Name _______________________
Worksheet 1.1
Introduction to Logic
In a nutshell, propositional logic helps us conclude something is true or false based on whether
its parts are true or false. It does not give us any insight into whether the building-block parts are
themselves true or false.
These are the five common “connectives” that you will be working with, along with the
respective “English translation.” You should memorize all of these.
Negation
Conjunction
Disjunction
Conditional
Biconditional
¬A
AB
A B
A→B
A↔B
“Not A”
“A and B”
“A or B (or both)”
“If A then B”
“A if and only if B”
1. Suppose A stands for “It is raining” and B stands for “The sun is shining.” Write each of the
following into meaningful (grammatically correct) sentences. For example, A  B is the same as
“it is raining or the sun is shining.”
(a) A  B
(b) (¬ A)  B
(c) (¬ A)  A
(d) B → (¬ A)
(e) Which one of the above statements is always true no matter what the actual meaning
of A and B? (You will later learn that such a statement is called a tautology)
2. We could write the expression, “If I make a driving error or forget some of the Highway Code
then I will fail my driving test” as (A  B) → C
(a) Interpret the meaning of A
(b) Interpret the meaning of B
(c) Interpret the meaning of C
(d) If you make a driving error, is A true or false?
(e) If you pass the driving test, is C true or false?
3. Write the following expressions symbolically given M stands for “the murderer has left the
country” and H stands for “someone is harboring him.”
(a) If the murderer has not left the country then someone is harboring him.
(b) If somebody harbors the murderer then he will not leave the country.
(c) The murderer has left the country and/or someone is harboring him.
(d) Which two of the above sentences are logically equivalent? Think about this one – it
is without doubt the toughest problem on this worksheet.
4. Write the following expressions in symbolic form. Let H = “She is happy” and R = “She is
rich.”
(a) If she is rich then she is unhappy.
(b) She is neither rich nor unhappy.
(c) If she is rich then she is happy and if she is happy then she is rich.
(d) She is never happy when she is rich.
(e) She is poor only if she is happy.
(f) She is poor or she is both rich and happy.
5. The logical connective “  “ meaning “implies” is used to translate several English
phrases: “ if…..then…..”, “therefore”, “then”, “means”, but the phrase “only if” creates more
problems. When the two directions of implication are used in the biconditional we say that the
condition is necessary and sufficient and we say “if and only if”. The interpretation of “only if
alone always causes issues.
Consider the following phrases:
(a) I will go to the movie with you only if I finish my homework
(b) If I don’t finish my homework I will not go to the movies with you
(c) If I come to the movies with you then it means I finished my homework.
Are these phrases equivalent? If A means “I finish my homework” and B means “ go to the
movies with you”. Represent these phrases symbolically.
6. A if and only if B translates to A  B . We say that A is a necessary and sufficient condition
for B. Break down each of these three forms into their component parts and match them
correctly.