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1 Propositional Logic
atomic propositions
compound propositions built using “extensional” connectives (and, or, not, . . . )
Example 1.0.1. Because is not an “extensional” connective. Indeed
A := 4 is an even number
B := 4 can be divided by 2
C := 4 is not prime
A, B and C are (indended to be) ”true” while
A because B is (indended to be) ”true”
A because C is (indended to be) ”false”
1.1 Syntax
and: ∧
or: ∨
if . . . then . . . : →
not: ¬
A, B, . . . : atomic propositions
P, Q, . . . : formulas
Definition 1.1.1 (Alphabet). Consists of the symbols:
• for atomic propositions
• connectives
• “(”, “)”
Definition 1.1.2 (Well Formed Formulas WFF). Is the minimal set X s.t.
• A, B, . . . ∈ X
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• If P ∈ X then (¬P ) ∈ X
• If P, Q ∈ X then (P ? Q) ∈ X, where ? ∈ {∧, ∨, . . . }
Priority to avoid parenthesis: ¬ > ∧ > ∨ > →
Example 1.1.1. A ∧ B → ¬C is interpreted as (A ∧ B) → (¬C).
One cannot remove the parenthesis in (A → B)∧C since the interpretation of A → B ∧C
is A → (B ∧ C).
1.2 Semantics
1
“true”
0
“false”
Definition 1.2.1 (Interpretation). v : WFF → {0, 1}
¬P :
¬P is true iff P is false: v(¬P ) = 1 iff v(P ) = 0
P
0
1
P ∧Q:
P
0
0
1
1
P ∨Q:
¬P
1
0
v(P ∧ Q) = 1 iff v(P ) = v(Q) = 1 v(P ∧ Q) = min{v(P ), v(Q)}
Q
0
1
0
1
P ∧Q
0
0
0
1
v(P ∨ Q) = 1 iff v(P ) = 1 or v(Q) = 1
v(P ∨ Q) = max{v(P ), v(Q)}
P
0
0
1
1
P →Q:
Q
0
1
0
1
P ∨Q
0
1
1
1
P
0
v(P → Q) = 0 iff v(P ) = 1 and v(Q) = 0 0
1
1
4
Q
0
1
0
1
P →Q
1
1
0
1
Definition 1.2.2 (Model). Let P ∈ WFF and v interpretation. If v(P ) = 1 we say
that P is satisfied by v, or that v is a model for P (denoted by v |= P ).
Definition 1.2.3. P is satisfiable if it has at least a model. P is unsatisfiable otherwise.
Definition 1.2.4. P is a tautology if every interpretation v is a model for P (denoted
by |= P ).
Theorem 1.2.1. P is a tautology iff ¬P is unsatisfiable.
Definition 1.2.5. Let Γ be a set of WFF and Q be a formula. Q is a (semantical)
consequence of Γ iff for each interpretation v ((∀P i ∈ Γ, v(Pi ) = 1) =⇒ v(Q) = 1)
(denoted by Γ |= Q).
Theorem 1.2.2. Γ |= Q iff Γ ∪ {¬Q} is unsatisfiable.
Proof.
(⇒) Since Γ |= Q it holds that v(Q) = 1 for all models v of Γ. Therefore v(¬Q) = 0 for
all models v of Γ and Γ ∪ {¬Q} has no model.
(⇐) Let v 0 be an interpretation. If v 0 is a model for Γ, being Γ ∪ {¬Q} unsatisfiable,
then necessarily v 0 (¬Q) = 0 iff v 0 (Q) = 1.
Example 1.2.1. (A → B) ∨ (B → A) is a tautology.
P
0
0
1
1
Q
0
1
0
1
(A → B) ∨ (B → A)
1
1
1
1
Definition 1.2.6. Let P and Q two WFF, P ≡ Q if for all interpretations v, v(P ) =
v(Q).
Each n-ary connective defines a function f : {0, 1} n → {0, 1}
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Example 1.2.2. n = 2: 22 = 16 functions from {0, 1}2 to {0, 1}
A connective is derivable if it is possible to define it from other connectives.
Proposition 1.2.1.
A → B ≡ ¬A ∨ B
A ∨ B ≡ ¬A → B
A ∨ B ≡ ¬(¬A ∧ ¬B)
A ∧ B ≡ ¬(¬A ∨ ¬B)
¬A ≡ A → ⊥
⊥ ≡ A ∧ ¬A
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Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and
v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with
P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there
exists i ∈ {1, . . . 2n } such that vP (a1 , . . . , an ) = vi
Definition 1.2.7. A set of connectives is functionally complete if every connective can
be derived from it.
Theorem 1.2.3. For each n-ary truth functions, there is a proposition P only using
the connectives ∧, ∨ and ¬, such that f = v P .
Proof. (See p. 47 of Gallier’s book). By induction on the arity n of f .
1
2
A
0
1
0
For n = 1 there are 4 possible truth functions:
1
0
1
A ∨ ¬A A ∧ ¬A
Assume the claim is true for n.
n + 1: Let f be a truth function with n + 1 arguments.
f1 (x1 , . . . , xn ) = f (x1 , . . . , xn , 1)
f2 (x1 , . . . , xn ) = f (x1 , . . . , xn , 0)
)
3
0
1
A
4
1
0
¬A
by i.h. f1 = vQ and f2 = vQ0
i.e. Q and Q0 represents f1 and f2 , resp.
Let P be (An+1 ∧ Q) ∨ (¬An+1 ∧ Q0 ) where An+1 is an atomic proposition not occurring
in Q, Q0 . It is easy to see that f = vP .
Corollary 1.2.1. {∧, ∨, ¬} is functionally complete.
1.3 Normal Forms
P1 ∨ · · · ∨ Pn . . . disjunction of formulas
P1 ∧ · · · ∧ Pn . . . conjunction of formulas
Definition 1.3.1. A literal is an atomic proposition or its negation.
Definition 1.3.2. A well-formed formula P is in conjunctive normal form (CNF) iff
P = P1 ∧ · · · ∧ Pn with n ≥ 1, where Pi , i ∈ {1, . . . , n}, is a disjunction of literals.
Theorem 1.3.1. For each WFF P , there exists a CNF P C and a DNF P D s.t. P ≡ P C
and P ≡ P D .
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Proof. Uses equivalences such as
)
¬(P ∨ Q) ≡ ¬P ∧ ¬Q
De Morgan laws
¬(P ∧ Q) ≡ ¬P ∨ ¬Q
¬¬P ≡ P
P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)
P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
)
Distributivity
Truth table method
A
0
0
1
1
B
0
1
0
1
P
1
0
1
0
→ P1D = (¬A ∧ ¬B)
→ P1C = ¬(¬A ∧ B)
→ P2D = (A ∧ ¬B)
→ P2C = ¬(A ∧ B)
P D = P1D ∨ P2D = (¬A ∧ ¬B) ∨ (A ∧ ¬B)
P C = P1C ∧ P2C = ¬(¬A ∧ B) ∧ ¬(A ∧ B) = (A ∨ ¬B) ∧ (¬A ∨ ¬B)
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