 •Contents
 •Preface
 •1 Introduction
 •1.1 Issues of Vagueness
 •1.3 The Problem of the Fringe
 •1.4 Preview of the Rest of the Book
 •1.5 History and Scope of Fuzzy Logic
 •1.6 Tall People
 •1.7 Exercises
 •2 Review of Classical Propositional Logic
 •2.1 The Language of Classical Propositional Logic
 •2.2 Semantics of Classical Propositional Logic
 •2.3 Normal Forms
 •2.4 An Axiomatic Derivation System for Classical Propositional Logic
 •2.5 Functional Completeness
 •2.6 Decidability
 •2.7 Exercises
 •3.2 Semantics of Classical FirstOrder Logic
 •3.3 An Axiomatic Derivation System for Classical FirstOrder Logic
 •3.4 Exercises
 •4.1 Numeric TruthValues for Classical Logic
 •4.2 Boolean Algebras and Classical Logic
 •4.3 More Results about Boolean Algebras
 •4.4 Exercises
 •5.2 Lukasiewicz’s ThreeValued Logic
 •5.3 Bochvar’s ThreeValued Logics
 •5.5 Normal Forms
 •5.7 Lukasiewicz’s System Expanded
 •5.8 Exercises
 •6.3 Exercises
 •7.3 Tautologies, Validity, and “Quasi”Semantic Concepts
 •7.4 Exercises
 •8.3 Exercises
 •9.3 MVAlgebras
 •9.4 Exercises
 •11 Fuzzy Propositional Logics: Semantics
 •11.1 Fuzzy Sets and Degrees of Truth
 •11.2 Lukasiewicz Fuzzy Propositional Logic
 •11.3 Tautologies, Contradictions, and Entailment in Fuzzy Logic
 •11.4 NTautologies, DegreeEntailment, and NDegreeEntailment
 •11.5 Fuzzy Consequence
 •11.7 TNorms, TConorms, and Implication in Fuzzy Logic
 •11.9 Product Fuzzy Propositional Logic
 •11.10 Fuzzy External Assertion and Negation
 •11.11 Exercises
 •12 Fuzzy Algebras
 •12.2 Residuated Lattices and BLAlgebras
 •12.3 Zero and Unit Projections in Algebraic Structures
 •12.4 Exercises
 •13 Derivation Systems for Fuzzy Propositional Logic
 •13.1 An Axiomatic System for Tautologies and Validity in Fuzzy
 •13.2 A PavelkaStyle Derivation System for Fuzzy
 •13.3 An Alternative Axiomatic System for Tautologies and Validity in FuzzyL Based on BLAlgebras
 •13.7 External Assertion Axioms
 •13.8 Exercises
 •14.1 Fuzzy Interpretations
 •14.2 Lukasiewicz Fuzzy FirstOrder Logic
 •14.3 Tautologies and Other Semantic Concepts
 •14.4 Lukasiewicz Fuzzy Logic and the Problems of Vagueness
 •14.6 Product Fuzzy FirstOrder Logic
 •14.8 Exercises
 •15.3 An Axiomatic Derivation System for Fuzzy
 •15.4 Combining Fuzzy FirstOrder Logical Systems; External Assertion
 •15.5 Exercises
 •16 Extensions of Fuzziness
 •16.2 Fuzzy “Linguistic” TruthValues
 •16.3 Other Fuzzy Extensions of Fuzzy Logic
 •16.4 Exercises
 •17 Fuzzy Membership Functions
 •17.2 Empirical Construction of Membership Functions
 •17.3 Logical Relevance?
 •17.4 Exercises
 •Bibliography
 •Index
12.3 Zero and Unit Projections in Algebraic Structures 
219 
Condition xiv gives us the Law of Noncontradiction for weak conjunction: for any formula P, at least one of P and P has the value 0, and so therefore does the weak conjunction of a formula and its negation. It is left as an exercise to prove that FuzzyPL is a product algebra, and hence that xiii is characteristic of FuzzyP’s adjoint bold conjunction and implication operations. Analogously to earlier results, we have
Result 12.4: A formula is a tautology of FuzzyP if and only if it is a Ptautology in every product algebra.
12.3 Zero and Unit Projections in Algebraic Structures
Recall that we introduced fuzzy Bochvarian external negation and assertion, known as 0 and 1projections in the fuzzy literature, in Section 11.10 of Chapter 11. We will call the corresponding algebraic operations zero projection and unit projection, and we will use the symbols ! and α, respectively, to denote these operations. The following algebraic conditions characterize unit projection in any BLalgebra:6
αi. αx (αx) = unit αii. α(x y) ≤ αx αy αiii. αx ≤ x
αiv. αx ≤ ααx
αv. αx α(x y) ≤ αy
αvi. α unit = unit
and we will call a BLalgebra that meets these additional conditions a unit projection algebra.
Although x and αx are not generally equivalent in a unit projection algebra, αx and ααx are:
αx = ααx
Proof:
αx ≤ ααx, by αiv, and ααx ≤ αx, by αiii, so αx = ααx.
The expressions x and αx are not generally equivalent because the inequality x ≤ αx does not hold in all unit projection algebras. This is as it should be, if unit projection is the algebraic counterpart to fuzzy external assertion: if V(P) = .5, for example,P has the value 0.
We can introduce zero projection in a unit projection algebra as the algebraic counterpart to fuzzy external negation with the deﬁnition
!x = def (αx) .
6 
Hajek´ (1998b) based these algebraic axioms on derivational axioms in Baaz (1996). 

220 



Fuzzy Algebras 
so that, for example, 



αx !x = unit 
(by αi) 


and 




!unit = zero 



Proof: 




α unit = unit 

(αvi) 

(α unit) = unit 

(same operation, both sides) 

(i) !unit 
= 
unit 

(deﬁnition) 



and 




unit (unit zero) = zero 
(BLii) 

unit zero = zero 

(identity for bold meet) 

(ii) unit 
= zero 

(deﬁnition) 

so 




!unit = zero 

(from (i) and (ii)) 
Adding the external assertion operator to FuzzyL, the MValgebraic structure of FuzzyL becomes a unit projection algebra. To establish this, we need to show that the following hold for all values x and y in [0. .1], where α stands for the external assertion operation:
α i. max (α x, 1 – α x) = 1
α ii. α (max (x, y)) ≤ max (α x, α y) α iii. α x ≤ x
α iv. α x ≤ α α x
α v. max (0, α x + α (min (1, 1 – x + y)) – 1) ≤ α y α vi. α 1 = 1
For (α i), it sufﬁces to point out that α x is always either 1 or 0. For (α iii), we note that if x =1 then α x = 0 and 0 is less than or equal to any value in the unit interval, while if x = 1 then α x = 1 and certainly 1 ≤ 1. We leave the remainder, as well as the proof that the Godel¨ and product algebraic structures for FuzzyG and FuzzyP augmented with the external assertion operator are unit projection algebras, as exercises.
12.4 Exercises
SECTION 12.1
1Prove that the algebra FuzzyLMV = <[0. .1], L, L, 1−, 1, 0>, where L, L, and 1− are FuzzyL’s bold disjunction, bold conjunction, and negation operations, is an MValgebra.
2Prove that it follows from the MValgebra deﬁnition x y =def (x ∩ y ) that x y = x (x y).
12.4 Exercises 
221 
SECTION 12.2
3Which MValgebra conditions other than Double Negation, if any, fail to hold for the algebra deﬁned by FuzzyG’s operations?
4Which MValgebra conditions other than Double Negation, if any, fail to hold for the algebra deﬁned by FuzzyP’s operations?
5Give a complete proof that every MValgebra is a residuated lattice in the sense described in Section 12.2. You are free to cite previous results along the way.
6Prove that for any formulas P and Q in a tnormbased fuzzy logical system, at least one of P → Q or Q → P has the value 1, by showing that this follows from the deﬁnition of the residuum operation
m tn n ≤ p if and only if m ≤ n p, for all m, n, p [0. .1]
for any tnorm tn. (Hint: consider the case where m = 1 and show that in this case either 1 ≤ n p or 1 ≤ p n. You will need to use tnorm properties to establish this.)
7Prove that every MValgebra is a BLalgebra.
8Prove that
(BLii) x (x y) = y holds in every BLalgebra.
9Complete the proof of BLiii:
a.Show that (x y) z ≤ x (y z) in every BLalgebra with Double Negation.
b.Show that if x ≤ y and y ≤ x in a BLalgebra, then x = y—so that the converse inequalities in the proof of BL3 in fact establish an equality.
10Prove that
(BLvi) x y = x y
holds in every BLalgebra with Double Negation.
11Complete the proof of Result 12.2, that every BLalgebra with Double Negation is an MValgebra, by showing that the following hold true in every BLalgebra with Double Negation:
a.condition i of MValgebras
b.condition ii of MValgebras
c.condition iii of MValgebras
d.the second half of condition iv of MValgebras: x zero = zero
e.condition v of MValgebras
f.condition vi of MValgebras
g.condition vii of MValgebras.
12Prove that FuzzyGL is a Godel¨ algebra.
13Prove that FuzzyPL is a product algebra.
14Why did we not include the condition xiv. (Godel¨ BLalgebra) x x = x
in the deﬁnition of product algebras?
222 
Fuzzy Algebras 
SECTION 12.3
15Prove that the following hold in every unit projection algebra:
a.α zero = zero
b.!zero = unit
c.!x = !!!x, for any x
16Complete the proof that the MValgebraic structure of FuzzyL augmented with the external assertion operator is a unit projection algebra, by showing that α ii, α iv, α v, and α vi all hold.
17Prove that the Godel¨ and product algebraic structures FuzzyGL and FuzzyPL augmented with the external assertion operator are unit projection algebras.