Logical Form And Logical Equivalence | Definitions | Compound Statements | Truth Tables
Logical Form And Logical Equivalence
The content of a statement is not the same as the logical form. For instance, consider the 2 following statements:
If Sally wakes up late or if she misses the bus, she will be late for work. Therefore, if Sally arrives at work on time, she did not wake up late and did not miss the bus.
If x is a real number such that x < -2 or x > 2, then x^{2} > 4. Therefore, if x^{2} < 4, then x > -2 and x < 2.
Logical analysis does not help determine the merit of an argument. Instead it helps to analyze the argument's form to determine if the truth of the conclusion follows from the truth of the preceding statements. While the content of the two above statements is different, their logical form is similar.
Let p stand for the statements "Sally wakes up late"
and "x is a real number such that x < -2".
Let q stand for the statements "Sally misses the bus"
and "x is a real number such that x > -2".
Let r stand for the statements "Sally is late for work"
and "x^{2} > 4".
Then the common form for both of the above arguments is:
If p or q, then r.
Therefore, if not r, then not p and not q.
Practice Exercises |
use the symbols (logical
AND), (logical OR),
and ~ (NOT) to build complicated logical statements out of simpler ones.
Let p = "It is hot" and let q = "It is sunny" Then
the statement "It is not hot but it is sunny." can be written
symbolically as:
~ p q |
Practice Exercises |
Truth Table for ~p | Truth Table for pq | Truth Table for pq | |||||
p | ~p | p | q | pq | p | q | pq |
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 |
Practice Exercises |
Two statements are logically equivalent if, and only if, their resulting forms are logically equivalent when identical statement variables are used to represent component statements.
Two statement forms are logically equivalent if, and only if, their resulting truth tables are identical for each variation of statement variables.
p | q | pq | qp |
T | T | T | T |
T | F | F | F |
F | T | F | F |
F | F | F | F |
Other logically equivalent statements include:
(p q) ~(p q) p xor q | Exclusive Or |
p ~(~p) | Double Negation |
To test for logical equivalence of 2 statements, construct a truth table that includes every variable to be evaluated, and then check to see if the resulting truth values of the 2 statements are equivalent.
The negation of a conjunction (logical AND) of 2 statements is logically equivalent to the disjunction (logical OR) of each statement's negation. That sounds like a mouthful, but what it means is that "not (A and B)" is logically equivalent to "not A or not B".
Similarly, the negation of a disjunction of 2 statements is logically equivalent to the conjunction of each statement's negation. Put simply, "not (A or B)" is logically equivalent to "not A and not B". Symbolically, this can be written as
~(pq) ~p ~q | . . . and . . . | ~(pq) ~p ~q |
These two statements are logically equivalent (click here for definition), and this can be verified by using a truth table.
Table of Logical Equivalencies:
The following table can be used to help reduce compound statements to simpler forms. Given statement variables p, q, and r, a tautology t and a contradiction c, the following rules of logic hold:
Commutative | p qq p | p q q p |
Associative | (p q) rp (q r) | (p q) rp (q r) |
Distributive | p (q r)(p q) (p r) | p (q r)(p q) (p r) |
Identity | p tp | p cp |
Negation | p ~pt | p ~pc |
Double Negation | ~(~p)p | |
Idempotent | p pp | p pp |
De Morgan's Laws | ~(p q)~p ~q | ~(p q)~p ~q |
Universal Bound | p tt | p cc |
Absorption | p (p q)p | p (p q)p |
Negations of t and c | ~tc | ~ct |
Logical equivalencies can be used to simplify statement forms, to confirm or disprove an equivalency, to create efficient and logically correct computer programs, or to aid in the design of digital logic circuits.
To simplify an equivalency, start with one side of the equation and attempt to replace sections of it with equivalent expressions. Continue doing this until you have achieved the desired statement form.
Practice Exercises |