Glossary

2.1 Statements and Quantifiers

4 min read•june 18, 2024

Logical statements form the foundation of mathematical reasoning. They're declarative sentences that are either true or false, but never both. Understanding how to identify and analyze these statements is crucial for building strong arguments and solving complex problems.

Logical connectives allow us to combine simple statements into more complex ones. By using symbols like (and), (or), and (if-then), we can create compound statements and explore their truth values using truth tables.

Logical Statements and Truth Values

Identify logical statements and their truth values in everyday language and symbolic form

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  • Logical statements are declarative sentences that are either true or false, but not both
    • "The sky is blue" is a because it can be either true or false (example)
    • "What time is it?" is not a logical because it is a question, not a declarative sentence (non-example)
  • Represent logical statements using symbols
    • Use lowercase letters (p, q, r) to represent simple statements
    • Combine simple statements into compound statements using logical connectives (∧, ∨, , →, ↔)
  • Determine the of a logical statement, which is either true (T) or false (F)
    • Simple statements' truth values are determined by their correspondence to reality
    • Compound statements' truth values depend on the truth values of their component statements and the logical connectives used
    • Use truth tables to systematically analyze the truth values of compound statements

Negation of simple statements

  • forms a new statement that is true when the original statement is false, and false when the original statement is true
    • Denote the negation of a simple statement pp by ¬p¬p, read as "not pp"
      • If pp is "The sky is blue," then ¬p¬p is "The sky is not blue" (example)
    • Form the negation of a compound statement by negating each component statement and replacing the logical connectives with their negations
      • The negation of p∧qp ∧ q is ¬p∨¬q¬p ∨ ¬q (example)
  • Express the quantity or extent of a statement's subject using quantifiers
    • Read the [∀](https://www.fiveableKeyTerm:∀)[∀](https://www.fiveableKeyTerm:∀) as "for all" or "for every," indicating that a statement holds for every element in a set
      • ∀x(x>0)∀x(x > 0) is read as "for all xx, xx is greater than 0" (example)
    • Read the [∃](https://www.fiveableKeyTerm:∃)[∃](https://www.fiveableKeyTerm:∃) as "there exists" or "for some," indicating that a statement holds for at least one element in a set
      • ∃x(x<0)∃x(x < 0) is read as "there exists an xx such that xx is less than 0" (example)
  • Negate a quantified statement by changing the and negating the predicate
    • The negation of ∀xP(x)∀x P(x) is ∃x¬P(x)∃x ¬P(x)
      • The negation of "All dogs are friendly" is "There exists a dog that is not friendly" (example)
    • The negation of ∃xP(x)∃x P(x) is ∀x¬P(x)∀x ¬P(x)
      • The negation of "Some cats are black" is "All cats are not black" (example)

Construction of inductive arguments

  • use specific examples or to draw a general
    • Premises provide evidence or support for the but do not guarantee its truth
    • The conclusion is a generalization that goes beyond the specific examples in the premises
  • Use quantifiers in the premises and conclusion of an
    • Premises may use the existential to provide specific examples
      • "Dog 1 is friendly," "Dog 2 is friendly," "Dog 3 is friendly" (example)
    • The conclusion may use the universal quantifier to make a generalization
      • "All dogs are friendly" (example)
  • Evaluate the strength of an inductive argument based on the quality and quantity of the premises
    • More diverse and representative examples in the premises strengthen the argument
    • More examples in the premises strengthen the argument
  • Classify inductive arguments as strong or weak, not valid or invalid
    • Strong inductive arguments have a high probability of a true conclusion given true premises
    • Weak inductive arguments have a low probability of a true conclusion given true premises

Logical Connectives and Compound Statements

  • Conditional statements (if-then statements) express a relationship between two statements
    • The statement "If p, then q" is denoted as p → q
    • The truth value of a is false only when the antecedent (p) is true and the consequent (q) is false
  • Biconditional statements (if and only if statements) express a two-way relationship between statements
    • The statement "p if and only if q" is denoted as p ↔ q
    • A is true when both components have the same truth value
  • Analyze compound statements using to determine their logical properties
    • A is a compound statement that is always true, regardless of the truth values of its components
    • A is a compound statement that is always false, regardless of the truth values of its components

Key Terms to Review (35)

→: The symbol '→' represents the logical conditional, which indicates that if one statement (the antecedent) is true, then another statement (the consequent) must also be true. This relationship is fundamental in logical reasoning and helps establish implications between statements, making it essential for understanding the structure of arguments and proofs.
∀: The symbol ∀ represents the universal quantifier in logic and mathematics. It is used to express that a certain property or statement applies to all elements within a specified set or domain. This notation is fundamental in forming statements that involve universality, allowing mathematicians and logicians to make broad claims about entire groups without listing each individual case.
∃: The symbol ∃ is known as the existential quantifier, used in logic and mathematics to assert that there exists at least one element in a specified set that satisfies a given property. It plays a critical role in forming statements that express the existence of certain conditions or entities within mathematical contexts.
¬: The symbol ¬ represents the logical negation operation, which is used to reverse the truth value of a given statement. When applied to a statement, if the statement is true, the negation makes it false, and vice versa. This fundamental concept plays a critical role in understanding logical reasoning, especially in creating compound statements and evaluating the truth of various propositions.
∧: The symbol ∧ represents the logical conjunction operator, also known as 'AND,' used to connect two statements. When two statements are combined with ∧, the result is true only if both statements are true. This operator plays a crucial role in constructing compound statements and analyzing the truth values of logical expressions.
∨: The symbol ∨ represents the logical operation known as 'disjunction,' which connects two statements and yields a true value if at least one of the statements is true. This operation is fundamental in constructing compound statements, enabling the combination of simple statements into more complex ones while determining the overall truth value based on individual components.
Biconditional Statement: A biconditional statement is a logical statement that connects two propositions with the phrase 'if and only if', indicating that both statements are true or both are false simultaneously. This concept is crucial in understanding the equivalence between statements and their conditions, making it a fundamental part of constructing compound statements and analyzing truth values through truth tables.
Conclusion: A conclusion is the statement that logically follows from the premises in an argument. It is the final part of a logical progression based on given statements or assumptions.
Conclusion: A conclusion is the statement or assertion that follows logically from the premises or assumptions in an argument or a logical expression. It serves as the result of deductive reasoning, encapsulating what can be inferred from given statements and quantifiers, and is essential in constructing logical proofs and arguments.
Conditional statement: A conditional statement is a logical statement that has the form 'if P, then Q', where P is called the hypothesis and Q is the conclusion. This type of statement establishes a relationship between two propositions and is fundamental in constructing more complex logical arguments, as well as in understanding how different statements interact with one another.
Conjunction: A conjunction is a logical connective that combines two or more statements into a single compound statement, which is true only when all the individual statements it connects are true. This concept is fundamental in understanding how to build complex logical expressions and analyze their truth values, especially in the context of logical reasoning and mathematical proofs.
Contradiction: A contradiction is a logical statement that asserts two or more propositions that cannot all be true at the same time. This concept is fundamental in logic, as identifying contradictions helps in evaluating the validity of arguments and statements. It often appears when dealing with quantifiers, compound statements, and truth tables, making it essential for understanding logical relationships and the structure of arguments.
Disjunction: A disjunction is a compound statement formed by combining two statements with the word 'or'. It is true if at least one of the statements is true.
Disjunction: Disjunction is a logical operation that connects two statements with the word 'or,' creating a compound statement that is true if at least one of the individual statements is true. This concept is essential for understanding how to combine statements logically, analyze their validity, and evaluate conditions in mathematical reasoning.
Existential Quantifier: The existential quantifier is a symbol used in logic and mathematics to express that there exists at least one element in a given set for which a certain property holds true. This quantifier, often denoted by the symbol ∃, plays a crucial role in forming logical statements that assert the existence of elements meeting specific conditions.
Implication: Implication is a logical relationship between two statements, where the truth of one statement (the antecedent) leads to the truth of another statement (the consequent). This concept is crucial in understanding how statements interact, especially in forming compound statements that combine multiple assertions. Implications help in reasoning and deriving conclusions from given premises, allowing for more complex logical expressions and arguments.
Inductive Argument: An inductive argument is a form of reasoning where the premises are viewed as supplying strong evidence for the truth of the conclusion. This type of argument makes generalizations based on specific observations or instances, leading to a conclusion that is probable but not guaranteed. Inductive arguments often involve statements and quantifiers, as they rely on patterns and trends observed in data or experiences.
Inductive arguments: Inductive arguments are reasoning processes that involve drawing general conclusions from specific observations or examples. They do not guarantee the truth of the conclusion but suggest that it is likely based on the premises.
Logical connective: A logical connective is a symbol or word used to connect two or more statements in a way that creates a new statement whose truth value depends on the truth values of the original statements. Logical connectives play a crucial role in forming compound statements and analyzing their validity through various logical structures. They help to establish relationships between statements such as conjunction, disjunction, and negation, making it easier to evaluate the overall truth of complex expressions.
Logical statement: A logical statement is a declarative sentence that is either true or false but not both. It forms the basic building block of logical reasoning and propositions.
Negation: Negation is the logical operation that takes a statement and turns it into its opposite. When we negate a statement, we assert that the original statement is false. This concept is crucial for understanding how to analyze statements, particularly when dealing with quantifiers, compound statements, and truth values.
Negation of a logical statement: A negation of a logical statement is the opposite of the original statement, often formed by adding 'not.' It changes a true statement to false and vice versa.
Predicate logic: Predicate logic is a formal system in mathematical logic that extends propositional logic by incorporating quantifiers and predicates, allowing for a more detailed expression of statements about objects and their properties. It focuses on the structure of logical sentences that involve quantifiers like 'for all' and 'there exists,' enabling deeper analysis and understanding of logical relationships.
Premise: A premise is a statement or proposition that serves as the foundation for a logical argument or reasoning. It provides the initial assumptions or evidence upon which conclusions are drawn. Understanding premises is essential for analyzing the structure of arguments and determining their validity, as they set the stage for what follows in the reasoning process.
Premises: Premises are statements or propositions that provide the basis for an argument's conclusion. They are assumed to be true within the context of the argument and support the logical derivation of the conclusion.
Quantifier: A quantifier is a logical operator that specifies the quantity of specimens in the domain of discourse that satisfy an open formula. The two most common quantifiers are the universal quantifier and the existential quantifier.
Quantifier: A quantifier is a logical construct used in mathematics and logic to specify the quantity of elements in a given set that satisfy a certain property. It helps to express statements about 'how many' or 'which' elements of a domain meet specific criteria, allowing for more precise communication in logical expressions.
Statement: A statement is a declarative sentence that is either true or false, but not both. In mathematics and logic, statements serve as the building blocks for more complex arguments and reasoning, allowing for the formulation of proofs and the application of logical principles. Understanding statements is crucial for working with quantifiers, which express the extent to which a property holds for elements within a certain set.
Symbolic form: Symbolic form is a way of representing logical statements using symbols and variables. It simplifies complex statements and makes them easier to analyze mathematically.
Tautology: A tautology is a logical statement that is true in every possible interpretation. It is a formula or assertion that cannot be false regardless of the truth values of its components.
Tautology: A tautology is a statement that is always true, regardless of the truth values of its components. This concept is essential in understanding logical reasoning and truth conditions, as it helps identify statements that remain valid under any circumstance. Tautologies play a significant role in constructing compound statements, creating truth tables, and establishing equivalent statements, as they ensure consistency in logical deductions.
Truth table: A truth table is a mathematical table used to determine if a logical expression is true or false under all possible interpretations. It lists all possible combinations of inputs and their corresponding output values for the expression.
Truth Table: A truth table is a mathematical table used to determine the truth values of a logical expression based on the possible combinations of truth values for its components. It provides a systematic way to evaluate complex statements and their relationships, which is essential for understanding how different logical operations interact with each other.
Truth value: Truth value refers to the classification of a statement as either true or false. This concept is fundamental in logic and mathematics, as it allows for the evaluation of statements and their validity. Understanding truth values is crucial when working with logical statements and quantifiers, enabling one to determine the overall truth of complex propositions through systematic analysis.
Universal Quantifier: The universal quantifier is a logical symbol used to indicate that a statement applies to all elements within a particular set. It is commonly represented by the symbol $$ orall$$ and is often used in mathematical expressions to assert that a given property holds true for every member of a specified group. This concept is fundamental in understanding logical statements and how they can be evaluated for their truth across different contexts.
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