Test 1 Review Flashcards
Which Law of Logical Equivalency?
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
De Morgan’s Laws
State the Identity Laws
p ∧ T ≡ p
p ∨ F ≡ p
What is the order of operations when simplifying complex logical expressions?
Operators:
¬ “not
∨ “or”
➜ “implies”
↔ “if and only if”
∧: “and”
- ¬ “not
- ∧: “and”
- ∨ “or”
- ➜ “implies”
- ↔ “if and only if”
Which Law of Logical Equivalency?
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Associative Laws
State the Logical Equivalences for Each Biconditional Statement
p ↔ q ≡
p ↔ q ≡
p ↔ q ≡
¬( p ↔ q ) ≡
p ↔ q ≡ ( p ➜ q ) ∧ (q ➜ p )
p ↔ q ≡ ¬p ↔ ¬q
p ↔ q ≡ ( p ∧ q ) ∨ (¬p ∧ ¬q)
¬( p ↔ q ) ≡ p ↔ ¬q
State the Idempotent Laws
p ∨ p ≡ p
p ∧ p ≡ p
List 8 common ways to express conditionals in English, p ➜ q, where p is the hypothesis and q is the conclusion.
- if p, then q
- q if p
- q follow from p
- p implies q
- q is necessary for p
- p is sufficient for q
- p only if q
- q unless ¬p
State the Double Negation Law
¬(¬p) ≡ p
State the Converse of “If I study, then I get an A on my exam.”
Converse: if q, then p
q ➜ p
If I get an A on my exam, I studied.
Which Law of Logical Equivalency?
p ∨ ¬p ≡ T
p ∧ ¬p ≡ F
Negation Laws
Which Law of Logical Equivalency?
p ∧ T ≡ p
p ∨ F ≡ p
Identity Laws
State the Converse, Inverse, Contrapostive, and Biconditional of if p, then q.
p ➜ q
Converse: if q, then p
q ➜ p
Inverse: if not p, then not q
¬p ➜ ¬q
Contrapositive: if not q, then not p
¬q ➜ ¬p
Biconditional: p if and only if q
p ↔ q
State the Associative Laws
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
State the Commutative Laws
p ∨ q ≡ q ∨ p
p ∧ q ≡ q ∧ p
State the De Morgan’s Laws
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
State the Absorption Laws
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Which Law of Logical Equivalency?
p ∨ q ≡ q ∨ p
p ∧ q ≡ q ∧ p
Commutative Laws
State the Logical Equivalences for Each Conditional Statement
p ➜ q ≡
p ➜ q ≡
p ∨ q ≡
p ∧ q ≡
¬( p ➜ q ) ≡
( p ➜ q ) ∧ (p ➜ r ) ≡
( p ➜ r ) ∧ (q ➜ r ) ≡
( p ➜ q ) ∨ (p ➜ r ) ≡
( p ➜ r ) ∨ (q ➜ r ) ≡
p ➜ q ≡ ¬p ∨ q
p ➜ q ≡ ¬q ➜ ¬p
p ∨ q ≡ ¬p ➜ q
p ∧ q ≡ ¬(p ➜ ¬q)
¬( p ➜ q ) ≡ p ∧ ¬q
( p ➜ q ) ∧ (p ➜ r ) ≡ p ➜ ( q ∧ r )
( p ➜ r ) ∧ (q ➜ r ) ≡ ( p ∨ q ) ➜ r
( p ➜ q ) ∨ (p ➜ r ) ≡ p ➜ ( q ∨ r )
( p ➜ r ) ∨ (q ➜ r ) ≡ ( p ∧ q ) ➜ r
State the Negation Laws
¬p ∨ ¬p ≡ T
p ∧ ¬p ≡ F
State the Domination Laws
p ∨ T ≡ T
p ∧ F ≡ F
Which Law of Logical Equivalency?
p ∨ T ≡ T
p ∧ F ≡ F
Domination Laws
Which Law of Logical Equivalency?
p ∨ p ≡ p
p ∧ p ≡ p
Idempotent Laws
Which Law of Logical Equivalency?
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Absorption Laws
Which Law of Logical Equivalency?
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Distributive Laws
State the Distributive Laws
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Which Law of Logical Equivalency?
¬(¬p) ≡ p
Double Negation
What is a Tautology?
A logical expression which is always true regardless of the truth values of the variables because it contains all possible outcomes.
p ∨ ¬p
What is a Contradiction?
A logical expression which is always false regardless of the truth value of the variables. It is self-contradicting.
p ∧ ¬p
What is a Contingency?
A logical expression which is neither a Tautology or Contradiction.
p
Modus Tollens
p ➜ q
¬q
———-
∴ ¬p
( ¬q ∧ ( p ➜ q )) ➜ ¬p
Hypothetical Syllogism
p ➜ q
q ➜ r
———-
∴ p ➜ r
(( p ➜ q ) ∧ ( q ➜ r )) ➜ ( p ➜ r )
Disjunctive Syllogism
p v q
¬p
———-
∴ q
(( p v q ) ∧ ¬p ) ➜ q
Resolution
p v q
¬p v r
———-
∴ q v r
(( p v q ) ∧ ( ¬p v r )) ➜ q v r
Addition
∴ p v q
p ➜ ( p v q )
Simplification
∴ p
( p ∧ q ) ➜ p
Conjunction
p
q
———-
∴ p ∧ q
( p ∧ q ) ➜ ( p ∧ q )
Modus Ponens
p ➜ q
p
———-
∴ q
( p ∧ ( p ➜ q)) ➜ q
When True? When False?
∀x∀y P(x, y)
∀y∀x P(x, y)
True:
P(x, y) is true for every pair x, y.
False:
There is a pair x , y for which P(x,y) is false.
When True? When False?
∀x∃y P(x, y)
True:
For every x there is a y for which P(x,y) is true.
False:
There is an x such that P(x,y) is false for every y.
When True? When False?
∃x∀y P(x, y)
True:
There is an x for which P(x,y) is true for every y.
False:
For every x there is a y for which P(x,y) is false.
When True? When False?
∃x∃y P(x, y)
∃y∃x P(x, y)
True:
There is a pair x,y for which P(x,y) is true.
False:
P(x,y) is false for every pair x,y.
Universal Specification / Instantiation (US or UI)
∴ P(c)
♡ for any c in the domain
Explanation: The universal quantifier ∀xP(x) states that P(x) is true for all x in the domain.
Therefore, you can select any c (a particular element in the domain) and conclude that P(c) is true because the statement applies universally.
Universal Generalization (UG)
∴ ∀xP(x)
♡ for an arbitrary c, not a particular one
Explanation: Here, P(c) holds for a specific c, but the conclusion ∀xP(x) requires P(x) to hold for every x in the domain.
This inference is only valid if c is chosen arbitrarily and represents any element in the domain (not a specific, particular one).
If c were a specific element, this conclusion would be invalid.
Existential Specification / Instantiation (ES or EI)
∴ P(c)
♡ for some specific c (unknown)
Explanation: The existential quantifier ∃xP(x) states that there exists at least one x such that P(x) is true.
Form this, you can infer that there is some specific c (though it may not be explicitly identified) such that P(c) holds true.
This c satisfies P(x), but we don’t know exactly which one it is.
Existential Generalization (EG)
∴ ∃xP(x)
♡ finding one c such that P(c)
Explanation: If P(c) holds for a specific element c, you can conclude that ∃xP(x), since you’ve identified at least one instance c in the domain where P(x) is true. This c serves as a witness to the truth of the existential quantifier ∃xP(x).
True or False?
Inverse and converse are logically equivalent.
True
True or False?
It is possible to define disjunction using only negations and conjunction.
True
¬( ¬p ∧ ¬q )
True of False?
Domain-restricted universal quantification is logically equivalent to the unrestricted universal quantification of a conditional.
True
True of False?
Domain-restricted existential quantification is logically equivalent to the unrestricted existential quantification of a conditional.
False
It is equal to the existential quantification of a conjunction.
The negation of a conjunction is the disjunction of the negations.
True
( p ∧ q)
≡ ¬( p ∧ q )
≡ ( ¬p ∨ ¬q )
