Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo

Key Concepts

2.1 Statements and Quantifiers

  • Logical statements have the form of a complete sentence and make claims that can be identified as true or false.
  • Logical statements are represented symbolically using a lowercase letter.
  • The negation of a logical statement has the opposite truth value of the original statement.
  • Be able to
    • Determine whether a sentence represents a logical statement.
    • Write and translate logical statements between words and symbols.
    • Negate logical statements, including logical statements containing quantifiers of all, some, and none.

2.2 Compound Statements

  • Logical connectives are used to form compound logical statements by using words such as and, or, and if …, then.
  • A conjunction is a compound logical statement formed by combining two statements with the words “and” or “but.” If the two independent clauses are represented by pp and qq, respectively, then the conjunction is written symbolically as pqpq. For the conjunction to be true, both pp and qq must be true.
  • A disjunction joins two logical statements with the or connective. In, logic or is inclusive. For an or statement to be true at least one statement must be true, but both may also be true.
  • A conditional statement has the form if pp, then qq, where pp and qq are logical statements. The only time the conditional statement is false is when pp is true, and qq is false.
  • The biconditional statement is formed using the connective if and only ifif and only if for the biconditional statement to be true, the true values of pp and qq, must match. If pp is true then qq must be true, if pp is false, then qq must be false.
  • Translate compound statements between words and symbolic form.
    Connective Symbol Name
    and
    but
    conjunction
    or disjunction, inclusive or
    not ~ negation
    if , then implies conditional, implication
    if and only if biconditional
  • The dominance of connectives explains the order in which compound logical statements containing multiple connectives should be interpreted.
  • The dominance of connectives should be applied in the following order
    • Parentheses
    • Negations
    • Disjunctions/Conjunctions, left to right
    • Conditionals
    • Biconditionals
    A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last.
    Figure 2.18

2.3 Constructing Truth Tables

  • Determine the true values of logical statements involving negations, conjunctions, and disjunctions.
    • The negation of a logical statement has the opposite true value of the original statement.
    • A conjunction is true when both pp and qq are true, otherwise it is false.
    • A disjunction is false when both pp and qq are false, otherwise it is true.
  • Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and disjunctions.
    Negation Conjunction (AND) Disjunction (OR)
    pp ~p~p pp qq pqpq pp qq pqpq
    T F T T T T T T
    F T T F F T F T
    F T F F T T
    F F F F F F
  • A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions.

2.4 Truth Tables for the Conditional and Biconditional

  • The conditional statement, if pp then qq, is like a contract. The only time it is false is when the contract has been broken. That is, when pp is true, and qq is false.
    Conditional
    pp qq pqpq
    T T T
    T F F
    F T T
    F F T
  • The biconditional statement, pp if and only if qq, it true whenever pp and qq have matching true values, otherwise it is false.
    Biconditional
    pp qq pqpq
    T T T
    T F F
    F T F
    F F T
  • Know how to construct truth tables involving conditional and biconditional statements.
  • Use truth tables to analyze conditional and biconditional statements and determine their validity.

2.5 Equivalent Statements

  • Two statements pp and qq are logically equivalent if the biconditional statement, pqpq is a valid argument. That is, the last column of the truth table consists of only true values. In other words, pqpq is a tautology. Symbolically, pp is logically equivalent to qq is written as: pq.pq.
  • A logical statement is a tautology if it is always true.
  • To be valid a local argument must be a tautology. It must always be true.
  • Know the variations of the conditional statement, be able to determine their truth values and compose statements with them.
  • The converse of a conditional statement, if pp then qq, is the statement formed by interchanging the hypothesis and conclusion. It is the statement if qq then pp.
  • The inverse of a conditional statement if formed by negating the hypothesis and the conclusion of the conditional statement.
  • The contrapositive negates and interchanges the hypothesis and the conclusion.
    Conditional Contrapositive Converse Inverse
    pp qq ~p~p ~q~q pqpq ~q~p~q~p qpqp ~p~q~p~q
    T T F F T T T T
    T F F T F F T T
    F T T F T T F F
    F F T T T T T T
  • The conditional statement is logically equivalent to the contrapositive.
  • The converse is logically equivalent to the inverse.
  • Know how to construct and use truth tables to determine whether statements are logically equivalent.

2.6 De Morgan’s Laws

  • De Morgan’s Law for the negation of a disjunction states that, ~(pq)~(pq) is logically equivalent to ~p~q.~p~q.
  • De Morgan’s Law The negation of a conjunction states that, ~(pq)~p~q.~(pq)~p~q.
  • Use De Morgan’s Laws to negate conjunctions and disjunctions.
  • The negation of a conditional statement, if pp then qq is logically equivalent to the statement pp and not qq. Use this property to write the negation of conditional statements.
  • Use truth tables to evaluate De Morgan’s Laws.

2.7 Logical Arguments

  • A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true.
  • The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, pqpq is true and the hypothesis, pp is true, then the conclusion qq must also be true.
    Law of Detachment
    Premise: pqpq
    Premise: pp
    Conclusion: q q
  • Know how to apply the law of detachment to determine the conclusion of a pair of statements.
  • The law of denying the consequent is a valid form of a conditional argument that asserts that if both the conditional, pqpq is true and the negation of the conclusion, ~q~q is true, then the negation of the hypothesis ~p~p must also be true.
    Law of Denying the Consequent
    Premise: pqpq
    Premise: ~q~q
    Conclusion: ~p ~p
  • Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements.
  • The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form, pqpq and qrqr, then it follows that pr.pr.
    Chain Rule for Conditional Arguments
    Premise: pqpq
    Premise: qrqr
    Conclusion: pr pr
  • Know how to apply the chain rule to determine valid conclusions for pairs of true statements.
Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/contemporary-mathematics/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/contemporary-mathematics/pages/1-introduction
Citation information

© Jul 25, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.