kurt gödel - selected topics intuitionistic logic versus...
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Faculty of Mathematics and Computer Science
Supervisor: Prof. Dr. Christoph Benzmüller
28.02.2019
Irina Makarenko
Kurt Gödel - Selected Topics
Intuitionistic Logic versus Classical LogicGödel’s Interpretation and Conjectures
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An Interpretation of the Intuitionistic Sentential Logic.““K. Gödel, 1933
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Structure
Intuitionism, Intuitionistic Logic & Heyting's Calculus
Classical Propositional Logic
Gödel’s Interpretation
Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
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Intuitionism
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Intuitionism is a philosophy of mathematics that was introduced by Luitzen Egbertus Jan Brouwer in 1908.
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⇛ Intuitionism centers on proof rather than truth.
Intuitionism
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
The truth of a mathematical statement can only be conceived via a mental construction (a proof or verification) that proves it to be true.
It does not make sense to think of truth or falsity of a mathematical statement independently of our knowledge concerning the statement.
A statement is true if we have proof of it, and false if we can show that the assumption that there is a proof for the statement leads to a contradiction.
A. S. Troelstra and D. van Dalen, Constructivism in Mathematics, 1988
““
Intuitionism is a philosophy of mathematics that was introduced by Luitzen Egbertus Jan Brouwer in 1908.
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Intuitionistic Propositional Calculus ( )
Introduced by Arend Heyting in 1930.
A logical calculus describing rules for the derivation of propositions that are valid from the point of view of intuitionism.
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Intuitionistic logic is most easily described as classical logic without the principle of excluded middle.
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
IPC
φ ∨ ¬φ
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The negation of a formula , denoted as , is abbreviated by .
Alphabet. Propositional variables ( ), logical connectives andconstant symbol .
Syntax of
⋅ , + , ⊃
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Formulas. The (well-formed) formulas of are defined inductively as follows: - Each atomic formula is a well-formed formula. - If and are well-formed formulas, so are , and . - Nothing else is a well-formed formula.
IPC
A, B, C ⋯
φ ψ φ ⋅ ψ φ + ψ φ ⊃ ψ
Atomic Formulas. Any propositional variable or is an atomic formula.
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
⊥
⊥
φ φ ⊃ ⊥∼ φ
IPC
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[ ]- A proof of is a construction which, given a proof of , would return a proof of .
Semantics of
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
- A proof of consists of a proof of and a proof of . - A proof of is given by presenting either a proof of or a proof of . - A proof of is a construction which, given a proof of , returns a proof of . - has no proof.
φ ⋅ ψ φ
states informally what is intended to be a proof of a given formula:
ψφ + ψ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
φ ψφ ⊃ ψ φ ψ
∼ φ⊥
φ ⊥
where are formulas in .φ, ψ IPC
IPC
The Brouwer-Heyting-Kolmogorov (BHK) interpretation
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Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Proof system for IPC
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Axiom schemes where are formulas in :
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
1. φ ⊃ (ψ ⊃ φ)2. (φ ⊃ (ψ ⊃ μ)) ⊃ ((φ ⊃ ψ) ⊃ (φ ⊃ μ))3. (φ ⋅ ψ) ⊃ φ4. (φ ⋅ ψ) ⊃ ψ5. φ ⊃ (ψ ⊃ (φ ⋅ ψ))6. φ ⊃ (φ + ψ)7. ψ ⊃ (φ + ψ)8. (φ ⊃ μ) ⊃ ((ψ ⊃ μ) ⊃ ((φ + ψ) ⊃ μ))9. ⊥ ⊃ φ
φ, ψ, μ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
IPC
Hilbert-style system for IPC
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9. ⊥ ⊃ φ8. (φ ⊃ μ) ⊃ ((ψ ⊃ μ) ⊃ ((φ + ψ) ⊃ μ))7. ψ ⊃ (φ + ψ)
From and , conclude . (Modus Ponens)
Inference rules where are formulas in :
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
⋮
Axiom schemes where are formulas in :φ, ψ, μ IPC
1. φ φ ⊃ ψ ψ
φ, ψ IPC
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Hilbert-style system for IPC
1. φ ⊃ (ψ ⊃ φ)
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Classical Propositional Logic ( )
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
CPL
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The negation of a formula , denoted as , is abbreviated by .φ φ → ⊥¬φ
Formulas. The (well-formed) formulas of are defined inductively as follows: - Each atomic formula is a well-formed formula. - If and are well-formed formulas, so are , and . - Nothing else is a well-formed formula.
Atomic Formulas. Any propositional variable or is an atomic formula.
Alphabet. Propositional variables ( ), logical connectives andconstant symbol .
Syntax of
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
∧ , ∨ , →
CPL
φ ∧ ψ φ ∨ ψ φ → ψ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
φ ψ
⊥
⊥
A, B, C ⋯
CPL
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Semantics of
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
- is true if and only if is true and is true. - is true if and only if is true or is true (or both). - is false if and only if is true and is false. - is false.
φ ∧ ψ
The semantics of is subject to the usual conditions (“truth tables”):
ψφ ∨ ψ φ ψφ → ψ φ ψ⊥
where are formulas in .φ, ψ CPL
φ
CPL
[ ]- is true if and only if is false. ¬φ φ
CPL
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Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Proof system for CPL
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Axiom schemes where are formulas in :
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
1. φ → (ψ → φ)2. (φ → (ψ → μ)) → ((φ → ψ) → (φ → μ))3. (φ ∧ ψ) → φ4. (φ ∧ ψ) → ψ5. φ → (ψ → (φ ∧ ψ))6. φ → (φ ∨ ψ)7. ψ → (φ ∨ ψ)8. (φ → μ) → ((ψ → μ) → ((φ ∨ ψ) → μ))9. ⊥ → φ
φ, ψ, μ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
CPL
Hilbert-style system for
10. φ ∨ (φ → ⊥ )
CPL
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Inference rules where are formulas in :
10. φ ∨ (φ → ⊥ )9. ⊥ → φ8. (φ → μ) → ((ψ → μ) → ((φ ∨ ψ) → μ))
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
⋮
Axiom schemes where are formulas in :φ, ψ, μ CPL
φ, ψ CPL
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
1. φ → (ψ → φ)
Hilbert-style system for CPL
From and , conclude . (Modus Ponens)1. φ φ → ψ ψ
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Additional inference rules where are formulas in :
Additional axiom schemes where are formulas in :
Expansion of into system
𝒢11. Bφ → φ12. Bφ → (B(φ → ψ) → Bψ)13. Bφ → BBφ
Additional concept ‘ is provable’ (denoted by with an additional unary operator ).φ Bφ B
𝒢
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
𝒢φ, ψ
φ, ψ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
CPL
From conclude .2. φ Bφ
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From conclude .2. φ Bφ
13. Bφ → BBφ12. Bφ → (B(φ → ψ) → Bψ)11. Bφ → φ
Hilbert-style system for
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
⋮
Axiom schemes where are formulas in :φ, ψ, μ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
1. φ → (ψ → φ)
10. φ ∨ (φ → ⊥ )
𝒢
𝒢
Inference rules where are formulas in :φ, ψ 𝒢 From and , conclude . (Modus Ponens)1. φ φ → ψ ψ
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g(φ ⋅ ψ) = g(φ) ∧ g(ψ)
[ ]
Interpretation function is defined as follows:
g(φ ⊃ ψ) = B g(φ) → B g(ψ)
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
g(A) = A
g : IPC → 𝒢
g( ⊥ ) = ⊥
g(φ + ψ) = B g(φ) ∨ B g(ψ)
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
g( ∼ φ) = ¬ B g(φ)
where is a propositional variable and are formulas in .φ, ψ IPCA
Interpretation
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[ ]
Variant
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
g( ∼ φ) = B ¬ B g(φ)
g(φ ⋅ ψ) = B g(φ) ∧ B g(ψ)
Interpretation
Interpretation function is defined as follows:
g(φ ⊃ ψ) = B g(φ) → B g(ψ)
g(A) = A
g : IPC → 𝒢
g( ⊥ ) = ⊥
g(φ + ψ) = B g(φ) ∨ B g(ψ)
where is a propositional variable and are formulas in .φ, ψ IPCA
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B g(A) ∨ B g(A ⊃ ⊥ )g(A + (A ⊃ ⊥ )) =
Interpretation function :
Exemplary
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
g : IPC → 𝒢
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
g(φ + ψ) = B g(φ) ∨ B g(ψ)g(φ ⊃ ψ) = B g(φ) → B g(ψ)
g(A) = A
⋮
= BA ∨ B g(A ⊃ ⊥ )
= BA ∨ B( B g(A) → B g( ⊥ ) )
= BA ∨ B( BA → B⊥ )
φ
g(φ) = ?
= A + (A ⊃ ⊥ )
Interpretation
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Interpretation function :
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
g : IPC → 𝒢
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
g(φ + ψ) = B g(φ) ∨ B g(ψ)g(φ ⊃ ψ) = B g(φ) → B g(ψ)
g(A) = A
⋮
= BA ∨ B g(A ⊃ ⊥ )
= BA ∨ B( B g(A) → B g( ⊥ ) )
= BA ∨ B( BA → B⊥ )
φ
g(φ) = BA ∨ B( BA → B⊥ )
= A + (A ⊃ ⊥ )
B g(A) ∨ B g(A ⊃ ⊥ )g(A + (A ⊃ ⊥ )) =
Exemplary Interpretation
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Remarks
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
No formula is derivable from , unless or is derivable from .Bφ ∨ Bψ 𝒢 𝒢Bφ Bψ⇛ The Law of Excluded Middle, , is not derivable from .Bφ ∨ B(φ → ⊥ )
The operator should be interpreted as ‘provable by any correct means’ and must not be
interpreted as ‘provable in a given formal system’ because this would contradict Gödel’s
second incompleteness theorem.
B
𝒢
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If , then .
Gödel’s Results
Gödel claims that if a formula is derivable from intuitionistic logic, then its ‘translation’ is derivable from , that is: 𝒢
⊢IPC φ
He conjectures that the converse also holds, and thus we should have:
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
⊢𝒢 g(φ)
if, and only if .⊢IPC φ ⊢𝒢 g(φ)
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Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Lewis Modal System S4
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
The system is a modal propositional logic.S4
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Modal logic extends classical propositional logic to include operators expressing modality,namely for necessity and for possibility.
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
What is Modal Logic?
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Modal logic describes the logical relations of modalities as necessities and possibilities.
□ ◊
is true
¬ □ EarthHasExactlyOneMoon The Earth has exactly one moon.
φ is necessarily trueφ
φ
is possibly true◊ φ φ□ φ
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Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
The system is a modal propositional logicS4
Lewis Modal System S4
.
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The system is a modal propositional logic
From , conclude . (Necessity Rule)
Additional inference rules where are formulas in :
φ □ φ
Additional axiom schemes where are formulas in :φ, ψ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
For and we take the rules and axioms of classical propositional logic as before.
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
□ φ → ( □ (φ → ψ) → □ ψ)□ φ → □ □ φ
→ , ∧ , ∨⊥
□ φ → φ
S4
φ, ψ S4
□
Lewis Modal System S4
.with necessity operator S4
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φ, ψ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
□ φ → ( □ (φ → ψ) → □ ψ)□ φ → □ □ φ
□ φ → φ
Additional axiom schemes where are formulas in :S4
Lewis Modal System S4
From , conclude . (Necessity Rule)
Additional inference rules where are formulas in :
φ □ φ
φ, ψ S4
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From , conclude .φ □ φ
Additional axiom schemes where are formulas in :φ, ψ
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
System
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
□ φ → ( □ (φ → ψ) → □ ψ)□ φ → □ □ φ
□ φ → φBBB
B
B BBB
𝒢
𝒢
𝒢
Additional inference rules where are formulas in :φ, ψ
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Relation of and𝒢
If is understood as ‘ is necessary’ the expanded system results as the Lewismodal system , with written for the necessity operator ︎ .
Bφ φS4 B □
Hence, Gödel’s result shows that there is an embedding of the intuitionistic propositional logic into the modal logic . Therefore,S4
S4
Intuitionistic Logic Classical Logic Gödel’s Interpretation Gödel’s Results
Kurt Gödel - Selected TopicsIntuitionistic Logic versus Classical Logic
Irina Makarenko
if, and only if .⊢IPC φ ⊢S4g(φ)
𝒢
IPC
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Thank you
Literature:
S. Artemov. On Two Models of Provability. In Mathematical Problems from Applied Logics, New Logics for the XXIst Century, II, Dov M. Gabbay (Eds.), International Mathematical Series, Springer, 2006.
K. Gödel. Eine Interpretation des intuitionistischen Aussagenkalküls. Ergebnisse Math. Kolloq. 4, 1933, pp. 39–40; English translation in: Kurt Gödel Collected Works. Vol. 1, S. Feferman et al. (Eds.), Oxford Univ. Press, Oxford, Clarendon Press, New York, 1986, pp. 301–303.
A. S. Troelstra, D. van Dalen. Constructivism in Mathematics: An Introduction. North-Holland Publishing, Amsterdam. A. Heyting. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der preussischen Akademie der Wissenschaften, 1930, pp. 42–71, 158–169.https://plato.stanford.edu/entries/logic-intuitionistic/. Retrieved February 27, 2019.
A. Akbar Tabatabai. Provability Interpretation of Propositional and Modal Logics. 2017.