Download Vydatná motivace jako predkrm, porádná porce Gödelových vet a

Něco málo o logice
Vydatná motivace jako předkrm, pořádná porce Gödelových vět
a trocha fuzzy logiky jako zákusek
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Akademie věd České Republiky
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Outline
1
What is logic? (And what is it good for?)
2
Classical propositional logic (the simplest logic there is)
3
Gödel’s theorems (a rough and dirty account)
4
Mathematical fuzzy logic (the stuff I do)
5
The future of fuzzy logic (and your role in it)
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Logic is not about solving logical puzzles
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Logic is not about solving logical puzzles
http://www.xkcd.com/246/
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What is logic? (paraphrasing Wikipedia entry)
Logic is generally held to be the systematic study of valid arguments
There is no universal agreement as to the exact scope and subject
matter of logic, but it has traditionally included
the classification of arguments,
the systematic exposition of the ’logical form’ common to all valid
arguments, and
the study of fallacies and paradoxes.
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Logic throughout the history
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Logic throughout the history
Logic started as a part of philosophy in IV century BCE
Aim: to recognize valid arguments in philosophical discussion
All men are mortal and Socrates in a man, therefore Socrates is mortal
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Logic throughout the history
Since 19th century a part of logic has evolved into mathematical logic
Aim: to ‘solve’ the crises of the foundation of mathematics
PA 6` ¬Pr(0 = S(0))
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Logic throughout the history
Nowadays logic is applied mainly in computer science
Aim: to describe and perform ‘reasoning’ in various formal settings
` [α](x = 4) → [α; (x := 2x)](x = 8)
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The three layers of logic
I Natural language and natural reasoning scenarios
cognitive science, psychology, linguistics, and philosophy
Aim: to understand valid reasoning in natural reasoning scenarios
and their transformations into formal one
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Logika
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The three layers of logic
I Natural language and natural reasoning scenarios
cognitive science, psychology, linguistics, and philosophy
Aim: to understand valid reasoning in natural reasoning scenarios
and their transformations into formal one
II Formal interpreted languages and artificial reasoning scenarios
mathematics and (theoretical) computer science
Aim: to develop various logical systems to describe and perform
reasoning in various formal scenarios
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Logika
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The three layers of logic
I Natural language and natural reasoning scenarios
cognitive science, psychology, linguistics, and philosophy
Aim: to understand valid reasoning in natural reasoning scenarios
and their transformations into formal one
II Formal interpreted languages and artificial reasoning scenarios
mathematics and (theoretical) computer science
Aim: to develop various logical systems to describe and perform
reasoning in various formal scenarios
III Formal abstract languages and mathematical logic
mathematics
Aim: to advance mathematical study of various logical systems
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Logika
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Outline
1
What is logic? (And what is it good for?)
2
Classical propositional logic (the simplest logic there is)
3
Gödel’s theorems (a rough and dirty account)
4
Mathematical fuzzy logic (the stuff I do)
5
The future of fuzzy logic (and your role in it)
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Logika
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What is a valid reasoning?
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What is a valid reasoning?
Example 1
If God exists, He must be good and omnipotent.
If God was good and omnipotent, He would not allow human
suffering.
There is human suffering.
Therefore, God does not exist.
Is this a valid reasoning?
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What is a valid reasoning?
Formalization
p:
q:
Atomic parts:
r:
s:
God exists
God is good
God is omnipotent
There is human suffering
p→q∧r
q ∧ r → ¬s
The form of the reasoning:
s
¬p
Is this a valid reasoning?
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Classical logic: Syntax
Primitive connectives: binary → and unary ¬
Defined connectives: binary ∨, ∧, ↔:
ϕ ∨ ψ = ¬ϕ → ψ
ϕ ∧ ψ = ¬(ϕ → ¬ψ)
ϕ ↔ ψ = (ϕ → ψ) ∧ (ψ → ϕ)
Formulas are built from fixed countable set of atoms using the
connectives
Let us by For denote the set of all formulas.
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Classical logic: Semantics
Bivalence Principle: Every formula is either true or false.
Definition 2
An evaluation is a mapping e from For to {0, 1} such that:
e(¬ϕ) = 1 − e(ϕ)
1 if e(ϕ) ≤ e(ψ)
e(ϕ → ψ) =
0 otherwise.
Note that:
e(ϕ ∧ ψ) = min{e(ϕ), e(ψ)}
e(ϕ ∨ ψ) = max{e(ϕ), e(ψ)}
(
1 if e(ϕ) = e(ψ)
e(ϕ ↔ ψ) =
0 otherwise
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Valid reasoning in classical logic
Definition 3
A formula ϕ is a logical consequence of set of formulas Γ,
Γ |= ϕ, if for every evaluation e:
if e(γ) = 1 for every γ ∈ Γ, then e(ϕ) = 1.
Valid reasoning = logical consequence
ψ1
..
.
ψn
ϕ
is a valid reasoning iff there is no interpretation making the premises
true and the conclusion false.
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Valid reasoning in classical logic
Example 4
Modus ponens:
p→q
p
q
It is a valid reasoning (if e(p → q) = e(p) = 1, then e(q) = 1).
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Valid reasoning in classical logic
Example 4
Modus ponens:
p→q
p
q
It is a valid reasoning (if e(p → q) = e(p) = 1, then e(q) = 1).
Example 5
Abduction:
p→q
q
p
It is NOT a valid reasoning (take: e(p) = 0 and e(q) = 1).
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Back to our ‘God’ example
Example 6
p→q∧r
q ∧ r → ¬s
s
¬p
Assume e(p → q ∧ r) = e(q ∧ r → ¬s) = e(s) = 1. Then e(¬s) = 0 and
so e(q ∧ r) = 0. Thus, we must have e(p) = 0, and therefore: e(¬p) = 1.
It is a valid reasoning!
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Back to our ‘God’ example
Example 6
p→q∧r
q ∧ r → ¬s
s
¬p
Assume e(p → q ∧ r) = e(q ∧ r → ¬s) = e(s) = 1. Then e(¬s) = 0 and
so e(q ∧ r) = 0. Thus, we must have e(p) = 0, and therefore: e(¬p) = 1.
It is a valid reasoning!
BUT, is this really a proof that God does not exist?
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Back to our ‘God’ example
Example 6
p→q∧r
q ∧ r → ¬s
s
¬p
Assume e(p → q ∧ r) = e(q ∧ r → ¬s) = e(s) = 1. Then e(¬s) = 0 and
so e(q ∧ r) = 0. Thus, we must have e(p) = 0, and therefore: e(¬p) = 1.
It is a valid reasoning!
BUT, is this really a proof that God does not exist?
OF COURSE NOT! We only know that if the premisses were true, then
the conclusion would be true as well.
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Back to our ‘God’ example
Example 6
p→q∧r
q ∧ r → ¬s
s
¬p
Assume e(p → q ∧ r) = e(q ∧ r → ¬s) = e(s) = 1. Then e(¬s) = 0 and
so e(q ∧ r) = 0. Thus, we must have e(p) = 0, and therefore: e(¬p) = 1.
It is a valid reasoning!
BUT, is this really a proof that God does not exist?
OF COURSE NOT! We only know that if the premisses were true, then
the conclusion would be true as well.
AND MORE! We would know that only if we would agree that we have
correctly formalized the original argument AND that logic is ‘valid’.
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Structurality of ‘logical’ reasoning
Compare our ‘God example’ with other ones of the same structure:
If God exists,
He must be good and omnipotent
If God was good and omnipotent,
He would not allow human suffering.
But, there is human suffering.
Therefore, God does not exist.
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Structurality of ‘logical’ reasoning
Compare our ‘God example’ with other ones of the same structure:
If our politicians were ideal,
they would be intelligent and honest.
If politicians were intelligent and honest, there would be no corruption.
But, there is corruption.
Therefore, our politicians are not ideal.
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Structurality of ‘logical’ reasoning
Compare our ‘God example’ with other ones of the same structure:
If X is the set of rationals,
X is denumerable and dense.
If a set X is denumerable and dense,
we can embed integers in X
But we cannot embed integers in X.
Therefore, X is not the set of rationals.
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Structurality of ‘logical’ reasoning
Compare our ‘God example’ with other ones of the same structure:
p→q∧r
q ∧ r → ¬s
s
¬p
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Classical proposition logic: Semantics
Definition 7
An evaluation is a mapping e from For to {0, 1} such that:
e(¬ϕ) = 1 − e(ϕ)
1 if e(ϕ) ≤ e(ψ)
e(ϕ → ψ) =
0 otherwise.
Definition 8
A formula ϕ is a logical consequence of set of formulas Γ,
Γ |= ϕ, if for every evaluation e:
if e(γ) = 1 for every γ ∈ Γ, then e(ϕ) = 1.
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Classical proposition logic: Axiomatic system
Axioms (for any ϕ, ψ, χ ∈ For):
A1 ϕ → (ψ → ϕ)
A2 (¬ϕ → ¬ψ) → (ψ → ϕ)
A3 (ϕ → ψ) → ((ψ → χ) → (ϕ → χ))
A4 ((ϕ → ψ) → ψ) → ((ψ → ϕ) → ϕ)
A5 ϕ ∨ ¬ϕ
Deduction rule Modus Ponens (for any ϕ, ψ ∈ For):
ϕ
ϕ→ψ
ψ
Proof: a proof of ϕ from a set of formulas Γ is a finite sequence of
formulas hψ1 , . . . , ψn i such that ψn = ϕ and for every i ≤ n, either
ψi is is an instance of an axiom or an element of Γ, or
there are j, k < i such that ψk = ψj → ψi .
We write Γ ` ϕ if there is a proof of ϕ from Γ.
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Classical proposition logic: The completeness
theorem
Theorem 9 (Completeness theorem)
For each set of formulas Γ ∪ {ϕ} we have:
Γ`ϕ
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Γ |= ϕ
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Outline
1
What is logic? (And what is it good for?)
2
Classical propositional logic (the simplest logic there is)
3
Gödel’s theorems (a rough and dirty account)
4
Mathematical fuzzy logic (the stuff I do)
5
The future of fuzzy logic (and your role in it)
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They seem to have far reaching consequences . . .
Amazon book description: Kurt Gödel was
an intellectual giant. His Incompleteness
Theorem turned not only mathematics but
also the whole world of science and
philosophy on its head. Shattering hopes
that logic would, in the end, allow us a
complete understanding of the universe,
Gödel’s theorem also raised many
provocative questions:
What are the limits of rational thought?
Can we ever fully understand the
machines we build?
Or the inner workings of our own
minds?
How should mathematicians proceed in
the absence of complete certainty about
their results?
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A word of warning . . .
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A word of warning . . .
Amazon book description: Probing
the life and work of Kurt Gödel,
Incompleteness indelibly portrays
the tortured genius whose vision
rocked the stability of mathematical
reasoning and brought him to the
edge of madness.
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Language of arithmetics
Object variables OV: x, y, z, . . .
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Language of arithmetics
Object variables OV: x, y, z, . . .
Terms:
Term = OV | 0 | Term + Term | Term · Term | S(Term)
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Language of arithmetics
Object variables OV: x, y, z, . . .
Terms:
Term = OV | 0 | Term + Term | Term · Term | S(Term)
Formulas:
For = Term ≤ Term | Term = Term | For → For | ¬For | ∀OV(For)
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Language of arithmetics
Object variables OV: x, y, z, . . .
Terms:
Term = OV | 0 | Term + Term | Term · Term | S(Term)
Formulas:
For = Term ≤ Term | Term = Term | For → For | ¬For | ∀OV(For)
Examples of formulas:
∀x∀y(x + y ≤ y)
∀x(∃y)¬(x = y)
∀x∀y(x ≤ y ↔ ∃v(v + x = y)))
Examples of non-formulas:
x→y
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S(∀x(x = x))
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∀S(x)(x ≤ y)
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Language of arithmetics
Object variables OV: x, y, z, . . .
Terms:
Term = OV | 0 | Term + Term | Term · Term | S(Term)
Formulas:
For = Term ≤ Term | Term = Term | For → For | ¬For | ∀OV(For)
Examples of formulas:
∀x∀y(x + y ≤ y)
∀x(∃y)¬(x = y)
∀x∀y(x ≤ y ↔ ∃v(v + x = y)))
Examples of non-formulas:
x→y
S(∀x(x = x))
∀S(x)(x ≤ y)
Sentence: a formula with all variables being ‘quantified’
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A standard model of arithmetics
Consider natural numbers {0, 1, . . . } equipped with usual
addition +
multiplication ·
successor function S(i) = i + 1
equality =
inequality ≤
We denote this model as N
We write N |= ϕ if a sentence ϕ is valid in N
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Axiomatizing N: Peano arithmetics PA
∀x∀y(S(x) = S(y) → x = y)
∀x(x · 0 = 0)
∀x(S(x) 6= 0)
∀x∀y(x · S(y) = x · y + x)
∀x(x 6= 0 → ∃y(x = S(y)))
∀x∀y(x ≤ y ↔ ∃v(v + xy))
∀x(x + 0 = x)
∀x∀y(x + S(y) = S(x + y))
ϕ(0) ∧ ∀x(ϕ(x) → ϕ(S(x))) → ∀xϕ(x) for arbitrary formula ϕ
We write PA ` ϕ if a sentence ϕ is provable in PA
Clearly all axioms of PA are valid in N
Actually: PA ` ϕ implies N |= ϕ
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Gödel completeness theorem
Note: there could be ‘non-standard’ models of PA, i.e., M |= ϕ
whenever PA ` ϕ
Theorem 10 (Completeness theorem)
PA ` ϕ IFF M |= ϕ for each model M of PA
He actually proved that for arbitrary language and arbitrary set
of sentences T
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1st incompleteness theorem
Theorem 11 (1st incompleteness theorem, simple, mode 1)
There is a sentence ϕ such that N |= ϕ but PA 6` ϕ
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1st incompleteness theorem
Theorem 11 (1st incompleteness theorem, simple, mode 1)
There is a sentence ϕ such that N |= ϕ but PA 6` ϕ
Theorem 13 (1st incompleteness theorem, simple, mode 2)
There is a sentence ϕ such that PA 6` ϕ and PA 6` ¬ϕ
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1st incompleteness theorem
Theorem 12 (1st incompleteness theorem, general, mode 1)
Let T be a set of sentences such that:
N is a model of T, i.e., N |= ϕ for each ϕ ∈ T
there is an algorithm to recognize for each ϕ whether ϕ ∈ T
There is a sentence ϕ such that N |= ϕ but T 6` ϕ
Theorem 13 (1st incompleteness theorem, simple, mode 2)
There is a sentence ϕ such that PA 6` ϕ and PA 6` ¬ϕ
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1st incompleteness theorem
Theorem 12 (1st incompleteness theorem, general, mode 1)
Let T be a set of sentences such that:
N is a model of T, i.e., N |= ϕ for each ϕ ∈ T
there is an algorithm to recognize for each ϕ whether ϕ ∈ T
There is a sentence ϕ such that N |= ϕ but T 6` ϕ
Theorem 14 (1st incompleteness theorem, general, mode 2)
Let T be a set of sentences such that:
T contains PA, i.e., T ` ϕ whenever PA ` ϕ
T is consistent, i.e., T 6` 0 = S(0)
there is an algorithm to recognize for each ϕ whether ϕ ∈ T
There is a sentence ϕ such that T 6` ϕ and T 6` ¬ϕ
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The main proof idea: coding things by natural numbers
Numerals - 1̄ = S(0) and (n + 1) = S(n̄)
Coding of sequences - sequence h2, 3, 0i is uniquely coded by 23 34 51
a more complex but ‘better’ coding is actually used
Coding of formulas - any formula is a sequence of symbols,
we write ϕ̄ for the numeral corresponding to the code of ϕ
Coding of proofs - any proof is a sequence of formulas, so . . .
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The main proof idea: coding things by natural numbers
Numerals - 1̄ = S(0) and (n + 1) = S(n̄)
Coding of sequences - sequence h2, 3, 0i is uniquely coded by 23 34 51
a more complex but ‘better’ coding is actually used
Coding of formulas - any formula is a sequence of symbols,
we write ϕ̄ for the numeral corresponding to the code of ϕ
Coding of proofs - any proof is a sequence of formulas, so . . .
Theorem 15 (Arithmetization of syntax)
For each sets of formulas describable by an algorithm there are
formulas τ (x) and Prτ (x) st.
N |= τ (m̄) iff m is a code of a sentence from T
N |= Prτ (ϕ̄) iff T ` ϕ
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The main proof idea (cont.)
Theorem 16 (Diagonalization)
For each formula ψ(x) there is a sentence ϕ such that
PA ` ϕ ↔ ψ(ϕ̄)
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The main proof idea (cont.)
Theorem 16 (Diagonalization)
For each formula ψ(x) there is a sentence ϕ such that
PA ` ϕ ↔ ψ(ϕ̄)
Theorem 17 (1st incompleteness theorem)
Let T be a set of sentences such that:
N is a model of T, i.e., N |= ϕ for each ϕ ∈ T
there is an algorithm to recognize for each ϕ whether ϕ ∈ T
There is a formula ϕ such that N |= ϕ but T 6` ϕ
Proof: take a sentence ϕ such that
T ` ϕ ↔ ¬Prτ (ϕ̄)
Assume that N |= ϕ then N 6|= Prτ (ϕ̄) thus T 6` ϕ
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The main proof idea (cont.)
Theorem 16 (Diagonalization)
For each formula ψ(x) there is a sentence ϕ such that
PA ` ϕ ↔ ψ(ϕ̄)
Theorem 17 (1st incompleteness theorem)
Let T be a set of sentences such that:
N is a model of T, i.e., N |= ϕ for each ϕ ∈ T
there is an algorithm to recognize for each ϕ whether ϕ ∈ T
There is a formula ϕ such that N |= ϕ but T 6` ϕ
Proof: take a sentence ϕ such that
T ` ϕ ↔ ¬Prτ (ϕ̄)
Assume that N 6|= ϕ then N |= Prτ (ϕ̄) thus T ` ϕ and so T 6` ¬ϕ
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2nd incompleteness theorem
Theorem 18 (2nd incompleteness theorem)
Let T be a set of formulas such that:
T contains PA, i.e., T ` ϕ whenever PA ` ϕ
there is an algorithm to recognize for each ϕ whether ϕ ∈ T
T is consistent, i.e., T 6` 0 = S(0)
Then T 6` ¬Prτ (0 = S(0))
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Are the consequences of Gödel’s theorems really so
earth-shattering?
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Are the consequences of Gödel’s theorems really so
earth-shattering?
Amazon book description: Among the
many expositions of Gödel’s
incompleteness theorems written for
non-specialists, this book stands apart.
With exceptional clarity, Franzén gives
careful, non-technical explanations both
of what those theorems say and, more
importantly, what they do not. No other
book aims, as his does, to address in
detail the misunderstandings and abuses
of the incompleteness theorems that are
so rife in popular discussions of their
significance. As an antidote to the many
spurious appeals to incompleteness in
theological, anti-mechanist and
post-modernist debates.
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Two more pieces of recommended literature
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Outline
1
What is logic? (And what is it good for?)
2
Classical propositional logic (the simplest logic there is)
3
Gödel’s theorems (a rough and dirty account)
4
Mathematical fuzzy logic (the stuff I do)
5
The future of fuzzy logic (and your role in it)
Petr Cintula (ÚI AV ČR)
Logika
32 / 50
Two valued logic is boring . . .
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33 / 50
Two valued logic is boring . . .
http://www.xkcd.com/74/
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33 / 50
Lets start from some serious observations . . .
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34 / 50
Lets start from some serious observations . . .
There are heaps of sand
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Lets start from some serious observations . . .
There are heaps of sand
If I remove one grain from a heap of sand, I still have a heap of sand.
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Lets start from some serious observations . . .
There are heaps of sand
If I remove one grain from a heap of sand, I still have a heap of sand.
Therefore, one grain of sand is a heap of sand.
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Sorites paradox [Eubulides of Miletus, IV century BCE]
If I remove one grain from any heap, I still have a heap. One million
grains of sand is a heap. Therefore, one grain of sand is a heap.
Formalization pn : n grains of sand is a heap of sand
p1000000
p999999
..
.
and
and
p1000000 → p999999
p999999 → p999998
thus
thus
p999999
p999998
p2
and
p2 → p1
thus
p1
There is no doubt that the premise p1000000 is true.
There is no doubt that the conclusion p1 is false.
For each i, the premise pi → pi−1 seems to be true.
The reasoning is logically valid.
We have a paradox!
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Vagueness and degrees of truth
The predicates that generate this kind of paradoxes are called vague.
Possible solution: assume that truth comes in degrees
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Vagueness and degrees of truth
The predicates that generate this kind of paradoxes are called vague.
Possible solution: assume that truth comes in degrees
Define an evaluation e as e(pn ) = nε, where ε = 10−6.
Note that e(p0 ) = 0 and e(p106 ) = 1, i.e. first premise is completely
true, the conclusion is completely false.
e(pn → pn−1 ) = min{1, 1 − e(pn ) + e(pn−1 )} = 1 − ε.
All premises have the same, almost completely true, truth value!
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Fuzzy logic in broad sense: fuzzy set theory
Fuzzy set a mapping µ : U → [0, 1]
Zadeh 1965
Fuzzy set theory a.k.a. fuzzy logic in a ‘broad sense’:
works with degrees of truth
combines them using operations corresponding to connectives
is a bunch of engineering methods which
I
I
are usually tailored to particular purposes
sometime are a major success at certain applications
but is NOT an area of mathematical logic
e.g., has no ‘special’ notions of truth or valid reasoning
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Mathematical Fuzzy Logic stems from those origins
It was established as a respectable branch of mathematical logic
by Petr Hájek
in Metamathematics of fuzzy logic
Kluwer,1998.
Citations: 1500+ (WOS) and 3800+ (Google Scholar)
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Recall the classical propositional logic (CL)
Formulas are built from atoms using connectives → and ¬
Semantics of CL: an evaluation is a mapping e : Form → {0, 1} st
e(¬ϕ) = 1 − e(ϕ)
e(ϕ → ψ) = min{1, 1 − e(ϕ) + e(ψ)}
Axiomatic system of CL: deduction rule modus ponens (from ϕ and
ϕ → ψ infer ψ) and axioms:
ϕ ∨ ¬ϕ
ϕ → (ψ → ϕ)
(ϕ → ψ) → ((ψ → χ) → (ϕ → χ))
(¬ϕ → ¬ψ) → (ψ → ϕ)
((ϕ → ψ) → ψ) → ((ψ → ϕ) → ϕ)
Theorem 19 (Completeness theorem)
Γ `CL ϕ
iff
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Γ |=CL ϕ.
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Łukasiewicz logic: An example of fuzzy logic
Łukasiewicz logic Š has the same formulas as classical logic
Semantics of Ł:
an evaluation is a mapping e : Form → [0, 1] st
e(¬ϕ) = 1 − e(ϕ)
e(ϕ → ψ) = min{1, 1 − e(ϕ) + e(ψ)}
Axiomatic system of Ł:
deduction rule modus ponens (from ϕ and
ϕ → ψ infer ψ) and axioms:
ϕ → (ψ → ϕ)
(ϕ → ψ) → ((ψ → χ) → (ϕ → χ))
(¬ϕ → ¬ψ) → (ψ → ϕ)
((ϕ → ψ) → ψ) → ((ψ → ϕ) → ϕ)
Theorem 20 (Completeness theorem)
Γ `Š ϕ
iff
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Γ |=Š ϕ.
Logika
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Since then the theory of MFL got ‘deeper’ . . .
The following disciplines
of mathematical logic
were, and still are,
developed in MFL
The connections of MFL
with the following areas
of mathematics were,
and still are, explored:
proof theory
lattice theory
model theory
group/field theory
set theory
geometry
recursion theory
game theory
complexity theory
topology
category theory
measure theory
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. . . and ‘broader’
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As witnessed by 1300 pages of . . .
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Outline
1
What is logic? (And what is it good for?)
2
Classical propositional logic (the simplest logic there is)
3
Gödel’s theorems (a rough and dirty account)
4
Mathematical fuzzy logic (the stuff I do)
5
The future of fuzzy logic (and your role in it)
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Logika
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The future of fuzzy logic
Problem: logic got divorced from the study of (human) reasoning
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Logika
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The future of fuzzy logic
Problem: logic got divorced from the study of (human) reasoning
There is a movement trying to amend the rift
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Logika
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The future of fuzzy logic
Problem: logic got divorced from the study of (human) reasoning
There is a movement trying to amend the rift
The key component: the transformation of natural reasoning scenarios
into formalized ones where various logics are directly applicable
Petr Cintula (ÚI AV ČR)
Logika
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The future of fuzzy logic
Problem: logic got divorced from the study of (human) reasoning
There is a movement trying to amend the rift
The key component: the transformation of natural reasoning scenarios
into formalized ones where various logics are directly applicable
My program: join the movement and concentrate on scenarios
of both kinds involving vague notions: tall, red, young, . . .
Petr Cintula (ÚI AV ČR)
Logika
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The future of fuzzy logic
Problem: logic got divorced from the study of (human) reasoning
There is a movement trying to amend the rift
The key component: the transformation of natural reasoning scenarios
into formalized ones where various logics are directly applicable
My program: join the movement and concentrate on scenarios
of both kinds involving vague notions: tall, red, young, . . .
The problem is that common transformations of natural scenarios
or designs of the formalized ones usually get rid of vagueness
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The three layers of my program
I To study natural reasoning scenarios involving graded notions
and their natural transformations into formalized scenarios
preserving the graduality
II To utilize logical methods to analyze and perform reasoning
in formalized scenarios
III To advance the logic of graded notions to meet the demands
of the previous goals
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Logika
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The three layers of my program and their three
aims (one more ambitious than the other :))
I To study natural reasoning scenarios involving graded notions
and their natural transformations into formalized scenarios
preserving the graduality
II To utilize logical methods to analyze and perform reasoning
in formalized scenarios
III To advance the logic of graded notions to meet the demands
of the previous goals
Petr Cintula (ÚI AV ČR)
Logika
46 / 50
The three layers of my program and their three
aims (one more ambitious than the other :))
I To study natural reasoning scenarios involving graded notions
and their natural transformations into formalized scenarios
preserving the graduality
II To utilize logical methods to analyze and perform reasoning
in formalized scenarios
III To advance the logic of graded notions to meet the demands
of the previous goals
Advancement of mathematical logic
Petr Cintula (ÚI AV ČR)
Logika
46 / 50
The three layers of my program and their three
aims (one more ambitious than the other :))
I To study natural reasoning scenarios involving graded notions
and their natural transformations into formalized scenarios
preserving the graduality
II To utilize logical methods to analyze and perform reasoning
in formalized scenarios
More powerful formal methods (mainly for CS)
III To advance the logic of graded notions to meet the demands
of the previous goals
Advancement of mathematical logic
Petr Cintula (ÚI AV ČR)
Logika
46 / 50
The three layers of my program and their three
aims (one more ambitious than the other :))
I To study natural reasoning scenarios involving graded notions
and their natural transformations into formalized scenarios
preserving the graduality
Better understanding of (human) reasoning
II To utilize logical methods to analyze and perform reasoning
in formalized scenarios
More powerful formal methods (mainly for CS)
III To advance the logic of graded notions to meet the demands
of the previous goals
Advancement of mathematical logic
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Logika
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I am not alone . . .
Akademie věd České republiky
Ústav Informatiky v.v.i.
Oddělení teoretické informatiky
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Our team
Marta Bílková: modal logics, proof theory, co-algebraic logic
Petr Cintula: abstract algebraic logic
Matěj Dostál (PhD student): modal logics, co-algebraic logic
Zuzana Haniková: computational complexity
Rostislav Horčík: logic in computer science, algebraic logic
Tomáš Lávička (PhD student): abstract algebraic logic
Ondrej Majer: game theory
Tommaso Moraschini: abstract algebraic logic
Adam Přenosil (PhD student): paraconsistent logic
Igor Sedlár: substructural logics, philosophical aspects
Amanda Vidal: modal logic
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Our close collaborators
Libor Běhounek, Ostrava: formal fuzzy mathematics
Rudolfo Ertola, Buenos Aires: paraconsistent logic
Chris Fermuller, Vienna: game theory, philosophical aspects
Nick Galatos, Denver: substructural logics
Lluis Godo, Barcelona: applied fuzzy logic
Tomáš Kroupa, Prague, game theory, probability
Carles Noguera: Prague, abstract algebraic logic
George Metcalfe, Bern: proof theory, automated deduction
James Raftery, Johannesburg: abstract algebraic logic
Nick Smith, Sydney: philosophical aspects
Kazushige Terui, Kyoto: substructural logics
..
.
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Logika
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We offer topics for bachelor, master and PhD theses
Petr Cintula (ÚI AV ČR)
Logika
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We offer topics for bachelor, master and PhD theses
On different levels of the hierarchy:
III Mostly in pure of mathematical fuzzy logic
II Recently also in applications of fuzzy logic (mainly in
computer science)
I In the future also in the quest for understanding human reasoning
Petr Cintula (ÚI AV ČR)
Logika
50 / 50
We offer topics for bachelor, master and PhD theses
On different levels of the hierarchy:
III Mostly in pure of mathematical fuzzy logic
II Recently also in applications of fuzzy logic (mainly in
computer science)
I In the future also in the quest for understanding human reasoning
If you are interested, please contact us . . .
[email protected]
http://www.ustavinformatiky.cz
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Logika
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