Propositional Logic Flashcards
T ^ T =
T ^ F =
F ^ T =
F ^ F =
T
F
F
F
Logical equivalency
Two propositions, say P and Q, are logically equivalent if P ⟺ Q is a TAUTOLOGY. In this case, the biconditional of P and Q (i.e. P ⟺ Q) is called a logical equivalency.
What is the no. of rows in a truth table for 3 propositions?
2^3 = 8
Premise
Proposition (atomic/compound) to the left of the implication operator. Also called antecedent or hypothesis.
Exclusive-Or
A compound proposition where ONLY the “XOR” operator is applied on 2 or more propositions.
Commutative properties (logical equivalencies)
P v Q ⟺ Q v P
P ^ Q ⟺ Q ^ P
Implication Law
(P ⇒ Q) ⟺ (¬P v Q)
Consequence
Proposition (atomic/compound) to the right of the implication operator. Also called conclusion.
Contingency
A compound proposition (statement) that is NEITHER a TAUTOLOGY NOR a CONTRADICTION.
Which words does conjunction (^) denote?
“and”, “but”, “also”, “as well as”
How is an implication of 2 statements/propositions, say P and Q written in English, where P ⇒ Q?
“if P then Q”
“If P, Q”
“P implies Q”
“Whenever P, Q”
“P is sufficient for Q”
“Q is necessary for P”
“Not P unless Q”
“P only if Q”
“Q if P”
“Q whenever P”
T ⇒ T
T ⇒ F
F ⇒ T
F ⇒ F
T
F
T
T
What is the no. of rows in a truth table for an n no. of propositions?
2^n
Negation operator
¬ or ~
Biconditional of P and Q
A compound proposition which is TRUE precisely when either P and Q are BOTH TRUE, or when P and Q are BOTH FALSE.
Implication
A compound proposition where ONLY the “IMPLIES/IF-THEN” (conditional) operator is applied on 2 propositions.
Conjunction
A compound proposition where ONLY the “AND” operator is applied on 2 or more propositions.
Biconditional symbol
↔ or ⟺
Conjunction symbol
(^)
Conclusion
Proposition (atomic/compound) to the right of the implication operator. Also called consequence.
Difference between disjunction/OR/v and or-exclusive/XOR/⊕ ?
In a disjunction of 2 statements, the truth value is “true” if both statements are true, but an exclusive-or of 2 statements is false if both statements are true.
truth value
The value of a proposition; it can be either “true” or “false”
Logical equivalency between Implication and Contrapositive
(P ⇒ Q) ⟺ (¬Q ⇒ ¬P)
T ⊕ T =
T ⊕ F =
F ⊕ T =
F ⊕ F =
F
T
T
F
DeMorgan’s Laws (logical equivalencies)
¬(P ^ Q) ⟺ ¬P v ¬Q
¬(P v Q) ⟺ ¬P ^ ¬Q
Tautology
A compound proposition (statement) that is ALWAYS TRUE IRRESPECTIVE of the TRUTH VALUES of the propositions in it.
compound proposition
A proposition that is a combination of two or more propositions through logical operators such as “and”, “or”, “not”, etc.
Order of operations (highest to lowest precedence)
Negation, Conjunction, Disjunction, Exclusive-or, Implication, Biconditional
Disjunction symbol
V
Associative properties (logical equivalencies)
(P v Q) v R ⟺ P v (Q v R)
(P ^ Q) ^ R ⟺ P ^ (Q ^ R)
What is the no. of rows in a truth table for 4 propositions?
2^4 = 16
proposition
A statement that can only be either true or false.
~T –>
~F –>
F
T
T v T =
T v F =
F v T =
F v F =
T
T
T
F
Converse of P ⇒ Q
Q ⇒ P
How is a biconditional of 2 statements/propositions, say P and Q written in English?
“P if and only if Q”, “Q if and only if P”, “P iff Q”, “Q iff P”, “P is necessary and sufficient for Q”, “Q is necessary and sufficient for P”, “if P then Q, and conversely”, “if Q then P, and conversely”.
Truth Table
A table that displays the possible values for a proposition or propositions as well as the results which followed when operators are applied.
Inverse of P ⇒ Q
¬P ⇒ ¬Q
Implication symbol
⇒ or →
Contrapositive of P ⇒ Q
¬Q ⇒ ¬P
How is a biconditional of 2 statements/propositions, say P and Q written in English?
“P if and only if Q”, “Q if and only if P”, “P iff Q”, “Q iff P”, “P is necessary and sufficient for Q”, “Q is necessary and sufficient for P”, “if P then Q, and conversely”, “if Q then P, and conversely”.
Exclusive-Or symbol
⊕
Antecedent
Proposition (atomic/compound) to the left of the implication operator. Also called premise or hypothesis.
Hypothesis
Proposition (atomic/compound) to the left of the implication operator. Also called premise or antecedent.
Distributive properties (logical equivalencies)
P v (Q ^ R) ⟺ (P v Q) ^ (P v R)
P ^ (Q v R) ⟺ (P ^ Q) v (P ^ R)
Double negation property (a logical equivalency)
¬ (¬ P) ⟺ P is a tautology.
Contradiction
A compound proposition (statement) that is ALWAYS FALSE IRRESPECTIVE of the TRUTH VALUES of the propositions in it.
Logical equivalency between Inverse and Converse
(Q ⇒ P) ⟺ (¬P ⇒ ¬Q)
T ⟺ T
T ⟺ F
F ⟺ T
F ⟺ F
T
F
F
T
Disjunction
A compound proposition where ONLY the “OR” operator is applied on 2 or more propositions.
