Many-valued Logic

Logical methods for AI

Many-valued Logic

Lecture 10

What you’ll be able to do

Bivalence

Every formula has value $0$ or $1$ .

Missing information: no language recorded for Zimbabwe.

Vagueness: is 20°C warm? And 19.999°C?

Missing values

SELECT country FROM LanguageOf
WHERE (language = 'Dutch');

SELECT country FROM LanguageOf
WHERE NOT (language = 'Dutch');

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The Kleene functions

NOT	NOT value
0	1
ω	ω
1	0

AND	ω	1
0	0	0
ω	ω	ω
1	ω	1

OR	0	ω	1
0	0	ω	1
ω	ω	ω	1
1	1	1	1

K₃ consequence

$1$ is the only designated value (glossary, opens in a new tab) here.

$3² = 9$ information states for $SUN$ and $RAIN$ .

The Kleene model space

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Excluded middle and explosion

$v(SUN) = ω$ :

v(SUN \lor \negSUN) = ω

All $ω$ gives $ω$ : no tautologies in $\neg$ , $\land$ , $\lor$ .

No model gives $A$ and $\negA$ both value $1$ .

Natural deduction for K₃

Classical rules that fail in K₃

Conditional introduction

$A \to A$ has value $ω$ when $A$ does.

Negation introduction

$\neg(B \land \negB)$ has value $ω$ when $B$ does.

Classical reductio

It would prove $C \lor \negC$ , even when $C$ is unknown.

K₃ derivations

Countermodels in SQL

Premises IS TRUE, conclusion IS NOT TRUE: a row is a countermodel.

-- SUN → RAIN, RAIN ∴ SUN
SELECT SUN, RAIN FROM Valuations
WHERE (NOT SUN OR RAIN) IS TRUE
  AND RAIN IS TRUE
  AND SUN IS NOT TRUE;

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Fuzzy predicates

$⟦IsWarm⟧ : ℝ \to [0,1]$ , $v(IsWarm(15°C)) = ⟦IsWarm⟧(15°C)$

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Combining degrees

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Fuzzy logic

Invalid with v(A) = 0.5 and v(B) = 0

Explosion

$v(A) = v(\negA) = 0.5 > 0 = v(B)$

Disjunctive syllogism

$v(A \lor B) = v(\negA) = 0.5 > 0 = v(B)$

Fuzzy rules

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Sugeno controllers

$IsCold(x) \mapsto 1$
$IsWarm(x) \mapsto 0$
$IsHot(x) \mapsto 0$

Weights $0.4$ , $0.6$ , $0$ : 40% power.

A Sugeno heating controller

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Mamdani controllers

Defuzzification

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Cooling a humid room

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