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 x² > 4. Therefore, if x² < 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 "x² > 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.

D E F I N I T I O N S

Argument:	a sequence of statements aimed at demonstrating the truth of a statement or assertion.
Statement:	a sentence that is either true or false, but not both. It is also called a proposition.
Negation:	if p is a statement variable, the negation of p is "not p", denoted by ~p. If p is true, then ~p is false.
Conjunction:	if p and q are statement variables, the conjunction of p and q is "p and q", denoted pq.  A conjunction is true only when both variables are true. If 1 or both variables are false, pq is false.
Disjunction:	if p and q are statement variables, the disjunction of p and q is "p or q", denoted pq.  A disjunction is true if one or both variables are true.  pq is false only if both variables are false.
Tautology:	A statement form which is always true. The truth does not rely upon the values of the individual statements substituted for the statement variables, but upon the logical structure of the statement itself. (i.e., You will get an A in this class, or you will not.).
Contradiction:	A statement form which is always false. Like a tautology, the falsity does not lie in the individual statement variables, but in the logical structure of the statement itself. (i.e., I always tell lies.).

~ p  q

However, statements must have well defined truth values - they must be either true or false. The truth of a statement can be expressed by a Truth Table.  A truth table for a given statement displays the resulting truth values for various combinations of truth values for the variables.  The truth of a compound statement can be logically derived by using the known truth values for various parts of a statement.

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

Logical Equivalence:

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.

pq and qp have the same truth values, so they are logically equivalent.

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.

DeMorgan's Laws

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.

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