments is assessed based on the logical structure and the application of valid rules of inference. The symbolic representation helps to express the arguments more formally and precisely.

**Recall Rules of Inference and their corresponding Tautological Form**

The Rules of Inference are logical rules that allow us to make valid deductions from given premises. Each rule of inference has a corresponding tautological form, which is a logical tautology that represents the rule. Here are some common rules of inference and their corresponding tautological forms:

- Modus Ponens:

Rule: If we have a premise of the form “If P, then Q” (P → Q) and another premise that is simply P, then we can validly conclude Q.

Tautological Form: ((P → Q) ∧ P) → Q

- Modus Tollens:

Rule: If we have a premise of the form “If P, then Q” (P → Q) and another premise that is the negation of the consequent (¬Q), then we can validly conclude the negation of the antecedent (¬P).

Tautological Form: ((P → Q) ∧ ¬Q) → ¬P

- Hypothetical Syllogism:

Rule: If we have premises of the form “If P, then Q” (P → Q) and “If Q, then R” (Q → R), then we can validly conclude “If P, then R” (P → R).

Tautological Form: ((P → Q) ∧ (Q → R)) → (P → R)

- Disjunctive Syllogism:

Rule: If we have a premise of the form “P or Q” (P ∨ Q) and another premise that is the negation of one of the disjuncts (¬P or ¬Q), then we can validly conclude the other disjunct.

Tautological Form: ((P ∨ Q) ∧ ¬P) → Q

- Constructive Dilemma:

Rule: If we have premises of the form “If P, then Q” (P → Q) and “If R, then S” (R → S), and a disjunction of P and R, then we can validly conclude the disjunction of Q and S.

Tautological Form: ((P → Q) ∧ (R → S) ∧ (P ∨ R)) → (Q ∨ S)

These are just a few examples of rules of inference and their corresponding tautological forms. Each rule of inference corresponds to a logical principle that can be represented as a tautology in propositional logic. These rules provide a formal basis for valid deductive reasoning.

**Test the validity of the given Arguments without using Truth Table**

To test the validity of an argument without using a truth table, we can use the following steps:

- Translate the premises and conclusion into symbolic logic.
- Use the Rules of Inference to derive the conclusion from the premises.
- Check if the argument is valid by verifying if the Tautological Form of the Rule of Inference used is valid. If it is valid, then the argument is valid.

Note that if the argument is invalid, it is possible that the conclusion can still be true. It only means that the premises do not necessarily imply the conclusion.

To test the validity of arguments without using a truth table, we can employ logical reasoning techniques and proof strategies. Here are five examples of arguments along with a procedure to test their validity:

- Argument:

Premise 1: P → Q

Premise 2: P

Conclusion: Q

Procedure to test validity:

- Assume the premises to be true.
- Use modus ponens: From P → Q and P, infer Q.
- If Q can be derived, the argument is valid. Otherwise, it is invalid.

- Argument:

Premise 1: P ∨ Q

Premise 2: ¬P

Conclusion: Q

Procedure to test validity:

- Assume the premises to be true.
- Use disjunctive syllogism: From P ∨ Q and ¬P, infer Q.
- If Q can be derived, the argument is valid. Otherwise, it is invalid.

- Argument:

Premise 1: (P → Q) ∧ (R → S)

Premise 2: P ∨ R

Conclusion: Q ∨ S

Procedure to test validity:

- Assume the premises to be true.
- Use constructive dilemma: From (P → Q) ∧ (R → S) and P ∨ R, infer Q ∨ S.
- If Q ∨ S can be derived, the argument is valid. Otherwise, it is invalid.

- Argument:

Premise 1: (P ∧ Q) → R

Premise 2: S → (¬R ∧ Q)

Conclusion: ¬P ∨ S

Procedure to test validity:

- Assume the premises to be true.
- Use a proof by contradiction:
- Assume ¬(¬P ∨ S).
- Derive P and ¬S using De Morgan’s laws and double negation elimination.
- Use modus tollens with (P ∧ Q) → R to derive ¬R ∧ ¬Q.
- Reach a contradiction by contradicting the premise S → (¬R ∧ Q).

- If a contradiction is reached, the argument is valid. Otherwise, it is invalid.

- Argument:

Premise 1: (P ∧ Q) ∨ (R ∧ S)

Premise 2: ¬(P ∧ Q)

Conclusion: R ∨ S

Procedure to test validity:

- Assume the premises to be true.
- Use proof by cases:
- Case 1: Assume (P ∧ Q). Reach a contradiction with the negation of (P ∧ Q).
- Case 2: Assume (R ∧ S). Infer R ∨ S using disjunction introduction.

- If R ∨ S can be derived in all cases, the argument is valid. Otherwise, it is invalid.

These procedures employ logical principles and proof techniques to establish the validity of the given arguments without resorting to truth tables.