Semantic Data

Chapter 2 : Introduction to first order logic

Jean-Louis Binot

12/02/2020Semantic Data1

Course content outline

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Sources and recommended readings

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Why study logic?

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Types of logics

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Agenda

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Model-theoretic semantics2

First order logic1

Decidability4

Satisfiability, validity, entailment3

Precursors of first order logic

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A very short reminder of proposition logic

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◼ ⊤ ⊥

◼ Π

◼ ∧ ∨ → ↔

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∈ Π

⊤ ⊥

∧ ∨ → ↔

∧ ∨ → ↔

∨ → ∧ ↔ ∨

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A very short reminder of proposition logic ./.

∧ ∨ → ↔

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Some standard equivalences in propositional logic

❑ ∧ ≡ ∧ ∧

❑ ∨ ≡ ∨ ∨

❑ ∧ ∧ ≡ ∧ ∧ ∧

❑ ∨ ∨ ≡ ∨ ∨ ∨

❑ ≡

❑ → ≡ →

❑ → ≡ ∨

❑ ↔ ≡ → ∧ →

❑ ∧ ≡ ∨

❑ ∨ ≡ ∧

❑ ∧ ∨ ≡ ∧ ∨ ∧ ∧ ∨

❑ ∨ ∧ ≡ ∨ ∧ ∨ ∨ ∧

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Limits of proposition logic

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∀ →

∃ ∧

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Vocabulary of first order logic

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◼ → ↔

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∀

∃

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Syntax of first order logic

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∨

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∀ → ∃ ∧

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First order logic: formal syntax

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◼ ⊤) ⊥)

◼ ¬ ∧ ∨ → ↔

◼ ∀ ∃

: ∀ ∃, ¬ ∧ ∨, →, ↔.

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Examples of terms and formulas

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∧

∃ ∧

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Intuitive understanding of quantifiers and common mistakes

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◼ ∀

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→

∧

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∃ ∧

∃ →

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Nested quantifiers

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∀ ∀ →

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∀ ∃

∃ ∀

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De Morgan’s rules for quantifiers

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❑ ∀ ∃

∨ ↔ ∧ ∀ ↔ ∃

∧ ↔ ∨ ∀ ↔ ∃

∧ ↔ ∨ ∀ ↔ ∃

∨ ↔ ∧ ∃ ↔ ∀

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Free and bound variables, quantifier scope

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∀ ∃

◼ ∀ ∃

◼ ∀ ∃

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∀ ∀

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Caution note

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∀ ∨ ∃

∀ ∨ ∃

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Why is it called first order logic ?

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∃

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Agenda

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Model-theoretic semantics2

First order logic1

Decidability4

Satisfiability, validity, entailment3

Model theoretic semantics

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I I

❑ I I

I ⊨23

I

IF

Ingredients of model-theoretic semantics

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Model theoretic semantics of propositional logic (reminder)

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❑ I

◼ I

❑ I I

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II

◼ ⊤I

⊥I

◼I I

◼ ∧ I I I

◼ ∨ I I I

◼ → I I I

◼ ↔ I I I

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Semantics of FOL: intuition

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Interpretation domain : example

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❑ I

◼I I I

I I

◼I

◼I

I1

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II

I2

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First order logic: semantics

I ΔI I α

❑ ΔI

❑I

◼I ∈ ΔI

◼I ΔI →

◼I ΔI → ΔI

❑ α α ∈ ΔI

❑ tI FI I

ΔI ΔI

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First order logic semantics : terms

I

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◼I

◼ α I α

◼I I I I I

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First order logic semantics : formulas

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◼I I I I I

❑ → ↔

◼

❑

I I ∈ ΔI

◼ ∀ I ∈ ΔI I

◼ ∃ I ∈ ΔI I

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Agenda

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Model-theoretic semantics2

First order logic1

Decidability4

Satisfiability, validity, entailment3

Fundamental semantic concepts

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Satisfiability and validity

❑ I I ⊨ I

◼ I

◼ I I I ⊭

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❑ ⊨

❑ ⊭

◼ ⊨ ⊭

∨

∨

∧

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Satisfiability for set of formulas

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I

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I ∈ ⊭

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∧ ∧

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Entailment and logical equivalence

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◼

❑ ⊨

◼ ∅ ⊨ ⊨

❑ ≡

I I I

⊨

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Some useful results

❑ ⊨ ⊨ → →

❑ ⊨ ⊭ ∧ ∧

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Logical consequence and knowledge-based inferences

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Updated framework of reference

Data integration component

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Monotonicity, closed-world and open-world assumptions

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⊨ ∧ ⊨

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◼ ⊨ ∧ ⊨

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Agenda

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Model-theoretic semantics2

First order logic1

Decidability4

Satisfiability, validity, entailment3

Questions regarding the use of FOL for reasoning

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1. Knowledge representation in FOL

∀ ∀ ∧ ∧ →

∀ → ∨

∀ ∀ ↔

∀ ∧ → ∨ ∨ ∨

∀ ∧ → ↔ ∃

∀ ∧ → ↔ ∃

∀ ∧ ⇒ ⇔

∀ →

∧

∧

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1. Knowledge representation in FOL

∀ ∀ ∧ ∧ →

∀ → ∨

∀ ∀ ↔

∀ ∧ → ∨ ∨ ∨

∀ ∧ → ↔ ∃

∀ ∧ → ↔ ∃

∀ ∧ ⇒ ⇔

∀ →

∧

∧

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✓

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▪

▪

2. Logical reasoning

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⊨

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A truth table enumeration algorithm for PL entailment

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Decision problems

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Decidability of propositional logic

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◼ ⊨ ⊭ ∧ ∧

◼ ⊨ ⊭

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◼ ≡ ⊨ ⊨

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Undecidability of first order logic

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Practical approaches for reasoning with first order logic

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∧ →

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Summary

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References

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THANK YOU

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