1. The Boolean Connectives
Language, Proof and
Logic
The Boolean Connectives
Chapter 3

2. Truth-functional connectives
3.0
The Boolean connectives are used to form compound sentences from
atomic ones.
They are called truth-functional connectives because the truth value
of a compound sentence is fully determined by the truth values of
its components.
Truth tables are the most straightforward way to define Boolean
connectives.
Another way is through Henkin-Hintikka games.

3. Committing to the truth of  P means committing to the falsity of P.
Negation 
3.1
P P
TRUE FALSE
FALSE TRUE
Truth table:
Game:
Committing to the truth of  P means committing to the falsity
of P.
Committing to the falsity of P means committing to the truth
Atomic sentences and their negations are called literals.
“You try it” on page 69.

4. If you commit to the truth, then your opponent chooses P or Q, and the
Conjunction 
3.2
P Q P  Q
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE FALSE
FALSE FALSE FALSE
Truth table:
Game:
If you commit to the truth, then your opponent chooses P or Q, and the
game continues as if you had committed to the truth of that simpler
component.
If you commit to the falsity, then you have to choose either P or Q and,
and the game continues as if you had committed to the falsity of that
“You try it” on page 72.

5. If you commit to the truth, then have to choose either P or Q, and the
Disjunction 
3.3
P Q P  Q
TRUE TRUE TRUE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE
Truth table:
Game:
If you commit to the truth, then have to choose either P or Q, and the
game continues as if you had committed to the truth of that simpler
component.
If you commit to the falsity, then your opponent chooses either P or Q,
and the game continues as if you had committed to the falsity of that
“You try it” on page 76.

6. Ambiguity and parentheses
3.4
Parentheses must be used whenever ambiguity would result from their
omission. In practice, this means that conjunctions and disjunctions
should be “wrapped” in parentheses whenever combined by means of
some other connective.
P  Q  R is ambiguous,
should be either P  (Q  R) or (P  Q)  R

7. Equivalent ways of saying things
3.5
PQ  QP
PQ  QP
PP  P
PP  P
Double negation: P  P
DeMorgan 1: (PQ)  PQ
DeMorgan 2: (PQ)  PQ

8. In order for a FOL sentence to be a good translation of an English
sentence, it is sufficient that the two sentences have the same truth
values in all possible circumstances.
It is not sufficient that they have the same truth value in some particular
world.

9. 1. The English expression “and” sometimes suggest temporal ordering;
Translation
3.6.b
1. The English expression “and” sometimes suggest temporal ordering;
the FOL expression  never does.
Bob went home and ate
2. The English expressions “but”, “however”, “yet”, “nonetheless”,
“moreover” are all stylistic variants of “and”.
Bob was sleepy but he did the homework
3. The English expressions “either” and “both” are often used like
parentheses to clarify an otherwise ambiguous sentence.
Either Bob is home and Jane is home or Max is happy
Max is happy or both Bob is home and Jane is home