Mathematical Logic: Lesson 2, propositional logic

1
Mathematical Logic Lesson 2, propositional
logic

• Marie Duží
• marie.duzi_at_vsb.cz

2
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
• 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.

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Examples of composed sentences

• It is raining in Prague and it is a sunshine in
  Brno.
• S1 connective S2
• It is not true that it is raining in Prague.
• 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
• Symbols for propositions p, q, r, ... (also
  with indexes p1, p2, )
• Symbols for logical connectives ?, ?, ?, ?, ?
• 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
• Symbols p, q, r, ... are (well-formed) formulas
  (the definition base).
• If A, B are formulas, then expressions
• ??A?, ?A ? B?, ?A ? B?, ?A ? B?, ?A ? B?
  
• are (well-formed) formulas (inductive definition
  step).
• 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 truthvalue functions

A B ?A A ? B A ? B A ? B A ? B
1 1 0 1 1 1 1
1 0 0 1 0 0 0
0 1 1 1 0 1 0
0 0 1 0 0 1 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 224 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 Ill go to the cinema
  or Ill stay at home
• Alternative or is a non-equivalence

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Implication be careful !!!

• if then ?
• (binary, non-commutative connective)A ? B A is
  the antecedent, B is the consequent.
• Implication (as well as any other connective of
  propositional logic) does not render any semantic
  connection between antecedent and consequent
• material implication (middle ages suppositio
  materialis).
• Hence implication does not render a causal or
  chronological connection
• If 112, then iron is a metal (a true
  proposition) p ? q
• If the UFOs (flying saucers) exist, then I am
  the Pope p ? r (What do I want to say?
  Since I am not the Pope, the UFOs do not exist)

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Implication be careful !!!

• 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

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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) Ill slap you if you cheat on me
  cheat ? slap
• b) Ill slap you if and only if you cheat on
  me 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 is 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) ? (p ? ?q) ?A
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 of
  atomic propositional variables 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 conclusion
• 1 1 0 1 0 1
• ? 1 0 1 1 1 1 is true
  in all
• 1 0 0 1 0 0
• ? 0 1 1 1 1 1 the
  four models
• ? 0 1 0 1 1 1 of
  premises
• 0 0 1 0 1 1
• 0 0 0 0 1 0

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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
• 1 0 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 sentences
  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

• The argument is valid
• P1,...,Pn Z
• iff the formula of the implicative form is a
  tautology
• (P1 ?...? Pn) ? Z.
• The proof that a formula is a tautology or that a
  conclusion Z logically follows from premises can
  be done
• 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 the
  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) commutative laws
• (p ? q) ? (q ? p)
• (p ? q) ? (q ? p)
• (p ? q) ? r ? p ? (q ? r) associative
  laws
• (p ? q) ? r ? p ? (q ? r)
• (p ? q) ? r ? p ? (q ? r)
• (p ? q) ? r ? (p ? r) ? (q ?
  r) distributive laws
• (p ? q) ? r ? (p ? r) ? (q ? r)

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Laws for implication

• p ? (q ? p) law of simplification
• (p ? ?p) ? q Duns Scots law
• (p ? q) ? (?q ? ?p) law of contra-position
• (p ? (q ? r)) ? ((p?q) ? r) premises joint
• (p ? (q ? r)) ? (q ? (p ? r)) order of
  premises does not matter
• (p ? q) ? ((q ? r) ? (p ? r)) hypothetic
  sylogism
• ((p ? q) ? (q ? r)) ? (p ? r) transitivity
  of implication
• (p ? (q ? r)) ? ((p ? q) ? (p ? r)) Freges
  law
• (?p ? p) ? p reductio ad absurdum
• ((p ? q) ? (p ? ?q)) ? ?p 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

• Implication works well in case of a promise.
• Example
• Parents If you behave well you will get a new
  iPhone at Christmas! (p ? q)
• Child I did behave well the whole year and there
  is no iPhone under the Christmas tree!
• p ? ?q
• (Did the parents fulfill their promise?)
• Public prosecutor
• If the accused man is guilty then he had an
  accomplice
• Defence lawyer
• It is not true !
• Question Did the advocate (defence lawyer) help
  the accused man? What did he actually say?
• (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 Ill kill you! (p ? q)
• It is not true I will steel it and yet you will
  not kill me. 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 (ecliptic) 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 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 first and second
  reading, because they are not equivalent. The
  conclusions will be different.

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Analysis of the 1. reading

• analysis (p ? q) ? r ? s, ?s, q ? ???
• 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

• analysis p ? (q ? r) ? s, ?s, q ? ???
• 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 must be 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
• 1. case by means of a table home work
• 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 1 0 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 transformations
• Indirect proof

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Propositional Satisfiability problem (the SAT
problem)

• In computer science, the Boolean satisfiability
  problem (sometimes called Propositional
  Satisfiability Problem and abbreviated as
  SATISFIABILITY or SAT) is the problem of
  determining if there exists an interpretation
  that satisfies a given propositional formula.
• In other words, it asks whether the propositional
  variables of a given formula can be consistently
  replaced by the values 1 or 0 in such a way that
  the formula evaluates to 1.
• If this is the case, the formula is called
  satisfiable. On the other hand, if no such
  assignment exists, the function expressed by the
  formula is FALSE for all possible variable
  assignments and the formula is unsatisfiable
  (contradiction).
• For example, the formula p ? ?q" is satisfiable
  because one can find the values p  1 and q  0,
  which make (p ? ?q)  1. In contrast, p ? ?p" is
  a contradiction (unsatisfiable).
• SAT is the first problem that was proven to be
  NP-complete see CookLevin theorem. This means
  that all problems in the complexity class NP,
  which includes a wide range of natural decision
  and optimization problems, are at most as
  difficult to solve as SAT. There is no known
  algorithm that efficiently solves each SAT
  problem, and it is generally believed that no
  such algorithm exists yet this belief has not
  been proven mathematically, and resolving the
  question whether SAT has a polynomial-time
  algorithm is equivalent to the P versus NP
  problem, which is a famous open problem in the
  theory of computing.
• Nevertheless, as of 2016, heuristical
  SAT-algorithms are able to solve problem
  instances involving tens of thousands of
  variables and formulas consisting of millions of
  symbols, which is sufficient for many practical
  SAT problems from e.g. artificial intelligence,
  circuit design, and automatic theorem proving.
• For details see, e.g. https//en.wikipedia.org/wik
  i/Boolean_satisfiability_problem