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Questions and Answers
Which of the following is NOT a logical connective?
Which of the following is NOT a logical connective?
The statement 'Questions and commands are examples of statements/propositions' is true.
The statement 'Questions and commands are examples of statements/propositions' is true.
False (B)
What is the primary difference between deductive and inductive reasoning?
What is the primary difference between deductive and inductive reasoning?
Deductive reasoning goes from general to specific, while inductive reasoning goes from specific to general.
A ________ is a declarative sentence that is either true or false, but not both.
A ________ is a declarative sentence that is either true or false, but not both.
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Match the following logical connectives with their descriptions:
Match the following logical connectives with their descriptions:
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Under what condition(s) is the biconditional statement (p ↔ q) considered true?
Under what condition(s) is the biconditional statement (p ↔ q) considered true?
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A tautology is a statement that is always false.
A tautology is a statement that is always false.
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When is considered false the conditional (p => q)?
When is considered false the conditional (p => q)?
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Flashcards
Logic
Logic
The study of reasoning and argumentation, structuring valid arguments.
Statement (Proposition)
Statement (Proposition)
A declarative sentence that is either true or false, but not both.
Non-Statements
Non-Statements
Expressions that are not true or false, like questions and commands.
Negation (¬p)
Negation (¬p)
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Conjunction (p ∧ q)
Conjunction (p ∧ q)
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Deductive Reasoning
Deductive Reasoning
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Logical Equivalence
Logical Equivalence
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Tautology
Tautology
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Study Notes
Logic: Definitions and Importance
- Logic is the study of reasoning and argumentation.
- It structures valid arguments and differentiates correct from incorrect reasoning.
- It's essential in mathematics, computer science, philosophy, and everyday decision-making.
Statements and Propositions
- A statement (proposition) is either true or false, but not both.
- It's a declarative sentence.
- Examples: "2 + 2 = 4" (True) and "The sun is blue" (False).
- Non-statements include questions, commands, and opinions (e.g., "How are you?").
Logical Connectives
- Logical connectives combine or modify statements.
Types of Logical Connectives
- Negation (¬p): The opposite of a statement. If p is true, ¬p is false.
- Conjunction (p ∧ q): "And" statement. True only when both p and q are true.
- Disjunction (p ∨ q): "Or" statement. True if at least one of p or q is true.
- Implication (p → q): "If p, then q." False only if p is true and q is false.
- Biconditional (p ↔ q): "p if and only if q." True when both p and q have the same truth value.
- Exclusive Or (p ⊕ q): True if and only if p and q have different truth values.
- NAND (p | q): True unless both p and q are true.
- NOR (p ↓ q): True only if both p and q are false.
- Conditional (p => q): False only when p is true and q is false, True if p is false.
Truth Tables and Reasoning
- Truth tables determine the truth value of compound statements.
- Example for p → q:
- p | q | p → q
- T | T | T
- T | F | F
- F | T | T
- F | F | T
Types of Reasoning
- Deductive Reasoning: Moves from general to specific; conclusions logically follow premises.
- Example: "All men are mortal; Socrates is a man; therefore, Socrates is mortal."
- Inductive Reasoning: Moves from specific to general; conclusions based on patterns or observations.
- Example: "The sun has risen every day, so it will rise tomorrow."
Logical Equivalence and Proofs
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Logical Equivalence: Two statements always have the same truth value.
-
Denoted as p ≡ q ("p is logically equivalent to q").
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Statements are equivalent if their truth tables match.
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Methods to Determine Logical Equivalence:
- Construct truth tables for both statements.
- Compare columns. If they match, the statements are equivalent.
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Types of Statements:
- Tautology: Always true.
- Contradiction: Always false.
- Contingency: Neither always true nor always false.
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Proof Techniques:
- Converse: q → p
- Inverse: ~p → ~q
- Contraposition: ~q → ~p
- Conditional Proof: Deduce the consequent from an additional premise.
- Indirect Proof: Reduce to absurdity by negating the conclusion.
Fallacies
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Fallacies are errors in reasoning, leading to invalid arguments.
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Types of Fallacies:
- Affirming the Consequent: p → q, q therefore p
- Denying the Antecedent: p → q, ~p therefore ~q
Arguments
- An argument is a series of statements that persuade.
- Deduce conclusions to establish validity.
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Description
Explore the fundamental concepts of logic, including statements and propositions, as well as various logical connectives and their significance. This quiz will help you understand how logical reasoning is applied in mathematics, computer science, and philosophy.
