Section 1.2: Propositional Equivalences
In the process of reasoning, we often replace
a known statement with an equivalent
statement that more closely addresses the
result that we are trying to establish. To
ensure that this process produces a valid
mathematical argument, we must be able to
verify that the statement we are replacing
does indeed have the very same meaning as
the one we replace it with. Two such
statements are said to be logically equivalent.
Tautologies and Contradictions
Def: A compound proposition that is always true,
regardless of the assignment of truth values to its
component propositions, is called a tautology. A
compound proposition that is always false is called a
contradiction. A proposition that is neither a
tautology nor a contradiction is called a contingency.
Ex:
1. “p  p” is a tautology.
[All T’s in the Truth Table.]
2. “p p” is a contradiction.
[All F’s in the Truth Table.]
3. “p  p” is a contingency.
[At least one of each in TT.]
Logical Equivalences
Def: Two compound propositions are called
logically equivalent if they have the same
truth value for every possible truth value
assignment to the component propositions.
Ex:
1. “p  q” is logically equivalent to “q  p” (as we have
seen).
[Truth Table] [What is the relationship?]
2. “(p  q)” is logically equivalent to “p q”. [T Table]
Note that if two propositions (which we label as p and q) are
logically equivalent, then the biconditional of the two
propositions (p  q) is a tautology [since it is true when p and q
have the same truth value, so it is always true if p and q have the
same truth value in all cases]. For this reason, we can denote
that two propositions p and q are logically equivalent by stating
that p  q is true. [What it means for a compound proposition
to be true is that the proposition is a tautology, true in all cases].
Ex: “p  q” is logically equivalent to “p  q”. [Why w/out TT?]
Recall that an implication is true whenever the hypothesis is false
(p) or when the consequence is true (q), or both. It is only false
when both the hypothesis is true and the consequence is false.
So what we have determined is that “p  q  p  q” is true.
Equivalence
Name
p  T  p; p  F  p
p  F  F; p  T  T
p  p  p; p  p  p
Identity Laws
Domination Laws
Idempotent Laws
(p)  p
p  q  q  p; p  q  q  p
(p  q)  r  p  (q  r)
(p  q)  r  p  (q  r)
p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p  r)
Double Negation Law
Commutative Laws
Associative Laws
[no need for parens]
Distributive Laws
(p  q)  p  q
(p  q)  p  q
De Morgan’s Laws
[generalizes]
p  (p  q)  p; p  (p  q)  p
p  p  T; p  p  F
Absorption Laws
Negation Laws
Logical Equivalences Involving 
p  q  p  q
p  q  q  p
p  q  p  q
p  q  (p  q)
[Transform]
(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
Duals
If we have a proposition involving only propositional variables (p,
q, r, s, t, etc.), negation(), disjunction(), conjunction(), and the
constants T and F, then we can form what is called the dual of the
proposition. The dual of the proposition is formed by replacing:
1. Each T with F and each F with T
2. Each  with  and each  with 
This results in a new proposition which is called the dual of the
original proposition.
Ex: The dual of “(p  q)  (q  p)” is “(p  q)  (q  p)”.
The dual of “p  T” is “p  F”.
The dual is important because of the following property. If we
have two logically equivalent compound propositions (say p and q)
then the duals of the two propositions are logically equivalent as
well. That is, if p  q then p*  q*.
Ex: We know that p  T  T. So p  F  F.
In fact, if we look back at our table of logical equivalences, we will
find that (with the exception of double negation) they come in pairs.
And if we take one of the equivalences from a pair, we find that it is
the dual of the other equivalence from the pair. The example above
illustrates this for the domination laws.
Taking the dual of both sides of a logical equivalence allows us to
generate another logical equivalence. This is a useful technique.
Aside: A truth table should be constructed for a compound
proposition, not simply filled in.
Summary
We have seen a great deal of logical equivalences. A very basic
logical equivalence can be established by using a truth table to see if
the two compound propositions have the same truth value in all cases.
This becomes very tedious, especially when we are dealing with
compound propositions that involve a large number of constituent
propositions. In general, to construct a truth table for a compound
proposition involving n constituent propositions, we must consider 2n
possibilities (rows of the truth table).
Another method that can be used to establish a logical equivalence is
to start with one compound proposition and then selectively replace
parts of the proposition with logically equivalent pieces by using
known logical equivalences. This becomes the method of choice
when we do deal with large propositions. It is also preferable because
by executing these transformations, we often understand the reason
for logical equivalence much better than we do by constructing a TT.
Homework problems from Section 1.2
Problems 6, 8, 10, 12, 15, 21, and 30 from this
section will be included on the next homework
assignment.