A Quick Overview of Conditionals
Definition Any sentence that can be put in the form “If p , then q ,” where p and q are
component statements, is called a conditional statement. It is denoted symbolically as p  q .
The component p is called the premise (or antecedent) and the component q is called the
conclusion (or consequent).
Example “If the cardinality of a set is 5, then the set has five distinct elements.” This is a
conditional statement in set theory whose premise is the statement “The cardinality of a set is
5” and whose conclusion is the statement “The set has five distinct elements.”
The above sentence could also be written in a number of other logically equivalent ways, such
as:
“The cardinality of a set is 5 implies that the set has five distinct elements,”
or
“All sets with cardinality 5 have five distinct elements.”
or
“A set has five distinct elements only if its cardinality is 5,”
or
“A set has five distinct elements provided its cardinality is 5.”
Note how in the last two examples above the premise and conclusion are switched within the
sentence! You must be careful when you read conditional statements to identify the premise
and conclusion properly. The rule of thumb is to always revert back to the explicit conditional
form “If p , then q .”
So it’s clear that there are many equivalent ways to write conditional statements.
Here are some of the most commonly used conditional forms, which are all equivalent to “If p ,
then q ”):
“ p implies q .”
“ p entails q .”
“Every A is B” / “All A are B.”
[Here the premise p is “It is A” and
the conclusion q is “It is B.”]
“ q , provided p .”
[Careful… This one switches p and q !]
“ q , (only) if p .”
[Careful… This one switches p and q !]
“ q , whenever p .”
[Careful… This one switches p and q !]
Basic Truth Table for p  q
Below is the truth table for the conditional statement p  q :
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
So a conditional statement “If p , then q ” is only FALSE when the premise p is
TRUE and the conclusion q is FALSE.
To illustrate this table, let’s examine the truth value of the conditional statement
“All Belgians love chocolate.”
First, we rewrite this sentence in the explicit form
“If the person is Belgian, then he/she loves chocolate.”
Symbolically, we have p  q , where the premise p is “You are Belgian” (or “You are from
Belgium”) and the conclusion q is “You love chocolate.”
Row 1
If both p and q are TRUE (i.e. you are Belgian and you love chocolate) then the
overall statement is clearly TRUE because the implication is not violated.
Row 2
If p is TRUE (i.e. you are Belgian) and q is FALSE (i.e. you don’t love chocolate),
then the overall statement is FALSE because the implication has been violated!
Rows 3 & 4
These two rows are a little counter-intuitive. However, in both rows the
premise is taken to be FALSE (i.e. you are not from Belgian), so the conclusion
becomes immediately irrelevant based on the implication contained within the
sentence. In other words, once the premise is false the implication is no longer
applicable!
In general, any conditional statement whose premise is FALSE is necessarily TRUE!
An immediate consequence of this truth-table with conditionals (which is, by the way, perfectly
validated in logic) is that sentences like “If I am immortal, then I can fly to Venus” are technically
TRUE, but are clearly absurd… We refer to such conditionals sentences as being vacuously true.
For a much more refined (mathematically speaking) argument as to why the truth table for
conditionals makes logical sense, read Timothy Gowers’s article titled “What is implied by
implies?” This article can be found as a link on my webpage. There, Gowers presents a
powerful set-theoretical argument as to why T  F  F .
Related Forms of p  q
There are three important forms related to the conditional “If p, then q.“ These are described
below.
The Converse
The Inverse
The Contrapositive
q p
~ p ~ q
~ q ~ p
Based on the basic truth table for p  q and its related statements (the converse, the inverse
and the contrapositive), we have the following two important equivalences:
p  q  ~ q ~ p
q  p  ~ p ~ q
So a conditional statement is logically equivalent to its contrapositive and its
converse is logically equivalent to its inverse.
Example 1 The statement “If the square of a natural is odd, then the natural is also odd” is
equivalent to the much simpler statement “If a natural number is even, then its square is also
even.” Can you see why? For natural numbers, the negation of being odd is being even and
vice-versa. Therefore, here you are looking at a conditional (the last statement) and its
contrapositive (the original statement) and so they must be equivalent.
Example 2 The statement “Every apple is green or red” is equivalent to “If it’s not green
and it’s not red, then it is not an apple.” Can you see why? Here you must write the original
statement as a conditional in the form “If a, then g or r,” where a is the statement “It’s an
apple,” r is the statement “It’s red,” and g is the statement “It’s green,” and then write the
contrapositive as “If (not g and not r), then not a” using De Morgan’s law.
Equivalent Disjunctive Form of a Conditional
Using basic truth tables, it turns out that the conditional “If p, then q” is equivalent to the
disjunction “Not p or q.”
Therefore, we have:
p  q ~ p  q
Example The statement “All squares are rectangles” is equivalent to the statement “It’s not a
square or it’s a rectangle.”
Negation of a Conditional
Using De Morgan’s laws and the disjunctive form of the conditional shown above, we then have
the following:
~  p  q  ~  ~ p  q   p ~ q
Example The negation of the statement “I will go to the beach if it doesn’t rain” is then given
by the statement “It rains and I don’t go to the beach.”