Marie Du

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Introduction to Logic

• Marie Duží
• marie.duzi_at_vsb.cz

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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.

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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
• ????????????? invalid
• Lecture on Logic takes place
• 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.
• ?????????????? induction, invalid
• (Therefore) All swans are white

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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

• 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. WellIf
  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
  wouldnt survive without logic
• 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.
• 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
• 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

• All men like football and beer.
• Some beer-lovers do not like football.
• Xaver likes only those who like football and
  beer.
• 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.
• 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.
• 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 No prime number is divisible by
  three.
• The number 9 is divisible by three.
• valid
• The number 9 is not a prime.
• Argument B No prime number is divisible by six.
• The number eight is not a prime.
• invalid
• The number eight is not divisible by six.
• 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 Ill pass the exam.
• I havent 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

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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.
• a, b, c ? a, d a, b, c, d
• odd numbers ? even numbers natural
  numbers denoted Nat
• Ui?I Ai x x ? Ai for some i ? I
• Let Ai x x 2.i for some i ? Nat
• Ui?Nat Ai the set of all even numbers

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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.
• a, b, c ? a, d a
• even numbers ? odd numbers ?
• ?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)
• 2a,b ?, a, b, a,b
• 2a,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 2A elements (hence the
  notation 2A)
• 2a,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)
• B P\(S?M) (P\S)?(P\M)
• P(x) ? ?(S(x) ? M(x)) ? P(x) ? ?S(x) ? ?M(x)
• C (S ? P) \ M
• S(x) ? P(x) ? ?M(x)
• D S ? P ? M
• S(x) ? P(x) ? M(x)
• E (S ? M) \ P
• S(x) ? M(x) ? ?P(x)
• F (P ? M) \ S
• P(x) ? M(x) ? ?S(x)
• G M\(P?S) (M\P)?(M\S)
• 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)

S
A
E
C
D
G
B
F
P
M
H
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Russells 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.

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• The end of lesson 1