First-Order Logic of Proofs - City University of New York · IPC+F tr(F), where tr(F) is obtained...

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First-Order Logic of Proofs Sergei Artemov & Tatiana Yavorskaya (Sidon) Kurt Gödel Research Center Vienna, April 27, 2011 1

Transcript of First-Order Logic of Proofs - City University of New York · IPC+F tr(F), where tr(F) is obtained...

Page 1: First-Order Logic of Proofs - City University of New York · IPC+F tr(F), where tr(F) is obtained from F by prefixing each subformula of F with a. ... constants as proofs of the concrete

First-Order Logic of ProofsSergei Artemov & Tatiana Yavorskaya (Sidon)

Kurt Gödel Research Center

Vienna, April 27, 2011

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BHK semantics

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Gödel’s embedding

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Logic of Proofs as BHK

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Lessons to learn from LP

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Lessons to learn from LP

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Quantification and LP

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First-order LP: format

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First-order LP: axioms

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First-order LP: example

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Internalization

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Realization

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Realization

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Realization

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Realization

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Realization

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Realization

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FOS4 = projection of FOLP

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Open variables in derivations

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Open variables in derivations

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Open variables in derivations

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Parametric semantics

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Existence of operations

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Interpretation

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Soundness

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Example

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Accidental tautologies

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Invariant semantics

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Invariant semantics works

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Barcan is valid

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The main thing is yet to come...

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Generic proof(X)

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Generic proofs

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Two-way Gödel’s Lemma

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Operations on proof forms

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Generic semantics

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Barcan fails

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BHK intuition seems to work too

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Completeness is not attainable

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Conclusions

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Thank You!

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