Podcast
Questions and Answers
Which of the following best describes the Law of Excluded Middle?
Which of the following best describes the Law of Excluded Middle?
- Things must be identical with themselves.
- A statement can be both true and false simultaneously.
- Every proposition must be either true or false, but not both. (correct)
- Logically correct propositions can affirm and deny the same thing.
The statement 'x > 10' is a proposition.
The statement 'x > 10' is a proposition.
False (B)
Explain the Law of Identity in propositional logic using the variable 'A'.
Explain the Law of Identity in propositional logic using the variable 'A'.
A = A
In propositional logic, a proposition is a declarative sentence that is either true or _____.
In propositional logic, a proposition is a declarative sentence that is either true or _____.
Match each of the following laws of thought with its description:
Match each of the following laws of thought with its description:
Which symbol represents the negation of a proposition?
Which symbol represents the negation of a proposition?
A conjunction is true if at least one of the propositions is true.
A conjunction is true if at least one of the propositions is true.
Which of the following statements violates the Law of Non-Contradiction?
Which of the following statements violates the Law of Non-Contradiction?
If proposition A is false and proposition B is true, what is the truth value of the disjunction A v B?
If proposition A is false and proposition B is true, what is the truth value of the disjunction A v B?
Which of the following uses numerical symbols to represent a false statement?
Which of the following uses numerical symbols to represent a false statement?
Given P = 'Today is Monday' and Q = 'The sky is blue', which of the following scenarios violates the Law of Excluded Middle?
Given P = 'Today is Monday' and Q = 'The sky is blue', which of the following scenarios violates the Law of Excluded Middle?
The combined proposition called a ______ will be true if one of the propositions is true.
The combined proposition called a ______ will be true if one of the propositions is true.
Given proposition P: 'The sky is blue' and proposition Q: 'Grass is green', which of the following correctly represents the conjunction of P and Q?
Given proposition P: 'The sky is blue' and proposition Q: 'Grass is green', which of the following correctly represents the conjunction of P and Q?
If proposition A is 'It is sunny' and proposition B is 'I am happy', then the disjunction A v B is only true if it is sunny.
If proposition A is 'It is sunny' and proposition B is 'I am happy', then the disjunction A v B is only true if it is sunny.
Provide the truth table result for A ^ B when A is true and B is false.
Provide the truth table result for A ^ B when A is true and B is false.
Match the logical operation with its corresponding condition for a TRUE result:
Match the logical operation with its corresponding condition for a TRUE result:
Given the implication 'If it is raining, then the ground is wet,' which of the following represents the converse?
Given the implication 'If it is raining, then the ground is wet,' which of the following represents the converse?
In a biconditional statement, both propositions must have the same truth value (either both true or both false) for the entire statement to be considered true.
In a biconditional statement, both propositions must have the same truth value (either both true or both false) for the entire statement to be considered true.
What is the symbol used to represent a biconditional statement?
What is the symbol used to represent a biconditional statement?
In the implication A → B, A is referred to as the ________ and B is the _________.
In the implication A → B, A is referred to as the ________ and B is the _________.
Match each type of implication with its correct description:
Match each type of implication with its correct description:
Which of the following statements accurately describes the truth conditions for an XOR (exclusive or) operation between two propositions?
Which of the following statements accurately describes the truth conditions for an XOR (exclusive or) operation between two propositions?
Given the implication: 'If a shape is a square, then it has four sides.' Which of the following is the contrapositive of this statement?
Given the implication: 'If a shape is a square, then it has four sides.' Which of the following is the contrapositive of this statement?
The antecedent in an implication is the 'then statement'.
The antecedent in an implication is the 'then statement'.
Which of the following best describes the purpose of a truth table?
Which of the following best describes the purpose of a truth table?
The statement x > 10 is always a proposition, regardless of the value of x.
The statement x > 10 is always a proposition, regardless of the value of x.
What is the fundamental difference between a statement and a proposition?
What is the fundamental difference between a statement and a proposition?
Combining propositions using the keyword not is an example of a logical connective called a ______.
Combining propositions using the keyword not is an example of a logical connective called a ______.
Match the logical scenario with its corresponding truth table result, given that 'A' represents 'age between 18-65' and 'B' represents 'fully vaccinated', and 'C' represents conditional to go out.
Match the logical scenario with its corresponding truth table result, given that 'A' represents 'age between 18-65' and 'B' represents 'fully vaccinated', and 'C' represents conditional to go out.
Which of the following is the correct interpretation of the logical connective 'and'?
Which of the following is the correct interpretation of the logical connective 'and'?
Which of the following is a compound proposition?
Which of the following is a compound proposition?
If proposition A is 'The sky is blue' and proposition B is 'Grass is green,' then the compound proposition 'A and B' is true.
If proposition A is 'The sky is blue' and proposition B is 'Grass is green,' then the compound proposition 'A and B' is true.
What is the result of $A \oplus B$ (A XOR B) when A is false and B is true?
What is the result of $A \oplus B$ (A XOR B) when A is false and B is true?
According to the truth table, if A is true and B is false, then $A \rightarrow B$ (if A then B) is true.
According to the truth table, if A is true and B is false, then $A \rightarrow B$ (if A then B) is true.
What is the result of $A \land B$ (A and B) when A is true and B is true?
What is the result of $A \land B$ (A and B) when A is true and B is true?
In a truth table, the symbol '~' represents the logical operator called ______.
In a truth table, the symbol '~' represents the logical operator called ______.
Match the logical operator to the description of its output:
Match the logical operator to the description of its output:
When is $A \rightarrow B$ (If A then B) false?
When is $A \rightarrow B$ (If A then B) false?
Which logical operator has the highest precedence?
Which logical operator has the highest precedence?
The biconditional operator ($A \leftrightarrow B$) is true when A and B have different truth values.
The biconditional operator ($A \leftrightarrow B$) is true when A and B have different truth values.
Which of the following statements accurately describes a tautology in logic?
Which of the following statements accurately describes a tautology in logic?
The statement 'B ∧ ¬B' is a contradiction.
The statement 'B ∧ ¬B' is a contradiction.
Define logical equivalence in your own words.
Define logical equivalence in your own words.
A statement that is neither a tautology nor a contradiction is called a __________.
A statement that is neither a tautology nor a contradiction is called a __________.
Match each type of logical statement with its defining characteristic:
Match each type of logical statement with its defining characteristic:
Which of the following accurately describes a 'contingency'?
Which of the following accurately describes a 'contingency'?
If two compound propositions, P and Q, are logically equivalent, which of the following statements must be true?
If two compound propositions, P and Q, are logically equivalent, which of the following statements must be true?
If a statement is a tautology, then its negation is a contradiction.
If a statement is a tautology, then its negation is a contradiction.
Flashcards
Proposition
Proposition
A declarative sentence that is either true or false, but not both.
Propositional Representation
Propositional Representation
Representing a proposition using a capital letter.
Truth Value: T/1
Truth Value: T/1
Representing a 'true' statement
Truth Value: F/0
Truth Value: F/0
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Law of Identity
Law of Identity
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Law of Excluded Middle
Law of Excluded Middle
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Law of Non-Contradiction
Law of Non-Contradiction
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Statement Analysis
Statement Analysis
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Truth Table
Truth Table
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Compound Proposition
Compound Proposition
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Negation
Negation
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Greater Than (>)
Greater Than (>)
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Less Than (<)
Less Than (<)
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Proposition truth value dependency
Proposition truth value dependency
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Logical Connectives
Logical Connectives
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Conjunction
Conjunction
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Disjunction
Disjunction
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Negation Symbols
Negation Symbols
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Conjunction Symbol
Conjunction Symbol
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Disjunction Symbol
Disjunction Symbol
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Proposition Value
Proposition Value
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Antecedent
Antecedent
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Consequence
Consequence
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Implication
Implication
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Converse
Converse
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Contrapositive
Contrapositive
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Inverse
Inverse
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Biconditional
Biconditional
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Exclusive Or (XOR)
Exclusive Or (XOR)
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Conjunction (A ∧ B)
Conjunction (A ∧ B)
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Disjunction (A ∨ B)
Disjunction (A ∨ B)
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Exclusive Disjunction (A XOR B)
Exclusive Disjunction (A XOR B)
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Implication (A → B)
Implication (A → B)
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Biconditional (A ↔ B)
Biconditional (A ↔ B)
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Precedence of Logical Operators
Precedence of Logical Operators
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Tautology
Tautology
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Contradiction
Contradiction
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Contingency
Contingency
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Logical Equivalence
Logical Equivalence
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A ^ ~A
A ^ ~A
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A v ~A
A v ~A
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Conditional Indicator (→)
Conditional Indicator (→)
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Neither Tautology nor Contradiction
Neither Tautology nor Contradiction
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Study Notes
- A proposition is a statement that can be proven, explained, or discussed, and is either true or false (but not both).
- Capital letters usually denote propositions in tables.
- T represents a true statement.
- F represents a false statement.
- 1 represents true, 0 represents false.
Bases for Propositional Logic
- Scholars in the 18th, 19th, and early 20th centuries considered the "laws of thought" as the basis of all logic.
- Law of Identity: Things are identical to themselves (P = P); water is water.
- Law of Excluded Middle: Every proposition is either true or false, not both, and not neither.
- Law of Non-Contradiction: Correct propositions cannot affirm and deny the same thing.
Truth Table
- Used to keep track of proposition possibilities.
- Each row lists a possible combination of true and false values for each variable, indicating whether the statement is true.
Compound Proposition
- Combining one or more existing propositions creates a compound proposition.
Types of Logical Connectives
- Negation: Uses "not" to state the opposite of a proposition, indicated by the symbol (~) or (-).
- Conjunction: Combines propositions using "and;" true only if both propositions are true, represented by the symbol (^).
- Disjunction: Combines propositions using "or;" true if at least one proposition is true, represented by the symbol (v).
- Implication: "If-then" statements; the first proposition is the antecedent, and the second is the consequence, symbolized by an arrow (→).
- Converse: Reverse of the implication (B → A).
- Contrapositive: Negates and interchanges propositions (~B → ~A).
- Inverse: Negates the propositions (~A → ~B).
- Biconditional: Combines a conditional statement with its converse, using "if and only if," symbolized by (↔); both propositions must have the same truth value (TT or FF) to be true.
- XOR (exclusive or): A version of disjunction that does not allow both propositions to be true simultaneously, symbolized by (⊕).
Truth Table Reference
- Can be used as a guide when working with truth tables.
- Summarizes the truth values for the logical connectives: negation, conjunction, disjunction, exclusive disjunction, implication and biconditional.
Precedence of Logical Operators
- Establishes an order for solving complex propositions.
- From highest to lowest precedence: Negation (~), Conjunction (^), Disjunction (v), Implication (→), and Biconditional (↔).
Tautology
- A statement that is always true, regardless of the truth values of its parts.
Contradiction
- A statement that is always false, regardless of the truth values of its parts.
Contingency
- A statement that is neither a tautology nor a contradiction.
Logical Equivalence
- Two compound propositions with the same truth value in every case, denoted by the symbol ≡.
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Description
Explore the fundamental principles of propositional logic, including the Law of Excluded Middle, the Law of Identity, and the Law of Non-Contradiction. This lesson covers propositions, conjunctions, disjunctions, and how to evaluate their truth values.
