CSCI 241 - Ch. 1 Logic
If a compound proposition has n variables, there are 2n rows. The truth table for compound proposition (p ∨ r) ∧ ¬q has 23 = ____ rows.
8
Laws of Propositional Logic p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p
Absorption
p ∴ p ∨ q
Addition
Laws of Propositional Logic ( p ∨ q ) ∨ r ≡ p ∨ ( q ∨ r ) ( p ∧ q ) ∧ r ≡ p ∧ ( q ∧ r )
Associative
Laws of Propositional Logic p ∨ q ≡ q ∨ p p ∧ q ≡ q ∧ p
Commutative
Laws of Propositional Logic p ∧ ¬p ≡ F ¬T ≡ F p ∨ ¬p ≡ T ¬F ≡ T
Complement
Laws of Propositional Logic p → q ≡ ¬p ∨ q p ↔ q ≡ ( p → q ) ∧ ( q → p )
Conditional
p q ∴ p ∧ q
Conjunction
Laws of Propositional Logic ¬( p ∨ q ) ≡ ¬p ∧ ¬q ¬( p ∧ q ) ≡ ¬p ∨ ¬q
De Morgan's
logical equivalences that show how to correctly distribute a negation operation inside a parenthesized expression
De Morgan's laws
p ∨ q ¬p ∴ q
Disjunctive Syllogism
Laws of Propositional Logic p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r )
Distributive
Laws of Propositional Logic p ∧ F ≡ F p ∨ T ≡ T
Domination
Laws of Propositional Logic ¬¬p ≡ p
Double negation
c is an element (arbitrary or particular) P(c) ∴ ∃x P(x)
Existential generalization
∃x P(x) ∴ (c is a particular element) ∧ P(c)
Existential instantiation
p → q q → r ∴ p → r
Hypothetical Syllogism
Laws of Propositional Logic p ∨ p ≡ p p ∧ p ≡ p
Idempotent
Laws of Propositional Logic p ∨ F ≡ p p ∧ T ≡ p
Identity
Nested Quantifiers As a Two Person Game
If the predicate is true after all the variables are set, then the quantified statement is true. If the predicate is false after all the variables are set, then the quantified statement is false.
p p → q ∴ q
Modus Ponens
¬q p → q ∴ ¬p
Modus Tollens
p ∨ q ¬p ∨ r ∴ q ∨ r
Resolution
p ∧ q ∴ p
Simplification
(T/F) A predicate can depend on more than one variable
T
(T/F) A proposition is still a proposition whether its truth value is known to be true, known to be false, unknown, or a matter of opinion
T
(T/F) A single counterexample is sufficient to show that a universally quantified statement is false.
T
(T/F) An existentially quantified statement evaluates to true even if there is more than one element in the domain that causes the predicate to evaluate to true. If the domain is a set of people who attend a meeting and the predicate L(x) indicates whether or not x came late to the meeting, then the statement ∃x L(x) is true if there are one, two or more people who came late.
T
(T/F) De Morgan's law can be applied to logical statements with more than one quantifier. Each time the negation sign moves past a quantifier, the quantifier changes type from universal to existential or from existential to universal
T
(T/F) If an element is introduced for the first time in the proof, the definition is labeled "Element definition" and must specify whether the element is arbitrary or particular.
T
(T/F) If two propositions are logically equivalent, then one can be substituted for the other within a more complex proposition. The compound proposition after the substitution is logically equivalent to the compound proposition before the substitution.
T
(T/F) In order to use a truth table to establish the validity of an argument, a truth table is constructed for all the hypotheses and the conclusion. Each row in which all the hypotheses are true is examined. If the conclusion is true in each of the examined rows, then the argument is valid. If there is any row in which all the hypotheses are true but the conclusion is false, then the argument is invalid.
T
(T/F) It is important to define a new particular element with a new name for each use of existential instantiation within the same logical proof in order to avoid a faulty proof that an invalid argument is valid.
T
(T/F) The logical statement ∀x P(x) is read "for all x, P(x)" or "for every x, P(x)".
T
(T/F) The logical statement ∃x P(x) is read "There exists an x, such that P(x)
T
(T/F) The only way for a conditional statement to be false is if the hypothesis is true and the conclusion is false
T
(T/F) The proposition p → q is false if p is true and q is false; otherwise, p → q is true.
T
(T/F) The proposition p ↔ q is true when p and q have the same truth value and is false when p and q have different truth values.
T
(T/F) The proposition p ∨ q is false only if both p and q are false
T
(T/F) The quantifiers ∀ and ∃ are applied before the logical operations (∧, ∨, →, and ↔) used for propositions. This means that the statement ∀x P(x) ∧ Q(x) is equivalent to (∀x P(x)) ∧ Q(x) as opposed to ∀x (P(x) ∧ Q(x)).
T
(T/F) Two quantified statements (whether they are expressed in English or the language of logic) have the same logical meaning if they have the same truth value regardless of value of the predicates for the elements in the domain.
T
(T/F) Universally and existentially quantified statements can also be constructed from logical operations.
T
(T/F) Using truth tables to establish the validity of an argument can become tedious, especially if an argument uses a large number of variables. Fortunately, some arguments can be shown to be valid by applying rules that are themselves arguments that have already been shown to be valid. The laws of propositional logic can also be used in establishing the validity of an argument.
T
(T/F) p ∧ q is true if both p is true and q is true. p ∧ q is false if p is false, q is false, or both are false.
T
(T/F) ∀x P(x) is a proposition because it is either true or false
T
(T/F) ∀x P(x) is true if and only if P(n) is true for every n in the domain.
T
(T/F) ∃x P(x) is a proposition because it is either true or false
T
(T/F) ∃x P(x) is true if and only if P(n) is true for at least one value n in the domain of variable x.
T
c is an arbitrary element P(c) ∴ ∀x P(x)
Universal generalization
c is an element (arbitrary or particular) ∀x P(x) ∴ P(c)
Universal instantiation
The proposition p ∧ q is read "p ____ q"
and
An ________ element of a domain has no special properties other than those shared by all the elements of the domain
arbitrary
An _________ is a sequence of propositions, called hypotheses, followed by a final proposition, called the conclusion.
argument
(T/F) If parentheses are not used to explicitly indicate the order in which the operations should be applied, then ∧, ∨, and ¬ should be applied _______ → or ↔
before
If p and q are propositions, the proposition "p if and only if q" is expressed with the ____________
biconditional operation
The _____________ is denoted p ↔ q
biconditional operation
The variable x in the statement ∀x P(x) is a _________ because the variable is bound to a quantifier.
bound variable
created by connecting individual propositions with logical operations
compound proposition
An argument is a sequence of propositions, called hypotheses, followed by a final proposition, called the __________.
conclusion
In p → q, the proposition q is called the ________
conclusion
denoted with the symbol →
conditional operation
A compound proposition that uses a conditional operation
conditional proposition
A conditional proposition expressed in English is sometimes referred to as a ___________
conditional statement
the __________ operation is denoted by ∧
conjunction
A compound proposition is a _________ if the proposition is always false, regardless of the truth value of the individual propositions that occur in it.
contradiction
The _________ of p → q is ¬q → ¬p
contrapositive
The _________ of p → q is q → p
converse
A __________ for a universally quantified statement is an element in the domain for which the predicate is false.
counterexample
Propositions are typically __________ sentences
declarative
The __________ operation is denoted by ∨
disjunction
An "arbitrary element" means nothing is assumed about the element other than the fact that it is in the ________
domain
The _______ of a variable in a predicate is the set of all possible values for the variable.
domain
A value that can be plugged in for variable x is called an _______ of the domain of x.
element
If s and r are two compound propositions, the notation s ≡ r is used to indicate that r and s are logically _________
equivalent
The symbol ∃ is an __________
existential quantifier
the statement ∃x P(x) is called a ______________
existentially quantified statement
An argument can be shown to be invalid by showing an assignment of truth values to its variables that makes all the hypotheses true and the conclusion ________
false
The _______ of an argument expressed in English is obtained by replacing each individual proposition with a variable.
form
A variable x in the predicate P(x) is called a _________ because the variable is free to take on any value in the domain
free variable
The rules existential and universal ____________ replace an element of the domain with a quantified variable.
generalization
An argument is a sequence of propositions, called ___________, followed by a final proposition, called the conclusion.
hypotheses
Elements of the domain can be defined in a _________ of an argument
hypothesis
In p → q, the proposition p is called the _______
hypothesis
The proposition p → q is read "____ p _____ q"
if, then
The rules existential and universal ___________ replace a quantified variable with an element of the domain.
instantiation
The __________ of p → q is ¬p → ¬q.
inverse
the study of formal reasoning
logic
combines propositions using a particular composition rule
logical operation
The validity of an argument can be established by applying the rules of inference and laws of propositional logic in a __________
logical proof
Two compound propositions are said to be __________ if they have the same truth value regardless of the truth values of their individual propositions
logically equivalent
acts on just one proposition and has the effect of reversing the truth value of the proposition
negation operation
A logical expression with more than one quantifier that bind different variables in the same predicate is said to have ___________
nested quantifiers
The negation of proposition p is denoted ¬p and is read as "____ p"
not
For compound propositions, what is the order of operations in the absence of parentheses?
not, and, or
The statement ∃x P(x) asserts that P(x) is true for at least ______ possible value for x in its domain
one
The proposition p ∨ q is read "p ____ q"
or
A ________ element of the domain may have properties that are not shared by all the elements of the domain
particular
A logical statement whose truth value is a function of one or more variables
predicate
A statement with no free variables is a ________ because the statement's truth value can be determined.
proposition
If all the variables in a predicate are assigned specific values from their domains, then the predicate becomes a ________ with a well defined truth value. Another way to turn a predicate into a proposition is to use a quantifier.
proposition
a statement that is either true or false
proposition
A logical proof of an argument is a sequence of steps, each of which consists of a _______ and a __________
proposition, justification
A logical statement that includes a universal or existential quantifier is called a _________
quantified statement
If a predicate has more than one variable, each variable must be bound by a separate ___________
quantifier
If all the variables in a predicate are assigned specific values from their domains, then the predicate becomes a proposition with a well defined truth value. Another way to turn a predicate into a proposition is to use a ________.
quantifier
The universal and existential quantifiers are generically called _________
quantifiers
A compound proposition is a ________ if the proposition is always true, regardless of the truth value of the individual propositions that occur in it.
tautology
An argument is valid if the conclusion is _______ whenever the hypotheses are all true, otherwise the argument is invalid
true
The language of logic allows us to formally establish the truth of logical statements, assuming that a set of hypotheses is _______
true
The statement ∀x P(x) asserts that P(x) is ______ for every possible value for x in its domain.
true
shows the truth value of a compound proposition for every possible combination of truth values for the variables contained in the compound proposition
truth table
a value indicating whether the proposition is actually true or false
truth value
In reasoning whether a quantified statement is true or false, it is useful to think of the statement as a ________ in which two players compete to set the statement's truth value
two player game
The symbol ∀ is a __________
universal quantifier
the statement ∀x P(x) is called a _________
universally quantified statement
De Morgan's Law for Quantified Statements
¬∀x P(x) ≡ ∃x ¬P(x) ¬∃x P(x) ≡ ∀x ¬P(x)
