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Introduction to Logic
Marie Duží
[email protected]
1
Introduction to Logic
Texts to study:
http://www.cs.vsb.cz/duzi
Courses
Introduction to Logic: Information for students
Chapters:
1. Introduction
2. propositional Logic
2.1. Semantic exposition
2.2. Resolution method
3. Predicate Logic
3.1. Semantic exposition
3.2. General resolution method
Presentation of lectures
Book: Gamut L.T.F., Logic, Language and Meaning, Chicago
Press 1991, Vol. 1.
Chapter 1, Chapter 2 (except of 2.7), Chapter 3, and
Chapter 4 – Sections: 4.1, 4.2, 4.4.
Introduction to Logic
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Requirements to pass the course
• Accreditation:
• Two written tests
– Propositional logic
- max. 15 grades, min. 5 grades.
– Predicate logic
- max. 15 grades, min. 5 grades.
– No repetitions for the tests!
+ 5 grades for being active
• Requirement: obtaining at least 15 grades.
• Exam: Written test (max. 65 grades, min. 30 grades)
• Total:
– at least 51 grades – good (3),
– at least 66 grades – very good (2),
– at least 86 grades – excellent (1)
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1. Introduction
What is logic about?
What is the subject of logic?
Logic is the science of correct, valid reasoning, or, in other words, the art of
a valid argumentation
What is an argument?
Argument: On the assumption of true premises P1,...,Pn it is possible to
reason that the conclusion Z is true as well:
P1, ..., Pn

Z
Example: On the assumption that it is Thursday I belief that today a lecture
on Introduction to logic takes place: Thursday  Lecture on Logic
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Introduction: valid arguments
In this course we deal only with deductively valid
arguments. Notation: P1,...,Pn |= Z
The conclusion Z logically follows from the
premises P1,..., Pn.
Definition 1:
The conclusion Z logically follows from the
premises P1,...,Pn , notation: P1,...,Pn |= Z, iff
under no circumstances it might happen that
the premises were true and the conclusion false.
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Introduction: valid arguments
Example: Because it is Thursday today I believe that the
lecture “Introduction to Logic” takes place:
It is Thursday

Lecture on Logic takes place
invalid
Is it a deductively valid argument? No, it is not: It might
happen that Duzi were sick and the lecture does not take
place though it is Thursday (a premise is missing, for
instance that Each Thursday the lecture takes place).
Each Thursday the lecture on Logic takes place.
It is Thursday today

valid 
Today the lecture on Logic takes place.
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(Deductively) invalid arguments:
generalization (induction), abduction
We will not deal with arguments that are not
deductively valid, like: generalization (induction),
abduction, and other –ductions  a subject of
Artificial Intelligence (non-monotonic reasoning)
Examples:
Till now logic always took place on Thursday.

induction, invalid
(Therefore) Logic will take place also this Thursday
All swans that I have seen till now are white.

(Therefore) All swans are white
Introduction to Logic
induction, invalid
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Deductively invalid arguments:
generalization (induction), abduction
Examples:
All rabbits in the hat are white.
These rabbits are from the hat.
 These rabbits are white.
Deduction, valid
These rabbits are from the hat.
These rabbits are white.
 (Probably) All rabbits in the hat are white.
Generalization, Induction, invalid
All rabbits in the hat are white.
These rabbits are white.
 (Probably because) These rabbits are from the hat.
Abduction, invalid
Seeking premises, causes of events, diagnosis of “malfunctions”
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Examples of deductively valid
arguments
1.
He is at home or he has gone to a pub.
If he is at home then he plays a piano.
But he did not play a piano.
------------------------------------------------ Hence
He has gone to the pub.
Sometimes the arguments are so obvious that it seems as if we did not need any logic.
Well:
If he did not play a piano (3. premise), then he was not at home (2. premise), and
according to the first premise he must have gone to the pub.
But, we all use logic in our everyday life, we wouldn’t survive without logic:
2.
All agarics (mushrooms) have a strong toxic effect.
The mushroom I have picked up is an agaric.
---------------------------------------------------------------------The mushroom I have picked up has a strong toxic effect.
Will you examine the mushroom by tasting it, or will you rely on logic?
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Examples of deductively valid
arguments
All agarics (mushrooms) have a strong toxic effect.
This apple is an agaric.
---------------------------------------------------------------------Hence  This apple has a strong toxic effect.
The argument is valid. But the conclusion is evidently not true (false).
Hence, at least one premise is false (obviously the second).
Circumstances according to Definition 1 are particular interpretations
(depending on the expressive power of the logical system).
Logical connectives (‘and’, ‘or’, ‘if …then …’) and quantifiers (‘all’,
‘some’, ‘every’, …) have a fixed interpretation; we interpret
elementary propositions and/or their parts.
In our example, if “this apple” and “agarics” were interpreted in such a
way that the second premise were true, the truth of the conclusion
is guaranteed.
We also say that the argument has a valid logical form.
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Deductively valid arguments
Logic is a tool that helps us to discover the relation of logical entailment,
to answer questions like „What follows from particular assumptions “?, etc.
1.
2.
3.
If the course is good then it is useful.
The lecturer is sharply demanding studiousness or the course is not useful.
But the lecturer is not demanding.
-------------------------------------------------------------------------- Hence
4.
•
The course is not good.
It helps our intuition that can sometimes fail.
–
–
–
The assumptions can be complicated, “enmeshed in negations and other
connectives”, so that the relation of entailment is not obvious at first sight.
Similarly as all the mother-tongue speakers use intuitively rules of grammar
without knowing the grammar explicitly (often not being able to formulate the
rules).
But sometimes it is useful to consult the grammar book or a dictionary (in
particular when taking part in a TV competition).
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Examples of valid arguments
1.
All men like football and beer.
2.
Some beer-lovers do not like football.
3.
Xaver likes only those who like football and beer.
––––––––––––––––––––––––––––––––––––
4.
Xaver does not like some women.
Necessarily, if premises are true the conclusion has to be true as well.
Is this argument valid?
Certainly, if Xaver likes only those who like football and beer (premise 3), then he does not like some
beer-lovers (namely those who do not like football – according to the premise 2). Hence,
(according to 1) he does not like some “no-men”, i.e., women.
But according to the Definition 1 the argument is not valid: the argument is valid if necessarily,
i.e., in all the circumstances (under all interpretations) in which the premises are true
the conclusion is true as well.
But: in our case those individuals that are not men would not have to be interpreted as women.
A premise is missing, viz. the premise “who is not a man is a woman”. Moreover, to be precise, we
should also specify that “who is a lover of something he likes that”.
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Examples of valid arguments
Hence: We have to state all the premises necessary for deriving the
conclusion.
1.
2.
3.
4.
5.
All men like football and beer.
Some beer-lovers do not like football.
Xaver likes only those who like football and beer.
Who is not a man is a woman.
Who is a lover of something he likes it.
––––––––––––––––––––––––––––––––––––
6.
Xaver does not like some women.
Now the argument is valid, it has a valid logical form.
The conclusion is logically entailed by (follows from) the premises. We
also say that the conclusion is informationally (deductively)
contained in the premises.
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Valid arguments in mathematics
Argument A:
Argument B:
No prime number is divisible by three.
The number 9 is divisible by three.
–––––––––––––––––––––––––––
The number 9 is not a prime.
No prime number is divisible by six.
The number eight is not a prime.
–––––––––––––––––––––––––––
The number eight is not divisible by six.
valid
invalid
Though in the second case B it can never happen that the premises were true
and the conclusion false, the argument is invalid. The conclusion is not
logically entailed by the premises.
If the expression “eight” were interpreted as the number 12, the premises
would be true and the conclusion false.
(The conclusion is not deductively contained in the premises)
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Theorem of deduction;
semantic variant
If the argument P1,...,Pn |= Z is valid, then the
statement of the form “if P1 and ... and Pn then Z”
P1 &...& Pn  Z
is analytically (necessarily) true.
Notation: |= P1  ...  Pn  Z.
Hence: P1,...,Pn |= Z
 (if and only if)
P1,...,Pn-1 |= (Pn  Z)

P1,...,Pn-2 |= ((Pn-1  Pn)  Z) 
P1,...,Pn-3 |= ((Pn-2  Pn-1  Pn)  Z)  …
|= (P1 ... Pn)  Z
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Logical analysis of language
Validness of an argument is determined by the meaning (interpretation)
of particular statements that are analyzed (formalized) in a less or
more fine-grained way according to the expressive power of a
logical system:
•
•
•
•
•
Propositional logic: makes it possible to analyze only the way in which a
complex statement is composed from elementary propositions. The
composition of elementary propositions is not examined, they contribute
only by its truth value:
True – 1, False – 0 (an algebra of truth values)
1st-order Predicate logic: makes it possible to analyze moreover the
composition of elementary propositions, namely the way in which
properties and/or relations are ascribed to (tuples of) individuals.
2nd-order Predicate logic: makes it possible to analyze moreover properties
of properties, propertied of functions and relations between them.
Modal logics (analyze “necessary” and “possible”), epistemic logics
(knowledge), doxastic logics (of hypotheses) deontic logics (of commands),
...
Transparent intensional logic (perhaps the most powerful system) –
see the course “Principles of logical analysis“.
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Properties of valid arguments

A valid argument may have a false conclusion:
 All primes are odd
 The number 2 is not odd
  The number 2 is not a prime
But then at least one premise has to be false
In such a case we also say that the argument is not sound. But a valid
argument that is not sound may also be useful: a proof ad absurdum.
If you want to show that your boss is not right, it is not diplomatic to
say it in an open way. Instead, you may argue by way of the proof
ad absurdum:
“Well, you say P – interesting, but P entails Q, and Q entails R,
which is obviously false.” (Hence, P must have been false as well.)

Monotonicity: if an argument is valid then extending the set of assumptions by
another premise does not change the validity of the argument.
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Properties of valid arguments
• From contradictory (inconsistent) assumptions (such that it can
never happen that all of them were simultanously true) any
conclusion follows.
• If I study hard then I’ll pass the exam.
• I haven’t passed the exam though I studied hard.
-------------------------------------------------------------------•  (e.g.) My dog plays a piano right now
• Reflexivity: If A is one of the assumptions P1,...,Pn, then
P1,...,Pn |= A.
• Transitivity: If P1, …, Pn |= Z and Q1, …, Qm, Z |= Z’, then
P1, …, Pn, Q1, …, Qm |= Z’ .
Break
Introduction to Logic
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Naïve theory of sets

What is it a set?




A set is a collection of elements,
and it is determined just by its elements;
a set consisting of elements a, b, c is denoted: {a, b, c}
An element of a set can be again a set,
a set may consist of no elements, it may be empty (denoted by ) !
Examples: , {a, b}, {b, a}, {a, b, a}, {{a, b}}, {a, {b, a}},
{, {}, {{}}}
Sets are identical if and only if (iff) they have exactly the
same elements (the principle of extensionality)



Notation: x  M – „x is an element of M“
a  {a, b}, a  {{a, b}}, {a, b}  {{a, b}},   {, {}, {{}}},
  {, {}}, but: x   for any (i.e., all) x.
{a, b} = {b, a} = {a, b, a}, but: {a, b}  {{a, b}}  {a, {b, a}}
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Set-theoretical operations
(create new sets from sets)

Union: A  B = {x | x  A or x  B}
read: „The set of all x such that x is an element of A
or x is an element of B.“
b, c}  {a, d} = {a, b, c, d}
 {odd numbers}  {even numbers} = {natural numbers}
– denoted Nat
 {a,

UiI Ai = {x | x  Ai for some i  I}
Ai = {x | x = 2.i for some i  Nat}
UiNat Ai = the set of all even numbers
 Let

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Set-theoretical operations
(create new sets from sets)

Intersection: A  B = {x | x  A and x  B}
read: „The set of all x such that x is an element of A
and x is an element of B as well.“
b, c}  {a, d} = {a}
 {even numbers}  {odd numbers} = 
 {a,
 iI Ai = {x | x  Ai for all i  I}
 Let

Ai = {x | x  Nat, x  i}
Then iNat Ai = 
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Relations between sets
A set A is a subset of a set B, denoted A  B,
iff each element of A is also an element of B.
 A set A is a proper subset of a set B, denoted
A  B, iff each element of A is also an
element of B but not vice versa.

{a}  {a}  {a, b}  {{a, b}} !!!
It holds: A  B, iff A  B and A  B
 It holds: A  B, iff A  B = B, iff A  B = A

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Some other set-theoretical operations

Difference: A \ B = {x | x  A and x  B}
 {a,






b, c} \ {a, b} = {c}
Complement: Let A  M. The complement of A with
respect to M is the set A’ = M \ A
Cartesian product: A  B = {a,b | aA, bB}
where a,b je an ordered couple
(the ordering is important:
a is the first, b is the second)
It holds: a,b = c,d iff a = c, b = d
But: a,b  b,a, though {a,b} = {b,a} !!!
generalization: A  …  A the set of n-tuples,
denoted also by An
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Some other set-theoretical operations

Potential set: 2A = {B | B  A},
denoted also by P(A)
2{a,b} = {, {a}, {b}, {a,b}}
2{a,b,c} = {, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}
How many elements are there in 2A ?
If |A| is the number of elements (cardinality) of a set
A, then 2A has 2|A| elements (hence the notation: 2A)
2{a,b}  {a} = {, {a,a}, {b,a}, {a,a, b,a}}
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Grafical picturing (in a universe U):
A: S\(PM) = (S\P)(S\M)
S(x)  (P(x)  M(x))  S(x)  P(x)  M(x)
S
B: P\(SM) = (P\S)(P\M)
P(x)  (S(x)  M(x))  P(x)  S(x)  M(x)
A
C: (S  P) \ M
S(x)  P(x)  M(x)
E
C
D: S  P  M
D
B
F
S(x)  P(x)  M(x)
G
E: (S  M) \ P
S(x)  M(x)  P(x)
F:
P
M
(P  M) \ S
P(x)  M(x)  S(x)
G: M\(PS) = (M\P)(M\S)
H
M(x)  (P(x)  S(x))  M(x)  P(x)  S(x)
H: U \ (S  P  M) = (U \ S  U \ P  U \ S)
(S(x)  P(x)  M(x))  S(x)  P(x)  M(x)
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Russell’s paradox




Is it true that any collection of elements (i.e., a collection
defined in an arbitrary way) can be considered to be a set?
It is normal that a set and its elements are entities of
different types. Hence a “normal set” is not an element of
itself.
Let N is a set of all normal sets:
N = {M | M  M}.
Question: Is N  N ? In other words, is N itself normal?



Yes?
But then according to the definition of N it holds that N is normal, i.e.,
NN.
No?
But then NN, hence N is normal, and therefore it belongs to N, i.e.,
NN.
Both the answers lead to a contradiction. N is not well defined. The
definition does not determine a collection of elements that could be
considered to be a set. Introduction to Logic
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The end of lesson 1
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