Truth values
!
2. Propositional Logic
Propositional formulas, like
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Truth tables
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It is raining
The train is moving
are true or false depending on the
circumstances.
One day it rains, another it does not. It
may rain in Helsinki but not in Warsaw.
! A true propositional formula is said to
have truth value 1.
! A false propositional formula is said to
have truth value 0.
The lecture
Jouko Väänänen: Propositional logic
Truth values (Contd.)
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Valuation
Propositional formulas of the simplest
form p0, p1,... can have truth value 1 or
0 according to our choice. But if we
give them truth values, then the truth
values of formulas built from them such
as p0 v p1 and p0 → p1 are completely
determined.
A choice of truth values for proposition
symbols is called a valuation.
Jouko Väänänen: Propositional logic
!
Valuations assign truth values 1 (true) or 0
(false) to proposition symbols.
!
Valuations are the building blocks of truth
tables, which are the main tool for analysing
complicated propositional formulas.
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Valuations mathematically defined
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Valuations map propositional symbols to the
two element set {0,1} of truth values.
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Valuation
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Valuations extend to all formulas by
means of truth tables.
The truth value v(A) of an arbitrary
propositional formula A can be easily
computed in terms of the truth values
of the immediate subformulas of A.
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Truth Tables
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Truth table of Conjunction
Truth table = table of all possible
valuations
Negation
Truth tables
reflect our
Implication
intuition of the
meaning of each
Conjunction
connective.
Disjunction
Equivalence
∧
The only true case
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Truth table of Disjunction
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Truth Table of Negation
The only false case
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Truth Table of Implication
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Interpretation of implication I
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The only false case
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Let p be the sentence “Jukka lives in Helsinki”.
Let q be the sentence “Jukka lives in Finland”.
Since Helsinki is in Finland, the formula p→q is
true.
This truth is based on Helsinki being in Finland,
and is unaffected if Jukka in fact does not live in
Helsinki.
The only thing that would shatter the truth of
p→q is if Jukka lived in Helsinki but not in
Finland, in which case we would have to review
our geography.
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Interpretation of implication II
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Truth table of Equivalence
Think of A and B as subsets of {0}.
There are just two subsets: and {0}.
Identify them with 0 and 1.
Think of A→B as “A is contained in B”
→
so 1st row is 1
!
so 2nd row is 0
so 3rd row is 1
!
so 4th row is 1.
!
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A truth table
!
A truth table
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
(p0
p1)
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p0
∧
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
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(p0
p1)
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p0
∧
p1)
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A truth table
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
!
p1)
A truth table
!
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(p0
p1)
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p0
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∧
p1)
!
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
(p0
p1)
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p0
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∧
p1)
A truth table
!
A truth table
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
(p0
p1)
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p0
∧
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(p0
p1)
p0
∧
!
p0
∧
p1)
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Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
p1)
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(p0
p1)
Jouko Väänänen: Propositional logic
A truth table
p0
∧
p1)
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A truth table
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
p1)
A truth table
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!
(p0
Jouko Väänänen: Propositional logic
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
p1)
A truth table
!
!
(p0
p1)
Jouko Väänänen: Propositional logic
p0
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∧
p1)
!
Every connective in a formula is the
main connective of a subformula. We
write under it the truth value of this
subformula.
p0 p1
(p0
p1)
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p0
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∧
p1)
A bigger truth table
p0
p1 p2
(p0
p1)
A bigger truth table
(p1
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p2))
(p0
p2))
p1 p2
(p0
p1)
(p0
p1)
p1)
(p1
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(p1
p2))
(p0
p2))
p0
p1 p2
(p0
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p1)
(p1
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A bigger truth table
p1 p2
(p0
p2))
(p0
p2))
(p0
p2))
(p0
p2))
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A bigger truth table
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p0
p1 p2
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A bigger truth table
p0
p0
p2))
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A bigger truth table
(p1
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p2))
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(p0
p2))
p0
p1 p2
(p0
p1)
(p1
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p2))
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A bigger truth table
p0
p1 p2
(p0
p1)
A bigger truth table
(p1
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p2))
(p0
p2))
p1 p2
(p0
p1)
(p0
p1)
p1)
(p1
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(p1
p2))
(p0
p2))
p0
p1 p2
(p0
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p1)
(p1
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A bigger truth table
p1 p2
(p0
p2))
(p0
p2))
(p0
p2))
(p0
p2))
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A bigger truth table
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p0
p1 p2
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A bigger truth table
p0
p0
p2))
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A bigger truth table
(p1
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p2))
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(p0
p2))
p0
p1 p2
(p0
p1)
(p1
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p2))
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A bigger truth table
p0
p1 p2
(p0
p1)
Inefficiency of Truth Tables
(p1
p2))
(p0
p2))
!
Truth tables become eventually too large
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"
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n propositional symbols
2n rows in the truth table
Truth table grows exponentially
Jouko Väänänen: Propositional logic
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