Propositional Logic Review
Computability and Logic
Boolean Connectives
Truth-Functional Connectives and
Boolean Connectives
• Connectives are usually called truth-functional
connectives:
– This is because the truth value of a complex claim
that has been constructed using a truth-functional
connective is considered to be a function of the truth
values of the claims that are being connected by that
connective.
– This is also why propositional logic is also called
truth-functional logic.
• For now, we will focus on three connectives:
and, or, not; these are called the Boolean
connectives.
Truth-Table for Negation
P P
T F
F T
Truth-Table for Conjunction
P Q PQ
T T
T
T F
F
F T
F
F F
F
Truth-Table for Disjunction
P
T
T
F
F
Q PQ
T
T
F
T
T
T
F
F
Combining Complex Claims:
Parentheses
• Using the truth-functional connectives, we
can combine complex claims to make
even more complex claims.
• We are going to use parentheses to
indicate the exact order in which claims
are being combined.
• Example: (P  Q)  (R  S) is a
conjunction of two disjunctions.
Parentheses and Ambiguity
• An ambiguous statements is a statement whose
meaning is not clear due to its syntax. Example :
”P or Q and R”
• In formal systems, an expression like P  Q  R
is simply not allowed and considered
unsyntactical.
• Claims in our formal language are therefore
never ambiguous.
• One important application of the use of formal
languages is exactly this: to avoid ambiguities!
Exclusive Disjunction vs
Inclusive Disjunction
• Notice that the disjunction as defined by
‘’ is considered to be true if both disjuncts
are true. This is called an inclusive
disjunction.
• However, when I say “a natural number is
either even or odd”, I mean to make a
claim that would be considered false if a
number turned out to be both even and
odd. Thus, I am trying to express an
exclusive disjunction.
How to express Exclusive
Disjunctions
• We could define a separate symbol for exclusive
disjunctions, but we are not going to do that.
• Fortunately, exclusive disjunctions can be
expressed using the symbols we already have:
(PQ)  (PQ)
P
T
T
F
F
Q (P  Q)  (PQ)
T
T F F T
F
T T T F
T
T T T F
F
F F T F
!
Conditionals
The Material Conditional
• Let us define the binary truth-functional
connective ‘’ according to the truth-table
below.
• The expression P  Q is called a conditional. In
here, P is the antecedent, and Q the
consequent.
P
T
T
F
F
Q PQ
T
T
F
F
T
T
F
T
‘If … then …’ Statements
• The conditional is used to capture ‘if … then …’ statements.
• However, the match isn’t perfect.
• For example, we don’t want to say that the claim “If grass is
green then elephants are big” is true just because grass is green
and elephants are big, nor that any ‘if … then’ statement is
automatically true once the ‘if’ part is false or the ‘then part true.
• The problem is that most English ‘if…then’ expressions aren’t
meant to make a claim that is truth-functional in nature.
• Still, any ‘if … then …’ statement will be false if the ‘if’ part is
true, but the ‘then’ part false, and the conditional captures at
least this important truth-functional aspect of any ‘if … then …’
statement.
• So, while we will from now on refer to the conditional as an ‘if …
then’ statement, we must be careful about the use of this, just as
care must be taken when applying Newtonian physics to some
situation.
Case in point: The Infamous
‘King-Ace’ problem
• The psychologist of reasoning gave the following
logic problem to Princeton undergraduates:
– Consider the following statement: “If there is a king in
the hand, then there is an ace in the hand, or else if
there is not a king in the hand, then there is an ace in
the hand”. What follows from this statement?
• Almost all students responded that it can be
inferred that there is an ace in the hand.
• Johnson-Laird, however, said that what can be
concluded is that there is not an ace in the hand,
and that this is evidence that people can easily
make logical reasoning mistakes! …. Really?
‘If and only if’ and the
Material Biconditional
• A statement of the form ‘P if and only if Q’ (or ‘P
iff Q’) is short for ‘P if Q, and P only if Q’. Hence,
we could translate this as (P  Q)  (Q  P).
However, since this is a common expression, we
define a new connective ‘’:
P
T
T
F
F
Q PQ
T
T
F
F
T
F
F
T
Logical Properties
Truth Tables
• Truth-tables can be used for:
– defining the truth-conditions of truth-functional
connectives
– evaluating the truth-conditions of any complex
statement
Tautologies
• A tautology is a statement that is
necessarily true.
• Example: P  P
P P  P
T TT F T
F F T TF
Contradictions
• A contradiction is a statement that is
necessarily false.
• Example: P  P
P P  P
T TF F T
F F F TF
Contingencies
• A contingency is a statement that can be
true as well as false
• Example: P
P P
T T
F F
Equivalences
• Two statements are equivalent if they
have the exact same truth-conditions.
• Example: P and P
P P P
T T TF T
F F F TF
Contradictories
• Two statements are contradictories if one
of them is false whenever the other one is
true and vice versa.
• Example: P and P
P P P
T T FT
F F TF
Implication
• One statement implies a second statement
if it is impossible for the second statement
to be false whenever the first statement is
true.
• Example: P implies P  Q
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
PQ
T
T
T
F
Consistency
• A set of statements is consistent if it is
possible for all of them to be true at the
same time.
• Example: {P, P  Q, Q}
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
PQ
T
T
T
F
Q
F
T
F
T
Consequence
• A statement is a consequence of a set of
statements if it is impossible for the
statement to be false while each statement
in the set of statements is true.
• Example: P is a consequence of {PQ, Q}
P
T
T
F
F
Q
T
F
T
F
PQ
T
T
T
F
Q
F
T
F
T
P
T
T
F
F
Validity
• An argument is valid if it is impossible for
the conclusion to be false whenever all of
its premises are true.
• Example: P  Q, Q  P
P
T
T
F
F
Q
T
F
T
F
PQ
T
T
T
F
Q
F
T
F
T
P
T
T
F
F
Implication, Consequence, Validity
• The notions of implication, consequence, and
validity are very closely related:
• A statement  implies a statement  if and only
if  is a consequence of the set of statements
{}
• For implication and consequence we use the
symbol ‘’:
– If statement  implies statement  we write   
– If statement  is a consequence of a set of
statements {1, …, n}, we write {1, …, n}  
• An argument consisting of premises 1, …, n
and conclusion  is valid iff {1, …, n}  
• The terms implication, consequence and validity
can therefore be used interchangeably.
Boolean Algebra:
Rewriting Statements
Logically Equivalent Statements
• To express that two statements P and Q
are logically equivalent, we will write:
PQ
• ‘’ is not a symbol of (the language of)
propositional logic!!
• Rather, it is a symbol used to say
something about (statements expressed in
the language of) propositional logic.
• It is a meta-logical symbol, expressing a
meta-logical claim.
Some Important Equivalences
• Double Negation:
– PP
• DeMorgan:
– (P  Q)  P  Q
– (P  Q)  P  Q
• Distribution:
–
–
–
–
P  (Q  R)  (P  Q)  (P  R)
P  (Q  R)  (P  Q)  (P  R)
(Q  R)  P  (Q  P)  (R  P)
(Q  R)  P  (Q  P)  (R  P)
More Equivalences
• Commutation:
– PQQP
– PQQP
• Association:
– P  (Q  R)  (P  Q)  R
– P  (Q  R)  (P  Q)  R
• Idempotence:
– PPP
– PPP
• Subsumption:
– P  (P  Q)  P
– P  (P  Q)  P
Even More Equivalences
• Implication:
– P  Q  P  Q
– (P  Q)  P  Q
• Transposition:
– P  Q  Q  P
• Exportation:
– P  (Q  R)  (P  Q)  R
• Absorption:
– P  Q  P  (P  Q)
• Equivalence:
– P  Q  (P  Q)  (Q  P)
– P  Q  (P  Q)  (P  Q)
Simplifying Statements I
• Using the principle of substitution of logical
equivalents, and using the logical equivalences
that we saw before (Double Negation,
Association, Commutation, Idempotence,
DeMorgan, Distribution, and Subsumption), we
can often simplify statements.
• Example:
(A  B)  A  (Commutation)
(B  A)  A  (Association)
B  (A  A)  (Idempotence)
BA
Generalized Conjunctions and
Generalized Disjunctions
• Recall the Association equivalences:
– P  (Q  R)  (P  Q)  R
– P  (Q  R)  (P  Q)  R
• Because of this, we’ll allow to drop brackets:
– PQR
– PQR
• Thus we can generalize conjunctions and
disjunctions
– A generalized conjunction (disjunction) can have any
number of conjuncts (disjuncts)
Simplifying Statements II
• The conjuncts (disjuncts) of a generalized
conjunction (disjunction) can be switched
around in any way you want. This really
helps with simplifying statements.
Example:
C  (A  (B  C))  (Distribution)
C  (A  B)  (A  C)  (Subsumption)
C  (A  B)
‘⊤’ and ‘⊥’
• A generalized conjunction is false if it has
at least one false conjunct, otherwise it is
true.
– So, a generalized conjunction with 0
conjuncts cannot have a false conjunct, and
hence cannot be false. Therefore, it is a
tautology! We will write this as ‘⊤’.
• A generalized disjunction is true if it has at
least one true disjunct, otherwise it is
false.
– Hence, a generalized disjunction with 0
disjuncts can never be true, and is therefore a
contradiction! We will write this as ‘⊥’.
Some equivalences involving ‘⊤’
and ‘⊥’
•
•
•
•
•
•
•
•
⊥  ⊤
⊤  ⊥
P⊥⊥
P⊤⊤
P⊤P
P⊥P
P  P  ⊥
P  P  ⊤
Simplifying Statements III
• Using ‘⊤’ and ‘⊥’, we can simplify
statements even more. Example:
(A  (B  (A  B))  (DeMorgan)
A  (B  (A  B))  (Double Neg.)
A  B  (A  B)  (Distribution)
(A  B  A)  (A  B  B) 
⊥⊥
⊥
Normal Forms and
Expressive Completeness
Negation Normal Form
• Literals: Atomic Sentences or negations thereof.
• Negation Normal Form: An expression built up
with ‘’, ‘’, and literals.
• Using repeated DeMorgan and Double
Negation, we can transform any truth-functional
expression built up with ‘’, ‘’, and ‘’ into an
expression that is in Negation Normal Form.
• Example:
((A  B)  C)  (DeMorgan)
(A  B)  C  (Double Neg, DeM)
(A  B)  C
Disjunctive Normal Form
• Disjunctive Normal Form: A disjunction of
conjunctions of literals.
• Using repeated distribution of  over , any
statement in Negation Normal Form can be
written in Disjunctive Normal Form.
• Example:
(AB)  (CD)  (Distribution)
[(AB)C]  [(AB)D]  (Distribution (2x))
(AC)  (BC)  (AD)  (BD)
Conjunctive Normal Form
• Conjunctive Normal Form: A conjunction of
disjunctions of literals.
• Using repeated distribution of  over , any
statement in Negation Normal Form can be
written in Conjunctive Normal Form.
• Example:
(AB)  (CD)  (Distribution)
[(AB)  C]  [(AB)  D]  (Distribution (2x))
(AC)  (BC)  (AD)  (BD)
Truth-Functional Connectives
• So far, we have seen one unary truth-functional
connective (‘’), and two binary truth-functional
connectives (‘’, ‘’).
• Later, we will see two more binary connectives
(‘’, ‘’)
• However, there are many more truth-functional
connectives possible:
– First of all, a connective can take any number of
arguments: 3 (ternary), 4, 5, etc.
– Second, there are unary and binary connectives other
than the ones listed above.
Unary Connectives
• What other unary connectives are there
besides ‘’?
• Thinking about this in terms of truth tables,
we see that there are 4 different unary
connectives:
P
*P
P
*P
P
*P
P
*P
T
F
T
T
T
F
T
F
T
F
F
T
T
F
F
F
Binary Connectives
• The truth table below shows that there are
24 = 16 binary connectives:
P Q
P*Q
T T
T/F
T F
T/F
F T
T/F
F F
T/F
In general:
n sentences 
2n truth value combinations
(i.e. 2n rows in truth table) 
2n
2 different n-ary connectives!
Expressing other connectives
using ‘and’, ‘or’, and ‘not’
• We saw that we can express the exclusive
disjunction using ‘and’, ‘or’, and ‘not’.
• Q: Can we express all other connectives as
well?
• A: Yes! We can generalize from this example:
P Q
P*Q
T T
F
Step 1:
T F
T
 PQ
F T
T
 PQ
F F
F
Step 2:
 (PQ)  (PQ)
Truth-Functional Expressive
Completeness
• Since I can express any truth function using ‘’,
‘’, and ‘’, we say that the set of operators {,
, } is (truth-functionally) expressively
complete.
• DeMorgan Laws:
– (P  Q)  P  Q
– (P  Q)  P  Q
• Hence, by the principle of substitution of logical
equivalents, since {, , } is expressively
complete, the sets {, } and {, } are
expressively complete as well!
The NAND
• Let us define the binary truth-functional
connective ‘NAND’ according to the truth-table
below.
• Obviously, P NAND Q  (P  Q) (hence the
name!)
P
T
T
F
F
Q P NAND Q
T
F
F
T
T
T
F
T
Expressive Completeness of the
NAND
• The NAND has a very interesting property, in
that it can express any truth-functional
connective, i.e. {NAND} is expressively
complete!
• Proof: We already know that we can express
every truth-functional connective using only 
and . Furthermore:
– P NAND P  (P  P)  P
– (P NAND P) NAND (Q NAND Q)  ((P NAND P) 
(Q NAND Q))  (P  Q)  P  Q
• In other words, we can build circuitry using only
one kind of logic gate!! Of course, the drawback
is that we need many of those gates.
Truth-Trees
Logical Possibility
• All logically interesting claims can be reduced to
questions about logical possibility:
– Logical Consistency: Is it possible for all statements
to be true?
– Logical Validity: Is it possible for all premises to be
true and the conclusion false?
– Logical Consequence: Is it possible for the implying
statements to be true and the implied statement to be
false?
– Logical Equivalence: Is it possible for the two
statements to have a different truth value?
– Logical Tautology: Is it possible for the statement to
be false?
Truth Table Method
• The truth table method systematically exhausts
all possible truth value combinations of the
statements involved.
• In the truth-table we look for a row that reflects a
certain possibility, and that will tell us the answer
to whatever question we had (e.g. if there is no
row where statement is false, then it is not
possible for that statement to be false, and
hence it is a tautology).
Drawback and Room for
Solution
• A drawback of the truth table method is that the
number of rows grows exponentially.
• Fortunately, there is room for a solution to this
problem. Since all we are interested in, is the
existence of a specific combination of truth
values of the statements involved, all we need to
find is one example of such a case. Once we
have found such a case, there is no need to
exhaust all other cases.
Simple Solution: Stopping Early
• One solution to the problem of big truth tables is
therefore to simply stop once you have found a
row that represents the combination of truth
values you are interested in.
• Thus, rather than working out a truth table
column by column, you may want to do it row by
row, so that you can stop as soon as you have
found a row of the kind you are looking for.
• A big drawback of this approach is that if no row
of the kind you are looking for exists, then you
have to complete the whole truth table after all.
A More Focused Search
• Consider the following argument:
P  (Q  R)
R  Q
R
• We are interested in whether all premises can be
true and the conclusion false:
– In order for the conclusion to be false, R must be false.
– In order for the second premise to be true while R is
false, Q must be false.
– In order for the first premise to be true while Q and R
are both false, P must be false.
The Short Truth Table Method
• The Short Truth Table Method assigns truth
values to the involved atomic and complex
statements in order to try and obtain a certain
combination of truth values:
P  (Q  R)
F T F FF
RQ
F TTF
/
R
F
• The Short Truth Table Method thus tries to
generate one row of the truth table that has
the combination of truth values you are
interested in.
Short Truth Table Method and
Indirect Proof
• As you assign truth values to certain statements,
the truth values of other statements can be
forced.
• If you are forced to make a statement both true
and false, then you know that the combination of
truth values you are looking for does not exist:
P  (Q  P)
TF TF F
 Contradiction, so the
statement is a tautology!
• The short truth table method is therefore a kind
of indirect proof (proof by contradiction), except
that you don’t always get a contradiction.
Drawback of the Short Truth
Table Method
• A drawback of the short truth table method is
that you are not always forced to assign any
further truth values:
R  (Q  P)
T T
(Q  R)  P
T
T
RP
T
Q
T
• At this point, you can choose to assign certain
truth values, but if your choice does not lead
to the row you are looking for, then you need
to try a different option, and the short truth
table method has no tools to do go through all
of your options in a systematic way.
Truth Trees
• The obvious solution to the drawback of the
short truth table method is to incorporate tools to
systematically keep track of multiple options.
• One method that does so is the truth tree
method:
– The truth tree method tests for the consistency of a
set of statements and, as such, can be used to
determine validity, tautologies, equivalence, etc.
– Like the short table method, it infers which other
statements are forced to be true under this
assumption.
– When nothing is forced, then the tree branches into
the possible options.
Truth Tree Example
(((PQ)R)  (P(QR))) 
(PQ)R 
(P(QR)) 
P
(QR) 
Q
((PQ)R) 
P(QR) 
PQ 
All branches close
R
 the original
P
statement cannot be
Q
false  tautology!
R
(PQ)  R
×
P Q
×
×
P
×
Q  R 
Q
×
R
×
Decomposition Rules for Truth
Trees
P 
P
PQ 
P
Q
PQ 
PQ 
P
Q
P
(PQ) 
P
Q
Q
PQ 
P
Q
P
Q
(PQ) 
P
Q
(PQ) 
P
Q
(PQ) 
P
Q
P
Q
Truth Trees as Decision
Procedures
• The truth tree method can easily be made
into a systematic procedure.
• As such, the truth tree method becomes a
decision procedure for truth-functional
consequence that is, on average, quite a
bit more efficient than the truth-table
method.