Download Lesson 2

Introduction to logic: Lesson 2,
propositional logic
Marie Duží
[email protected]
1
Introduction to Logic
Some more arguments
• An argument is valid iff it is necessary that
under all interpretations (valuations in
propositional logic), in which the premises are
true the conclusion is true as well: P1,...,Pn |= Z
• P1,...,Pn |= Z if and only if
The statement of the form P1 and ... and Pn
implies Z is necessarily true (a tautology):
|= (P1 & … & Pn)  Z
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Arguments
P1, ..., Pn |= Z iff
|= (P1 & … & Pn)  Z
• BUT !!!
• It does not mean that the conclusion is (or
must be) true. We are dealing with a valid
logical form, a necessary relation
between premises and the conclusion.
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Arguments
No prime is divisible by 3
9 is divisible by 3
--------------------------------- 9 is not a prime
• It is a valid argument though the first premise is
not true (3 is a prime divisible by 3). Another
interpretation:
All men are rational.
A stone is not rational.
------------------------------- A stone is not a man.
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Arguments
• Or, by substituting:
If the number 12 is a prime then it is not divisible by 3.
12 is divisible by 3.
 12 is not a prime.
• Or:
12 is not a prime number or it is not divisible by 3.
12 is divisible by 3.
 12 is not a prime number.
Valid argument schemes (examples of logical forms):
• A  B, A |= B
modus ponens
• A  B, B |= A,
modus ponens + transposition
• A  B, B |= A
modus ponens + transposition
• A  B, B |= A
elimination of disjunction
(disjunctive syllogism)
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Arguments
• Hence if we prove that the conclusion logically follows from the
assumptions, then by virtue of it we do not prove that the conclusion
is true
• It is true, provided the premises are true
• The argument the premises of which are true is called sound.
• Truthfulness or Falseness of premises can be a contingent matter.
But the relation of logical entailment is a necessary relation (“in all
the circumstances ...“).
• Similarly a tautology is a logically, necessarily true formula.
• If a tautology is of an implication form, then according to the definition
of the implication it is true also in case of the antecedent being false,
and false only in case the antecedent is true and consequent false,
which corresponds to the definition of logical entailment:
• A1,…,An |= C
iff
|= A1 …  An  C
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Propositional (Sententional) Logic
• The simplest logical system. It analyzes a way of composing a
complex sentence (proposition) from elementary propositions by
means of logical connectives.
• What is a proposition? A proposition (sentence) is a statement that can
be said to be true or false.
• The Two-Value Principle – tercium non datur – two-valued logic (but
there are many-valued logical systems, logics of partial functions,
fuzzy logics, etc.)
• Is the definition of a sentence trivial? Are all the statements sentences,
or in other words, do all the statements denote a proposition?
No, it is not so:
– The (current) King of France is bald.
• True? But then the King of France exists. False? But then it is true that the King
of France is not bald, hence the King of France exists as well. The statement is
neither true nor false, it is not a sentence.
– Did you stop beating your wife?
(try to answer in case you have never been married or never beat your wife)
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Propositional logic: semantic
exposition
(Semantics = meaning)

There are two kinds of Sentences:
–
–



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Atomic (Elementary) – no proper part of the sentence is a
sentence as well
Molecular (Composed) – the sentence has its own part(s)
that is (are) a sentence(s) as well
The Compositionality Principle: meaning of a
composed sentence is a function (depends only on)
the meanings of its components.
The meaning of sentences is in propositional logic
reduces to: True (1), False (0).
An algebra of truth values.
Introduction to Logic
Examples of composed sentences
It is raining in Prague and it is a sunshine in Brno.
S1

S2
It is not true that it is raining in Prague.
connective
9
connective
S
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Definition: language of PL
•
•
A formal language is defined by an alphabet (a
set of symbols) and a grammar (a set of rules
that define the way of forming “Well Formed
Formulas” - WFF)
Language of Propositional Logic (PL)
•
alphabet:
a) Symbols for propositions: p, q, r, ... (also with
indexes p1, p2, …)
b) Symbols for logical connectives: , , , , 
c) Auxiliary symbols: (, ), [, ], {, }
•
•
Symbols ad a) stand for elementary sentences
Symbols ad b), i.e., , , , ,  are called:
negation (), disjunction (), conjunction (), implication
(), equivalence ().
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Definition: language of PL
Grammar (defines inductively well-formed-formulas)
Inductive definition of an infinite set of WFF:
1. Symbols p, q, r, ... are (well-formed) formulas
(the definition base).
2. If A, B are formulas, then expressions
A, A  B, A  B, A  B, A  B
are (well-formed) formulas (inductive definition step).
3. Only expressions due to 1. and 2. are WFFs.
(the definition closure).
• The language of PL is the set of well-formed formulas.
Note: Formulas according to 1. are atomic formulas
Formulas according to 2. are composed formulas
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Well-formed formulas
Notes:
Symbols A, B are metasymbols. We can substitute for them any WFF
created according to the definition.
The outermost parentheses can be omitted.
For the logical connectives other symbols are sometimes used:
Symbol
alternate
-------------------------------
, 

, 

&

~
Example:
(p  q)  p is a WFF (the outer parentheses omitted)
(p )   q is not a WFF
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Definition: semantics (meaning) of formulas
The truth-value valuation of propositional symbols is a
mapping v that to each propositional symbol p assigns a
truth value, i.e., a value from the set {1,0}, which codes the
set {True, False}: {pi}  {1,0}
The truth-value function of a PL formula is a function w,
which for each valuation v of propositional symbols pi
associates the formula with its truth value in the following
way:
• The truth value of an elementary formula p: wpv = vp for
any propositional variable p.
• If the truth values of formulas A, B are given, then the truth
value of the formulas
A, A  B, A  B, A  B, A  B
are defined by the table 2.1.:
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Table 2.1.:
the truth–value functions
A
1
1
B
1
0
0
0
1
0
A A  B A  B A  B A  B
0
1
1
1
1
0
1
0
0
0
1
1
1
0
0
0
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1
1
0
1
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Transforming natural language to the
PL language
 Elementary sentences: by the propositional variables p, q, r, ...
 Connectives of natural language: by means of the symbols for logical
connectives:
 Negation:
 “it is not true that”:
 (unary connective)
 Conjunction:
 “and”:
 (binary, commutative connection)
 Prague is a capital and 2+2=4:
pq
 Note: not every “and” denotes a logical connective!
Example: “Peter came home and opened the window”.
 Disjunction:
 “or”:
 (binary, commutative connection)
 Prague or Brno is a great city.
pq
 non-alternative
 In a natural language we often use “or” as an alternative “either, or”:
“I’ll go to the cinema or I’ll stay at home”

Alternative “or” is a non-equivalence
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Implication
• Implication
• “if … then …“:
 (binary, non-commutative connective)
A  B; A is the antecedent, B is the consequent.
• Implication (as well as any other connective) does not render any
semantic connection between antecendent and consequent: material
implication (middle ages: ”suppositio materialis”).
• Hence implication does not render a causal or chronological
connection:
”If 1+1=2, then iron is a metal” (a true proposition):
pq
”If the UFOs (flying saucers) exist, then I am a pope”:
pr
(What does the sentence entail?
Since I am not a pope, the UFOs do not exist)
Note: The connectives “because”, “therefore”, “since”, etc. do not
correspond to the logical implication!
– “The ice-hockey team lost the match, therefore the players came home from
the world championship earlier”.
“Because I am sick, I stay at home”.
– “sick”  “home”?
But then it would have to be true even if I am not sick (see the table 2.1 – the
definition of implication)
– We might analyze it by means of the modus ponens: [p  (p  q)]  q 16
The equivalence connective
 Equivalence:



”if and only if” (iff)
”The Greek army used to win if and only if the result of the battle
depended on their physical strength”:
pq
It is most frequently used in mathematics (in definitions), in a natural
language its use is seldom
 Example:
a) “I’ll slap you if you cheat on me”
b) “I’ll slap you if and only if you cheat on me”
cheat  slap
cheat  slap
Situation: You did not cheat. When can you be slapped?
Ad a) – You may be slapped,
Ad b) – You might not be slapped.
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Definition. Satisfiable formulas,
tautology, contradiction, model
 A model of a formula A: valuation v such that A is




true in v: w(A)v = 1.
A formula is satisfiable iff it has at least one model
A formula is a contradiction iff it has no model
A formula is a tautology iff any valuation v is its
model.
A set of formulas {A1,…,An} is satisfiable iff there
is a valuation v such that v is a model of every
formula Ai, i = 1,...,n. The valuation v is then a model
of the set {A1,…,An}.
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Satisfiable formulas,
tautology, contradiction, model
 Example. Formula A is a tautology, A is a contradiction,
formulas (p  q), (p  q) are satisfiable.
 Formula A: (p  q)  (p  q)
p  q p  q (p  q) (p  q)  (p  q) A
p
q
1
1
1
0
0
1
0
1
0
0
1
1
1
0
0
1
1
0
0
1
0
0
0
1
0
0
1
0
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Logical entailment in PL
 A formula A logically follows from a set of
formulas M, denoted M |= A, iff A is true in
every model of the set M.
 Note: Mind the Definition 1 (slide 5 of Lesson
1). The circumstances are in propositional
logic mapped as valuations (True – 1, False
- 0) of elementary atomic sentences:

Under all the circumstances (means
valuations in PL) such that the premises are
true the conclusion must be true as well.
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Examples: Logical entailment
 He is at home (h) or he has gone to a pub (p)
 If he is at home (h) then he is waiting for us (w)
  If he is not waiting (w) for us then he has gone to the pub (p).




h, p, w | h  p, h  w | w  p
1 1 1
1
1
1
1 1 0
1
0
1
1 0 1
1
1
1
1 0 0
1
0
0
0 1 1
1
1
1
0 1 0
1
1
1
0 0 1
0
1
1
0 0 0
0
1
0
Introduction to Logic
conclusion
is true in all
the four models
of premises
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Examples: Logical entailment
 He is at home (h) or he has gone to a pub (p)
 If he is at home (h) then he is waiting for us (w)
  If he is not waiting (w) for us then he has gone to the pub (p).
h  p, h  w | w  p
 The table has 2n lines!
Hence, an indirect proof is more effective:
 Assume that the argument is not valid. But then all the
premises may be true and the conclusion false:

h  p, h  w | w  p
1
1
0
10 0
1 0 1 0
0
contradiction
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Examples: Logical entailment
 All the arguments with the same logical form
as a valid argument are valid:
h  p, h  w |= w  p
For variables h, p, w any elementary sentence can
be substituted:
He plays a piano or studies logic.
If he plays a piano then he is a virtuous.
Hence 
If he is not a virtuous then he studies logic.
Valid argument – the same valid logical form
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Logical entailment
 If the argument is valid,
P1,...,Pn |= Z,
then the formula of the implication form is a
tautology:

|= (P1 ... Pn)  Z.
 A proof that a formula is a tautology or that a
conclusion Z logically follows from premises can
be performed:
a)
b)
In the direct way – for instance by a truth-value table
In the indirect way: P1 ... Pn  Z is a
contradiction; hence the set of premises + negated
conclusion {P1, ..., Pn, Z} is contradictory, i.e., does
not have a model: there is no valuation under which
all the formulas – its elements were true.
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A proof of a tautology
|= ((p  q)  q)  p
Indirect:
((p  q)  q)  p negated f., must be a contradiction
1
1 attempt whether it can be 1
1
1
1
1
0
contradiction
There is no valuation under which the negated formula
were true. Therefore, the original formula is a tautology
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The most important tautologies
Tautologies with one propositional variable:
|= p  p
|= p  p
the law of excluded middle
|= (p  p) the law of contradiction
|= p  p
the law of double negation
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Algebraic laws for conjunction,
disjunction and equivalence
• |= (p  q)  (q  p)
• |= (p  q)  (q  p)
• |= (p  q)  (q  p)
commutative laws
• |= [(p  q)  r]  [p  (q  r)]
• |= [(p  q)  r]  [p  (q  r)]
• |= [(p  q)  r]  [p  (q  r)]
associative laws
• |= [(p  q)  r]  [(p  r)  (q  r)]
• |= [(p  q)  r]  [(p  r)  (q  r)]
distributive laws
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Laws of implication
|=
|=
|=
|=
|=
p  (q  p)
(p  p)  q
(p  q)  (q  p)
(p  (q  r))  ((pq)  r)
(p  (q  r))  (q  (p  r))
law of simplification
Duns Scot’s law
law of contra-position
premises joint
order of premises does not matter
|= (p  q)  ((q  r)  (p  r))
|= ((p  q)  (q  r))  (p  r)
|= (p  (q  r))  ((p  q)  (p  r))
hypothetic sylogism
transitivity of implication
Frege’s law
|= (p  p)  p
|= ((p  q)  (p  q))  p
reductio ad absurdum
reductio ad absurdum
|= (p  q)  p , |= (p  q)  q
|= p  (p  q) , |= q  (p  q)
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Laws of transformation
|=
|=
|=
|=
|=
|=
|=
(p  q)  (p  q)  (q  p)
(p  q)  (p  q)  (q  p)
(p  q)  (p  q)  (q  p)
(p  q)  (p  q)
(p  q)  (p  q)
Negation of implication
(p  q)  (p  q)
De Morgan law
(p  q)  (p  q)
De Morgan law
These laws define a method for negating
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Negation of implication
Parents: If you are good you well get a new ski at Christmas!
(p  q)
Child: I have been good all the year and there is no ski under the
Christmas tree!
p  q
(Did the parents fulfill their promise?)
Attorney general:
If the accused man is guilty then there was an accessory in the fact
Defence lawyer:
It is not true !
Question: Did the advocate (defence lawyer) help the
accused man; what did he actually claim?
(The man is guilty and he performed the illegal act
alone!)
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Negation of implication
Sentence in the future tense:
If you steel it I’ll kill you!
It is not true: I will steel it and yet you will not kill me.
(p  q)
p  q
OK, but:
If the 3rd world war breaks out tomorrow then more than three million
people will be killed.
It is not true: The 3rd world war will break out tomorrow and less than three
million people will be killed ???
Probably by negating the sentence we did not intend to claim that
(certainly) the 3rd world war will break out tomorrow:
There is an unsaid modality: Necessarily, if the 3rd world war breaks
out tomorrow then more than three million people will be killed.
It is not true: Possibly the 3rd world war breaks out tomorrow but at
that case less than three million people will be killed.
Handled by modal logics – not a subject of this course.
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Some more arguments
• Transformation from natural language may be
ambiguous:
If a man has high blood pressure and breathes
with difficulties or he has a fever then he is sick.
p – ”X has high blood pressure”
q – ”X breathes with difficulties”
r – ”X has a fever”
s – ”X is sick”
1. possible analysis:
[(p  q)  r]  s
2. possible analysis:
[p  (q  r)]  s
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Some more arguments
If Charles has high blood pressure and
breathes with difficulties or he has a fever
then he is sick.
Charles is not sick but he breathes with
difficulties.
 What can be deduced from these facts?
We have to distinguish the 1. reading and
the 2. reading because they are not
equivalent. The conclusions will be
different.
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Analysis of the 1. reading
1. analysis: [(p  q)  r]  s, s, q  ???
a) By means of equivalent transformations:
[(p  q)  r]  s, s   [(p  q)  r]  (de
transposition
Morgan)
(p  q)  r  (p  q), r, but q holds 
p, r (consequences)
Hence  Charles does not have a high blood
pressure and does not have a fever.
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Analysis of the 2. reading
2. analysis: [p  (q  r)]  s, s, q  ???
a) reasoning with equivalent transformations:
[p  (q  r)]  s, s   [p  (q  r)] 
transposition de Morgan:
p  (q  r)  but q is true  the second disjunct
cannot be true  the first is true:
p (consequence)
Hence  Charles does not have a high blood pressure
(we cannot conclude anything about his
temperature r)
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A proof of both cases
1. analysis: [(p  q)  r]  s, s, q |= p,r
2. analysis: [p  (q  r)]  s, s, q |= p
home work
a) 1. case – by means of a table: home work
b) Indirect: premises + negated conclusion (p  r) 
(p  r) and we assume that every f. is true:
•
[(p  q)  r]  s, s, q, p  r
•
1 10 1 1
•
0
0
•
0
0
•
0 1
pr=0
contradiction
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Summary
• Typical tasks:
–
–
–
–
Verifying a valid argument
What can be deduced from given assumptions?
Add the missing assumptions
Is a given formula a tautology, contradiction,
satisfiable?
– Find models of a formula, find a model of a set of
formulas
• Methods we have learnt till now:
– Table method
– reasoning and equivalent transformation
– Indirect proof
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Thank you for your attention
See you next week
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