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Přednáška 5
Sémantický výklad PL1
(pokračování a opakování)
Prázdné universum ?
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Představme si interpretaci nad universem U = 
x P(x) je pravda či nepravda?
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dle definice kvantifikátoru je to nepravda, neboť nenajdeme žádné
individuum, které splňuje P, pak je ale pravda x P(x), tj.
x P(x), ale to je nepravda – spor
Nebo je to pravda, protože neexistuje prvek univerza, který by
neměl vlastnost P, pak ale má být také pravda x P(x), což je
nepravda – spor
Podobně pro x P(x) dojdeme ke sporu, ať to
chápeme jako pravda či nepravda
Proto volíme vždy neprázdné universum
Logika pro pustý svět by byla nesmyslná
Přednáška 5
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Existenční kvantifikátor +
implikace ?
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Existuje někdo takový, že je-li to génius,
pak jsou všichni géniové
Tato věta nemůže být nepravdivá !
|= x (G(x)  x G(x))
V každé interpretaci I bude platit:
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Je-li obor pravdivosti GU predikátu G roven celému
universu (GU = U), pak je formule v I pravdivá, neboť je
pravdivá podformule x G(x), tedy i G(x)  x G(x).
Je-li GU vlastní podmnožinou U (GU  U), pak stačí
nalézt jeden prvek (valuaci x), který neleží v GU a
formule G(x)  x G(x) je v I pravdivá, neboť není
pravdivý antecedent G(x).
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Existenční kvantifikátor +
konjunkce !
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Podobně formule x (P(x)  Q(x)) je „skoro
tautologie“. Je pravdivá v každé interpretaci I
takové, že
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PU  U, neboť pak je |=I P(x)  Q(x) [e] pro e(x)  PU
nebo QU = U, neboť pak je |=I P(x)  Q(x) pro všechny
valuace e
Tedy tato formule je nepravdivá pouze v takové I,
kde PU = U a QU  U.
Proto věty typu „Některá P jsou Q“ analyzujeme
jako x (P(x)  Q(x)).
Přednáška 5
4
Všeobecný kvantifikátor +
konjunkce ? Implikace !
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x [P(x)  Q(x)] je „skoro nesplnitelná“ !
Je nepravdivá v každé interpretaci I, ve které je
PU  U nebo QU  U.
Tedy pravdivá je pouze v takové I, kde
PU = U a QU = U !
Proto věty typu „Všechna P jsou Q“ analyzujeme
jako x [P(x)  Q(x)]
Pro všechna individua platí, že je-li to P, pak je to
také Q. (Pak je PU  QU – definice podmnožiny)
Přednáška 5
5
Relace a vztahy
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Výroky s jedno-argumentovým predikátem
(charakterizujícím nějakou vlastnost) zkoumal již
ve starověku Aristoteles.
Teprve Gottlob Frege (zakladatel moderní
logiky) však zavedl formální predikátovou logiku
(s poněkud jiným jazykem, než používáme
dnes) s více-argumentovými predikáty
(charakterizujícími vztahy) a kvantifikátory.
Přednáška 5
6
Sémantické ověření
správnosti úsudku
Kdo zná Marii a Karla, ten Marii lituje.
x [(Z(x,m)  Z(x,k))  L(x,m)]
Někteří nelitují Marii, ačkoliv ji znají. x [L(x,m)  Z(x,m)]
|= Někdo zná Marii, ale ne Karla.
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Znázorníme, jaké budou obory pravdivosti predikátů Z a L, tj.
relace ZU a LU, aby byly pravdivé premisy:
ZU = {…, i1,m,  i1,k, i2,m, i2,k,…,,m,… }
1. premisa
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x [Z(x,m)  Z(x,k)]
2. premisa
LU = {…, i1,m, ...., i2,m,…........., ,m,… }
Přednáška 5
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Sémantické ověření
správnosti úsudku
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Kdo zná Marii a Karla, ten Marii lituje.
Někteří nelitují Marii, ačkoliv ji znají.
x [(Z(x,m)  Z(x,k))  L(x,m)]
x [L(x,m)  Z(x,m)]
x [Z(x,m)  Z(x,k)]

|= Někdo zná Marii, ale ne Karla.
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Nyní dokážeme platnost sporem: předpokládejme, že všichni, kdo jsou ve
vztahu Z k m (tedy i prvek ), jsou také v Z ke k (negace závěru).
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ZU = {…, i1,m,  i1,k, i2,m, i2,k,…,,m, ,k, … }
1. p.
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2. p.
1.p.
LU = {…, i1,m, ..., i2,m,…................, ,m, ,m, … }
spor
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Gottlob Frege
1848 – 1925
Německý
matematik,
logik a filosof,
působil na
universitě v Jeně.
Zakladatel
moderní logiky
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Gottlob Frege
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Friedrich Ludwig Gottlob Frege (b. 1848, d. 1925) was a German
mathematician, logician, and philosopher who worked at the University of
Jena.
Frege essentially reconceived the discipline of logic by constructing a
formal system which, in effect, constituted the first ‘predicate calculus’. In
this formal system, Frege developed an analysis of quantified statements
and formalized the notion of a ‘proof’ in terms that are still accepted today.
Frege then demonstrated that one could use his system to resolve
theoretical mathematical statements in terms of simpler logical and
mathematical notions. Bertrand Russell showed that of the axioms that
Frege later added to his system, in the attempt to derive significant parts of
mathematics from logic, proved to be inconsistent.
Nevertheless, his definitions (of the predecessor relation and of the
concept of natural number) and methods (for deriving the axioms of
number theory) constituted a significant advance. To ground his views about
the relationship of logic and mathematics, Frege conceived a
comprehensive philosophy of language that many philosophers still find
insightful. However, his lifelong project, of showing that mathematics was
reducible to logic, was not successful.
Stanford Encyclopedia of Philosophy
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http://plato.stanford.edu/entries/frege/
Bertrand Russell
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1872-1970
Britský filosof, logik,
esejista
Přednáška 5
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Bertrand Russell
Bertrand Arthur William Russell (b.1872 - d.1970) was
a British philosopher, logician, essayist, and social
critic, best known for his work in mathematical logic
and analytic philosophy. His most influential
contributions include his defense of logicism (the
view that mathematics is in some important sense
reducible to logic), and his theories of definite
descriptions and logical atomism. Along with G.E.
Moore, Russell is generally recognized as one of the
founders of analytic philosophy. Along with Kurt
Gödel, he is also regularly credited with being one
of the two most important logicians of the twentieth
century.
Přednáška 5
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Kurt Gödel (1906-Brno, 1978-Princeton)
Největší logik 20. století, přítel A. Einsteina,
proslavil se větami o neúplnosti teorie aritmetiky
Russell's Paradox
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Russell's paradox is the most famous of the
logical or set-theoretical paradoxes. The
paradox arises within naive set theory by
considering the set of all sets that are not
members of themselves. Such a set appears
to be a member of itself if and only if it is not
a member of itself, hence the paradox.
http://plato.stanford.edu/entries/russellparadox/
Přednáška 5
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Russell's Paradox
Some sets, such as the set of all teacups, are not
members of themselves. Other sets, such as the set
of all non-teacups, are members of themselves. Call
the set of all sets that are not members of
themselves "R." If R is a member of itself, then by
definition it must not be a member of itself. Similarly,
if R is not a member of itself, then by definition it
must be a member of itself. Discovered by Bertrand
Russell in 1901, the paradox has prompted much
work in logic, set theory and the philosophy and
foundations of mathematics.
Přednáška 5
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Russell's Paradox
R – množina všech normálních množin, které nejsou
prvky sebe sama
Otázka: Je R  R? Je R normální? vede ke sporu.
 Symbolicky: xR  (xx) – definice množiny R
 Položená otázka Je R  R? vede ke sporné formuli
(kontradikci):
 RR  RR, neboť:
 Odpověď Ano – R není normální (RR), pak dle
definice množiny R nemá být R prvkem R, tj. RR
 Odpověď Ne – R je normální (RR), pak je ale dle
definice RR
(R je množina všech normálních množin)
Přednáška 5
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Russell wrote to Gottlob Frege with news of his
paradox on June 16, 1902. The paradox was of
significance to Frege's logical work since, in effect, it
showed that the axioms Frege was using to
formalize his logic were inconsistent.
Specifically, Frege's Rule V, which states that two sets
are equal if and only if their corresponding functions
coincide in values for all possible arguments,
requires that an expression such as f(x) be
considered both a function of the argument x and a
function of the argument f. In effect, it was this
ambiguity that allowed Russell to construct R in
such a way that it could both be and not be a
member of itself.
Přednáška 5
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Russell's Paradox
Russell's letter arrived just as the second volume of Frege's Grundgesetze der
Arithmetik (The Basic Laws of Arithmetic, 1893, 1903) was in press.
Immediately appreciating the difficulty the paradox posed, Frege added to the
Grundgesetze a hastily composed appendix discussing Russell's discovery.
In the appendix Frege observes that the consequences of Russell's paradox
are not immediately clear. For example, "Is it always permissible to speak of
the extension of a concept, of a class? And if not, how do we recognize the
exceptional cases? Can we always infer from the extension of one concept's
coinciding with that of a second, that every object which falls under the first
concept also falls under the second? These are the questions," Frege notes,
"raised by Mr Russell's communication." Because of these worries, Frege
eventually felt forced to abandon many of his views about logic and
mathematics.
Of course, Russell also was concerned about the contradiction. Upon learning
that Frege agreed with him about the significance of the result, he
immediately began writing an appendix for his own soon-to-be-released
Principles of Mathematics. Entitled "Appendix B: The Doctrine of Types," the
appendix represents Russell's first detailed attempt at providing a principled
method for avoiding what was soon to become known as "Russell's paradox."
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Russell's Paradox
The significance of Russell's paradox can be seen once it is realized that, using classical logic, all
sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary
proposition, Q, can be proved as follows: from P we obtain P Q by the rule of Addition; then from
P Q and ~P we obtain Q by the rule of Disjunctive Syllogism. Because of this, and because set
theory underlies all branches of mathematics, many people began to worry that, if set theory was
inconsistent, no mathematical proof could be trusted completely.
Russell's paradox ultimately stems from the idea that any coherent condition may be used to
determine a set. As a result, most attempts at resolving the paradox have concentrated on various
ways of restricting the principles governing set existence found within naive set theory, particularly
the so-called Comprehension (or Abstraction) axiom. This axiom in effect states that any
propositional function, P(x), containing x as a free variable can be used to determine a set. In other
words, corresponding to every propositional function, P(x), there will exist a set whose members
are exactly those things, x, that have property P. It is now generally, although not universally,
agreed that such an axiom must either be abandoned or modified.
Russell's own response to the paradox was his aptly named theory of types. Recognizing that selfreference lies at the heart of the paradox, Russell's basic idea is that we can avoid commitment to
R (the set of all sets that are not members of themselves) by arranging all sentences (or,
equivalently, all propositional functions) into a hierarchy. The lowest level of this hierarchy will
consist of sentences about individuals. The next lowest level will consist of sentences about sets of
individuals. The next lowest level will consist of sentences about sets of sets of individuals, and so
on. It is then possible to refer to all objects for which a given condition (or predicate) holds only if
they are all at the same level or of the same "type."
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Russell’s paradox – 3 solutions
Russell's own response to the paradox was his aptly named theory of types.
Recognizing that self-reference lies at the heart of the paradox, Russell's basic
idea is that we can avoid commitment to R (the set of all sets that are not
members of themselves) by arranging all sentences (or, equivalently, all
propositional functions) into a hierarchy. The lowest level of this hierarchy will
consist of sentences about individuals. The next lowest level will consist of
sentences about sets of individuals. The next lowest level will consist of
sentences about sets of sets of individuals, and so on. It is then possible to
refer to all objects for which a given condition (or predicate) holds only if they
are all at the same level or of the same "type."
This solution to Russell's paradox is motivated in large part by the so-called
vicious circle principle, a principle which, in effect, states that no
propositional function can be defined prior to specifying the function's range. In
other words, before a function can be defined, one first has to specify exactly
those objects to which the function will apply. (For example, before defining the
predicate "is a prime number," one first needs to define the range of objects
that this predicate might be said to satisfy, namely the set, N, of natural
numbers.) From this it follows that no function's range will ever be able to
include any object defined in terms of the function itself. As a result,
propositional functions (along with their corresponding propositions) will end up
being arranged in a hierarchy of exactly the kind Russell proposes.
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Russell’s paradox – 3 solutions
Although Russell first introduced his theory of types in his 1903 Principles of Mathematics,
type theory found its mature expression five years later in his 1908 article, "Mathematical
Logic as Based on the Theory of Types," and in the monumental work he co-authored
with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913). Russell's
type theory thus appears in two versions: the "simple theory" of 1903 and the "ramified
theory" of 1908. Both versions have been criticized for being too ad hoc to eliminate the
paradox successfully. In addition, even if type theory is successful in eliminating
Russell's paradox, it is likely to be ineffective at resolving other, unrelated paradoxes.
Other responses to Russell's paradox have included those of David Hilbert and the
formalists (whose basic idea was to allow the use of only finite, well-defined and
constructible objects, together with rules of inference deemed to be absolutely certain),
and of Luitzen Brouwer and the intuitionists (whose basic idea was that one cannot
assert the existence of a mathematical object unless one can also indicate how to go
about constructing it).
Yet a fourth response was embodied in Ernst Zermelo's 1908 axiomatization of set theory.
Zermelo's axioms were designed to resolve Russell's paradox by again restricting the
Comprehension axiom in a manner not dissimilar to that proposed by Russell. ZF and
ZFC (i.e., ZF supplemented by the Axiom of Choice), the two axiomatizations generally
used today, are modifications of Zermelo's theory developed primarily by Abraham
Fraenkel.
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